63633 lines
2.8 MiB
63633 lines
2.8 MiB
$( nf.mm - Version of 22-Feb-2021. $)
|
|
|
|
$(
|
|
~~ PUBLIC DOMAIN ~~
|
|
This work is waived of all rights, including copyright, according to the CC0
|
|
Public Domain Dedication. http://creativecommons.org/publicdomain/zero/1.0/
|
|
|
|
Principal curator: Scott Fenton
|
|
|
|
Partly based on the set.mm database, itself dedicated to public domain
|
|
by mean of the CC0 Public Domain Dedication.
|
|
$)
|
|
|
|
$( Begin $[ set-pred.mm $] $)
|
|
|
|
$(
|
|
###############################################################################
|
|
CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
|
|
###############################################################################
|
|
|
|
Logic can be defined as the "study of the principles of correct reasoning"
|
|
(Merrilee H. Salmon's 1991 "Informal Reasoning and Informal Logic" in
|
|
_Informal Reasoning and Education_ ) or as "a formal system using symbolic
|
|
techniques and mathematical methods to establish truth-values" (the Oxford
|
|
English Dictionary).
|
|
|
|
This section formally defines the logic system we will use. In particular,
|
|
it defines symbols for declaring truthful statements, along with rules for
|
|
deriving truthful statements from other truthful statements. The system
|
|
defined here is classical first order logic with equality (the most common
|
|
logic system used by mathematicians).
|
|
|
|
We begin with a few housekeeping items in pre-logic, and then introduce
|
|
propositional calculus (both its axioms and important theorems that can be
|
|
derived from them). Propositional calculus deals with general truths about
|
|
well-formed formulas (wffs) regardless of how they are constructed. This is
|
|
followed by proofs that other axiomatizations of classical propositional
|
|
calculus can be derived from the axioms we have chosen to use.
|
|
|
|
We then define predicate calculus, which adds additional symbols and rules
|
|
useful for discussing objects (beyond simply true or false). In particular,
|
|
it introduces the symbols ` = ` ("equals"), ` e. ` ("is a member of"), and `
|
|
A. ` ("for all"). The first two are called "predicates." A predicate
|
|
specifies a true or false relationship between its two arguments.
|
|
|
|
$)
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Pre-logic
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
|
|
This section includes a few "housekeeping" mechanisms before we begin
|
|
defining the basics of logic.
|
|
|
|
$)
|
|
|
|
$( Declare the primitive constant symbols for propositional calculus. $)
|
|
$c ( $. $( Left parenthesis $)
|
|
$c ) $. $( Right parenthesis $)
|
|
$c -> $. $( Right arrow (read: "implies") $)
|
|
$c -. $. $( Right handle (read: "not") $)
|
|
$c wff $. $( Well-formed formula symbol (read: "the following symbol
|
|
sequence is a wff") $)
|
|
$c |- $. $( Turnstile (read: "the following symbol sequence is provable" or
|
|
'a proof exists for") $)
|
|
|
|
$( Define the syntax and logical typecodes, and declare that our grammar is
|
|
unambiguous (verifiable using the KLR parser, with compositing depth 5).
|
|
(This $ j comment need not be read by verifiers, but is useful for parsers
|
|
like mmj2.) $)
|
|
$( $j
|
|
syntax 'wff';
|
|
syntax '|-' as 'wff';
|
|
unambiguous 'klr 5';
|
|
$)
|
|
|
|
$( wff variable sequence: ph ps ch th ta et ze si rh mu la ka $)
|
|
$( Introduce some variable names we will use to represent well-formed
|
|
formulas (wff's). $)
|
|
$v ph $. $( Greek phi $)
|
|
$v ps $. $( Greek psi $)
|
|
$v ch $. $( Greek chi $)
|
|
$v th $. $( Greek theta $)
|
|
$v ta $. $( Greek tau $)
|
|
$v et $. $( Greek eta $)
|
|
$v ze $. $( Greek zeta $)
|
|
$v si $. $( Greek sigma $)
|
|
$v rh $. $( Greek rho $)
|
|
$v mu $. $( Greek mu $)
|
|
$v la $. $( Greek lambda $)
|
|
$v ka $. $( Greek kappa $)
|
|
|
|
$( Specify some variables that we will use to represent wff's.
|
|
The fact that a variable represents a wff is relevant only to a theorem
|
|
referring to that variable, so we may use $f hypotheses. The symbol
|
|
` wff ` specifies that the variable that follows it represents a wff. $)
|
|
$( Let variable ` ph ` be a wff. $)
|
|
wph $f wff ph $.
|
|
$( Let variable ` ps ` be a wff. $)
|
|
wps $f wff ps $.
|
|
$( Let variable ` ch ` be a wff. $)
|
|
wch $f wff ch $.
|
|
$( Let variable ` th ` be a wff. $)
|
|
wth $f wff th $.
|
|
$( Let variable ` ta ` be a wff. $)
|
|
wta $f wff ta $.
|
|
$( Let variable ` et ` be a wff. $)
|
|
wet $f wff et $.
|
|
$( Let variable ` ze ` be a wff. $)
|
|
wze $f wff ze $.
|
|
$( Let variable ` si ` be a wff. $)
|
|
wsi $f wff si $.
|
|
$( Let variable ` rh ` be a wff. $)
|
|
wrh $f wff rh $.
|
|
$( Let variable ` mu ` be a wff. $)
|
|
wmu $f wff mu $.
|
|
$( Let variable ` la ` be a wff. $)
|
|
wla $f wff la $.
|
|
$( Let variable ` ka ` be a wff. $)
|
|
wka $f wff ka $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Inferences for assisting proof development
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The inference rules in this section will normally never appear in a completed
|
|
proof. They can be ignored if you are using this database to assist learning
|
|
logic - please start with the statement ~ wn instead.
|
|
|
|
$)
|
|
|
|
${
|
|
a1ii.1 $e |- ph $.
|
|
a1ii.2 $e |- ps $.
|
|
$( (_Note_: This inference rule and the next one, ~ idi , will normally
|
|
never appear in a completed proof. It can be ignored if you are using
|
|
this database to assist learning logic - please start with the statement
|
|
~ wn instead.)
|
|
|
|
This is a technical inference to assist proof development. It provides
|
|
a temporary way to add an independent subproof to a proof under
|
|
development, for later assignment to a normal proof step.
|
|
|
|
The metamath program's Proof Assistant requires proofs to be developed
|
|
backwards from the conclusion with no gaps, and it has no mechanism that
|
|
lets the user to work on isolated subproofs. This inference provides a
|
|
workaround for this limitation. It can be inserted at any point in a
|
|
proof to allow an independent subproof to be developed on the side, for
|
|
later use as part of the final proof.
|
|
|
|
_Instructions_: (1) Assign this inference to any unknown step in the
|
|
proof. Typically, the last unknown step is the most convenient, since
|
|
'assign last' can be used. This step will be replicated in hypothesis
|
|
a1ii.1, from where the development of the main proof can continue. (2)
|
|
Develop the independent subproof backwards from hypothesis a1ii.2. If
|
|
desired, use a 'let' command to pre-assign the conclusion of the
|
|
independent subproof to a1ii.2. (3) After the independent subproof is
|
|
complete, use 'improve all' to assign it automatically to an unknown
|
|
step in the main proof that matches it. (4) After the entire proof is
|
|
complete, use 'minimize *' to clean up (discard) all ~ a1ii references
|
|
automatically.
|
|
|
|
This inference was originally designed to assist importing partially
|
|
completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can
|
|
also be useful on its own. Interestingly, no axioms are required for
|
|
its proof. (Contributed by NM, 7-Feb-2006.) $)
|
|
a1ii $p |- ph $=
|
|
( ) C $.
|
|
$}
|
|
|
|
${
|
|
idi.1 $e |- ph $.
|
|
$( Inference form of ~ id . This inference rule, which requires no axioms
|
|
for its proof, is useful as a copy-paste mechanism during proof
|
|
development in mmj2. It is normally not referenced in the final version
|
|
of a proof, since it is always redundant and can be removed using the
|
|
'minimize *' command in the metamath program's Proof Assistant.
|
|
(Contributed by Alan Sare, 31-Dec-2011.) $)
|
|
idi $p |- ph $=
|
|
( ) B $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Propositional calculus
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
|
|
Propositional calculus deals with general truths about well-formed formulas
|
|
(wffs) regardless of how they are constructed. The simplest propositional
|
|
truth is ` ( ph -> ph ) ` , which can be read "if something is true, then it
|
|
is true" - rather trivial and obvious, but nonetheless it must be proved from
|
|
the axioms (see theorem ~ id ).
|
|
|
|
Our system of propositional calculus consists of three basic axioms and
|
|
another axiom that defines the modus-ponens inference rule. It is attributed
|
|
to Jan Lukasiewicz (pronounced woo-kah-SHAY-vitch) and was popularized by
|
|
Alonzo Church, who called it system P2. (Thanks to Ted Ulrich for this
|
|
information.) These axioms are ~ ax-1 , ~ ax-2 , ~ ax-3 , and (for modus
|
|
ponens) ~ ax-mp . Some closely followed texts include [Margaris] for the
|
|
axioms and [WhiteheadRussell] for the theorems.
|
|
|
|
The propositional calculus used here is the classical system widely used by
|
|
mathematicians. In particular, this logic system accepts the "law of the
|
|
excluded middle" as proven in ~ exmid , which says that a logical statement
|
|
is either true or not true. This is an essential distinction of classical
|
|
logic and is not a theorem of intuitionistic logic.
|
|
|
|
All 194 axioms, definitions, and theorems for propositional calculus in
|
|
_Principia Mathematica_ (specifically *1.2 through *5.75) are axioms or
|
|
formally proven. See the Bibliographic Cross-References at
|
|
~ http://us.metamath.org/mpeuni/mmbiblio.html for a complete
|
|
cross-reference from sources used to its formalization in the Metamath
|
|
Proof Explorer.
|
|
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Recursively define primitive wffs for propositional calculus
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( If ` ph ` is a wff, so is ` -. ph ` or "not ` ph ` ." Part of the
|
|
recursive definition of a wff (well-formed formula). In classical logic
|
|
(which is our logic), a wff is interpreted as either true or false. So if
|
|
` ph ` is true, then ` -. ph ` is false; if ` ph ` is false, then
|
|
` -. ph ` is true. Traditionally, Greek letters are used to represent
|
|
wffs, and we follow this convention. In propositional calculus, we define
|
|
only wffs built up from other wffs, i.e. there is no starting or "atomic"
|
|
wff. Later, in predicate calculus, we will extend the basic wff
|
|
definition by including atomic wffs ( ~ weq and ~ wel ). $)
|
|
wn $a wff -. ph $.
|
|
|
|
$( If ` ph ` and ` ps ` are wff's, so is ` ( ph -> ps ) ` or " ` ph ` implies
|
|
` ps ` ." Part of the recursive definition of a wff. The resulting wff
|
|
is (interpreted as) false when ` ph ` is true and ` ps ` is false; it is
|
|
true otherwise. Think of the truth table for an OR gate with input ` ph `
|
|
connected through an inverter. After we define the axioms of
|
|
propositional calculus ( ~ ax-1 , ~ ax-2 , ~ ax-3 , and ~ ax-mp ), the
|
|
biconditional ( ~ df-bi ), the constant true ` T. ` ( ~ df-tru ), and the
|
|
constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth
|
|
table values: ` ( ( T. -> T. ) <-> T. ) ` ( ~ truimtru ),
|
|
` ( ( T. -> F. ) <-> F. ) ` ( ~ truimfal ), ` ( ( F. -> T. ) <-> T. ) `
|
|
( ~ falimtru ), and ` ( ( F. -> F. ) <-> T. ) ` ( ~ falimfal ). These
|
|
have straightforward meanings, for example, ` ( ( T. -> T. ) <-> T. ) `
|
|
just means "the value of ` T. -> T. ` is ` T. ` ".
|
|
|
|
The left-hand wff is called the antecedent, and the right-hand wff is
|
|
called the consequent. In the case of ` ( ph -> ( ps -> ch ) ) ` , the
|
|
middle ` ps ` may be informally called either an antecedent or part of the
|
|
consequent depending on context. Contrast with ` <-> ` ( ~ df-bi ),
|
|
` /\ ` ( ~ df-an ), and ` \/ ` ( ~ df-or ).
|
|
|
|
This is called "material implication" and the arrow is usually read as
|
|
"implies." However, material implication is not identical to the meaning
|
|
of "implies" in natural language. For example, the word "implies" may
|
|
suggest a causal relationship in natural language. Material implication
|
|
does not require any causal relationship. Also, note that in material
|
|
implication, if the consequent is true then the wff is always true (even
|
|
if the antecedent is false). Thus, if "implies" means material
|
|
implication, it is true that "if the moon is made of green cheese that
|
|
implies that 5=5" (because 5=5). Similarly, if the antecedent is false,
|
|
the wff is always true. Thus, it is true that, "if the moon made of green
|
|
cheese that implies that 5=7" (because the moon is not actually made of
|
|
green cheese). A contradiction implies anything ( ~ pm2.21i ). In short,
|
|
material implication has a very specific technical definition, and
|
|
misunderstandings of it are sometimes called "paradoxes of logical
|
|
implication." $)
|
|
wi $a wff ( ph -> ps ) $.
|
|
|
|
$( Register '-.' and '->' as primitive expressions (lacking definitions). $)
|
|
$( $j primitive 'wn' 'wi'; $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The axioms of propositional calculus
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Postulate the three axioms of classical propositional calculus.
|
|
|
|
Propositional calculus (axioms ~ ax-1 through ~ ax-3 and rule ~ ax-mp ) can
|
|
be thought of as asserting formulas that are universally "true" when their
|
|
variables are replaced by any combination of "true" and "false."
|
|
Propositional calculus was first formalized by Frege in 1879, using as his
|
|
axioms (in addition to rule ~ ax-mp ) the wffs ~ ax-1 , ~ ax-2 , ~ pm2.04 ,
|
|
~ con3 , ~ notnot2 , and ~ notnot1 . Around 1930, Lukasiewicz simplified the
|
|
system by eliminating the third (which follows from the first two, as you can
|
|
see by looking at the proof of ~ pm2.04 ) and replacing the last three with
|
|
our ~ ax-3 . (Thanks to Ted Ulrich for this information.)
|
|
|
|
The theorems of propositional calculus are also called _tautologies_.
|
|
Tautologies can be proved very simply using truth tables, based on the
|
|
true/false interpretation of propositional calculus. To do this, we assign
|
|
all possible combinations of true and false to the wff variables and verify
|
|
that the result (using the rules described in ~ wi and ~ wn ) always
|
|
evaluates to true. This is called the _semantic_ approach. Our approach is
|
|
called the _syntactic_ approach, in which everything is derived from axioms.
|
|
A metatheorem called the Completeness Theorem for Propositional Calculus
|
|
shows that the two approaches are equivalent and even provides an algorithm
|
|
for automatically generating syntactic proofs from a truth table. Those
|
|
proofs, however, tend to be long, since truth tables grow exponentially with
|
|
the number of variables, and the much shorter proofs that we show here were
|
|
found manually.
|
|
|
|
$)
|
|
|
|
$( Axiom _Simp_. Axiom A1 of [Margaris] p. 49. One of the 3 axioms of
|
|
propositional calculus. The 3 axioms are also given as Definition 2.1 of
|
|
[Hamilton] p. 28. This axiom is called _Simp_ or "the principle of
|
|
simplification" in _Principia Mathematica_ (Theorem *2.02 of
|
|
[WhiteheadRussell] p. 100) because "it enables us to pass from the joint
|
|
assertion of ` ph ` and ` ps ` to the assertion of ` ph ` simply."
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ax-1 $a |- ( ph -> ( ps -> ph ) ) $.
|
|
|
|
$( Axiom _Frege_. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of
|
|
propositional calculus. It "distributes" an antecedent over two
|
|
consequents. This axiom was part of Frege's original system and is known
|
|
as _Frege_ in the literature. It is also proved as Theorem *2.77 of
|
|
[WhiteheadRussell] p. 108. The other direction of this axiom also turns
|
|
out to be true, as demonstrated by ~ pm5.41 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ax-2 $a |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $.
|
|
|
|
$( Axiom _Transp_. Axiom A3 of [Margaris] p. 49. One of the 3 axioms of
|
|
propositional calculus. It swaps or "transposes" the order of the
|
|
consequents when negation is removed. An informal example is that the
|
|
statement "if there are no clouds in the sky, it is not raining" implies
|
|
the statement "if it is raining, there are clouds in the sky." This axiom
|
|
is called _Transp_ or "the principle of transposition" in _Principia
|
|
Mathematica_ (Theorem *2.17 of [WhiteheadRussell] p. 103). We will also
|
|
use the term "contraposition" for this principle, although the reader is
|
|
advised that in the field of philosophical logic, "contraposition" has a
|
|
different technical meaning. (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-3 $a |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $.
|
|
|
|
$(
|
|
Postulate the modus ponens rule of inference.
|
|
$)
|
|
|
|
${
|
|
$( Minor premise for modus ponens. $)
|
|
min $e |- ph $.
|
|
$( Major premise for modus ponens. $)
|
|
maj $e |- ( ph -> ps ) $.
|
|
$( Rule of Modus Ponens. The postulated inference rule of propositional
|
|
calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if
|
|
` ph ` is true, and ` ph ` implies ` ps ` , then ` ps ` must also be
|
|
true." This rule is sometimes called "detachment," since it detaches
|
|
the minor premise from the major premise. "Modus ponens" is short for
|
|
"modus ponendo ponens," a Latin phrase that means "the mood that by
|
|
affirming affirms" [Sanford] p. 39. This rule is similar to the rule of
|
|
modus tollens ~ mto .
|
|
|
|
Note: In some web page displays such as the Statement List, the symbols
|
|
"&" and "=>" informally indicate the relationship between the hypotheses
|
|
and the assertion (conclusion), abbreviating the English words "and" and
|
|
"implies." They are not part of the formal language. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
ax-mp $a |- ps $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical implication
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The results in this section are based on implication only, and avoid ax-3.
|
|
In an implication, the wff before the arrow is called the "antecedent" and
|
|
the wff after the arrow is called the "consequent."
|
|
|
|
We will use the following descriptive terms very loosely: A "closed form" or
|
|
"tautology" has no $e hypotheses. An "inference" has one or more $e
|
|
hypotheses. A "deduction" is an inference in which the hypotheses and the
|
|
conclusion share the same antecedent.
|
|
|
|
$)
|
|
|
|
${
|
|
mp2b.1 $e |- ph $.
|
|
mp2b.2 $e |- ( ph -> ps ) $.
|
|
mp2b.3 $e |- ( ps -> ch ) $.
|
|
$( A double modus ponens inference. (Contributed by Mario Carneiro,
|
|
24-Jan-2013.) $)
|
|
mp2b $p |- ch $=
|
|
( ax-mp ) BCABDEGFG $.
|
|
$}
|
|
|
|
${
|
|
$( Premise for ~ a1i . $)
|
|
a1i.1 $e |- ph $.
|
|
$( Inference derived from axiom ~ ax-1 . See ~ a1d for an explanation of
|
|
our informal use of the terms "inference" and "deduction." See also the
|
|
comment in ~ syld . (Contributed by NM, 5-Aug-1993.) $)
|
|
a1i $p |- ( ps -> ph ) $=
|
|
( wi ax-1 ax-mp ) ABADCABEF $.
|
|
$}
|
|
|
|
${
|
|
mp1i.a $e |- ph $.
|
|
mp1i.b $e |- ( ph -> ps ) $.
|
|
$( Drop and replace an antecedent. (Contributed by Stefan O'Rear,
|
|
29-Jan-2015.) $)
|
|
mp1i $p |- ( ch -> ps ) $=
|
|
( ax-mp a1i ) BCABDEFG $.
|
|
$}
|
|
|
|
${
|
|
$( Premise for ~ a2i . $)
|
|
a2i.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference derived from axiom ~ ax-2 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
a2i $p |- ( ( ph -> ps ) -> ( ph -> ch ) ) $=
|
|
( wi ax-2 ax-mp ) ABCEEABEACEEDABCFG $.
|
|
$}
|
|
|
|
${
|
|
imim2i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding common antecedents in an implication. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
imim2i $p |- ( ( ch -> ph ) -> ( ch -> ps ) ) $=
|
|
( wi a1i a2i ) CABABECDFG $.
|
|
$}
|
|
|
|
${
|
|
mpd.1 $e |- ( ph -> ps ) $.
|
|
mpd.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( A modus ponens deduction. A translation of natural deduction rule
|
|
` -> ` E ( ` -> ` elimination). (Contributed by NM, 5-Aug-1993.) $)
|
|
mpd $p |- ( ph -> ch ) $=
|
|
( wi a2i ax-mp ) ABFACFDABCEGH $.
|
|
$}
|
|
|
|
${
|
|
$( First of 2 premises for ~ syl . $)
|
|
syl.1 $e |- ( ph -> ps ) $.
|
|
$( Second of 2 premises for ~ syl . $)
|
|
syl.2 $e |- ( ps -> ch ) $.
|
|
$( An inference version of the transitive laws for implication ~ imim2 and
|
|
~ imim1 , which Russell and Whitehead call "the principle of the
|
|
syllogism...because...the syllogism in Barbara is derived from them"
|
|
(quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Some authors
|
|
call this law a "hypothetical syllogism."
|
|
|
|
(A bit of trivia: this is the most commonly referenced assertion in our
|
|
database. In second place is ~ eqid , followed by ~ syl2anc ,
|
|
~ adantr , ~ syl3anc , and ~ ax-mp . The Metamath program command 'show
|
|
usage' shows the number of references.) (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by O'Cat, 20-Oct-2011.) (Proof shortened
|
|
by Wolf Lammen, 26-Jul-2012.) $)
|
|
syl $p |- ( ph -> ch ) $=
|
|
( wi a1i mpd ) ABCDBCFAEGH $.
|
|
$}
|
|
|
|
${
|
|
mpi.1 $e |- ps $.
|
|
mpi.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( A nested modus ponens inference. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Stefan Allan, 20-Mar-2006.) $)
|
|
mpi $p |- ( ph -> ch ) $=
|
|
( a1i mpd ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
mp2.1 $e |- ph $.
|
|
mp2.2 $e |- ps $.
|
|
mp2.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( A double modus ponens inference. (Contributed by NM, 5-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
|
|
mp2 $p |- ch $=
|
|
( mpi ax-mp ) ACDABCEFGH $.
|
|
$}
|
|
|
|
${
|
|
3syl.1 $e |- ( ph -> ps ) $.
|
|
3syl.2 $e |- ( ps -> ch ) $.
|
|
3syl.3 $e |- ( ch -> th ) $.
|
|
$( Inference chaining two syllogisms. (Contributed by NM, 5-Aug-1993.) $)
|
|
3syl $p |- ( ph -> th ) $=
|
|
( syl ) ACDABCEFHGH $.
|
|
$}
|
|
|
|
$( Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For
|
|
another version of the proof directly from axioms, see ~ idALT .
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan,
|
|
20-Mar-2006.) $)
|
|
id $p |- ( ph -> ph ) $=
|
|
( wi ax-1 mpd ) AAABZAAACAECD $.
|
|
|
|
$( Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. This
|
|
version is proved directly from the axioms for demonstration purposes.
|
|
This proof is a popular example in the literature and is identical, step
|
|
for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a)
|
|
of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of
|
|
[Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's _A
|
|
Primer for Logic and Proof_ p. 17 (PDF p. 23) at
|
|
~ http://www.mathsci.appstate.edu/~~hirstjl/primer/hirst.pdf . For a
|
|
shorter version of the proof that takes advantage of previously proved
|
|
theorems, see ~ id . (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is discouraged.) Use ~ id instead.
|
|
(New usage is discouraged.) $)
|
|
idALT $p |- ( ph -> ph ) $=
|
|
( wi ax-1 ax-2 ax-mp ) AAABZBZFAACAFABBGFBAFCAFADEE $.
|
|
|
|
$( Principle of identity with antecedent. (Contributed by NM,
|
|
26-Nov-1995.) $)
|
|
idd $p |- ( ph -> ( ps -> ps ) ) $=
|
|
( wi id a1i ) BBCABDE $.
|
|
|
|
${
|
|
a1d.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction introducing an embedded antecedent.
|
|
|
|
_Naming convention_: We often call a theorem a "deduction" and suffix
|
|
its label with "d" whenever the hypotheses and conclusion are each
|
|
prefixed with the same antecedent. This allows us to use the theorem in
|
|
places where (in traditional textbook formalizations) the standard
|
|
Deduction Theorem would be used; here ` ph ` would be replaced with a
|
|
conjunction ( ~ df-an ) of the hypotheses of the would-be deduction. By
|
|
contrast, we tend to call the simpler version with no common antecedent
|
|
an "inference" and suffix its label with "i"; compare theorem ~ a1i .
|
|
Finally, a "theorem" would be the form with no hypotheses; in this case
|
|
the "theorem" form would be the original axiom ~ ax-1 . We usually show
|
|
the theorem form without a suffix on its label (e.g. ~ pm2.43 vs.
|
|
~ pm2.43i vs. ~ pm2.43d ). When an inference is converted to a theorem
|
|
by eliminating an "is a set" hypothesis, we sometimes suffix the theorem
|
|
form with "g" (for "more general") as in ~ uniex vs. ~ uniexg .
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan,
|
|
20-Mar-2006.) $)
|
|
a1d $p |- ( ph -> ( ch -> ps ) ) $=
|
|
( wi ax-1 syl ) ABCBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
a2d.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Deduction distributing an embedded antecedent. (Contributed by NM,
|
|
23-Jun-1994.) $)
|
|
a2d $p |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $=
|
|
( wi ax-2 syl ) ABCDFFBCFBDFFEBCDGH $.
|
|
$}
|
|
|
|
${
|
|
2a1i.1 $e |- ch $.
|
|
$( Add two antecedents to a wff. (Contributed by Jeff Hankins,
|
|
4-Aug-2009.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
|
|
2a1i $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( a1i a1d ) ACBCADEF $.
|
|
$}
|
|
|
|
${
|
|
sylcom.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylcom.2 $e |- ( ps -> ( ch -> th ) ) $.
|
|
$( Syllogism inference with commutation of antecedents. (Contributed by
|
|
NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof
|
|
shortened by Stefan Allan, 23-Feb-2006.) $)
|
|
sylcom $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a2i syl ) ABCGBDGEBCDFHI $.
|
|
$}
|
|
|
|
${
|
|
syl5com.1 $e |- ( ph -> ps ) $.
|
|
syl5com.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( Syllogism inference with commuted antecedents. (Contributed by NM,
|
|
24-May-2005.) $)
|
|
syl5com $p |- ( ph -> ( ch -> th ) ) $=
|
|
( a1d sylcom ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
$( Premise for ~ com12 . See ~ pm2.04 for the theorem form. $)
|
|
com12.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference that swaps (commutes) antecedents in an implication.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
4-Aug-2012.) $)
|
|
com12 $p |- ( ps -> ( ph -> ch ) ) $=
|
|
( id syl5com ) BBACBEDF $.
|
|
$}
|
|
|
|
${
|
|
syl5.1 $e |- ( ph -> ps ) $.
|
|
syl5.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( A syllogism rule of inference. The first premise is used to replace the
|
|
second antecedent of the second premise. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-May-2013.) $)
|
|
syl5 $p |- ( ch -> ( ph -> th ) ) $=
|
|
( syl5com com12 ) ACDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6.2 $e |- ( ch -> th ) $.
|
|
$( A syllogism rule of inference. The second premise is used to replace
|
|
the consequent of the first premise. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 30-Jul-2012.) $)
|
|
syl6 $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a1i sylcom ) ABCDECDGBFHI $.
|
|
$}
|
|
|
|
${
|
|
syl56.1 $e |- ( ph -> ps ) $.
|
|
syl56.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
syl56.3 $e |- ( th -> ta ) $.
|
|
$( Combine ~ syl5 and ~ syl6 . (Contributed by NM, 14-Nov-2013.) $)
|
|
syl56 $p |- ( ch -> ( ph -> ta ) ) $=
|
|
( syl6 syl5 ) ABCEFCBDEGHIJ $.
|
|
$}
|
|
|
|
${
|
|
syl6com.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6com.2 $e |- ( ch -> th ) $.
|
|
$( Syllogism inference with commuted antecedents. (Contributed by NM,
|
|
25-May-2005.) $)
|
|
syl6com $p |- ( ps -> ( ph -> th ) ) $=
|
|
( syl6 com12 ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
mpcom.1 $e |- ( ps -> ph ) $.
|
|
mpcom.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Modus ponens inference with commutation of antecedents. (Contributed by
|
|
NM, 17-Mar-1996.) $)
|
|
mpcom $p |- ( ps -> ch ) $=
|
|
( com12 mpd ) BACDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
syli.1 $e |- ( ps -> ( ph -> ch ) ) $.
|
|
syli.2 $e |- ( ch -> ( ph -> th ) ) $.
|
|
$( Syllogism inference with common nested antecedent. (Contributed by NM,
|
|
4-Nov-2004.) $)
|
|
syli $p |- ( ps -> ( ph -> th ) ) $=
|
|
( com12 sylcom ) BACDECADFGH $.
|
|
$}
|
|
|
|
${
|
|
syl2im.1 $e |- ( ph -> ps ) $.
|
|
syl2im.2 $e |- ( ch -> th ) $.
|
|
syl2im.3 $e |- ( ps -> ( th -> ta ) ) $.
|
|
$( Replace two antecedents. Implication-only version of ~ syl2an .
|
|
(Contributed by Wolf Lammen, 14-May-2013.) $)
|
|
syl2im $p |- ( ph -> ( ch -> ta ) ) $=
|
|
( wi syl5 syl ) ABCEIFCDBEGHJK $.
|
|
$}
|
|
|
|
$( This theorem, called "Assertion," can be thought of as closed form of
|
|
modus ponens ~ ax-mp . Theorem *2.27 of [WhiteheadRussell] p. 104.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
pm2.27 $p |- ( ph -> ( ( ph -> ps ) -> ps ) ) $=
|
|
( wi id com12 ) ABCZABFDE $.
|
|
|
|
${
|
|
mpdd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
mpdd.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A nested modus ponens deduction. (Contributed by NM, 12-Dec-2004.) $)
|
|
mpdd $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a2d mpd ) ABCGBDGEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
mpid.1 $e |- ( ph -> ch ) $.
|
|
mpid.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A nested modus ponens deduction. (Contributed by NM, 14-Dec-2004.) $)
|
|
mpid $p |- ( ph -> ( ps -> th ) ) $=
|
|
( a1d mpdd ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
mpdi.1 $e |- ( ps -> ch ) $.
|
|
mpdi.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.)
|
|
(Proof shortened by O'Cat, 15-Jan-2008.) $)
|
|
mpdi $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a1i mpdd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpii.1 $e |- ch $.
|
|
mpii.2 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A doubly nested modus ponens inference. (Contributed by NM,
|
|
31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.) $)
|
|
mpii $p |- ( ph -> ( ps -> th ) ) $=
|
|
( a1i mpdi ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syld.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syld.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( Syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened
|
|
by O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
|
|
|
|
Notice that ~ syld has the same form as ~ syl with ` ph ` added in front
|
|
of each hypothesis and conclusion. When all theorems referenced in a
|
|
proof are converted in this way, we can replace ` ph ` with a hypothesis
|
|
of the proof, allowing the hypothesis to be eliminated with ~ id and
|
|
become an antecedent. The Deduction Theorem for propositional calculus,
|
|
e.g. Theorem 3 in [Margaris] p. 56, tells us that this procedure is
|
|
always possible. $)
|
|
syld $p |- ( ph -> ( ps -> th ) ) $=
|
|
( wi a1d mpdd ) ABCDEACDGBFHI $.
|
|
$}
|
|
|
|
${
|
|
mp2d.1 $e |- ( ph -> ps ) $.
|
|
mp2d.2 $e |- ( ph -> ch ) $.
|
|
mp2d.3 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( A double modus ponens deduction. (Contributed by NM, 23-May-2013.)
|
|
(Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
|
|
mp2d $p |- ( ph -> th ) $=
|
|
( mpid mpd ) ABDEABCDFGHI $.
|
|
$}
|
|
|
|
${
|
|
a1dd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction introducing a nested embedded antecedent. (Contributed by NM,
|
|
17-Dec-2004.) (Proof shortened by O'Cat, 15-Jan-2008.) $)
|
|
a1dd $p |- ( ph -> ( ps -> ( th -> ch ) ) ) $=
|
|
( wi ax-1 syl6 ) ABCDCFECDGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.43i.1 $e |- ( ph -> ( ph -> ps ) ) $.
|
|
$( Inference absorbing redundant antecedent. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) $)
|
|
pm2.43i $p |- ( ph -> ps ) $=
|
|
( id mpd ) AABADCE $.
|
|
$}
|
|
|
|
${
|
|
pm2.43d.1 $e |- ( ph -> ( ps -> ( ps -> ch ) ) ) $.
|
|
$( Deduction absorbing redundant antecedent. (Contributed by NM,
|
|
18-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) $)
|
|
pm2.43d $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( id mpdi ) ABBCBEDF $.
|
|
$}
|
|
|
|
${
|
|
pm2.43a.1 $e |- ( ps -> ( ph -> ( ps -> ch ) ) ) $.
|
|
$( Inference absorbing redundant antecedent. (Contributed by NM,
|
|
7-Nov-1995.) (Proof shortened by O'Cat, 28-Nov-2008.) $)
|
|
pm2.43a $p |- ( ps -> ( ph -> ch ) ) $=
|
|
( id mpid ) BABCBEDF $.
|
|
$}
|
|
|
|
${
|
|
pm2.43b.1 $e |- ( ps -> ( ph -> ( ps -> ch ) ) ) $.
|
|
$( Inference absorbing redundant antecedent. (Contributed by NM,
|
|
31-Oct-1995.) $)
|
|
pm2.43b $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( pm2.43a com12 ) BACABCDEF $.
|
|
$}
|
|
|
|
$( Absorption of redundant antecedent. Also called the "Contraction" or
|
|
"Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat,
|
|
15-Aug-2004.) $)
|
|
pm2.43 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
|
|
( wi pm2.27 a2i ) AABCBABDE $.
|
|
|
|
${
|
|
imim2d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction adding nested antecedents. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
imim2d $p |- ( ph -> ( ( th -> ps ) -> ( th -> ch ) ) ) $=
|
|
( wi a1d a2d ) ADBCABCFDEGH $.
|
|
$}
|
|
|
|
$( A closed form of syllogism (see ~ syl ). Theorem *2.05 of
|
|
[WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 6-Sep-2012.) $)
|
|
imim2 $p |- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) ) $=
|
|
( wi id imim2d ) ABDZABCGEF $.
|
|
|
|
${
|
|
embantd.1 $e |- ( ph -> ps ) $.
|
|
embantd.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( Deduction embedding an antecedent. (Contributed by Wolf Lammen,
|
|
4-Oct-2013.) $)
|
|
embantd $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
|
|
( wi imim2d mpid ) ABCGBDEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
3syld.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3syld.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
3syld.3 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( Triple syllogism deduction. (Contributed by Jeff Hankins,
|
|
4-Aug-2009.) $)
|
|
3syld $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( syld ) ABDEABCDFGIHI $.
|
|
$}
|
|
|
|
${
|
|
sylsyld.1 $e |- ( ph -> ps ) $.
|
|
sylsyld.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
sylsyld.3 $e |- ( ps -> ( th -> ta ) ) $.
|
|
$( Virtual deduction rule e12 without virtual deduction symbols.
|
|
(Contributed by Alan Sare, 20-Apr-2011.) $)
|
|
sylsyld $p |- ( ph -> ( ch -> ta ) ) $=
|
|
( wi syl syld ) ACDEGABDEIFHJK $.
|
|
$}
|
|
|
|
${
|
|
imim12i.1 $e |- ( ph -> ps ) $.
|
|
imim12i.2 $e |- ( ch -> th ) $.
|
|
$( Inference joining two implications. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by O'Cat, 29-Oct-2011.) $)
|
|
imim12i $p |- ( ( ps -> ch ) -> ( ph -> th ) ) $=
|
|
( wi imim2i syl5 ) ABBCGDECDBFHI $.
|
|
$}
|
|
|
|
${
|
|
imim1i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding common consequents in an implication, thereby
|
|
interchanging the original antecedent and consequent. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) $)
|
|
imim1i $p |- ( ( ps -> ch ) -> ( ph -> ch ) ) $=
|
|
( id imim12i ) ABCCDCEF $.
|
|
$}
|
|
|
|
${
|
|
imim3i.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference adding three nested antecedents. (Contributed by NM,
|
|
19-Dec-2006.) $)
|
|
imim3i $p |- ( ( th -> ph ) -> ( ( th -> ps ) -> ( th -> ch ) ) ) $=
|
|
( wi imim2i a2d ) DAFDBCABCFDEGH $.
|
|
$}
|
|
|
|
${
|
|
sylc.1 $e |- ( ph -> ps ) $.
|
|
sylc.2 $e |- ( ph -> ch ) $.
|
|
sylc.3 $e |- ( ps -> ( ch -> th ) ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by NM,
|
|
4-May-1994.) (Revised by NM, 13-Jul-2013.) $)
|
|
sylc $p |- ( ph -> th ) $=
|
|
( syl2im pm2.43i ) ADABACDEFGHI $.
|
|
$}
|
|
|
|
${
|
|
syl3c.1 $e |- ( ph -> ps ) $.
|
|
syl3c.2 $e |- ( ph -> ch ) $.
|
|
syl3c.3 $e |- ( ph -> th ) $.
|
|
syl3c.4 $e |- ( ps -> ( ch -> ( th -> ta ) ) ) $.
|
|
$( A syllogism inference combined with contraction. e111 without virtual
|
|
deductions. (Contributed by Alan Sare, 7-Jul-2011.) $)
|
|
syl3c $p |- ( ph -> ta ) $=
|
|
( wi sylc mpd ) ADEHABCDEJFGIKL $.
|
|
$}
|
|
|
|
${
|
|
syl6mpi.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6mpi.2 $e |- th $.
|
|
syl6mpi.3 $e |- ( ch -> ( th -> ta ) ) $.
|
|
$( e20 without virtual deductions. (Contributed by Alan Sare,
|
|
8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.) $)
|
|
syl6mpi $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( mpi syl6 ) ABCEFCDEGHIJ $.
|
|
$}
|
|
|
|
${
|
|
mpsyl.1 $e |- ph $.
|
|
mpsyl.2 $e |- ( ps -> ch ) $.
|
|
mpsyl.3 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( Modus ponens combined with a syllogism inference. (Contributed by Alan
|
|
Sare, 20-Apr-2011.) $)
|
|
mpsyl $p |- ( ps -> th ) $=
|
|
( a1i sylc ) BACDABEHFGI $.
|
|
$}
|
|
|
|
${
|
|
syl6c.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6c.2 $e |- ( ph -> ( ps -> th ) ) $.
|
|
syl6c.3 $e |- ( ch -> ( th -> ta ) ) $.
|
|
$( Inference combining ~ syl6 with contraction. (Contributed by Alan Sare,
|
|
2-May-2011.) $)
|
|
syl6c $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( wi syl6 mpdd ) ABDEGABCDEIFHJK $.
|
|
$}
|
|
|
|
${
|
|
syldd.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
syldd.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.
|
|
$( Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof
|
|
shortened by Wolf Lammen, 11-May-2013.) $)
|
|
syldd $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
|
|
( wi imim2 syl6c ) ABDEHCDHCEHGFDECIJ $.
|
|
$}
|
|
|
|
${
|
|
syl5d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl5d.2 $e |- ( ph -> ( th -> ( ch -> ta ) ) ) $.
|
|
$( A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat,
|
|
2-Feb-2006.) $)
|
|
syl5d $p |- ( ph -> ( th -> ( ps -> ta ) ) ) $=
|
|
( wi a1d syldd ) ADBCEABCHDFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl7.1 $e |- ( ph -> ps ) $.
|
|
syl7.2 $e |- ( ch -> ( th -> ( ps -> ta ) ) ) $.
|
|
$( A syllogism rule of inference. The first premise is used to replace the
|
|
third antecedent of the second premise. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) $)
|
|
syl7 $p |- ( ch -> ( th -> ( ph -> ta ) ) ) $=
|
|
( wi a1i syl5d ) CABDEABHCFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl6d.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
syl6d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat,
|
|
2-Feb-2006.) $)
|
|
syl6d $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
|
|
( wi a1d syldd ) ABCDEFADEHBGIJ $.
|
|
$}
|
|
|
|
${
|
|
syl8.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
syl8.2 $e |- ( th -> ta ) $.
|
|
$( A syllogism rule of inference. The second premise is used to replace
|
|
the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)
|
|
(Proof shortened by Wolf Lammen, 3-Aug-2012.) $)
|
|
syl8 $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
|
|
( wi a1i syl6d ) ABCDEFDEHAGIJ $.
|
|
$}
|
|
|
|
${
|
|
syl9.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl9.2 $e |- ( th -> ( ch -> ta ) ) $.
|
|
$( A nested syllogism inference with different antecedents. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) $)
|
|
syl9 $p |- ( ph -> ( th -> ( ps -> ta ) ) ) $=
|
|
( wi a1i syl5d ) ABCDEFDCEHHAGIJ $.
|
|
$}
|
|
|
|
${
|
|
syl9r.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl9r.2 $e |- ( th -> ( ch -> ta ) ) $.
|
|
$( A nested syllogism inference with different antecedents. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
syl9r $p |- ( th -> ( ph -> ( ps -> ta ) ) ) $=
|
|
( wi syl9 com12 ) ADBEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
imim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
imim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( Deduction combining antecedents and consequents. (Contributed by NM,
|
|
7-Aug-1994.) (Proof shortened by O'Cat, 30-Oct-2011.) $)
|
|
imim12d $p |- ( ph -> ( ( ch -> th ) -> ( ps -> ta ) ) ) $=
|
|
( wi imim2d syl5d ) ABCCDHEFADECGIJ $.
|
|
$}
|
|
|
|
${
|
|
imim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction adding nested consequents. (Contributed by NM, 3-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 12-Sep-2012.) $)
|
|
imim1d $p |- ( ph -> ( ( ch -> th ) -> ( ps -> th ) ) ) $=
|
|
( idd imim12d ) ABCDDEADFG $.
|
|
$}
|
|
|
|
$( A closed form of syllogism (see ~ syl ). Theorem *2.06 of
|
|
[WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 25-May-2013.) $)
|
|
imim1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( wi id imim1d ) ABDZABCGEF $.
|
|
|
|
$( Theorem *2.83 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.83 $p |- ( ( ph -> ( ps -> ch ) )
|
|
-> ( ( ph -> ( ch -> th ) ) -> ( ph -> ( ps -> th ) ) ) ) $=
|
|
( wi imim1 imim3i ) BCECDEBDEABCDFG $.
|
|
|
|
${
|
|
com3.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Commutation of antecedents. Swap 2nd and 3rd. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) $)
|
|
com23 $p |- ( ph -> ( ch -> ( ps -> th ) ) ) $=
|
|
( wi pm2.27 syl9 ) ABCDFCDECDGH $.
|
|
|
|
$( Commutation of antecedents. Rotate right. (Contributed by NM,
|
|
25-Apr-1994.) $)
|
|
com3r $p |- ( ch -> ( ph -> ( ps -> th ) ) ) $=
|
|
( wi com23 com12 ) ACBDFABCDEGH $.
|
|
|
|
$( Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
|
|
com13 $p |- ( ch -> ( ps -> ( ph -> th ) ) ) $=
|
|
( com3r com23 ) CABDABCDEFG $.
|
|
|
|
$( Commutation of antecedents. Rotate left. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
|
|
com3l $p |- ( ps -> ( ch -> ( ph -> th ) ) ) $=
|
|
( com3r ) CABDABCDEFF $.
|
|
$}
|
|
|
|
$( Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
12-Sep-2012.) $)
|
|
pm2.04 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
|
|
( wi id com23 ) ABCDDZABCGEF $.
|
|
|
|
${
|
|
com4.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
|
|
$( Commutation of antecedents. Swap 3rd and 4th. (Contributed by NM,
|
|
25-Apr-1994.) $)
|
|
com34 $p |- ( ph -> ( ps -> ( th -> ( ch -> ta ) ) ) ) $=
|
|
( wi pm2.04 syl6 ) ABCDEGGDCEGGFCDEHI $.
|
|
|
|
$( Commutation of antecedents. Rotate left. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by O'Cat, 15-Aug-2004.) $)
|
|
com4l $p |- ( ps -> ( ch -> ( th -> ( ph -> ta ) ) ) ) $=
|
|
( wi com3l com34 ) BCADEABCDEGFHI $.
|
|
|
|
$( Commutation of antecedents. Rotate twice. (Contributed by NM,
|
|
25-Apr-1994.) $)
|
|
com4t $p |- ( ch -> ( th -> ( ph -> ( ps -> ta ) ) ) ) $=
|
|
( com4l ) BCDAEABCDEFGG $.
|
|
|
|
$( Commutation of antecedents. Rotate right. (Contributed by NM,
|
|
25-Apr-1994.) $)
|
|
com4r $p |- ( th -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) $=
|
|
( com4t com4l ) CDABEABCDEFGH $.
|
|
|
|
$( Commutation of antecedents. Swap 2nd and 4th. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
|
|
com24 $p |- ( ph -> ( th -> ( ch -> ( ps -> ta ) ) ) ) $=
|
|
( wi com4t com13 ) CDABEGABCDEFHI $.
|
|
|
|
$( Commutation of antecedents. Swap 1st and 4th. (Contributed by NM,
|
|
25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) $)
|
|
com14 $p |- ( th -> ( ps -> ( ch -> ( ph -> ta ) ) ) ) $=
|
|
( wi com4l com3r ) BCDAEGABCDEFHI $.
|
|
$}
|
|
|
|
${
|
|
com5.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $.
|
|
$( Commutation of antecedents. Swap 4th and 5th. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com45 $p |- ( ph -> ( ps -> ( ch -> ( ta -> ( th -> et ) ) ) ) ) $=
|
|
( wi pm2.04 syl8 ) ABCDEFHHEDFHHGDEFIJ $.
|
|
|
|
$( Commutation of antecedents. Swap 3rd and 5th. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com35 $p |- ( ph -> ( ps -> ( ta -> ( th -> ( ch -> et ) ) ) ) ) $=
|
|
( wi com34 com45 ) ABDECFHABDCEFABCDEFHGIJI $.
|
|
|
|
$( Commutation of antecedents. Swap 2nd and 5th. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com25 $p |- ( ph -> ( ta -> ( ch -> ( th -> ( ps -> et ) ) ) ) ) $=
|
|
( wi com24 com45 ) ADCEBFHADCBEFABCDEFHGIJI $.
|
|
|
|
$( Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins,
|
|
28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.) $)
|
|
com5l $p |- ( ps -> ( ch -> ( th -> ( ta -> ( ph -> et ) ) ) ) ) $=
|
|
( wi com4l com45 ) BCDAEFABCDEFHGIJ $.
|
|
|
|
$( Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen,
|
|
29-Jul-2012.) $)
|
|
com15 $p |- ( ta -> ( ps -> ( ch -> ( th -> ( ph -> et ) ) ) ) ) $=
|
|
( wi com5l com4r ) BCDEAFHABCDEFGIJ $.
|
|
|
|
$( Commutation of antecedents. Rotate left twice. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com52l $p |- ( ch -> ( th -> ( ta -> ( ph -> ( ps -> et ) ) ) ) ) $=
|
|
( com5l ) BCDEAFABCDEFGHH $.
|
|
|
|
$( Commutation of antecedents. Rotate right twice. (Contributed by Jeff
|
|
Hankins, 28-Jun-2009.) $)
|
|
com52r $p |- ( th -> ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) ) ) $=
|
|
( com52l com5l ) CDEABFABCDEFGHI $.
|
|
|
|
$( Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen,
|
|
29-Jul-2012.) $)
|
|
com5r $p |- ( ta -> ( ph -> ( ps -> ( ch -> ( th -> et ) ) ) ) ) $=
|
|
( com52l ) CDEABFABCDEFGHH $.
|
|
$}
|
|
|
|
$( Elimination of a nested antecedent as a kind of reversal of inference
|
|
~ ja . (Contributed by Wolf Lammen, 9-May-2013.) $)
|
|
jarr $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $=
|
|
( wi ax-1 imim1i ) BABDCBAEF $.
|
|
|
|
${
|
|
pm2.86i.1 $e |- ( ( ph -> ps ) -> ( ph -> ch ) ) $.
|
|
$( Inference based on ~ pm2.86 . (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 3-Apr-2013.) $)
|
|
pm2.86i $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wi ax-1 syl com12 ) BACBABEACEBAFDGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.86d.1 $e |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $.
|
|
$( Deduction based on ~ pm2.86 . (Contributed by NM, 29-Jun-1995.) (Proof
|
|
shortened by Wolf Lammen, 3-Apr-2013.) $)
|
|
pm2.86d $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wi ax-1 syl5 com23 ) ACBDCBCFABDFCBGEHI $.
|
|
$}
|
|
|
|
$( Converse of axiom ~ ax-2 . Theorem *2.86 of [WhiteheadRussell] p. 108.
|
|
(Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen,
|
|
3-Apr-2013.) $)
|
|
pm2.86 $p |- ( ( ( ph -> ps )
|
|
-> ( ph -> ch ) ) -> ( ph -> ( ps -> ch ) ) ) $=
|
|
( wi id pm2.86d ) ABDACDDZABCGEF $.
|
|
|
|
$( The Linearity Axiom of the infinite-valued sentential logic (L-infinity)
|
|
of Lukasiewicz. This version of ~ loolin does not use ~ ax-3 , meaning
|
|
that this theorem is intuitionistically valid. (Contributed by O'Cat,
|
|
12-Aug-2004.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
loolinALT $p |- ( ( ( ph -> ps ) -> ( ps -> ph ) ) -> ( ps -> ph ) ) $=
|
|
( wi jarr pm2.43d ) ABCBACZCBAABFDE $.
|
|
|
|
$( An alternate for the Linearity Axiom of the infinite-valued sentential
|
|
logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, _Reports
|
|
on Mathematical Logic_ 10, 129-137 (1978). (Contributed by O'Cat,
|
|
8-Aug-2004.) $)
|
|
loowoz $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) )
|
|
-> ( ( ps -> ph ) -> ( ps -> ch ) ) ) $=
|
|
( wi jarr a2d ) ABDACDZDBACABGEF $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical negation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
This section makes our first use of the third axiom of propositional
|
|
calculus, ~ ax-3 .
|
|
|
|
$)
|
|
|
|
${
|
|
con4d.1 $e |- ( ph -> ( -. ps -> -. ch ) ) $.
|
|
$( Deduction derived from axiom ~ ax-3 . (Contributed by NM,
|
|
26-Mar-1995.) $)
|
|
con4d $p |- ( ph -> ( ch -> ps ) ) $=
|
|
( wn wi ax-3 syl ) ABECEFCBFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.21d.1 $e |- ( ph -> -. ps ) $.
|
|
$( A contradiction implies anything. Deduction from ~ pm2.21 .
|
|
(Contributed by NM, 10-Feb-1996.) $)
|
|
pm2.21d $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wn a1d con4d ) ACBABECEDFG $.
|
|
$}
|
|
|
|
${
|
|
pm2.21dd.1 $e |- ( ph -> ps ) $.
|
|
pm2.21dd.2 $e |- ( ph -> -. ps ) $.
|
|
$( A contradiction implies anything. Deduction from ~ pm2.21 .
|
|
(Contributed by Mario Carneiro, 9-Feb-2017.) $)
|
|
pm2.21dd $p |- ( ph -> ch ) $=
|
|
( pm2.21d mpd ) ABCDABCEFG $.
|
|
$}
|
|
|
|
$( From a wff and its negation, anything is true. Theorem *2.21 of
|
|
[WhiteheadRussell] p. 104. Also called the Duns Scotus law. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Sep-2012.) $)
|
|
pm2.21 $p |- ( -. ph -> ( ph -> ps ) ) $=
|
|
( wn id pm2.21d ) ACZABFDE $.
|
|
|
|
$( Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.24 $p |- ( ph -> ( -. ph -> ps ) ) $=
|
|
( wn pm2.21 com12 ) ACABABDE $.
|
|
|
|
$( Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also
|
|
called the Law of Clavius. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm2.18 $p |- ( ( -. ph -> ph ) -> ph ) $=
|
|
( wn wi pm2.21 a2i con4d pm2.43i ) ABZACZAIAIHAIBZAJDEFG $.
|
|
|
|
${
|
|
pm2.18d.1 $e |- ( ph -> ( -. ps -> ps ) ) $.
|
|
$( Deduction based on reductio ad absurdum. (Contributed by FL,
|
|
12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
pm2.18d $p |- ( ph -> ps ) $=
|
|
( wn wi pm2.18 syl ) ABDBEBCBFG $.
|
|
$}
|
|
|
|
$( Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by David Harvey,
|
|
5-Sep-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) $)
|
|
notnot2 $p |- ( -. -. ph -> ph ) $=
|
|
( wn pm2.21 pm2.18d ) ABZBAEACD $.
|
|
|
|
${
|
|
notnotrd.1 $e |- ( ph -> -. -. ps ) $.
|
|
$( Deduction converting double-negation into the original wff, aka the
|
|
double negation rule. A translation of natural deduction rule ` -. -. `
|
|
-C, Gamma ` |- -. -. ps ` => Gamma ` |- ps ` ; see natded in set.mm.
|
|
This is definition NNC in [Pfenning] p. 17. This rule is valid in
|
|
classical logic (which MPE uses), but not intuitionistic logic.
|
|
(Contributed by DAW, 8-Feb-2017.) $)
|
|
notnotrd $p |- ( ph -> ps ) $=
|
|
( wn notnot2 syl ) ABDDBCBEF $.
|
|
$}
|
|
|
|
${
|
|
notnotri.1 $e |- -. -. ph $.
|
|
$( Inference from double negation. (Contributed by NM, 27-Feb-2008.) $)
|
|
notnotri $p |- ph $=
|
|
( wn notnot2 ax-mp ) ACCABADE $.
|
|
$}
|
|
|
|
${
|
|
con2d.1 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
$( A contraposition deduction. (Contributed by NM, 19-Aug-1993.) $)
|
|
con2d $p |- ( ph -> ( ch -> -. ps ) ) $=
|
|
( wn notnot2 syl5 con4d ) ABEZCIEBACEBFDGH $.
|
|
$}
|
|
|
|
$( Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
|
|
con2 $p |- ( ( ph -> -. ps ) -> ( ps -> -. ph ) ) $=
|
|
( wn wi id con2d ) ABCDZABGEF $.
|
|
|
|
${
|
|
mt2d.1 $e |- ( ph -> ch ) $.
|
|
mt2d.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
$( Modus tollens deduction. (Contributed by NM, 4-Jul-1994.) $)
|
|
mt2d $p |- ( ph -> -. ps ) $=
|
|
( wn con2d mpd ) ACBFDABCEGH $.
|
|
$}
|
|
|
|
${
|
|
mt2i.1 $e |- ch $.
|
|
mt2i.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
$( Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof
|
|
shortened by Wolf Lammen, 15-Sep-2012.) $)
|
|
mt2i $p |- ( ph -> -. ps ) $=
|
|
( a1i mt2d ) ABCCADFEG $.
|
|
$}
|
|
|
|
${
|
|
nsyl3.1 $e |- ( ph -> -. ps ) $.
|
|
nsyl3.2 $e |- ( ch -> ps ) $.
|
|
$( A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) $)
|
|
nsyl3 $p |- ( ch -> -. ph ) $=
|
|
( wn wi a1i mt2d ) CABEABFGCDHI $.
|
|
$}
|
|
|
|
${
|
|
con2i.a $e |- ( ph -> -. ps ) $.
|
|
$( A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen,
|
|
13-Jun-2013.) $)
|
|
con2i $p |- ( ps -> -. ph ) $=
|
|
( id nsyl3 ) ABBCBDE $.
|
|
$}
|
|
|
|
${
|
|
nsyl.1 $e |- ( ph -> -. ps ) $.
|
|
nsyl.2 $e |- ( ch -> ps ) $.
|
|
$( A negated syllogism inference. (Contributed by NM, 31-Dec-1993.)
|
|
(Proof shortened by Wolf Lammen, 2-Mar-2013.) $)
|
|
nsyl $p |- ( ph -> -. ch ) $=
|
|
( nsyl3 con2i ) CAABCDEFG $.
|
|
$}
|
|
|
|
$( Converse of double negation. Theorem *2.12 of [WhiteheadRussell] p. 101.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
2-Mar-2013.) $)
|
|
notnot1 $p |- ( ph -> -. -. ph ) $=
|
|
( wn id con2i ) ABZAECD $.
|
|
|
|
${
|
|
negbi.1 $e |- ph $.
|
|
$( Infer double negation. (Contributed by NM, 27-Feb-2008.) $)
|
|
notnoti $p |- -. -. ph $=
|
|
( wn notnot1 ax-mp ) AACCBADE $.
|
|
$}
|
|
|
|
${
|
|
con1d.1 $e |- ( ph -> ( -. ps -> ch ) ) $.
|
|
$( A contraposition deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
con1d $p |- ( ph -> ( -. ch -> ps ) ) $=
|
|
( wn notnot1 syl6 con4d ) ABCEZABECIEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
mt3d.1 $e |- ( ph -> -. ch ) $.
|
|
mt3d.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
|
|
$( Modus tollens deduction. (Contributed by NM, 26-Mar-1995.) $)
|
|
mt3d $p |- ( ph -> ps ) $=
|
|
( wn con1d mpd ) ACFBDABCEGH $.
|
|
$}
|
|
|
|
${
|
|
mt3i.1 $e |- -. ch $.
|
|
mt3i.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
|
|
$( Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof
|
|
shortened by Wolf Lammen, 15-Sep-2012.) $)
|
|
mt3i $p |- ( ph -> ps ) $=
|
|
( wn a1i mt3d ) ABCCFADGEH $.
|
|
$}
|
|
|
|
${
|
|
nsyl2.1 $e |- ( ph -> -. ps ) $.
|
|
nsyl2.2 $e |- ( -. ch -> ps ) $.
|
|
$( A negated syllogism inference. (Contributed by NM, 26-Jun-1994.) $)
|
|
nsyl2 $p |- ( ph -> ch ) $=
|
|
( wn wi a1i mt3d ) ACBDCFBGAEHI $.
|
|
$}
|
|
|
|
$( Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
|
|
con1 $p |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) $=
|
|
( wn wi id con1d ) ACBDZABGEF $.
|
|
|
|
${
|
|
con1i.a $e |- ( -. ph -> ps ) $.
|
|
$( A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen,
|
|
19-Jun-2013.) $)
|
|
con1i $p |- ( -. ps -> ph ) $=
|
|
( wn id nsyl2 ) BDZBAGECF $.
|
|
$}
|
|
|
|
${
|
|
con4i.1 $e |- ( -. ph -> -. ps ) $.
|
|
$( Inference rule derived from axiom ~ ax-3 . (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Jun-2013.) $)
|
|
con4i $p |- ( ps -> ph ) $=
|
|
( wn notnot1 nsyl2 ) BBDABECF $.
|
|
$}
|
|
|
|
${
|
|
pm2.21i.1 $e |- -. ph $.
|
|
$( A contradiction implies anything. Inference from ~ pm2.21 .
|
|
(Contributed by NM, 16-Sep-1993.) $)
|
|
pm2.21i $p |- ( ph -> ps ) $=
|
|
( wn a1i con4i ) BAADBDCEF $.
|
|
$}
|
|
|
|
${
|
|
pm2.24ii.1 $e |- ph $.
|
|
pm2.24ii.2 $e |- -. ph $.
|
|
$( A contradiction implies anything. Inference from ~ pm2.24 .
|
|
(Contributed by NM, 27-Feb-2008.) $)
|
|
pm2.24ii $p |- ps $=
|
|
( pm2.21i ax-mp ) ABCABDEF $.
|
|
$}
|
|
|
|
${
|
|
con3d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( A contraposition deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
con3d $p |- ( ph -> ( -. ch -> -. ps ) ) $=
|
|
( wn notnot2 syl5 con1d ) ABEZCIEBACBFDGH $.
|
|
$}
|
|
|
|
$( Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.) $)
|
|
con3 $p |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) ) $=
|
|
( wi id con3d ) ABCZABFDE $.
|
|
|
|
${
|
|
con3i.a $e |- ( ph -> ps ) $.
|
|
$( A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 20-Jun-2013.) $)
|
|
con3i $p |- ( -. ps -> -. ph ) $=
|
|
( wn id nsyl ) BDZBAGECF $.
|
|
$}
|
|
|
|
${
|
|
con3rr3.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Rotate through consequent right. (Contributed by Wolf Lammen,
|
|
3-Nov-2013.) $)
|
|
con3rr3 $p |- ( -. ch -> ( ph -> -. ps ) ) $=
|
|
( wn con3d com12 ) ACEBEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
mt4.1 $e |- ph $.
|
|
mt4.2 $e |- ( -. ps -> -. ph ) $.
|
|
$( The rule of modus tollens. (Contributed by Wolf Lammen,
|
|
12-May-2013.) $)
|
|
mt4 $p |- ps $=
|
|
( con4i ax-mp ) ABCBADEF $.
|
|
$}
|
|
|
|
${
|
|
mt4d.1 $e |- ( ph -> ps ) $.
|
|
mt4d.2 $e |- ( ph -> ( -. ch -> -. ps ) ) $.
|
|
$( Modus tollens deduction. (Contributed by NM, 9-Jun-2006.) $)
|
|
mt4d $p |- ( ph -> ch ) $=
|
|
( con4d mpd ) ABCDACBEFG $.
|
|
$}
|
|
|
|
${
|
|
mt4i.1 $e |- ch $.
|
|
mt4i.2 $e |- ( ph -> ( -. ps -> -. ch ) ) $.
|
|
$( Modus tollens inference. (Contributed by Wolf Lammen, 12-May-2013.) $)
|
|
mt4i $p |- ( ph -> ps ) $=
|
|
( a1i mt4d ) ACBCADFEG $.
|
|
$}
|
|
|
|
${
|
|
nsyld.1 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
nsyld.2 $e |- ( ph -> ( ta -> ch ) ) $.
|
|
$( A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.) $)
|
|
nsyld $p |- ( ph -> ( ps -> -. ta ) ) $=
|
|
( wn con3d syld ) ABCGDGEADCFHI $.
|
|
$}
|
|
|
|
${
|
|
nsyli.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
nsyli.2 $e |- ( th -> -. ch ) $.
|
|
$( A negated syllogism inference. (Contributed by NM, 3-May-1994.) $)
|
|
nsyli $p |- ( ph -> ( th -> -. ps ) ) $=
|
|
( wn con3d syl5 ) DCGABGFABCEHI $.
|
|
$}
|
|
|
|
${
|
|
nsyl4.1 $e |- ( ph -> ps ) $.
|
|
nsyl4.2 $e |- ( -. ph -> ch ) $.
|
|
$( A negated syllogism inference. (Contributed by NM, 15-Feb-1996.) $)
|
|
nsyl4 $p |- ( -. ch -> ps ) $=
|
|
( wn con1i syl ) CFABACEGDH $.
|
|
$}
|
|
|
|
${
|
|
pm2.24d.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction version of ~ pm2.24 . (Contributed by NM, 30-Jan-2006.) $)
|
|
pm2.24d $p |- ( ph -> ( -. ps -> ch ) ) $=
|
|
( wn a1d con1d ) ACBABCEDFG $.
|
|
$}
|
|
|
|
${
|
|
pm2.24i.1 $e |- ph $.
|
|
$( Inference version of ~ pm2.24 . (Contributed by NM, 20-Aug-2001.) $)
|
|
pm2.24i $p |- ( -. ph -> ps ) $=
|
|
( wn a1i con1i ) BAABDCEF $.
|
|
$}
|
|
|
|
$( Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive
|
|
connectives. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh
|
|
Purinton, 29-Dec-2000.) $)
|
|
pm3.2im $p |- ( ph -> ( ps -> -. ( ph -> -. ps ) ) ) $=
|
|
( wn wi pm2.27 con2d ) AABCZDBAGEF $.
|
|
|
|
$( Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Josh Purinton, 29-Dec-2000.) $)
|
|
mth8 $p |- ( ph -> ( -. ps -> -. ( ph -> ps ) ) ) $=
|
|
( wi pm2.27 con3d ) AABCBABDE $.
|
|
|
|
${
|
|
jc.1 $e |- ( ph -> ps ) $.
|
|
jc.2 $e |- ( ph -> ch ) $.
|
|
$( Inference joining the consequents of two premises. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
jc $p |- ( ph -> -. ( ps -> -. ch ) ) $=
|
|
( wn wi pm3.2im sylc ) ABCBCFGFDEBCHI $.
|
|
$}
|
|
|
|
${
|
|
impi.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( An importation inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 20-Jul-2013.) $)
|
|
impi $p |- ( -. ( ph -> -. ps ) -> ch ) $=
|
|
( wn wi con3rr3 con1i ) CABEFABCDGH $.
|
|
$}
|
|
|
|
${
|
|
expi.1 $e |- ( -. ( ph -> -. ps ) -> ch ) $.
|
|
$( An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by O'Cat, 28-Nov-2008.) $)
|
|
expi $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wn wi pm3.2im syl6 ) ABABEFECABGDH $.
|
|
$}
|
|
|
|
$( Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell]
|
|
p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 13-Nov-2012.) $)
|
|
simprim $p |- ( -. ( ph -> -. ps ) -> ps ) $=
|
|
( idd impi ) ABBABCD $.
|
|
|
|
$( Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell]
|
|
p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 21-Jul-2012.) $)
|
|
simplim $p |- ( -. ( ph -> ps ) -> ph ) $=
|
|
( wi pm2.21 con1i ) AABCABDE $.
|
|
|
|
$( Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 9-Oct-2012.) $)
|
|
pm2.5 $p |- ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) $=
|
|
( wi wn simplim pm2.24d ) ABCDABABEF $.
|
|
|
|
$( Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.51 $p |- ( -. ( ph -> ps ) -> ( ph -> -. ps ) ) $=
|
|
( wi wn ax-1 con3i a1d ) ABCZDBDABHBAEFG $.
|
|
|
|
$( Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.) $)
|
|
pm2.521 $p |- ( -. ( ph -> ps ) -> ( ps -> ph ) ) $=
|
|
( wi wn simplim a1d ) ABCDABABEF $.
|
|
|
|
$( Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.) $)
|
|
pm2.52 $p |- ( -. ( ph -> ps ) -> ( -. ph -> -. ps ) ) $=
|
|
( wi wn pm2.521 con3d ) ABCDBAABEF $.
|
|
|
|
$( Exportation theorem expressed with primitive connectives. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
expt $p |- ( ( -. ( ph -> -. ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) $=
|
|
( wn wi pm3.2im imim1d com12 ) AABDEDZCEBCEABICABFGH $.
|
|
|
|
$( Importation theorem expressed with primitive connectives. (Contributed by
|
|
NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2013.) $)
|
|
impt $p |- ( ( ph -> ( ps -> ch ) ) -> ( -. ( ph -> -. ps ) -> ch ) ) $=
|
|
( wi wn simprim simplim imim1i mpdi ) ABCDZDABEZDEZBCABFLAJAKGHI $.
|
|
|
|
${
|
|
pm2.61d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
pm2.61d.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
|
|
$( Deduction eliminating an antecedent. (Contributed by NM, 27-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 12-Sep-2013.) $)
|
|
pm2.61d $p |- ( ph -> ch ) $=
|
|
( wn con1d syld pm2.18d ) ACACFBCABCEGDHI $.
|
|
$}
|
|
|
|
${
|
|
pm2.61d1.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
pm2.61d1.2 $e |- ( -. ps -> ch ) $.
|
|
$( Inference eliminating an antecedent. (Contributed by NM,
|
|
15-Jul-2005.) $)
|
|
pm2.61d1 $p |- ( ph -> ch ) $=
|
|
( wn wi a1i pm2.61d ) ABCDBFCGAEHI $.
|
|
$}
|
|
|
|
${
|
|
pm2.61d2.1 $e |- ( ph -> ( -. ps -> ch ) ) $.
|
|
pm2.61d2.2 $e |- ( ps -> ch ) $.
|
|
$( Inference eliminating an antecedent. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
pm2.61d2 $p |- ( ph -> ch ) $=
|
|
( wi a1i pm2.61d ) ABCBCFAEGDH $.
|
|
$}
|
|
|
|
${
|
|
ja.1 $e |- ( -. ph -> ch ) $.
|
|
ja.2 $e |- ( ps -> ch ) $.
|
|
$( Inference joining the antecedents of two premises. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by O'Cat, 19-Feb-2008.) $)
|
|
ja $p |- ( ( ph -> ps ) -> ch ) $=
|
|
( wi imim2i pm2.61d1 ) ABFACBCAEGDH $.
|
|
$}
|
|
|
|
${
|
|
jad.1 $e |- ( ph -> ( -. ps -> th ) ) $.
|
|
jad.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( Deduction form of ~ ja . (Contributed by Scott Fenton, 13-Dec-2010.)
|
|
(Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
jad $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
|
|
( wi wn com12 ja ) BCGADBCADGABHDEIACDFIJI $.
|
|
$}
|
|
|
|
$( Elimination of a nested antecedent as a kind of reversal of inference
|
|
~ ja . (Contributed by Wolf Lammen, 10-May-2013.) $)
|
|
jarl $p |- ( ( ( ph -> ps ) -> ch ) -> ( -. ph -> ch ) ) $=
|
|
( wn wi pm2.21 imim1i ) ADABECABFG $.
|
|
|
|
${
|
|
pm2.61i.1 $e |- ( ph -> ps ) $.
|
|
pm2.61i.2 $e |- ( -. ph -> ps ) $.
|
|
$( Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 12-Sep-2013.) $)
|
|
pm2.61i $p |- ps $=
|
|
( wi id ja ax-mp ) AAEBAFAABDCGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.61ii.1 $e |- ( -. ph -> ( -. ps -> ch ) ) $.
|
|
pm2.61ii.2 $e |- ( ph -> ch ) $.
|
|
pm2.61ii.3 $e |- ( ps -> ch ) $.
|
|
$( Inference eliminating two antecedents. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) $)
|
|
pm2.61ii $p |- ch $=
|
|
( wn pm2.61d2 pm2.61i ) ACEAGBCDFHI $.
|
|
$}
|
|
|
|
${
|
|
pm2.61nii.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
pm2.61nii.2 $e |- ( -. ph -> ch ) $.
|
|
pm2.61nii.3 $e |- ( -. ps -> ch ) $.
|
|
$( Inference eliminating two antecedents. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof
|
|
shortened by Wolf Lammen, 13-Nov-2012.) $)
|
|
pm2.61nii $p |- ch $=
|
|
( pm2.61d1 pm2.61i ) ACABCDFGEH $.
|
|
$}
|
|
|
|
${
|
|
pm2.61iii.1 $e |- ( -. ph -> ( -. ps -> ( -. ch -> th ) ) ) $.
|
|
pm2.61iii.2 $e |- ( ph -> th ) $.
|
|
pm2.61iii.3 $e |- ( ps -> th ) $.
|
|
pm2.61iii.4 $e |- ( ch -> th ) $.
|
|
$( Inference eliminating three antecedents. (Contributed by NM,
|
|
2-Jan-2002.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)
|
|
pm2.61iii $p |- th $=
|
|
( wn wi a1d pm2.61ii pm2.61i ) CDHABCIZDJEADNFKBDNGKLM $.
|
|
$}
|
|
|
|
$( Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100.
|
|
(Contributed by NM, 18-Aug-1993.) (Proof shortened by O'Cat,
|
|
21-Nov-2008.) (Proof shortened by Wolf Lammen, 31-Oct-2012.) $)
|
|
pm2.01 $p |- ( ( ph -> -. ph ) -> -. ph ) $=
|
|
( wn id ja ) AABZEECZFD $.
|
|
|
|
${
|
|
pm2.01d.1 $e |- ( ph -> ( ps -> -. ps ) ) $.
|
|
$( Deduction based on reductio ad absurdum. (Contributed by NM,
|
|
18-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Mar-2013.) $)
|
|
pm2.01d $p |- ( ph -> -. ps ) $=
|
|
( wn id pm2.61d1 ) ABBDZCGEF $.
|
|
$}
|
|
|
|
$( Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.6 $p |- ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) $=
|
|
( wn wi id idd jad ) ACBDZABBHEHBFG $.
|
|
|
|
$( Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an
|
|
antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 22-Sep-2013.) $)
|
|
pm2.61 $p |- ( ( ph -> ps ) -> ( ( -. ph -> ps ) -> ps ) ) $=
|
|
( wn wi pm2.6 com12 ) ACBDABDBABEF $.
|
|
|
|
$( Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
8-Mar-2013.) $)
|
|
pm2.65 $p |- ( ( ph -> ps ) -> ( ( ph -> -. ps ) -> -. ph ) ) $=
|
|
( wi wn idd con3 jad ) ABCZABDADZHIEABFG $.
|
|
|
|
${
|
|
pm2.65i.1 $e |- ( ph -> ps ) $.
|
|
pm2.65i.2 $e |- ( ph -> -. ps ) $.
|
|
$( Inference rule for proof by contradiction. (Contributed by NM,
|
|
18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) $)
|
|
pm2.65i $p |- -. ph $=
|
|
( wn con2i con3i pm2.61i ) BAEABDFABCGH $.
|
|
$}
|
|
|
|
${
|
|
pm2.65d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
pm2.65d.2 $e |- ( ph -> ( ps -> -. ch ) ) $.
|
|
$( Deduction rule for proof by contradiction. (Contributed by NM,
|
|
26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.) $)
|
|
pm2.65d $p |- ( ph -> -. ps ) $=
|
|
( nsyld pm2.01d ) ABABCBEDFG $.
|
|
$}
|
|
|
|
${
|
|
mto.1 $e |- -. ps $.
|
|
mto.2 $e |- ( ph -> ps ) $.
|
|
$( The rule of modus tollens. The rule says, "if ` ps ` is not true, and
|
|
` ph ` implies ` ps ` , then ` ps ` must also be not true." Modus
|
|
tollens is short for "modus tollendo tollens," a Latin phrase that means
|
|
"the mood that by denying affirms" [Sanford] p. 39. It is also called
|
|
denying the consequent. Modus tollens is closely related to modus
|
|
ponens ~ ax-mp . (Contributed by NM, 19-Aug-1993.) (Proof shortened by
|
|
Wolf Lammen, 11-Sep-2013.) $)
|
|
mto $p |- -. ph $=
|
|
( wn a1i pm2.65i ) ABDBEACFG $.
|
|
$}
|
|
|
|
${
|
|
mtod.1 $e |- ( ph -> -. ch ) $.
|
|
mtod.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 11-Sep-2013.) $)
|
|
mtod $p |- ( ph -> -. ps ) $=
|
|
( wn a1d pm2.65d ) ABCEACFBDGH $.
|
|
$}
|
|
|
|
${
|
|
mtoi.1 $e |- -. ch $.
|
|
mtoi.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof
|
|
shortened by Wolf Lammen, 15-Sep-2012.) $)
|
|
mtoi $p |- ( ph -> -. ps ) $=
|
|
( wn a1i mtod ) ABCCFADGEH $.
|
|
$}
|
|
|
|
${
|
|
mt2.1 $e |- ps $.
|
|
mt2.2 $e |- ( ph -> -. ps ) $.
|
|
$( A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 10-Sep-2013.) $)
|
|
mt2 $p |- -. ph $=
|
|
( a1i pm2.65i ) ABBACEDF $.
|
|
$}
|
|
|
|
${
|
|
mt3.1 $e |- -. ps $.
|
|
mt3.2 $e |- ( -. ph -> ps ) $.
|
|
$( A rule similar to modus tollens. (Contributed by NM, 18-May-1994.)
|
|
(Proof shortened by Wolf Lammen, 11-Sep-2013.) $)
|
|
mt3 $p |- ph $=
|
|
( wn mto notnotri ) AAEBCDFG $.
|
|
$}
|
|
|
|
$( Peirce's axiom. This odd-looking theorem is the "difference" between an
|
|
intuitionistic system of propositional calculus and a classical system and
|
|
is not accepted by intuitionists. When Peirce's axiom is added to an
|
|
intuitionistic system, the system becomes equivalent to our classical
|
|
system ~ ax-1 through ~ ax-3 . A curious fact about this theorem is that
|
|
it requires ~ ax-3 for its proof even though the result has no negation
|
|
connectives in it. (Contributed by NM, 5-Aug-1993.) (Proof shortened by
|
|
Wolf Lammen, 9-Oct-2012.) $)
|
|
peirce $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
|
|
( wi simplim id ja ) ABCAAABDAEF $.
|
|
|
|
$( The Linearity Axiom of the infinite-valued sentential logic (L-infinity)
|
|
of Lukasiewicz. For a version not using ~ ax-3 , see ~ loolinALT .
|
|
(Contributed by O'Cat, 12-Aug-2004.) (Proof shortened by Wolf Lammen,
|
|
2-Nov-2012.) $)
|
|
loolin $p |- ( ( ( ph -> ps ) -> ( ps -> ph ) ) -> ( ps -> ph ) ) $=
|
|
( wi pm2.521 id ja ) ABCBACZGABDGEF $.
|
|
|
|
$( The Inversion Axiom of the infinite-valued sentential logic (L-infinity)
|
|
of Lukasiewicz. Using ~ dfor2 , we can see that this essentially
|
|
expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell]
|
|
p. 108. (Contributed by NM, 12-Aug-2004.) $)
|
|
looinv $p |- ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) $=
|
|
( wi imim1 peirce syl6 ) ABCZBCBACGACAGBADABEF $.
|
|
|
|
$( Theorem used to justify definition of biconditional ~ df-bi .
|
|
(Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton,
|
|
29-Dec-2000.) $)
|
|
bijust $p |- -. ( ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) )
|
|
-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
|
|
-> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) )
|
|
-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) ) $=
|
|
( wi wn id pm2.01 mt2 ) ABCBACDCDZHCZIDCIHEIFG $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical equivalence
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The definition ~ df-bi in this section is our first definition, which
|
|
introduces and defines the biconditional connective ` <-> ` . We define a wff
|
|
of the form ` ( ph <-> ps ) ` as an abbreviation for
|
|
` -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ` .
|
|
|
|
Unlike most traditional developments, we have chosen not to have a separate
|
|
symbol such as "Df." to mean "is defined as." Instead, we will later use the
|
|
biconditional connective for this purpose ( ~ df-or is its first use), as it
|
|
allows us to use logic to manipulate definitions directly. This greatly
|
|
simplifies many proofs since it eliminates the need for a separate mechanism
|
|
for introducing and eliminating definitions.
|
|
$)
|
|
|
|
$( Declare the biconditional connective. $)
|
|
$c <-> $. $( Double arrow (read: 'if and only if' or
|
|
'is logically equivalent to') $)
|
|
|
|
$( Extend our wff definition to include the biconditional connective. $)
|
|
wb $a wff ( ph <-> ps ) $.
|
|
|
|
$( Define the biconditional (logical 'iff').
|
|
|
|
The definition ~ df-bi in this section is our first definition, which
|
|
introduces and defines the biconditional connective ` <-> ` . We define a
|
|
wff of the form ` ( ph <-> ps ) ` as an abbreviation for
|
|
` -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ` .
|
|
|
|
Unlike most traditional developments, we have chosen not to have a
|
|
separate symbol such as "Df." to mean "is defined as." Instead, we will
|
|
later use the biconditional connective for this purpose ( ~ df-or is its
|
|
first use), as it allows us to use logic to manipulate definitions
|
|
directly. This greatly simplifies many proofs since it eliminates the
|
|
need for a separate mechanism for introducing and eliminating
|
|
definitions. Of course, we cannot use this mechanism to define the
|
|
biconditional itself, since it hasn't been introduced yet. Instead, we
|
|
use a more general form of definition, described as follows.
|
|
|
|
In its most general form, a definition is simply an assertion that
|
|
introduces a new symbol (or a new combination of existing symbols, as in
|
|
~ df-3an ) that is eliminable and does not strengthen the existing
|
|
language. The latter requirement means that the set of provable
|
|
statements not containing the new symbol (or new combination) should
|
|
remain exactly the same after the definition is introduced. Our
|
|
definition of the biconditional may look unusual compared to most
|
|
definitions, but it strictly satisfies these requirements.
|
|
|
|
The justification for our definition is that if we mechanically replace
|
|
` ( ph <-> ps ) ` (the definiendum i.e. the thing being defined) with
|
|
` -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ` (the definiens i.e. the
|
|
defining expression) in the definition, the definition becomes the
|
|
previously proved theorem ~ bijust . It is impossible to use ~ df-bi to
|
|
prove any statement expressed in the original language that can't be
|
|
proved from the original axioms, because if we simply replace each
|
|
instance of ~ df-bi in the proof with the corresponding ~ bijust instance,
|
|
we will end up with a proof from the original axioms.
|
|
|
|
Note that from Metamath's point of view, a definition is just another
|
|
axiom - i.e. an assertion we claim to be true - but from our high level
|
|
point of view, we are not strengthening the language. To indicate this
|
|
fact, we prefix definition labels with "df-" instead of "ax-". (This
|
|
prefixing is an informal convention that means nothing to the Metamath
|
|
proof verifier; it is just a naming convention for human readability.)
|
|
|
|
After we define the constant true ` T. ` ( ~ df-tru ) and the constant
|
|
false ` F. ` ( ~ df-fal ), we will be able to prove these truth table
|
|
values: ` ( ( T. <-> T. ) <-> T. ) ` ( ~ trubitru ),
|
|
` ( ( T. <-> F. ) <-> F. ) ` ( ~ trubifal ), ` ( ( F. <-> T. ) <-> F. ) `
|
|
( ~ falbitru ), and ` ( ( F. <-> F. ) <-> T. ) ` ( ~ falbifal ).
|
|
|
|
See ~ dfbi1 , ~ dfbi2 , and ~ dfbi3 for theorems suggesting typical
|
|
textbook definitions of ` <-> ` , showing that our definition has the
|
|
properties we expect. Theorem ~ dfbi1 is particularly useful if we want
|
|
to eliminate ` <-> ` from an expression to convert it to primitives.
|
|
Theorem ~ dfbi shows this definition rewritten in an abbreviated form
|
|
after conjunction is introduced, for easier understanding.
|
|
|
|
Contrast with ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), ` -/\ ` ( ~ df-nan ),
|
|
and ` \/_ ` ( ~ df-xor ) . In some sense ` <-> ` returns true if two
|
|
truth values are equal; ` = ` ( ~ df-cleq ) returns true if two classes
|
|
are equal. (Contributed by NM, 5-Aug-1993.) $)
|
|
df-bi $a |- -. ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
|
|
-> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) ) $.
|
|
|
|
$( $j justification 'bijust' for 'df-bi'; $)
|
|
|
|
$( Property of the biconditional connective. (Contributed by NM,
|
|
11-May-1999.) $)
|
|
bi1 $p |- ( ( ph <-> ps ) -> ( ph -> ps ) ) $=
|
|
( wb wi wn df-bi simplim ax-mp syl ) ABCZABDZBADEZDEZKJMDZMJDEZDENABFNOGHKL
|
|
GI $.
|
|
|
|
$( Property of the biconditional connective. (Contributed by NM,
|
|
11-May-1999.) $)
|
|
bi3 $p |- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) ) $=
|
|
( wi wb wn df-bi simprim ax-mp expi ) ABCZBACZABDZLJKECEZCZMLCZECEOABFNOGHI
|
|
$.
|
|
|
|
${
|
|
impbii.1 $e |- ( ph -> ps ) $.
|
|
impbii.2 $e |- ( ps -> ph ) $.
|
|
$( Infer an equivalence from an implication and its converse. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
impbii $p |- ( ph <-> ps ) $=
|
|
( wi wb bi3 mp2 ) ABEBAEABFCDABGH $.
|
|
$}
|
|
|
|
${
|
|
impbidd.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
impbidd.2 $e |- ( ph -> ( ps -> ( th -> ch ) ) ) $.
|
|
$( Deduce an equivalence from two implications. (Contributed by Rodolfo
|
|
Medina, 12-Oct-2010.) $)
|
|
impbidd $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
|
|
( wi wb bi3 syl6c ) ABCDGDCGCDHEFCDIJ $.
|
|
$}
|
|
|
|
${
|
|
impbid21d.1 $e |- ( ps -> ( ch -> th ) ) $.
|
|
impbid21d.2 $e |- ( ph -> ( th -> ch ) ) $.
|
|
$( Deduce an equivalence from two implications. (Contributed by Wolf
|
|
Lammen, 12-May-2013.) $)
|
|
impbid21d $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
|
|
( wi a1i a1d impbidd ) ABCDBCDGGAEHADCGBFIJ $.
|
|
$}
|
|
|
|
${
|
|
impbid.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
impbid.2 $e |- ( ph -> ( ch -> ps ) ) $.
|
|
$( Deduce an equivalence from two implications. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.) $)
|
|
impbid $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wb impbid21d pm2.43i ) ABCFAABCDEGH $.
|
|
$}
|
|
|
|
$( Relate the biconditional connective to primitive connectives. See
|
|
~ dfbi1gb for an unusual version proved directly from axioms.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
dfbi1 $p |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) $=
|
|
( wb wi wn df-bi simplim ax-mp bi3 impi impbii ) ABCZABDZBADZEDEZLODZOLDEZD
|
|
EPABFPQGHMNLABIJK $.
|
|
|
|
$( This proof of ~ dfbi1 , discovered by Gregory Bush on 8-Mar-2004, has
|
|
several curious properties. First, it has only 17 steps directly from the
|
|
axioms and ~ df-bi , compared to over 800 steps were the proof of ~ dfbi1
|
|
expanded into axioms. Second, step 2 demands only the property of "true";
|
|
any axiom (or theorem) could be used. It might be thought, therefore,
|
|
that it is in some sense redundant, but in fact no proof is shorter than
|
|
this (measured by number of steps). Third, it illustrates how
|
|
intermediate steps can "blow up" in size even in short proofs. Fourth,
|
|
the compressed proof is only 182 bytes (or 17 bytes in D-proof notation),
|
|
but the generated web page is over 200kB with intermediate steps that are
|
|
essentially incomprehensible to humans (other than Gregory Bush). If
|
|
there were an obfuscated code contest for proofs, this would be a
|
|
contender. This "blowing up" and incomprehensibility of the intermediate
|
|
steps vividly demonstrate the advantages of using many layered
|
|
intermediate theorems, since each theorem is easier to understand.
|
|
(Contributed by Gregory Bush, 10-Mar-2004.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
dfbi1gb $p |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) $=
|
|
( wch wth wb wi wn df-bi ax-1 ax-mp ax-3 ax-2 ) ABEZABFBAFGFGZFNMFGFGZMNEZA
|
|
BHCDCFFZOPFZCDIRGZQGZFZQRFSPOFZSFZFZUASUBISUCTFZFZUDUAFUEUFTGZUCGZFZUEUHUIM
|
|
NHUHUGIJTUCKJUESIJSUCTLJJRQKJJJ $.
|
|
|
|
${
|
|
biimpi.1 $e |- ( ph <-> ps ) $.
|
|
$( Infer an implication from a logical equivalence. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
biimpi $p |- ( ph -> ps ) $=
|
|
( wb wi bi1 ax-mp ) ABDABECABFG $.
|
|
$}
|
|
|
|
${
|
|
sylbi.1 $e |- ( ph <-> ps ) $.
|
|
sylbi.2 $e |- ( ps -> ch ) $.
|
|
$( A mixed syllogism inference from a biconditional and an implication.
|
|
Useful for substituting an antecedent with a definition. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
sylbi $p |- ( ph -> ch ) $=
|
|
( biimpi syl ) ABCABDFEG $.
|
|
$}
|
|
|
|
${
|
|
sylib.1 $e |- ( ph -> ps ) $.
|
|
sylib.2 $e |- ( ps <-> ch ) $.
|
|
$( A mixed syllogism inference from an implication and a biconditional.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sylib $p |- ( ph -> ch ) $=
|
|
( biimpi syl ) ABCDBCEFG $.
|
|
$}
|
|
|
|
$( Property of the biconditional connective. (Contributed by NM,
|
|
11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) $)
|
|
bi2 $p |- ( ( ph <-> ps ) -> ( ps -> ph ) ) $=
|
|
( wb wi wn dfbi1 simprim sylbi ) ABCABDZBADZEDEJABFIJGH $.
|
|
|
|
$( Commutative law for equivalence. (Contributed by Wolf Lammen,
|
|
10-Nov-2012.) $)
|
|
bicom1 $p |- ( ( ph <-> ps ) -> ( ps <-> ph ) ) $=
|
|
( wb bi2 bi1 impbid ) ABCBAABDABEF $.
|
|
|
|
$( Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell]
|
|
p. 117. (Contributed by NM, 5-Aug-1993.) $)
|
|
bicom $p |- ( ( ph <-> ps ) <-> ( ps <-> ph ) ) $=
|
|
( wb bicom1 impbii ) ABCBACABDBADE $.
|
|
|
|
${
|
|
bicomd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Commute two sides of a biconditional in a deduction. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
bicomd $p |- ( ph -> ( ch <-> ps ) ) $=
|
|
( wb bicom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
bicomi.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference from commutative law for logical equivalence. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
bicomi $p |- ( ps <-> ph ) $=
|
|
( wb bicom1 ax-mp ) ABDBADCABEF $.
|
|
$}
|
|
|
|
${
|
|
impbid1.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
impbid1.2 $e |- ( ch -> ps ) $.
|
|
$( Infer an equivalence from two implications. (Contributed by NM,
|
|
6-Mar-2007.) $)
|
|
impbid1 $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wi a1i impbid ) ABCDCBFAEGH $.
|
|
$}
|
|
|
|
${
|
|
impbid2.1 $e |- ( ps -> ch ) $.
|
|
impbid2.2 $e |- ( ph -> ( ch -> ps ) ) $.
|
|
$( Infer an equivalence from two implications. (Contributed by NM,
|
|
6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) $)
|
|
impbid2 $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( impbid1 bicomd ) ACBACBEDFG $.
|
|
$}
|
|
|
|
${
|
|
impcon4bid.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
impcon4bid.2 $e |- ( ph -> ( -. ps -> -. ch ) ) $.
|
|
$( A variation on ~ impbid with contraposition. (Contributed by Jeff
|
|
Hankins, 3-Jul-2009.) $)
|
|
impcon4bid $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( con4d impbid ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
biimpri.1 $e |- ( ph <-> ps ) $.
|
|
$( Infer a converse implication from a logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.) $)
|
|
biimpri $p |- ( ps -> ph ) $=
|
|
( bicomi biimpi ) BAABCDE $.
|
|
$}
|
|
|
|
${
|
|
biimpd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduce an implication from a logical equivalence. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
biimpd $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wb wi bi1 syl ) ABCEBCFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
mpbi.min $e |- ph $.
|
|
mpbi.maj $e |- ( ph <-> ps ) $.
|
|
$( An inference from a biconditional, related to modus ponens.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
mpbi $p |- ps $=
|
|
( biimpi ax-mp ) ABCABDEF $.
|
|
$}
|
|
|
|
${
|
|
mpbir.min $e |- ps $.
|
|
mpbir.maj $e |- ( ph <-> ps ) $.
|
|
$( An inference from a biconditional, related to modus ponens.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
mpbir $p |- ph $=
|
|
( biimpri ax-mp ) BACABDEF $.
|
|
$}
|
|
|
|
${
|
|
mpbid.min $e |- ( ph -> ps ) $.
|
|
mpbid.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, related to modus ponens. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
mpbid $p |- ( ph -> ch ) $=
|
|
( biimpd mpd ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mpbii.min $e |- ps $.
|
|
mpbii.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An inference from a nested biconditional, related to modus ponens.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
25-Oct-2012.) $)
|
|
mpbii $p |- ( ph -> ch ) $=
|
|
( a1i mpbid ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
sylibr.1 $e |- ( ph -> ps ) $.
|
|
sylibr.2 $e |- ( ch <-> ps ) $.
|
|
$( A mixed syllogism inference from an implication and a biconditional.
|
|
Useful for substituting a consequent with a definition. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sylibr $p |- ( ph -> ch ) $=
|
|
( biimpri syl ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
sylbir.1 $e |- ( ps <-> ph ) $.
|
|
sylbir.2 $e |- ( ps -> ch ) $.
|
|
$( A mixed syllogism inference from a biconditional and an implication.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sylbir $p |- ( ph -> ch ) $=
|
|
( biimpri syl ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
sylibd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylibd.2 $e |- ( ph -> ( ch <-> th ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)
|
|
sylibd $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpd syld ) ABCDEACDFGH $.
|
|
$}
|
|
|
|
${
|
|
sylbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
sylbid.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)
|
|
sylbid $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpd syld ) ABCDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
mpbidi.min $e |- ( th -> ( ph -> ps ) ) $.
|
|
mpbidi.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, related to modus ponens. (Contributed
|
|
by NM, 9-Aug-1994.) $)
|
|
mpbidi $p |- ( th -> ( ph -> ch ) ) $=
|
|
( biimpd sylcom ) DABCEABCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5bi.1 $e |- ( ph <-> ps ) $.
|
|
syl5bi.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( A mixed syllogism inference from a nested implication and a
|
|
biconditional. Useful for substituting an embedded antecedent with a
|
|
definition. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5bi $p |- ( ch -> ( ph -> th ) ) $=
|
|
( biimpi syl5 ) ABCDABEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5bir.1 $e |- ( ps <-> ph ) $.
|
|
syl5bir.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( A mixed syllogism inference from a nested implication and a
|
|
biconditional. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5bir $p |- ( ch -> ( ph -> th ) ) $=
|
|
( biimpri syl5 ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5ib.1 $e |- ( ph -> ps ) $.
|
|
syl5ib.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5ib $p |- ( ch -> ( ph -> th ) ) $=
|
|
( biimpd syl5 ) ABCDECBDFGH $.
|
|
|
|
$( A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.) $)
|
|
syl5ibcom $p |- ( ph -> ( ch -> th ) ) $=
|
|
( syl5ib com12 ) CADABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5ibr.1 $e |- ( ph -> th ) $.
|
|
syl5ibr.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) $)
|
|
syl5ibr $p |- ( ch -> ( ph -> ps ) ) $=
|
|
( bicomd syl5ib ) ADCBECBDFGH $.
|
|
|
|
$( A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) $)
|
|
syl5ibrcom $p |- ( ph -> ( ch -> ps ) ) $=
|
|
( syl5ibr com12 ) CABABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
biimprd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduce a converse implication from a logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)
|
|
biimprd $p |- ( ph -> ( ch -> ps ) ) $=
|
|
( id syl5ibr ) CBACCEDF $.
|
|
$}
|
|
|
|
${
|
|
biimpcd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduce a commuted implication from a logical equivalence. (Contributed
|
|
by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)
|
|
biimpcd $p |- ( ps -> ( ph -> ch ) ) $=
|
|
( id syl5ibcom ) BBACBEDF $.
|
|
|
|
$( Deduce a converse commuted implication from a logical equivalence.
|
|
(Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen,
|
|
20-Dec-2013.) $)
|
|
biimprcd $p |- ( ch -> ( ph -> ps ) ) $=
|
|
( id syl5ibrcom ) CBACCEDF $.
|
|
$}
|
|
|
|
${
|
|
syl6ib.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6ib.2 $e |- ( ch <-> th ) $.
|
|
$( A mixed syllogism inference from a nested implication and a
|
|
biconditional. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl6ib $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpi syl6 ) ABCDECDFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6ibr.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl6ibr.2 $e |- ( th <-> ch ) $.
|
|
$( A mixed syllogism inference from a nested implication and a
|
|
biconditional. Useful for substituting an embedded consequent with a
|
|
definition. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl6ibr $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpri syl6 ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6bi.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6bi.2 $e |- ( ch -> th ) $.
|
|
$( A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.) $)
|
|
syl6bi $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimpd syl6 ) ABCDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6bir.1 $e |- ( ph -> ( ch <-> ps ) ) $.
|
|
syl6bir.2 $e |- ( ch -> th ) $.
|
|
$( A mixed syllogism inference. (Contributed by NM, 18-May-1994.) $)
|
|
syl6bir $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimprd syl6 ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl7bi.1 $e |- ( ph <-> ps ) $.
|
|
syl7bi.2 $e |- ( ch -> ( th -> ( ps -> ta ) ) ) $.
|
|
$( A mixed syllogism inference from a doubly nested implication and a
|
|
biconditional. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl7bi $p |- ( ch -> ( th -> ( ph -> ta ) ) ) $=
|
|
( biimpi syl7 ) ABCDEABFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl8ib.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
syl8ib.2 $e |- ( th <-> ta ) $.
|
|
$( A syllogism rule of inference. The second premise is used to replace
|
|
the consequent of the first premise. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
syl8ib $p |- ( ph -> ( ps -> ( ch -> ta ) ) ) $=
|
|
( biimpi syl8 ) ABCDEFDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mpbird.min $e |- ( ph -> ch ) $.
|
|
mpbird.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, related to modus ponens. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
mpbird $p |- ( ph -> ps ) $=
|
|
( biimprd mpd ) ACBDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mpbiri.min $e |- ch $.
|
|
mpbiri.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An inference from a nested biconditional, related to modus ponens.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
25-Oct-2012.) $)
|
|
mpbiri $p |- ( ph -> ps ) $=
|
|
( a1i mpbird ) ABCCADFEG $.
|
|
$}
|
|
|
|
${
|
|
sylibrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylibrd.2 $e |- ( ph -> ( th <-> ch ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)
|
|
sylibrd $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimprd syld ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
sylbird.1 $e |- ( ph -> ( ch <-> ps ) ) $.
|
|
sylbird.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 3-Aug-1994.) $)
|
|
sylbird $p |- ( ph -> ( ps -> th ) ) $=
|
|
( biimprd syld ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
$( Principle of identity for logical equivalence. Theorem *4.2 of
|
|
[WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) $)
|
|
biid $p |- ( ph <-> ph ) $=
|
|
( id impbii ) AAABZDC $.
|
|
|
|
$( Principle of identity with antecedent. (Contributed by NM,
|
|
25-Nov-1995.) $)
|
|
biidd $p |- ( ph -> ( ps <-> ps ) ) $=
|
|
( wb biid a1i ) BBCABDE $.
|
|
|
|
$( Two propositions are equivalent if they are both true. Closed form of
|
|
~ 2th . Equivalent to a ~ bi1 -like version of the xor-connective. This
|
|
theorem stays true, no matter how you permute its operands. This is
|
|
evident from its sharper version
|
|
` ( ph <-> ( ps <-> ( ph <-> ps ) ) ) ` . (Contributed by Wolf Lammen,
|
|
12-May-2013.) $)
|
|
pm5.1im $p |- ( ph -> ( ps -> ( ph <-> ps ) ) ) $=
|
|
( ax-1 impbid21d ) ABABBACABCD $.
|
|
|
|
${
|
|
2th.1 $e |- ph $.
|
|
2th.2 $e |- ps $.
|
|
$( Two truths are equivalent. (Contributed by NM, 18-Aug-1993.) $)
|
|
2th $p |- ( ph <-> ps ) $=
|
|
( a1i impbii ) ABBADEABCEF $.
|
|
$}
|
|
|
|
${
|
|
2thd.1 $e |- ( ph -> ps ) $.
|
|
2thd.2 $e |- ( ph -> ch ) $.
|
|
$( Two truths are equivalent (deduction rule). (Contributed by NM,
|
|
3-Jun-2012.) $)
|
|
2thd $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wb pm5.1im sylc ) ABCBCFDEBCGH $.
|
|
$}
|
|
|
|
${
|
|
ibi.1 $e |- ( ph -> ( ph <-> ps ) ) $.
|
|
$( Inference that converts a biconditional implied by one of its arguments,
|
|
into an implication. (Contributed by NM, 17-Oct-2003.) $)
|
|
ibi $p |- ( ph -> ps ) $=
|
|
( biimpd pm2.43i ) ABAABCDE $.
|
|
$}
|
|
|
|
${
|
|
ibir.1 $e |- ( ph -> ( ps <-> ph ) ) $.
|
|
$( Inference that converts a biconditional implied by one of its arguments,
|
|
into an implication. (Contributed by NM, 22-Jul-2004.) $)
|
|
ibir $p |- ( ph -> ps ) $=
|
|
( bicomd ibi ) ABABACDE $.
|
|
$}
|
|
|
|
${
|
|
ibd.1 $e |- ( ph -> ( ps -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction that converts a biconditional implied by one of its arguments,
|
|
into an implication. (Contributed by NM, 26-Jun-2004.) $)
|
|
ibd $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wb bi1 syli ) BABCECDBCFG $.
|
|
$}
|
|
|
|
$( Distribution of implication over biconditional. Theorem *5.74 of
|
|
[WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof
|
|
shortened by Wolf Lammen, 11-Apr-2013.) $)
|
|
pm5.74 $p |- ( ( ph -> ( ps <-> ch ) ) <->
|
|
( ( ph -> ps ) <-> ( ph -> ch ) ) ) $=
|
|
( wb wi bi1 imim3i bi2 impbid pm2.86d impbidd impbii ) ABCDZEZABEZACEZDZNOP
|
|
MBCABCFGMCBABCHGIQABCQABCOPFJQACBOPHJKL $.
|
|
|
|
${
|
|
pm5.74i.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Distribution of implication over biconditional (inference rule).
|
|
(Contributed by NM, 1-Aug-1994.) $)
|
|
pm5.74i $p |- ( ( ph -> ps ) <-> ( ph -> ch ) ) $=
|
|
( wb wi pm5.74 mpbi ) ABCEFABFACFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
pm5.74ri.1 $e |- ( ( ph -> ps ) <-> ( ph -> ch ) ) $.
|
|
$( Distribution of implication over biconditional (reverse inference
|
|
rule). (Contributed by NM, 1-Aug-1994.) $)
|
|
pm5.74ri $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wb wi pm5.74 mpbir ) ABCEFABFACFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
pm5.74d.1 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 21-Mar-1996.) $)
|
|
pm5.74d $p |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $=
|
|
( wb wi pm5.74 sylib ) ABCDFGBCGBDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
pm5.74rd.1 $e |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 19-Mar-1997.) $)
|
|
pm5.74rd $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
|
|
( wi wb pm5.74 sylibr ) ABCFBDFGBCDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
bitri.1 $e |- ( ph <-> ps ) $.
|
|
bitri.2 $e |- ( ps <-> ch ) $.
|
|
$( An inference from transitive law for logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) $)
|
|
bitri $p |- ( ph <-> ch ) $=
|
|
( biimpi sylib biimpri sylibr impbii ) ACABCABDFEGCBABCEHDIJ $.
|
|
$}
|
|
|
|
${
|
|
bitr2i.1 $e |- ( ph <-> ps ) $.
|
|
bitr2i.2 $e |- ( ps <-> ch ) $.
|
|
$( An inference from transitive law for logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
bitr2i $p |- ( ch <-> ph ) $=
|
|
( bitri bicomi ) ACABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
bitr3i.1 $e |- ( ps <-> ph ) $.
|
|
bitr3i.2 $e |- ( ps <-> ch ) $.
|
|
$( An inference from transitive law for logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
bitr3i $p |- ( ph <-> ch ) $=
|
|
( bicomi bitri ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
bitr4i.1 $e |- ( ph <-> ps ) $.
|
|
bitr4i.2 $e |- ( ch <-> ps ) $.
|
|
$( An inference from transitive law for logical equivalence. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
bitr4i $p |- ( ph <-> ch ) $=
|
|
( bicomi bitri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
$( Register '<->' as an equality for its type (wff). $)
|
|
$( $j
|
|
equality 'wb' from 'biid' 'bicomi' 'bitri';
|
|
definition 'dfbi1' for 'wb';
|
|
$)
|
|
|
|
${
|
|
bitrd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $.
|
|
$( Deduction form of ~ bitri . (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 14-Apr-2013.) $)
|
|
bitrd $p |- ( ph -> ( ps <-> th ) ) $=
|
|
( wi pm5.74i bitri pm5.74ri ) ABDABGACGADGABCEHACDFHIJ $.
|
|
$}
|
|
|
|
${
|
|
bitr2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bitr2d.2 $e |- ( ph -> ( ch <-> th ) ) $.
|
|
$( Deduction form of ~ bitr2i . (Contributed by NM, 9-Jun-2004.) $)
|
|
bitr2d $p |- ( ph -> ( th <-> ps ) ) $=
|
|
( bitrd bicomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
bitr3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bitr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
|
|
$( Deduction form of ~ bitr3i . (Contributed by NM, 5-Aug-1993.) $)
|
|
bitr3d $p |- ( ph -> ( ch <-> th ) ) $=
|
|
( bicomd bitrd ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
bitr4d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bitr4d.2 $e |- ( ph -> ( th <-> ch ) ) $.
|
|
$( Deduction form of ~ bitr4i . (Contributed by NM, 5-Aug-1993.) $)
|
|
bitr4d $p |- ( ph -> ( ps <-> th ) ) $=
|
|
( bicomd bitrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5bb.1 $e |- ( ph <-> ps ) $.
|
|
syl5bb.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl5bb $p |- ( ch -> ( ph <-> th ) ) $=
|
|
( wb a1i bitrd ) CABDABGCEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl5rbb.1 $e |- ( ph <-> ps ) $.
|
|
syl5rbb.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl5rbb $p |- ( ch -> ( th <-> ph ) ) $=
|
|
( syl5bb bicomd ) CADABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5bbr.1 $e |- ( ps <-> ph ) $.
|
|
syl5bbr.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl5bbr $p |- ( ch -> ( ph <-> th ) ) $=
|
|
( bicomi syl5bb ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5rbbr.1 $e |- ( ps <-> ph ) $.
|
|
syl5rbbr.2 $e |- ( ch -> ( ps <-> th ) ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
25-Nov-1994.) $)
|
|
syl5rbbr $p |- ( ch -> ( th <-> ph ) ) $=
|
|
( bicomi syl5rbb ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6bb.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6bb.2 $e |- ( ch <-> th ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl6bb $p |- ( ph -> ( ps <-> th ) ) $=
|
|
( wb a1i bitrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6rbb.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6rbb.2 $e |- ( ch <-> th ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl6rbb $p |- ( ph -> ( th <-> ps ) ) $=
|
|
( syl6bb bicomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6bbr.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6bbr.2 $e |- ( th <-> ch ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
syl6bbr $p |- ( ph -> ( ps <-> th ) ) $=
|
|
( bicomi syl6bb ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6rbbr.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl6rbbr.2 $e |- ( th <-> ch ) $.
|
|
$( A syllogism inference from two biconditionals. (Contributed by NM,
|
|
25-Nov-1994.) $)
|
|
syl6rbbr $p |- ( ph -> ( th <-> ps ) ) $=
|
|
( bicomi syl6rbb ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
3imtr3.1 $e |- ( ph -> ps ) $.
|
|
3imtr3.2 $e |- ( ph <-> ch ) $.
|
|
3imtr3.3 $e |- ( ps <-> th ) $.
|
|
$( A mixed syllogism inference, useful for removing a definition from both
|
|
sides of an implication. (Contributed by NM, 10-Aug-1994.) $)
|
|
3imtr3i $p |- ( ch -> th ) $=
|
|
( sylbir sylib ) CBDCABFEHGI $.
|
|
$}
|
|
|
|
${
|
|
3imtr4.1 $e |- ( ph -> ps ) $.
|
|
3imtr4.2 $e |- ( ch <-> ph ) $.
|
|
3imtr4.3 $e |- ( th <-> ps ) $.
|
|
$( A mixed syllogism inference, useful for applying a definition to both
|
|
sides of an implication. (Contributed by NM, 5-Aug-1993.) $)
|
|
3imtr4i $p |- ( ch -> th ) $=
|
|
( sylbi sylibr ) CBDCABFEHGI $.
|
|
$}
|
|
|
|
${
|
|
3imtr3d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3imtr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
|
|
3imtr3d.3 $e |- ( ph -> ( ch <-> ta ) ) $.
|
|
$( More general version of ~ 3imtr3i . Useful for converting conditional
|
|
definitions in a formula. (Contributed by NM, 8-Apr-1996.) $)
|
|
3imtr3d $p |- ( ph -> ( th -> ta ) ) $=
|
|
( sylibd sylbird ) ADBEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3imtr4d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3imtr4d.2 $e |- ( ph -> ( th <-> ps ) ) $.
|
|
3imtr4d.3 $e |- ( ph -> ( ta <-> ch ) ) $.
|
|
$( More general version of ~ 3imtr4i . Useful for converting conditional
|
|
definitions in a formula. (Contributed by NM, 26-Oct-1995.) $)
|
|
3imtr4d $p |- ( ph -> ( th -> ta ) ) $=
|
|
( sylibrd sylbid ) ADBEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3imtr3g.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3imtr3g.2 $e |- ( ps <-> th ) $.
|
|
3imtr3g.3 $e |- ( ch <-> ta ) $.
|
|
$( More general version of ~ 3imtr3i . Useful for converting definitions
|
|
in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by
|
|
Wolf Lammen, 20-Dec-2013.) $)
|
|
3imtr3g $p |- ( ph -> ( th -> ta ) ) $=
|
|
( syl5bir syl6ib ) ADCEDBACGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3imtr4g.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3imtr4g.2 $e |- ( th <-> ps ) $.
|
|
3imtr4g.3 $e |- ( ta <-> ch ) $.
|
|
$( More general version of ~ 3imtr4i . Useful for converting definitions
|
|
in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by
|
|
Wolf Lammen, 20-Dec-2013.) $)
|
|
3imtr4g $p |- ( ph -> ( th -> ta ) ) $=
|
|
( syl5bi syl6ibr ) ADCEDBACGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3bitri.1 $e |- ( ph <-> ps ) $.
|
|
3bitri.2 $e |- ( ps <-> ch ) $.
|
|
3bitri.3 $e |- ( ch <-> th ) $.
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
3bitri $p |- ( ph <-> th ) $=
|
|
( bitri ) ABDEBCDFGHH $.
|
|
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 4-Aug-2006.) $)
|
|
3bitrri $p |- ( th <-> ph ) $=
|
|
( bitr2i bitr3i ) DCAGABCEFHI $.
|
|
$}
|
|
|
|
${
|
|
3bitr2i.1 $e |- ( ph <-> ps ) $.
|
|
3bitr2i.2 $e |- ( ch <-> ps ) $.
|
|
3bitr2i.3 $e |- ( ch <-> th ) $.
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 4-Aug-2006.) $)
|
|
3bitr2i $p |- ( ph <-> th ) $=
|
|
( bitr4i bitri ) ACDABCEFHGI $.
|
|
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 4-Aug-2006.) $)
|
|
3bitr2ri $p |- ( th <-> ph ) $=
|
|
( bitr4i bitr2i ) ACDABCEFHGI $.
|
|
$}
|
|
|
|
${
|
|
3bitr3i.1 $e |- ( ph <-> ps ) $.
|
|
3bitr3i.2 $e |- ( ph <-> ch ) $.
|
|
3bitr3i.3 $e |- ( ps <-> th ) $.
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 19-Aug-1993.) $)
|
|
3bitr3i $p |- ( ch <-> th ) $=
|
|
( bitr3i bitri ) CBDCABFEHGI $.
|
|
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
3bitr3ri $p |- ( th <-> ch ) $=
|
|
( bitr3i ) DBCGBACEFHH $.
|
|
$}
|
|
|
|
${
|
|
3bitr4i.1 $e |- ( ph <-> ps ) $.
|
|
3bitr4i.2 $e |- ( ch <-> ph ) $.
|
|
3bitr4i.3 $e |- ( th <-> ps ) $.
|
|
$( A chained inference from transitive law for logical equivalence. This
|
|
inference is frequently used to apply a definition to both sides of a
|
|
logical equivalence. (Contributed by NM, 5-Aug-1993.) $)
|
|
3bitr4i $p |- ( ch <-> th ) $=
|
|
( bitr4i bitri ) CADFABDEGHI $.
|
|
|
|
$( A chained inference from transitive law for logical equivalence.
|
|
(Contributed by NM, 2-Sep-1995.) $)
|
|
3bitr4ri $p |- ( th <-> ch ) $=
|
|
( bitr4i bitr2i ) CADFABDEGHI $.
|
|
$}
|
|
|
|
${
|
|
3bitrd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $.
|
|
3bitrd.3 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
13-Aug-1999.) $)
|
|
3bitrd $p |- ( ph -> ( ps <-> ta ) ) $=
|
|
( bitrd ) ABDEABCDFGIHI $.
|
|
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitrrd $p |- ( ph -> ( ta <-> ps ) ) $=
|
|
( bitr2d bitr3d ) ADEBHABCDFGIJ $.
|
|
$}
|
|
|
|
${
|
|
3bitr2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr2d.2 $e |- ( ph -> ( th <-> ch ) ) $.
|
|
3bitr2d.3 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitr2d $p |- ( ph -> ( ps <-> ta ) ) $=
|
|
( bitr4d bitrd ) ABDEABCDFGIHJ $.
|
|
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitr2rd $p |- ( ph -> ( ta <-> ps ) ) $=
|
|
( bitr4d bitr2d ) ABDEABCDFGIHJ $.
|
|
$}
|
|
|
|
${
|
|
3bitr3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr3d.2 $e |- ( ph -> ( ps <-> th ) ) $.
|
|
3bitr3d.3 $e |- ( ph -> ( ch <-> ta ) ) $.
|
|
$( Deduction from transitivity of biconditional. Useful for converting
|
|
conditional definitions in a formula. (Contributed by NM,
|
|
24-Apr-1996.) $)
|
|
3bitr3d $p |- ( ph -> ( th <-> ta ) ) $=
|
|
( bitr3d bitrd ) ADCEABDCGFIHJ $.
|
|
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitr3rd $p |- ( ph -> ( ta <-> th ) ) $=
|
|
( bitr3d ) ACEDHABCDFGII $.
|
|
$}
|
|
|
|
${
|
|
3bitr4d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr4d.2 $e |- ( ph -> ( th <-> ps ) ) $.
|
|
3bitr4d.3 $e |- ( ph -> ( ta <-> ch ) ) $.
|
|
$( Deduction from transitivity of biconditional. Useful for converting
|
|
conditional definitions in a formula. (Contributed by NM,
|
|
18-Oct-1995.) $)
|
|
3bitr4d $p |- ( ph -> ( th <-> ta ) ) $=
|
|
( bitr4d bitrd ) ADBEGABCEFHIJ $.
|
|
|
|
$( Deduction from transitivity of biconditional. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3bitr4rd $p |- ( ph -> ( ta <-> th ) ) $=
|
|
( bitr4d ) AEBDAECBHFIGI $.
|
|
$}
|
|
|
|
${
|
|
3bitr3g.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr3g.2 $e |- ( ps <-> th ) $.
|
|
3bitr3g.3 $e |- ( ch <-> ta ) $.
|
|
$( More general version of ~ 3bitr3i . Useful for converting definitions
|
|
in a formula. (Contributed by NM, 4-Jun-1995.) $)
|
|
3bitr3g $p |- ( ph -> ( th <-> ta ) ) $=
|
|
( syl5bbr syl6bb ) ADCEDBACGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3bitr4g.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3bitr4g.2 $e |- ( th <-> ps ) $.
|
|
3bitr4g.3 $e |- ( ta <-> ch ) $.
|
|
$( More general version of ~ 3bitr4i . Useful for converting definitions
|
|
in a formula. (Contributed by NM, 5-Aug-1993.) $)
|
|
3bitr4g $p |- ( ph -> ( th <-> ta ) ) $=
|
|
( syl5bb syl6bbr ) ADCEDBACGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
bi3ant.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Construct a bi-conditional in antecedent position. (Contributed by Wolf
|
|
Lammen, 14-May-2013.) $)
|
|
bi3ant $p |- ( ( ( th -> ta ) -> ph )
|
|
-> ( ( ( ta -> th ) -> ps ) -> ( ( th <-> ta ) -> ch ) ) ) $=
|
|
( wi wb bi1 imim1i bi2 imim3i syl2im ) DEGZAGDEHZAGEDGZBGOBGOCGONADEIJOPB
|
|
DEKJABCOFLM $.
|
|
$}
|
|
|
|
$( Express symmetries of theorems in terms of biconditionals. (Contributed
|
|
by Wolf Lammen, 14-May-2013.) $)
|
|
bisym $p |- ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ( ps -> ph )
|
|
-> ( th -> ch ) ) -> ( ( ph <-> ps ) -> ( ch <-> th ) ) ) ) $=
|
|
( wi wb bi3 bi3ant ) CDEDCECDFABCDGH $.
|
|
|
|
$( Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
notnot $p |- ( ph <-> -. -. ph ) $=
|
|
( wn notnot1 notnot2 impbii ) AABBACADE $.
|
|
|
|
$( Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
con34b $p |- ( ( ph -> ps ) <-> ( -. ps -> -. ph ) ) $=
|
|
( wi wn con3 ax-3 impbii ) ABCBDADCABEBAFG $.
|
|
|
|
${
|
|
con4bid.1 $e |- ( ph -> ( -. ps <-> -. ch ) ) $.
|
|
$( A contraposition deduction. (Contributed by NM, 21-May-1994.) $)
|
|
con4bid $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wn biimprd con4d biimpd impcon4bid ) ABCACBABEZCEZDFGAJKDHI $.
|
|
$}
|
|
|
|
${
|
|
notbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction negating both sides of a logical equivalence. (Contributed by
|
|
NM, 21-May-1994.) $)
|
|
notbid $p |- ( ph -> ( -. ps <-> -. ch ) ) $=
|
|
( wn notnot 3bitr3g con4bid ) ABEZCEZABCIEJEDBFCFGH $.
|
|
$}
|
|
|
|
$( Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed
|
|
by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.) $)
|
|
notbi $p |- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) ) $=
|
|
( wb wn id notbid con4bid impbii ) ABCZADBDCZIABIEFJABJEGH $.
|
|
|
|
${
|
|
notbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Negate both sides of a logical equivalence. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) $)
|
|
notbii $p |- ( -. ph <-> -. ps ) $=
|
|
( wb wn notbi mpbi ) ABDAEBEDCABFG $.
|
|
|
|
$( Theorem notbii is the congruence law for negation. $)
|
|
$( $j congruence 'notbii'; $)
|
|
$}
|
|
|
|
${
|
|
con4bii.1 $e |- ( -. ph <-> -. ps ) $.
|
|
$( A contraposition inference. (Contributed by NM, 21-May-1994.) $)
|
|
con4bii $p |- ( ph <-> ps ) $=
|
|
( wb wn notbi mpbir ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
mtbi.1 $e |- -. ph $.
|
|
mtbi.2 $e |- ( ph <-> ps ) $.
|
|
$( An inference from a biconditional, related to modus tollens.
|
|
(Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen,
|
|
25-Oct-2012.) $)
|
|
mtbi $p |- -. ps $=
|
|
( biimpri mto ) BACABDEF $.
|
|
$}
|
|
|
|
${
|
|
mtbir.1 $e |- -. ps $.
|
|
mtbir.2 $e |- ( ph <-> ps ) $.
|
|
$( An inference from a biconditional, related to modus tollens.
|
|
(Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen,
|
|
14-Oct-2012.) $)
|
|
mtbir $p |- -. ph $=
|
|
( bicomi mtbi ) BACABDEF $.
|
|
$}
|
|
|
|
${
|
|
mtbid.min $e |- ( ph -> -. ps ) $.
|
|
mtbid.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, similar to modus tollens.
|
|
(Contributed by NM, 26-Nov-1995.) $)
|
|
mtbid $p |- ( ph -> -. ch ) $=
|
|
( biimprd mtod ) ACBDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mtbird.min $e |- ( ph -> -. ch ) $.
|
|
mtbird.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( A deduction from a biconditional, similar to modus tollens.
|
|
(Contributed by NM, 10-May-1994.) $)
|
|
mtbird $p |- ( ph -> -. ps ) $=
|
|
( biimpd mtod ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mtbii.min $e |- -. ps $.
|
|
mtbii.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An inference from a biconditional, similar to modus tollens.
|
|
(Contributed by NM, 27-Nov-1995.) $)
|
|
mtbii $p |- ( ph -> -. ch ) $=
|
|
( biimprd mtoi ) ACBDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mtbiri.min $e |- -. ch $.
|
|
mtbiri.maj $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An inference from a biconditional, similar to modus tollens.
|
|
(Contributed by NM, 24-Aug-1995.) $)
|
|
mtbiri $p |- ( ph -> -. ps ) $=
|
|
( biimpd mtoi ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
sylnib.1 $e |- ( ph -> -. ps ) $.
|
|
sylnib.2 $e |- ( ps <-> ch ) $.
|
|
$( A mixed syllogism inference from an implication and a biconditional.
|
|
(Contributed by Wolf Lammen, 16-Dec-2013.) $)
|
|
sylnib $p |- ( ph -> -. ch ) $=
|
|
( wb a1i mtbid ) ABCDBCFAEGH $.
|
|
$}
|
|
|
|
${
|
|
sylnibr.1 $e |- ( ph -> -. ps ) $.
|
|
sylnibr.2 $e |- ( ch <-> ps ) $.
|
|
$( A mixed syllogism inference from an implication and a biconditional.
|
|
Useful for substituting a consequent with a definition. (Contributed by
|
|
Wolf Lammen, 16-Dec-2013.) $)
|
|
sylnibr $p |- ( ph -> -. ch ) $=
|
|
( bicomi sylnib ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
sylnbi.1 $e |- ( ph <-> ps ) $.
|
|
sylnbi.2 $e |- ( -. ps -> ch ) $.
|
|
$( A mixed syllogism inference from a biconditional and an implication.
|
|
Useful for substituting an antecedent with a definition. (Contributed
|
|
by Wolf Lammen, 16-Dec-2013.) $)
|
|
sylnbi $p |- ( -. ph -> ch ) $=
|
|
( wn notbii sylbi ) AFBFCABDGEH $.
|
|
$}
|
|
|
|
${
|
|
sylnbir.1 $e |- ( ps <-> ph ) $.
|
|
sylnbir.2 $e |- ( -. ps -> ch ) $.
|
|
$( A mixed syllogism inference from a biconditional and an implication.
|
|
(Contributed by Wolf Lammen, 16-Dec-2013.) $)
|
|
sylnbir $p |- ( -. ph -> ch ) $=
|
|
( bicomi sylnbi ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
xchnxbi.1 $e |- ( -. ph <-> ps ) $.
|
|
xchnxbi.2 $e |- ( ph <-> ch ) $.
|
|
$( Replacement of a subexpression by an equivalent one. (Contributed by
|
|
Wolf Lammen, 27-Sep-2014.) $)
|
|
xchnxbi $p |- ( -. ch <-> ps ) $=
|
|
( wn notbii bitr3i ) CFAFBACEGDH $.
|
|
$}
|
|
|
|
${
|
|
xchnxbir.1 $e |- ( -. ph <-> ps ) $.
|
|
xchnxbir.2 $e |- ( ch <-> ph ) $.
|
|
$( Replacement of a subexpression by an equivalent one. (Contributed by
|
|
Wolf Lammen, 27-Sep-2014.) $)
|
|
xchnxbir $p |- ( -. ch <-> ps ) $=
|
|
( bicomi xchnxbi ) ABCDCAEFG $.
|
|
$}
|
|
|
|
${
|
|
xchbinx.1 $e |- ( ph <-> -. ps ) $.
|
|
xchbinx.2 $e |- ( ps <-> ch ) $.
|
|
$( Replacement of a subexpression by an equivalent one. (Contributed by
|
|
Wolf Lammen, 27-Sep-2014.) $)
|
|
xchbinx $p |- ( ph <-> -. ch ) $=
|
|
( wn notbii bitri ) ABFCFDBCEGH $.
|
|
$}
|
|
|
|
${
|
|
xchbinxr.1 $e |- ( ph <-> -. ps ) $.
|
|
xchbinxr.2 $e |- ( ch <-> ps ) $.
|
|
$( Replacement of a subexpression by an equivalent one. (Contributed by
|
|
Wolf Lammen, 27-Sep-2014.) $)
|
|
xchbinxr $p |- ( ph <-> -. ch ) $=
|
|
( bicomi xchbinx ) ABCDCBEFG $.
|
|
$}
|
|
|
|
$( The next three rules are useful for building up wff's around a
|
|
definition, in order to make use of the definition. $)
|
|
|
|
${
|
|
bi.a $e |- ( ph <-> ps ) $.
|
|
$( Introduce an antecedent to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
6-Feb-2013.) $)
|
|
imbi2i $p |- ( ( ch -> ph ) <-> ( ch -> ps ) ) $=
|
|
( wb a1i pm5.74i ) CABABECDFG $.
|
|
$}
|
|
|
|
${
|
|
bibi.a $e |- ( ph <-> ps ) $.
|
|
$( Inference adding a biconditional to the left in an equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.) $)
|
|
bibi2i $p |- ( ( ch <-> ph ) <-> ( ch <-> ps ) ) $=
|
|
( wb id syl6bb syl6bbr impbii ) CAEZCBEZJCABJFDGKCBAKFDHI $.
|
|
|
|
$( Inference adding a biconditional to the right in an equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
bibi1i $p |- ( ( ph <-> ch ) <-> ( ps <-> ch ) ) $=
|
|
( wb bicom bibi2i 3bitri ) ACECAECBEBCEACFABCDGCBFH $.
|
|
|
|
${
|
|
bibi12.2 $e |- ( ch <-> th ) $.
|
|
$( The equivalence of two equivalences. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
bibi12i $p |- ( ( ph <-> ch ) <-> ( ps <-> th ) ) $=
|
|
( wb bibi2i bibi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
imbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction adding an antecedent to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
imbi2d $p |- ( ph -> ( ( th -> ps ) <-> ( th -> ch ) ) ) $=
|
|
( wb a1d pm5.74d ) ADBCABCFDEGH $.
|
|
|
|
$( Deduction adding a consequent to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
17-Sep-2013.) $)
|
|
imbi1d $p |- ( ph -> ( ( ps -> th ) <-> ( ch -> th ) ) ) $=
|
|
( wi biimprd imim1d biimpd impbid ) ABDFCDFACBDABCEGHABCDABCEIHJ $.
|
|
|
|
$( Deduction adding a biconditional to the left in an equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
19-May-2013.) $)
|
|
bibi2d $p |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) ) $=
|
|
( wb wi pm5.74i bibi2i pm5.74 3bitr4i pm5.74ri ) ADBFZDCFZADGZABGZFOACGZF
|
|
AMGANGPQOABCEHIADBJADCJKL $.
|
|
|
|
$( Deduction adding a biconditional to the right in an equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
bibi1d $p |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> th ) ) ) $=
|
|
( wb bibi2d bicom 3bitr4g ) ADBFDCFBDFCDFABCDEGBDHCDHI $.
|
|
$}
|
|
|
|
${
|
|
imbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
imbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction joining two equivalences to form equivalence of implications.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
imbi12d $p |- ( ph -> ( ( ps -> th ) <-> ( ch -> ta ) ) ) $=
|
|
( wi imbi1d imbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
|
|
$( Deduction joining two equivalences to form equivalence of
|
|
biconditionals. (Contributed by NM, 5-Aug-1993.) $)
|
|
bibi12d $p |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> ta ) ) ) $=
|
|
( wb bibi1d bibi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
$( Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
imbi1 $p |- ( ( ph <-> ps ) -> ( ( ph -> ch ) <-> ( ps -> ch ) ) ) $=
|
|
( wb id imbi1d ) ABDZABCGEF $.
|
|
|
|
$( Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) $)
|
|
imbi2 $p |- ( ( ph <-> ps ) -> ( ( ch -> ph ) <-> ( ch -> ps ) ) ) $=
|
|
( wb id imbi2d ) ABDZABCGEF $.
|
|
|
|
${
|
|
imbi1i.1 $e |- ( ph <-> ps ) $.
|
|
$( Introduce a consequent to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
17-Sep-2013.) $)
|
|
imbi1i $p |- ( ( ph -> ch ) <-> ( ps -> ch ) ) $=
|
|
( wb wi imbi1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
imbi12i.1 $e |- ( ph <-> ps ) $.
|
|
imbi12i.2 $e |- ( ch <-> th ) $.
|
|
$( Join two logical equivalences to form equivalence of implications.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
imbi12i $p |- ( ( ph -> ch ) <-> ( ps -> th ) ) $=
|
|
( wi imbi2i imbi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
|
|
$( Theorem imbi12i is the congruence law for implication. $)
|
|
$( $j congruence 'imbi12i'; $)
|
|
$}
|
|
|
|
$( Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
bibi1 $p |- ( ( ph <-> ps ) -> ( ( ph <-> ch ) <-> ( ps <-> ch ) ) ) $=
|
|
( wb id bibi1d ) ABDZABCGEF $.
|
|
|
|
$( Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed
|
|
by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.) $)
|
|
con2bi $p |- ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) $=
|
|
( wn wb notbi notnot bibi2i bicom 3bitr2i ) ABCZDACZJCZDKBDBKDAJEBLKBFGKBHI
|
|
$.
|
|
|
|
${
|
|
con2bid.1 $e |- ( ph -> ( ps <-> -. ch ) ) $.
|
|
$( A contraposition deduction. (Contributed by NM, 15-Apr-1995.) $)
|
|
con2bid $p |- ( ph -> ( ch <-> -. ps ) ) $=
|
|
( wn wb con2bi sylibr ) ABCEFCBEFDCBGH $.
|
|
$}
|
|
|
|
${
|
|
con1bid.1 $e |- ( ph -> ( -. ps <-> ch ) ) $.
|
|
$( A contraposition deduction. (Contributed by NM, 9-Oct-1999.) $)
|
|
con1bid $p |- ( ph -> ( -. ch <-> ps ) ) $=
|
|
( wn bicomd con2bid ) ABCEACBABECDFGF $.
|
|
$}
|
|
|
|
${
|
|
con1bii.1 $e |- ( -. ph <-> ps ) $.
|
|
$( A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 13-Oct-2012.) $)
|
|
con1bii $p |- ( -. ps <-> ph ) $=
|
|
( wn notnot xchbinx bicomi ) ABDAADBAECFG $.
|
|
$}
|
|
|
|
${
|
|
con2bii.1 $e |- ( ph <-> -. ps ) $.
|
|
$( A contraposition inference. (Contributed by NM, 5-Aug-1993.) $)
|
|
con2bii $p |- ( ps <-> -. ph ) $=
|
|
( wn bicomi con1bii ) ADBBAABDCEFE $.
|
|
$}
|
|
|
|
$( Contraposition. Bidirectional version of ~ con1 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
con1b $p |- ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) $=
|
|
( wn wi con1 impbii ) ACBDBCADABEBAEF $.
|
|
|
|
$( Contraposition. Bidirectional version of ~ con2 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
con2b $p |- ( ( ph -> -. ps ) <-> ( ps -> -. ph ) ) $=
|
|
( wn wi con2 impbii ) ABCDBACDABEBAEF $.
|
|
|
|
$( A wff is equivalent to itself with true antecedent. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
biimt $p |- ( ph -> ( ps <-> ( ph -> ps ) ) ) $=
|
|
( wi ax-1 pm2.27 impbid2 ) ABABCBADABEF $.
|
|
|
|
$( Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.5 $p |- ( ph -> ( ( ph -> ps ) <-> ps ) ) $=
|
|
( wi biimt bicomd ) ABABCABDE $.
|
|
|
|
${
|
|
a1bi.1 $e |- ph $.
|
|
$( Inference rule introducing a theorem as an antecedent. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) $)
|
|
a1bi $p |- ( ps <-> ( ph -> ps ) ) $=
|
|
( wi wb biimt ax-mp ) ABABDECABFG $.
|
|
$}
|
|
|
|
${
|
|
mt2bi.1 $e |- ph $.
|
|
$( A false consequent falsifies an antecedent. (Contributed by NM,
|
|
19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) $)
|
|
mt2bi $p |- ( -. ps <-> ( ps -> -. ph ) ) $=
|
|
( wn wi a1bi con2b bitri ) BDZAIEBADEAICFABGH $.
|
|
$}
|
|
|
|
$( Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof
|
|
shortened by Wolf Lammen, 12-Nov-2012.) $)
|
|
mtt $p |- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) ) $=
|
|
( wn wi biimt con34b syl6bbr ) ACZBCZHIDBADHIEBAFG $.
|
|
|
|
$( Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.501 $p |- ( ph -> ( ps <-> ( ph <-> ps ) ) ) $=
|
|
( wb pm5.1im bi1 com12 impbid ) ABABCZABDHABABEFG $.
|
|
|
|
$( Implication in terms of implication and biconditional. (Contributed by
|
|
NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.) $)
|
|
ibib $p |- ( ( ph -> ps ) <-> ( ph -> ( ph <-> ps ) ) ) $=
|
|
( wb pm5.501 pm5.74i ) ABABCABDE $.
|
|
|
|
$( Implication in terms of implication and biconditional. (Contributed by
|
|
NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.) $)
|
|
ibibr $p |- ( ( ph -> ps ) <-> ( ph -> ( ps <-> ph ) ) ) $=
|
|
( wb pm5.501 bicom syl6bb pm5.74i ) ABBACZABABCHABDABEFG $.
|
|
|
|
${
|
|
tbt.1 $e |- ph $.
|
|
$( A wff is equivalent to its equivalence with truth. (Contributed by NM,
|
|
18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
tbt $p |- ( ps <-> ( ps <-> ph ) ) $=
|
|
( wb ibibr pm5.74ri ax-mp ) ABBADZDCABHABEFG $.
|
|
$}
|
|
|
|
$( The negation of a wff is equivalent to the wff's equivalence to
|
|
falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Proof
|
|
shortened by Wolf Lammen, 28-Jan-2013.) $)
|
|
nbn2 $p |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) ) $=
|
|
( wn wb pm5.501 notbi syl6bbr ) ACZBCZHIDABDHIEABFG $.
|
|
|
|
$( Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.)
|
|
(Proof shortened by Wolf Lammen, 28-Jan-2013.) $)
|
|
bibif $p |- ( -. ps -> ( ( ph <-> ps ) <-> -. ph ) ) $=
|
|
( wn wb nbn2 bicom syl6rbb ) BCACBADABDBAEBAFG $.
|
|
|
|
${
|
|
nbn.1 $e |- -. ph $.
|
|
$( The negation of a wff is equivalent to the wff's equivalence to
|
|
falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 3-Oct-2013.) $)
|
|
nbn $p |- ( -. ps <-> ( ps <-> ph ) ) $=
|
|
( wb wn bibif ax-mp bicomi ) BADZBEZAEIJDCBAFGH $.
|
|
$}
|
|
|
|
${
|
|
nbn3.1 $e |- ph $.
|
|
$( Transfer falsehood via equivalence. (Contributed by NM,
|
|
11-Sep-2006.) $)
|
|
nbn3 $p |- ( -. ps <-> ( ps <-> -. ph ) ) $=
|
|
( wn notnoti nbn ) ADBACEF $.
|
|
$}
|
|
|
|
$( Two propositions are equivalent if they are both false. Closed form of
|
|
~ 2false . Equivalent to a ~ bi2 -like version of the xor-connective.
|
|
(Contributed by Wolf Lammen, 13-May-2013.) $)
|
|
pm5.21im $p |- ( -. ph -> ( -. ps -> ( ph <-> ps ) ) ) $=
|
|
( wn wb nbn2 biimpd ) ACBCABDABEF $.
|
|
|
|
${
|
|
2false.1 $e |- -. ph $.
|
|
2false.2 $e |- -. ps $.
|
|
$( Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof
|
|
shortened by Wolf Lammen, 19-May-2013.) $)
|
|
2false $p |- ( ph <-> ps ) $=
|
|
( wn 2th con4bii ) ABAEBECDFG $.
|
|
$}
|
|
|
|
${
|
|
2falsed.1 $e |- ( ph -> -. ps ) $.
|
|
2falsed.2 $e |- ( ph -> -. ch ) $.
|
|
$( Two falsehoods are equivalent (deduction rule). (Contributed by NM,
|
|
11-Oct-2013.) $)
|
|
2falsed $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( pm2.21d impbid ) ABCABCDFACBEFG $.
|
|
$}
|
|
|
|
${
|
|
pm5.21ni.1 $e |- ( ph -> ps ) $.
|
|
pm5.21ni.2 $e |- ( ch -> ps ) $.
|
|
$( Two propositions implying a false one are equivalent. (Contributed by
|
|
NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.) $)
|
|
pm5.21ni $p |- ( -. ps -> ( ph <-> ch ) ) $=
|
|
( wn con3i 2falsed ) BFACABDGCBEGH $.
|
|
|
|
${
|
|
pm5.21nii.3 $e |- ( ps -> ( ph <-> ch ) ) $.
|
|
$( Eliminate an antecedent implied by each side of a biconditional.
|
|
(Contributed by NM, 21-May-1999.) $)
|
|
pm5.21nii $p |- ( ph <-> ch ) $=
|
|
( wb pm5.21ni pm2.61i ) BACGFABCDEHI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
pm5.21ndd.1 $e |- ( ph -> ( ch -> ps ) ) $.
|
|
pm5.21ndd.2 $e |- ( ph -> ( th -> ps ) ) $.
|
|
pm5.21ndd.3 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
|
|
$( Eliminate an antecedent implied by each side of a biconditional,
|
|
deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Proof
|
|
shortened by Wolf Lammen, 6-Oct-2013.) $)
|
|
pm5.21ndd $p |- ( ph -> ( ch <-> th ) ) $=
|
|
( wb wn con3d pm5.21im syl6c pm2.61d ) ABCDHZGABICIDINACBEJADBFJCDKLM $.
|
|
$}
|
|
|
|
${
|
|
bija.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
bija.2 $e |- ( -. ph -> ( -. ps -> ch ) ) $.
|
|
$( Combine antecedents into a single bi-conditional. This inference,
|
|
reminiscent of ~ ja , is reversible: The hypotheses can be deduced from
|
|
the conclusion alone (see ~ pm5.1im and ~ pm5.21im ). (Contributed by
|
|
Wolf Lammen, 13-May-2013.) $)
|
|
bija $p |- ( ( ph <-> ps ) -> ch ) $=
|
|
( wb bi2 syli wn bi1 con3d pm2.61d ) ABFZBCBMACABGDHBIMAICMABABJKEHL $.
|
|
$}
|
|
|
|
$( Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that
|
|
logical equivalence is the same as negated "exclusive-or." (Contributed
|
|
by NM, 28-Jun-2002.) (Proof shortened by Andrew Salmon, 20-Jun-2011.)
|
|
(Proof shortened by Wolf Lammen, 15-Oct-2013.) $)
|
|
pm5.18 $p |- ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) $=
|
|
( wb wn pm5.501 con1bid bitr2d nbn2 pm2.61i ) AABCZABDZCZDZCAMBJABLAKEFABEG
|
|
ADZMKJNKLAKHFABHGI $.
|
|
|
|
$( Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.) $)
|
|
xor3 $p |- ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) $=
|
|
( wn wb pm5.18 con2bii bicomi ) ABCDZABDZCIHABEFG $.
|
|
|
|
$( Move negation outside of biconditional. Compare Theorem *5.18 of
|
|
[WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof
|
|
shortened by Wolf Lammen, 20-Sep-2013.) $)
|
|
nbbn $p |- ( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) $=
|
|
( wb wn xor3 con2bi bicom 3bitrri ) ABCDABDCBADZCIBCABEABFBIGH $.
|
|
|
|
$( Associative law for the biconditional. An axiom of system DS in Vladimir
|
|
Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic,
|
|
113:207-224, 2002,
|
|
~ http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805 .
|
|
Interestingly, this law was not included in _Principia Mathematica_ but
|
|
was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by
|
|
NM, 8-Jan-2005.) (Proof shortened by Juha Arpiainen, 19-Jan-2006.)
|
|
(Proof shortened by Wolf Lammen, 21-Sep-2013.) $)
|
|
biass $p |- ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) $=
|
|
( wb pm5.501 bibi1d bitr3d wn nbbn nbn2 syl5bbr pm2.61i ) AABDZCDZABCDZDZDA
|
|
ONPABMCABEFAOEGAHZOHZNPRBHZCDQNBCIQSMCABJFKAOJGL $.
|
|
|
|
$( Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.19 $p |- -. ( ph <-> -. ph ) $=
|
|
( wb wn biid pm5.18 mpbi ) AABAACBCADAAEF $.
|
|
|
|
$( Logical equivalence of commuted antecedents. Part of Theorem *4.87 of
|
|
[WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.) $)
|
|
bi2.04 $p |- ( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) ) $=
|
|
( wi pm2.04 impbii ) ABCDDBACDDABCEBACEF $.
|
|
|
|
$( Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell]
|
|
p. 125. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm5.4 $p |- ( ( ph -> ( ph -> ps ) ) <-> ( ph -> ps ) ) $=
|
|
( wi pm2.43 ax-1 impbii ) AABCZCGABDGAEF $.
|
|
|
|
$( Distributive law for implication. Compare Theorem *5.41 of
|
|
[WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) $)
|
|
imdi $p |- ( ( ph -> ( ps -> ch ) ) <->
|
|
( ( ph -> ps ) -> ( ph -> ch ) ) ) $=
|
|
( wi ax-2 pm2.86 impbii ) ABCDDABDACDDABCEABCFG $.
|
|
|
|
$( Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.) $)
|
|
pm5.41 $p |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) <->
|
|
( ph -> ( ps -> ch ) ) ) $=
|
|
( wi imdi bicomi ) ABCDDABDACDDABCEF $.
|
|
|
|
$( Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.8 $p |- ( ( ph -> -. ph ) <-> -. ph ) $=
|
|
( wn wi pm2.01 ax-1 impbii ) AABZCGADGAEF $.
|
|
|
|
$( Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.81 $p |- ( ( -. ph -> ph ) <-> ph ) $=
|
|
( wn wi pm2.18 pm2.24 impbii ) ABACAADAAEF $.
|
|
|
|
$( Simplify an implication between two implications when the antecedent of
|
|
the first is a consequence of the antecedent of the second. The reverse
|
|
form is useful in producing the successor step in induction proofs.
|
|
(Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf
|
|
Lammen, 14-Sep-2013.) $)
|
|
imim21b $p |- ( ( ps -> ph ) -> ( ( ( ph -> ch ) -> ( ps -> th ) ) <->
|
|
( ps -> ( ch -> th ) ) ) ) $=
|
|
( wi bi2.04 wb pm5.5 imbi1d imim2i pm5.74d syl5bb ) ACEZBDEEBMDEZEBAEZBCDEZ
|
|
EMBDFOBNPANPGBAMCDACHIJKL $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical disjunction and conjunction
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Here we define disjunction (logical 'or') ` \/ ` ( ~ df-or ) and conjunction
|
|
(logical 'and') ` /\ ` ( ~ df-an ). We also define various rules for
|
|
simplifying and applying them, e.g., ~ olc , ~ orc , and ~ orcom .
|
|
|
|
$)
|
|
|
|
$( Declare connectives for disjunction ('or') and conjunction ('and'). $)
|
|
$c \/ $. $( Vee (read: 'or') $)
|
|
$c /\ $. $( Wedge (read: 'and') $)
|
|
|
|
$( Extend wff definition to include disjunction ('or'). $)
|
|
wo $a wff ( ph \/ ps ) $.
|
|
$( Extend wff definition to include conjunction ('and'). $)
|
|
wa $a wff ( ph /\ ps ) $.
|
|
|
|
$( Define disjunction (logical 'or'). Definition of [Margaris] p. 49. When
|
|
the left operand, right operand, or both are true, the result is true;
|
|
when both sides are false, the result is false. For example, it is true
|
|
that (2 = 3 ` \/ ` 4 = 4) (see ex-or in set.mm). After we define the
|
|
constant true ` T. ` ( ~ df-tru ) and the constant false ` F. `
|
|
( ~ df-fal ), we will be able to prove these truth table values:
|
|
` ( ( T. \/ T. ) <-> T. ) ` ( ~ truortru ), ` ( ( T. \/ F. ) <-> T. ) `
|
|
( ~ truorfal ), ` ( ( F. \/ T. ) <-> T. ) ` ( ~ falortru ), and
|
|
` ( ( F. \/ F. ) <-> F. ) ` ( ~ falorfal ).
|
|
|
|
This is our first use of the biconditional connective in a definition; we
|
|
use the biconditional connective in place of the traditional "<=def=>",
|
|
which means the same thing, except that we can manipulate the
|
|
biconditional connective directly in proofs rather than having to rely on
|
|
an informal definition substitution rule. Note that if we mechanically
|
|
substitute ` ( -. ph -> ps ) ` for ` ( ph \/ ps ) ` , we end up with an
|
|
instance of previously proved theorem ~ biid . This is the justification
|
|
for the definition, along with the fact that it introduces a new symbol
|
|
` \/ ` . Contrast with ` /\ ` ( ~ df-an ), ` -> ` ( ~ wi ), ` -/\ `
|
|
( ~ df-nan ), and ` \/_ ` ( ~ df-xor ) . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
df-or $a |- ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) $.
|
|
|
|
$( Define conjunction (logical 'and'). Definition of [Margaris] p. 49. When
|
|
both the left and right operand are true, the result is true; when either
|
|
is false, the result is false. For example, it is true that (2 = 2 ` /\ `
|
|
3 = 3). After we define the constant true ` T. ` ( ~ df-tru ) and the
|
|
constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth
|
|
table values: ` ( ( T. /\ T. ) <-> T. ) ` ( ~ truantru ),
|
|
` ( ( T. /\ F. ) <-> F. ) ` ( ~ truanfal ), ` ( ( F. /\ T. ) <-> F. ) `
|
|
( ~ falantru ), and ` ( ( F. /\ F. ) <-> F. ) ` ( ~ falanfal ).
|
|
|
|
Contrast with ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), ` -/\ ` ( ~ df-nan ),
|
|
and ` \/_ ` ( ~ df-xor ) . (Contributed by NM, 5-Aug-1993.) $)
|
|
df-an $a |- ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) $.
|
|
|
|
$( Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.64 $p |- ( ( -. ph -> ps ) <-> ( ph \/ ps ) ) $=
|
|
( wo wn wi df-or bicomi ) ABCADBEABFG $.
|
|
|
|
$( Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.53 $p |- ( ( ph \/ ps ) -> ( -. ph -> ps ) ) $=
|
|
( wo wn wi df-or biimpi ) ABCADBEABFG $.
|
|
|
|
$( Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.54 $p |- ( ( -. ph -> ps ) -> ( ph \/ ps ) ) $=
|
|
( wo wn wi df-or biimpri ) ABCADBEABFG $.
|
|
|
|
${
|
|
ori.1 $e |- ( ph \/ ps ) $.
|
|
$( Infer implication from disjunction. (Contributed by NM,
|
|
11-Jun-1994.) $)
|
|
ori $p |- ( -. ph -> ps ) $=
|
|
( wo wn wi df-or mpbi ) ABDAEBFCABGH $.
|
|
$}
|
|
|
|
${
|
|
orri.1 $e |- ( -. ph -> ps ) $.
|
|
$( Infer implication from disjunction. (Contributed by NM,
|
|
11-Jun-1994.) $)
|
|
orri $p |- ( ph \/ ps ) $=
|
|
( wo wn wi df-or mpbir ) ABDAEBFCABGH $.
|
|
$}
|
|
|
|
${
|
|
ord.1 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
$( Deduce implication from disjunction. (Contributed by NM,
|
|
18-May-1994.) $)
|
|
ord $p |- ( ph -> ( -. ps -> ch ) ) $=
|
|
( wo wn wi df-or sylib ) ABCEBFCGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
orrd.1 $e |- ( ph -> ( -. ps -> ch ) ) $.
|
|
$( Deduce implication from disjunction. (Contributed by NM,
|
|
27-Nov-1995.) $)
|
|
orrd $p |- ( ph -> ( ps \/ ch ) ) $=
|
|
( wn wi wo pm2.54 syl ) ABECFBCGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
jaoi.1 $e |- ( ph -> ps ) $.
|
|
jaoi.2 $e |- ( ch -> ps ) $.
|
|
$( Inference disjoining the antecedents of two implications. (Contributed
|
|
by NM, 5-Apr-1994.) $)
|
|
jaoi $p |- ( ( ph \/ ch ) -> ps ) $=
|
|
( wo wn pm2.53 syl6 pm2.61d2 ) ACFZABKAGCBACHEIDJ $.
|
|
$}
|
|
|
|
${
|
|
jaod.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jaod.2 $e |- ( ph -> ( th -> ch ) ) $.
|
|
$( Deduction disjoining the antecedents of two implications. (Contributed
|
|
by NM, 18-Aug-1994.) $)
|
|
jaod $p |- ( ph -> ( ( ps \/ th ) -> ch ) ) $=
|
|
( wo wi com12 jaoi ) BDGACBACHDABCEIADCFIJI $.
|
|
|
|
jaod.3 $e |- ( ph -> ( ps \/ th ) ) $.
|
|
$( Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro,
|
|
29-May-2016.) $)
|
|
mpjaod $p |- ( ph -> ch ) $=
|
|
( wo jaod mpd ) ABDHCGABCDEFIJ $.
|
|
$}
|
|
|
|
$( Elimination of disjunction by denial of a disjunct. Theorem *2.55 of
|
|
[WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof
|
|
shortened by Wolf Lammen, 21-Jul-2012.) $)
|
|
orel1 $p |- ( -. ph -> ( ( ph \/ ps ) -> ps ) ) $=
|
|
( wo wn pm2.53 com12 ) ABCADBABEF $.
|
|
|
|
$( Elimination of disjunction by denial of a disjunct. Theorem *2.56 of
|
|
[WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof
|
|
shortened by Wolf Lammen, 5-Apr-2013.) $)
|
|
orel2 $p |- ( -. ph -> ( ( ps \/ ph ) -> ps ) ) $=
|
|
( wn idd pm2.21 jaod ) ACZBBAGBDABEF $.
|
|
|
|
$( Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96.
|
|
(Contributed by NM, 30-Aug-1993.) $)
|
|
olc $p |- ( ph -> ( ps \/ ph ) ) $=
|
|
( wn ax-1 orrd ) ABAABCDE $.
|
|
|
|
$( Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104.
|
|
(Contributed by NM, 30-Aug-1993.) $)
|
|
orc $p |- ( ph -> ( ph \/ ps ) ) $=
|
|
( pm2.24 orrd ) AABABCD $.
|
|
|
|
$( Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm1.4 $p |- ( ( ph \/ ps ) -> ( ps \/ ph ) ) $=
|
|
( wo olc orc jaoi ) ABACBABDBAEF $.
|
|
|
|
$( Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 15-Nov-2012.) $)
|
|
orcom $p |- ( ( ph \/ ps ) <-> ( ps \/ ph ) ) $=
|
|
( wo pm1.4 impbii ) ABCBACABDBADE $.
|
|
|
|
${
|
|
orcomd.1 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
$( Commutation of disjuncts in consequent. (Contributed by NM,
|
|
2-Dec-2010.) $)
|
|
orcomd $p |- ( ph -> ( ch \/ ps ) ) $=
|
|
( wo orcom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
orcoms.1 $e |- ( ( ph \/ ps ) -> ch ) $.
|
|
$( Commutation of disjuncts in antecedent. (Contributed by NM,
|
|
2-Dec-2012.) $)
|
|
orcoms $p |- ( ( ps \/ ph ) -> ch ) $=
|
|
( wo pm1.4 syl ) BAEABECBAFDG $.
|
|
$}
|
|
|
|
${
|
|
orci.1 $e |- ph $.
|
|
$( Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.)
|
|
(Proof shortened by Wolf Lammen, 14-Nov-2012.) $)
|
|
orci $p |- ( ph \/ ps ) $=
|
|
( pm2.24i orri ) ABABCDE $.
|
|
|
|
$( Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.)
|
|
(Proof shortened by Wolf Lammen, 14-Nov-2012.) $)
|
|
olci $p |- ( ps \/ ph ) $=
|
|
( wn a1i orri ) BAABDCEF $.
|
|
$}
|
|
|
|
${
|
|
orcd.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction introducing a disjunct. A translation of natural deduction
|
|
rule ` \/ ` IR ( ` \/ ` insertion right), see natded in set.mm.
|
|
(Contributed by NM, 20-Sep-2007.) $)
|
|
orcd $p |- ( ph -> ( ps \/ ch ) ) $=
|
|
( wo orc syl ) ABBCEDBCFG $.
|
|
|
|
$( Deduction introducing a disjunct. A translation of natural deduction
|
|
rule ` \/ ` IL ( ` \/ ` insertion left), see natded in set.mm.
|
|
(Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen,
|
|
3-Oct-2013.) $)
|
|
olcd $p |- ( ph -> ( ch \/ ps ) ) $=
|
|
( orcd orcomd ) ABCABCDEF $.
|
|
$}
|
|
|
|
${
|
|
orcs.1 $e |- ( ( ph \/ ps ) -> ch ) $.
|
|
$( Deduction eliminating disjunct. _Notational convention_: We sometimes
|
|
suffix with "s" the label of an inference that manipulates an
|
|
antecedent, leaving the consequent unchanged. The "s" means that the
|
|
inference eliminates the need for a syllogism ( ~ syl ) -type inference
|
|
in a proof. (Contributed by NM, 21-Jun-1994.) $)
|
|
orcs $p |- ( ph -> ch ) $=
|
|
( wo orc syl ) AABECABFDG $.
|
|
$}
|
|
|
|
${
|
|
olcs.1 $e |- ( ( ph \/ ps ) -> ch ) $.
|
|
$( Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.)
|
|
(Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
|
|
olcs $p |- ( ps -> ch ) $=
|
|
( orcoms orcs ) BACABCDEF $.
|
|
$}
|
|
|
|
$( Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.07 $p |- ( ph -> ( ph \/ ph ) ) $=
|
|
( olc ) AAB $.
|
|
|
|
$( Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.45 $p |- ( -. ( ph \/ ps ) -> -. ph ) $=
|
|
( wo orc con3i ) AABCABDE $.
|
|
|
|
$( Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.46 $p |- ( -. ( ph \/ ps ) -> -. ps ) $=
|
|
( wo olc con3i ) BABCBADE $.
|
|
|
|
$( Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.47 $p |- ( -. ( ph \/ ps ) -> ( -. ph \/ ps ) ) $=
|
|
( wo wn pm2.45 orcd ) ABCDADBABEF $.
|
|
|
|
$( Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.48 $p |- ( -. ( ph \/ ps ) -> ( ph \/ -. ps ) ) $=
|
|
( wo wn pm2.46 olcd ) ABCDBDAABEF $.
|
|
|
|
$( Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.49 $p |- ( -. ( ph \/ ps ) -> ( -. ph \/ -. ps ) ) $=
|
|
( wo wn pm2.46 olcd ) ABCDBDADABEF $.
|
|
|
|
$( Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107.
|
|
(Contributed by NM, 3-Jan-2005.) $)
|
|
pm2.67-2 $p |- ( ( ( ph \/ ch ) -> ps ) -> ( ph -> ps ) ) $=
|
|
( wo orc imim1i ) AACDBACEF $.
|
|
|
|
$( Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.67 $p |- ( ( ( ph \/ ps ) -> ps ) -> ( ph -> ps ) ) $=
|
|
( pm2.67-2 ) ABBC $.
|
|
|
|
$( Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.25 $p |- ( ph \/ ( ( ph \/ ps ) -> ps ) ) $=
|
|
( wo wi orel1 orri ) AABCBDABEF $.
|
|
|
|
$( A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of
|
|
[WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof
|
|
shortened by Wolf Lammen, 18-Nov-2012.) $)
|
|
biorf $p |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) ) $=
|
|
( wn wo olc orel1 impbid2 ) ACBABDBAEABFG $.
|
|
|
|
$( A wff is equivalent to its negated disjunction with falsehood.
|
|
(Contributed by NM, 9-Jul-2012.) $)
|
|
biortn $p |- ( ph -> ( ps <-> ( -. ph \/ ps ) ) ) $=
|
|
( wn wo wb notnot1 biorf syl ) AACZCBIBDEAFIBGH $.
|
|
|
|
${
|
|
biorfi.1 $e |- -. ph $.
|
|
$( A wff is equivalent to its disjunction with falsehood. (Contributed by
|
|
NM, 23-Mar-1995.) $)
|
|
biorfi $p |- ( ps <-> ( ps \/ ph ) ) $=
|
|
( wn wo wb orc orel2 impbid2 ax-mp ) ADZBBAEZFCKBLBAGABHIJ $.
|
|
$}
|
|
|
|
$( Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.621 $p |- ( ( ph -> ps ) -> ( ( ph \/ ps ) -> ps ) ) $=
|
|
( wi id idd jaod ) ABCZABBGDGBEF $.
|
|
|
|
$( Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.) $)
|
|
pm2.62 $p |- ( ( ph \/ ps ) -> ( ( ph -> ps ) -> ps ) ) $=
|
|
( wi wo pm2.621 com12 ) ABCABDBABEF $.
|
|
|
|
$( Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.68 $p |- ( ( ( ph -> ps ) -> ps ) -> ( ph \/ ps ) ) $=
|
|
( wi jarl orrd ) ABCBCABABBDE $.
|
|
|
|
$( Logical 'or' expressed in terms of implication only. Theorem *5.25 of
|
|
[WhiteheadRussell] p. 124. (Contributed by NM, 12-Aug-2004.) (Proof
|
|
shortened by Wolf Lammen, 20-Oct-2012.) $)
|
|
dfor2 $p |- ( ( ph \/ ps ) <-> ( ( ph -> ps ) -> ps ) ) $=
|
|
( wo wi pm2.62 pm2.68 impbii ) ABCABDBDABEABFG $.
|
|
|
|
$( Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell]
|
|
p. 120. (Contributed by NM, 5-Aug-1993.) $)
|
|
imor $p |- ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) $=
|
|
( wi wn wo notnot imbi1i df-or bitr4i ) ABCADZDZBCJBEAKBAFGJBHI $.
|
|
|
|
${
|
|
imori.1 $e |- ( ph -> ps ) $.
|
|
$( Infer disjunction from implication. (Contributed by NM,
|
|
12-Mar-2012.) $)
|
|
imori $p |- ( -. ph \/ ps ) $=
|
|
( wi wn wo imor mpbi ) ABDAEBFCABGH $.
|
|
$}
|
|
|
|
${
|
|
imorri.1 $e |- ( -. ph \/ ps ) $.
|
|
$( Infer implication from disjunction. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
imorri $p |- ( ph -> ps ) $=
|
|
( wi wn wo imor mpbir ) ABDAEBFCABGH $.
|
|
$}
|
|
|
|
$( Law of excluded middle, also called the principle of _tertium non datur_.
|
|
Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is
|
|
either true or not true; there are no in-between values of truth. This is
|
|
an essential distinction of our classical logic and is not a theorem of
|
|
intuitionistic logic. (Contributed by NM, 5-Aug-1993.) $)
|
|
exmid $p |- ( ph \/ -. ph ) $=
|
|
( wn id orri ) AABZECD $.
|
|
|
|
$( Law of excluded middle in a context. (Contributed by Mario Carneiro,
|
|
9-Feb-2017.) $)
|
|
exmidd $p |- ( ph -> ( ps \/ -. ps ) ) $=
|
|
( wn wo exmid a1i ) BBCDABEF $.
|
|
|
|
$( Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) $)
|
|
pm2.1 $p |- ( -. ph \/ ph ) $=
|
|
( id imori ) AAABC $.
|
|
|
|
$( Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.13 $p |- ( ph \/ -. -. -. ph ) $=
|
|
( wn notnot1 orri ) AABZBBECD $.
|
|
|
|
$( Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.62 $p |- ( ( ph -> -. ps ) <-> ( -. ph \/ -. ps ) ) $=
|
|
( wn imor ) ABCD $.
|
|
|
|
$( Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.66 $p |- ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) ) $=
|
|
( wn pm4.64 ) ABCD $.
|
|
|
|
$( Theorem *4.63 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.63 $p |- ( -. ( ph -> -. ps ) <-> ( ph /\ ps ) ) $=
|
|
( wa wn wi df-an bicomi ) ABCABDEDABFG $.
|
|
|
|
$( Express implication in terms of conjunction. (Contributed by NM,
|
|
9-Apr-1994.) $)
|
|
imnan $p |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) ) $=
|
|
( wa wn wi df-an con2bii ) ABCABDEABFG $.
|
|
|
|
${
|
|
imnani.1 $e |- -. ( ph /\ ps ) $.
|
|
$( Express implication in terms of conjunction. (Contributed by Mario
|
|
Carneiro, 28-Sep-2015.) $)
|
|
imnani $p |- ( ph -> -. ps ) $=
|
|
( wn wi wa imnan mpbir ) ABDEABFDCABGH $.
|
|
$}
|
|
|
|
$( Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll]
|
|
p. 176. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 30-Oct-2012.) $)
|
|
iman $p |- ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) $=
|
|
( wi wn wa notnot imbi2i imnan bitri ) ABCABDZDZCAJEDBKABFGAJHI $.
|
|
|
|
$( Express conjunction in terms of implication. (Contributed by NM,
|
|
2-Aug-1994.) $)
|
|
annim $p |- ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) $=
|
|
( wi wn wa iman con2bii ) ABCABDEABFG $.
|
|
|
|
$( Theorem *4.61 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.61 $p |- ( -. ( ph -> ps ) <-> ( ph /\ -. ps ) ) $=
|
|
( wn wa wi annim bicomi ) ABCDABECABFG $.
|
|
|
|
$( Theorem *4.65 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.65 $p |- ( -. ( -. ph -> ps ) <-> ( -. ph /\ -. ps ) ) $=
|
|
( wn pm4.61 ) ACBD $.
|
|
|
|
$( Theorem *4.67 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.67 $p |- ( -. ( -. ph -> -. ps ) <-> ( -. ph /\ ps ) ) $=
|
|
( wn pm4.63 ) ACBD $.
|
|
|
|
${
|
|
imp.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Eric Schmidt, 22-Dec-2006.) $)
|
|
imp $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wa wn wi df-an impi sylbi ) ABEABFGFCABHABCDIJ $.
|
|
|
|
$( Importation inference with commuted antecedents. (Contributed by NM,
|
|
25-May-2005.) $)
|
|
impcom $p |- ( ( ps /\ ph ) -> ch ) $=
|
|
( com12 imp ) BACABCDEF $.
|
|
$}
|
|
|
|
${
|
|
imp3.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Importation deduction. (Contributed by NM, 31-Mar-1994.) $)
|
|
imp3a $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
|
|
( wa wi com3l imp com12 ) BCFADBCADGABCDEHIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp31 $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
|
|
( wa wi imp ) ABFCDABCDGEHH $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp32 $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
|
|
( wa imp3a imp ) ABCFDABCDEGH $.
|
|
$}
|
|
|
|
${
|
|
exp.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Exportation inference. (This theorem used to be labeled "exp" but was
|
|
changed to "ex" so as not to conflict with the math token "exp", per the
|
|
June 2006 Metamath spec change.) A translation of natural deduction
|
|
rule ` -> ` I ( ` -> ` introduction), see natded in set.mm.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt,
|
|
22-Dec-2006.) $)
|
|
ex $p |- ( ph -> ( ps -> ch ) ) $=
|
|
( wn wi wa df-an sylbir expi ) ABCABEFEABGCABHDIJ $.
|
|
|
|
$( Exportation inference with commuted antecedents. (Contributed by NM,
|
|
25-May-2005.) $)
|
|
expcom $p |- ( ps -> ( ph -> ch ) ) $=
|
|
( ex com12 ) ABCABCDEF $.
|
|
$}
|
|
|
|
${
|
|
exp3a.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Exportation deduction. (Contributed by NM, 20-Aug-1993.) $)
|
|
exp3a $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wi wa com12 ex com3r ) BCADBCADFABCGDEHIJ $.
|
|
|
|
$( A deduction version of exportation, followed by importation.
|
|
(Contributed by NM, 6-Sep-2008.) $)
|
|
expdimp $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
|
|
( wi exp3a imp ) ABCDFABCDEGH $.
|
|
$}
|
|
|
|
${
|
|
impancom.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Mixed importation/commutation inference. (Contributed by NM,
|
|
22-Jun-2013.) $)
|
|
impancom $p |- ( ( ph /\ ch ) -> ( ps -> th ) ) $=
|
|
( wi ex com23 imp ) ACBDFABCDABCDFEGHI $.
|
|
$}
|
|
|
|
${
|
|
con3and.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Variant of ~ con3d with importation. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
con3and $p |- ( ( ph /\ -. ch ) -> -. ps ) $=
|
|
( wn con3d imp ) ACEBEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
pm2.01da.1 $e |- ( ( ph /\ ps ) -> -. ps ) $.
|
|
$( Deduction based on reductio ad absurdum. (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) $)
|
|
pm2.01da $p |- ( ph -> -. ps ) $=
|
|
( wn ex pm2.01d ) ABABBDCEF $.
|
|
$}
|
|
|
|
${
|
|
pm2.18da.1 $e |- ( ( ph /\ -. ps ) -> ps ) $.
|
|
$( Deduction based on reductio ad absurdum. (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) $)
|
|
pm2.18da $p |- ( ph -> ps ) $=
|
|
( wn ex pm2.18d ) ABABDBCEF $.
|
|
$}
|
|
|
|
$( Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
|
|
pm3.3 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) $=
|
|
( wa wi id exp3a ) ABDCEZABCHFG $.
|
|
|
|
$( Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
|
|
pm3.31 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph /\ ps ) -> ch ) ) $=
|
|
( wi id imp3a ) ABCDDZABCGEF $.
|
|
|
|
$( Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell]
|
|
p. 122. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 24-Mar-2013.) $)
|
|
impexp $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) $=
|
|
( wa wi pm3.3 pm3.31 impbii ) ABDCEABCEEABCFABCGH $.
|
|
|
|
$( Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell]
|
|
p. 111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 12-Nov-2012.) $)
|
|
pm3.2 $p |- ( ph -> ( ps -> ( ph /\ ps ) ) ) $=
|
|
( wa id ex ) ABABCZFDE $.
|
|
|
|
$( Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell]
|
|
p. 111. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm3.21 $p |- ( ph -> ( ps -> ( ps /\ ph ) ) ) $=
|
|
( wa pm3.2 com12 ) BABACBADE $.
|
|
|
|
$( Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) $)
|
|
pm3.22 $p |- ( ( ph /\ ps ) -> ( ps /\ ph ) ) $=
|
|
( wa pm3.21 imp ) ABBACABDE $.
|
|
|
|
$( Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Wolf
|
|
Lammen, 4-Nov-2012.) $)
|
|
ancom $p |- ( ( ph /\ ps ) <-> ( ps /\ ph ) ) $=
|
|
( wa pm3.22 impbii ) ABCBACABDBADE $.
|
|
|
|
${
|
|
ancomd.1 $e |- ( ph -> ( ps /\ ch ) ) $.
|
|
$( Commutation of conjuncts in consequent. (Contributed by Jeff Hankins,
|
|
14-Aug-2009.) $)
|
|
ancomd $p |- ( ph -> ( ch /\ ps ) ) $=
|
|
( wa ancom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
ancoms.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Inference commuting conjunction in antecedent. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
ancoms $p |- ( ( ps /\ ph ) -> ch ) $=
|
|
( expcom imp ) BACABCDEF $.
|
|
$}
|
|
|
|
${
|
|
ancomsd.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Deduction commuting conjunction in antecedent. (Contributed by NM,
|
|
12-Dec-2004.) $)
|
|
ancomsd $p |- ( ph -> ( ( ch /\ ps ) -> th ) ) $=
|
|
( wa ancom syl5bi ) CBFBCFADCBGEH $.
|
|
$}
|
|
|
|
${
|
|
pm3.2i.1 $e |- ph $.
|
|
pm3.2i.2 $e |- ps $.
|
|
$( Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm3.2i $p |- ( ph /\ ps ) $=
|
|
( wa pm3.2 mp2 ) ABABECDABFG $.
|
|
$}
|
|
|
|
$( Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.) $)
|
|
pm3.43i $p |- ( ( ph -> ps )
|
|
-> ( ( ph -> ch ) -> ( ph -> ( ps /\ ch ) ) ) ) $=
|
|
( wa pm3.2 imim3i ) BCBCDABCEF $.
|
|
|
|
$( Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell]
|
|
p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 13-Nov-2012.) $)
|
|
simpl $p |- ( ( ph /\ ps ) -> ph ) $=
|
|
( ax-1 imp ) ABAABCD $.
|
|
|
|
${
|
|
simpli.1 $e |- ( ph /\ ps ) $.
|
|
$( Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) $)
|
|
simpli $p |- ph $=
|
|
( wa simpl ax-mp ) ABDACABEF $.
|
|
$}
|
|
|
|
${
|
|
simpld.1 $e |- ( ph -> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. A translation of natural deduction
|
|
rule ` /\ ` EL ( ` /\ ` elimination left), see natded in set.mm.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
simpld $p |- ( ph -> ps ) $=
|
|
( wa simpl syl ) ABCEBDBCFG $.
|
|
$}
|
|
|
|
${
|
|
simplbi.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.) $)
|
|
simplbi $p |- ( ph -> ps ) $=
|
|
( wa biimpi simpld ) ABCABCEDFG $.
|
|
$}
|
|
|
|
$( Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell]
|
|
p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 13-Nov-2012.) $)
|
|
simpr $p |- ( ( ph /\ ps ) -> ps ) $=
|
|
( idd imp ) ABBABCD $.
|
|
|
|
${
|
|
simpri.1 $e |- ( ph /\ ps ) $.
|
|
$( Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) $)
|
|
simpri $p |- ps $=
|
|
( wa simpr ax-mp ) ABDBCABEF $.
|
|
$}
|
|
|
|
${
|
|
simprd.1 $e |- ( ph -> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) A
|
|
translation of natural deduction rule ` /\ ` ER ( ` /\ ` elimination
|
|
right), see natded in set.mm. (Proof shortened by Wolf Lammen,
|
|
3-Oct-2013.) $)
|
|
simprd $p |- ( ph -> ch ) $=
|
|
( ancomd simpld ) ACBABCDEF $.
|
|
$}
|
|
|
|
${
|
|
simprbi.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.) $)
|
|
simprbi $p |- ( ph -> ch ) $=
|
|
( wa biimpi simprd ) ABCABCEDFG $.
|
|
$}
|
|
|
|
${
|
|
adantr.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding a conjunct to the right of an antecedent. (Contributed
|
|
by NM, 30-Aug-1993.) $)
|
|
adantr $p |- ( ( ph /\ ch ) -> ps ) $=
|
|
( a1d imp ) ACBABCDEF $.
|
|
$}
|
|
|
|
${
|
|
adantl.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding a conjunct to the left of an antecedent. (Contributed
|
|
by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) $)
|
|
adantl $p |- ( ( ch /\ ph ) -> ps ) $=
|
|
( adantr ancoms ) ACBABCDEF $.
|
|
$}
|
|
|
|
${
|
|
adantld.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction adding a conjunct to the left of an antecedent. (Contributed
|
|
by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.) $)
|
|
adantld $p |- ( ph -> ( ( th /\ ps ) -> ch ) ) $=
|
|
( wa simpr syl5 ) DBFBACDBGEH $.
|
|
$}
|
|
|
|
${
|
|
adantrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction adding a conjunct to the right of an antecedent. (Contributed
|
|
by NM, 4-May-1994.) $)
|
|
adantrd $p |- ( ph -> ( ( ps /\ th ) -> ch ) ) $=
|
|
( wa simpl syl5 ) BDFBACBDGEH $.
|
|
$}
|
|
|
|
${
|
|
mpan9.1 $e |- ( ph -> ps ) $.
|
|
mpan9.2 $e |- ( ch -> ( ps -> th ) ) $.
|
|
$( Modus ponens conjoining dissimilar antecedents. (Contributed by NM,
|
|
1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
mpan9 $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( syl5 impcom ) CADABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syldan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
syldan.2 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
$( A syllogism deduction with conjoined antecedents. (Contributed by NM,
|
|
24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.) $)
|
|
syldan $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa expcom adantrd mpcom ) CABGDECADBACDFHIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan.1 $e |- ( ph -> ps ) $.
|
|
sylan.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 22-Nov-2012.) $)
|
|
sylan $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( expcom mpan9 ) ABCDEBCDFGH $.
|
|
$}
|
|
|
|
${
|
|
sylanb.1 $e |- ( ph <-> ps ) $.
|
|
sylanb.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 18-May-1994.) $)
|
|
sylanb $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( biimpi sylan ) ABCDABEGFH $.
|
|
$}
|
|
|
|
${
|
|
sylanbr.1 $e |- ( ps <-> ph ) $.
|
|
sylanbr.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 18-May-1994.) $)
|
|
sylanbr $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( biimpri sylan ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
sylan2.1 $e |- ( ph -> ch ) $.
|
|
sylan2.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 22-Nov-2012.) $)
|
|
sylan2 $p |- ( ( ps /\ ph ) -> th ) $=
|
|
( adantl syldan ) BACDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
sylan2b.1 $e |- ( ph <-> ch ) $.
|
|
sylan2b.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) $)
|
|
sylan2b $p |- ( ( ps /\ ph ) -> th ) $=
|
|
( biimpi sylan2 ) ABCDACEGFH $.
|
|
$}
|
|
|
|
${
|
|
sylan2br.1 $e |- ( ch <-> ph ) $.
|
|
sylan2br.2 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference. (Contributed by NM, 21-Apr-1994.) $)
|
|
sylan2br $p |- ( ( ps /\ ph ) -> th ) $=
|
|
( biimpri sylan2 ) ABCDCAEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl2an.1 $e |- ( ph -> ps ) $.
|
|
syl2an.2 $e |- ( ta -> ch ) $.
|
|
syl2an.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A double syllogism inference. (Contributed by NM, 31-Jan-1997.) $)
|
|
syl2an $p |- ( ( ph /\ ta ) -> th ) $=
|
|
( sylan sylan2 ) EACDGABCDFHIJ $.
|
|
|
|
$( A double syllogism inference. (Contributed by NM, 17-Sep-2013.) $)
|
|
syl2anr $p |- ( ( ta /\ ph ) -> th ) $=
|
|
( syl2an ancoms ) AEDABCDEFGHIJ $.
|
|
$}
|
|
|
|
${
|
|
syl2anb.1 $e |- ( ph <-> ps ) $.
|
|
syl2anb.2 $e |- ( ta <-> ch ) $.
|
|
syl2anb.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A double syllogism inference. (Contributed by NM, 29-Jul-1999.) $)
|
|
syl2anb $p |- ( ( ph /\ ta ) -> th ) $=
|
|
( sylanb sylan2b ) EACDGABCDFHIJ $.
|
|
$}
|
|
|
|
${
|
|
syl2anbr.1 $e |- ( ps <-> ph ) $.
|
|
syl2anbr.2 $e |- ( ch <-> ta ) $.
|
|
syl2anbr.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A double syllogism inference. (Contributed by NM, 29-Jul-1999.) $)
|
|
syl2anbr $p |- ( ( ph /\ ta ) -> th ) $=
|
|
( sylanbr sylan2br ) EACDGABCDFHIJ $.
|
|
$}
|
|
|
|
${
|
|
syland.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syland.2 $e |- ( ph -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 15-Dec-2004.) $)
|
|
syland $p |- ( ph -> ( ( ps /\ th ) -> ta ) ) $=
|
|
( wi exp3a syld imp3a ) ABDEABCDEHFACDEGIJK $.
|
|
$}
|
|
|
|
${
|
|
sylan2d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylan2d.2 $e |- ( ph -> ( ( th /\ ch ) -> ta ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 15-Dec-2004.) $)
|
|
sylan2d $p |- ( ph -> ( ( th /\ ps ) -> ta ) ) $=
|
|
( ancomsd syland ) ABDEABCDEFADCEGHIH $.
|
|
$}
|
|
|
|
${
|
|
syl2and.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
syl2and.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
syl2and.3 $e |- ( ph -> ( ( ch /\ ta ) -> et ) ) $.
|
|
$( A syllogism deduction. (Contributed by NM, 15-Dec-2004.) $)
|
|
syl2and $p |- ( ph -> ( ( ps /\ th ) -> et ) ) $=
|
|
( sylan2d syland ) ABCDFGADECFHIJK $.
|
|
$}
|
|
|
|
${
|
|
biimpa.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Inference from a logical equivalence. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
biimpa $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( biimpd imp ) ABCABCDEF $.
|
|
|
|
$( Inference from a logical equivalence. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
biimpar $p |- ( ( ph /\ ch ) -> ps ) $=
|
|
( biimprd imp ) ACBABCDEF $.
|
|
|
|
$( Inference from a logical equivalence. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
biimpac $p |- ( ( ps /\ ph ) -> ch ) $=
|
|
( biimpcd imp ) BACABCDEF $.
|
|
|
|
$( Inference from a logical equivalence. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
biimparc $p |- ( ( ch /\ ph ) -> ps ) $=
|
|
( biimprcd imp ) CABABCDEF $.
|
|
$}
|
|
|
|
$( Negated conjunction in terms of disjunction (De Morgan's law). Theorem
|
|
*4.51 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
ianor $p |- ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) $=
|
|
( wa wn wi wo imnan pm4.62 bitr3i ) ABCDABDZEADJFABGABHI $.
|
|
|
|
$( Conjunction in terms of disjunction (De Morgan's law). Theorem *4.5 of
|
|
[WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 3-Nov-2012.) $)
|
|
anor $p |- ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) $=
|
|
( wn wo wa ianor bicomi con2bii ) ACBCDZABEZJCIABFGH $.
|
|
|
|
$( Negated disjunction in terms of conjunction (De Morgan's law). Compare
|
|
Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
ioran $p |- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) ) $=
|
|
( wn wi wa wo pm4.65 pm4.64 xchnxbi ) ACZBDJBCEABFABGABHI $.
|
|
|
|
$( Theorem *4.52 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.) $)
|
|
pm4.52 $p |- ( ( ph /\ -. ps ) <-> -. ( -. ph \/ ps ) ) $=
|
|
( wn wa wi wo annim imor xchbinx ) ABCDABEACBFABGABHI $.
|
|
|
|
$( Theorem *4.53 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.53 $p |- ( -. ( ph /\ -. ps ) <-> ( -. ph \/ ps ) ) $=
|
|
( wn wo wa pm4.52 con2bii bicomi ) ACBDZABCEZCJIABFGH $.
|
|
|
|
$( Theorem *4.54 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.) $)
|
|
pm4.54 $p |- ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) $=
|
|
( wn wa wi wo df-an pm4.66 xchbinx ) ACZBDJBCZEAKFJBGABHI $.
|
|
|
|
$( Theorem *4.55 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.55 $p |- ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) $=
|
|
( wn wo wa pm4.54 con2bii bicomi ) ABCDZACBEZCJIABFGH $.
|
|
|
|
$( Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.56 $p |- ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) ) $=
|
|
( wo wn wa ioran bicomi ) ABCDADBDEABFG $.
|
|
|
|
$( Disjunction in terms of conjunction (De Morgan's law). Compare Theorem
|
|
*4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
oran $p |- ( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) $=
|
|
( wn wa wo pm4.56 con2bii ) ACBCDABEABFG $.
|
|
|
|
$( Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.57 $p |- ( -. ( -. ph /\ -. ps ) <-> ( ph \/ ps ) ) $=
|
|
( wo wn wa oran bicomi ) ABCADBDEDABFG $.
|
|
|
|
$( Theorem *3.1 of [WhiteheadRussell] p. 111. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.1 $p |- ( ( ph /\ ps ) -> -. ( -. ph \/ -. ps ) ) $=
|
|
( wa wn wo anor biimpi ) ABCADBDEDABFG $.
|
|
|
|
$( Theorem *3.11 of [WhiteheadRussell] p. 111. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.11 $p |- ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) $=
|
|
( wa wn wo anor biimpri ) ABCADBDEDABFG $.
|
|
|
|
$( Theorem *3.12 of [WhiteheadRussell] p. 111. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.12 $p |- ( ( -. ph \/ -. ps ) \/ ( ph /\ ps ) ) $=
|
|
( wn wo wa pm3.11 orri ) ACBCDABEABFG $.
|
|
|
|
$( Theorem *3.13 of [WhiteheadRussell] p. 111. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.13 $p |- ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) $=
|
|
( wn wo wa pm3.11 con1i ) ACBCDABEABFG $.
|
|
|
|
$( Theorem *3.14 of [WhiteheadRussell] p. 111. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.14 $p |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) ) $=
|
|
( wa wn wo pm3.1 con2i ) ABCADBDEABFG $.
|
|
|
|
$( Introduction of antecedent as conjunct. Theorem *4.73 of
|
|
[WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) $)
|
|
iba $p |- ( ph -> ( ps <-> ( ps /\ ph ) ) ) $=
|
|
( wa pm3.21 simpl impbid1 ) ABBACABDBAEF $.
|
|
|
|
$( Introduction of antecedent as conjunct. (Contributed by NM,
|
|
5-Dec-1995.) $)
|
|
ibar $p |- ( ph -> ( ps <-> ( ph /\ ps ) ) ) $=
|
|
( wa pm3.2 simpr impbid1 ) ABABCABDABEF $.
|
|
|
|
${
|
|
biantru.1 $e |- ph $.
|
|
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
biantru $p |- ( ps <-> ( ps /\ ph ) ) $=
|
|
( wa wb iba ax-mp ) ABBADECABFG $.
|
|
$}
|
|
|
|
${
|
|
biantrur.1 $e |- ph $.
|
|
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
|
|
3-Aug-1994.) $)
|
|
biantrur $p |- ( ps <-> ( ph /\ ps ) ) $=
|
|
( wa wb ibar ax-mp ) ABABDECABFG $.
|
|
$}
|
|
|
|
${
|
|
biantrud.1 $e |- ( ph -> ps ) $.
|
|
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
|
|
2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.) $)
|
|
biantrud $p |- ( ph -> ( ch <-> ( ch /\ ps ) ) ) $=
|
|
( wa wb iba syl ) ABCCBEFDBCGH $.
|
|
|
|
$( A wff is equivalent to its conjunction with truth. (Contributed by NM,
|
|
1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
biantrurd $p |- ( ph -> ( ch <-> ( ps /\ ch ) ) ) $=
|
|
( wa wb ibar syl ) ABCBCEFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
jaao.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jaao.2 $e |- ( th -> ( ta -> ch ) ) $.
|
|
$( Inference conjoining and disjoining the antecedents of two
|
|
implications. (Contributed by NM, 30-Sep-1999.) $)
|
|
jaao $p |- ( ( ph /\ th ) -> ( ( ps \/ ta ) -> ch ) ) $=
|
|
( wa wi adantr adantl jaod ) ADHBCEABCIDFJDECIAGKL $.
|
|
|
|
$( Inference disjoining and conjoining the antecedents of two
|
|
implications. (Contributed by Stefan Allan, 1-Nov-2008.) $)
|
|
jaoa $p |- ( ( ph \/ th ) -> ( ( ps /\ ta ) -> ch ) ) $=
|
|
( wa wi adantrd adantld jaoi ) ABEHCIDABCEFJDECBGKL $.
|
|
$}
|
|
|
|
$( Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
|
|
pm3.44 $p |- ( ( ( ps -> ph ) /\ ( ch -> ph ) )
|
|
-> ( ( ps \/ ch ) -> ph ) ) $=
|
|
( wi id jaao ) BADZBACADZCGEHEF $.
|
|
|
|
$( Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell]
|
|
p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf
|
|
Lammen, 4-Apr-2013.) $)
|
|
jao $p |- ( ( ph -> ps ) -> ( ( ch -> ps ) -> ( ( ph \/ ch ) -> ps ) ) ) $=
|
|
( wi wo pm3.44 ex ) ABDCBDACEBDBACFG $.
|
|
|
|
$( Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut".
|
|
(Contributed by NM, 3-Jan-2005.) $)
|
|
pm1.2 $p |- ( ( ph \/ ph ) -> ph ) $=
|
|
( id jaoi ) AAAABZDC $.
|
|
|
|
$( Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell]
|
|
p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.) $)
|
|
oridm $p |- ( ( ph \/ ph ) <-> ph ) $=
|
|
( wo pm1.2 pm2.07 impbii ) AABAACADE $.
|
|
|
|
$( Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.25 $p |- ( ph <-> ( ph \/ ph ) ) $=
|
|
( wo oridm bicomi ) AABAACD $.
|
|
|
|
${
|
|
orim12i.1 $e |- ( ph -> ps ) $.
|
|
orim12i.2 $e |- ( ch -> th ) $.
|
|
$( Disjoin antecedents and consequents of two premises. (Contributed by
|
|
NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.) $)
|
|
orim12i $p |- ( ( ph \/ ch ) -> ( ps \/ th ) ) $=
|
|
( wo orcd olcd jaoi ) ABDGCABDEHCDBFIJ $.
|
|
$}
|
|
|
|
${
|
|
orim1i.1 $e |- ( ph -> ps ) $.
|
|
$( Introduce disjunct to both sides of an implication. (Contributed by NM,
|
|
6-Jun-1994.) $)
|
|
orim1i $p |- ( ( ph \/ ch ) -> ( ps \/ ch ) ) $=
|
|
( id orim12i ) ABCCDCEF $.
|
|
|
|
$( Introduce disjunct to both sides of an implication. (Contributed by NM,
|
|
6-Jun-1994.) $)
|
|
orim2i $p |- ( ( ch \/ ph ) -> ( ch \/ ps ) ) $=
|
|
( id orim12i ) CCABCEDF $.
|
|
$}
|
|
|
|
${
|
|
orbi2i.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding a left disjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 12-Dec-2012.) $)
|
|
orbi2i $p |- ( ( ch \/ ph ) <-> ( ch \/ ps ) ) $=
|
|
( wo biimpi orim2i biimpri impbii ) CAECBEABCABDFGBACABDHGI $.
|
|
|
|
$( Inference adding a right disjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) $)
|
|
orbi1i $p |- ( ( ph \/ ch ) <-> ( ps \/ ch ) ) $=
|
|
( wo orcom orbi2i 3bitri ) ACECAECBEBCEACFABCDGCBFH $.
|
|
$}
|
|
|
|
${
|
|
orbi12i.1 $e |- ( ph <-> ps ) $.
|
|
orbi12i.2 $e |- ( ch <-> th ) $.
|
|
$( Infer the disjunction of two equivalences. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
orbi12i $p |- ( ( ph \/ ch ) <-> ( ps \/ th ) ) $=
|
|
( wo orbi2i orbi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
$}
|
|
|
|
$( Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm1.5 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ps \/ ( ph \/ ch ) ) ) $=
|
|
( wo orc olcd olc orim2i jaoi ) ABACDZDBCDAJBACEFCJBCAGHI $.
|
|
|
|
$( Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by
|
|
Wolf Lammen, 14-Nov-2012.) $)
|
|
or12 $p |- ( ( ph \/ ( ps \/ ch ) ) <-> ( ps \/ ( ph \/ ch ) ) ) $=
|
|
( wo pm1.5 impbii ) ABCDDBACDDABCEBACEF $.
|
|
|
|
$( Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
orass $p |- ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) ) $=
|
|
( wo orcom or12 orbi2i 3bitri ) ABDZCDCIDACBDZDABCDZDICECABFJKACBEGH $.
|
|
|
|
$( Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.31 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ( ph \/ ps ) \/ ch ) ) $=
|
|
( wo orass biimpri ) ABDCDABCDDABCEF $.
|
|
|
|
$( Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.32 $p |- ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ( ps \/ ch ) ) ) $=
|
|
( wo orass biimpi ) ABDCDABCDDABCEF $.
|
|
|
|
$( A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
or32 $p |- ( ( ( ph \/ ps ) \/ ch ) <-> ( ( ph \/ ch ) \/ ps ) ) $=
|
|
( wo orass or12 orcom 3bitri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.
|
|
|
|
$( Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.) $)
|
|
or4 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <->
|
|
( ( ph \/ ch ) \/ ( ps \/ th ) ) ) $=
|
|
( wo or12 orbi2i orass 3bitr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHAC
|
|
LHI $.
|
|
|
|
$( Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.) $)
|
|
or42 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <->
|
|
( ( ph \/ ch ) \/ ( th \/ ps ) ) ) $=
|
|
( wo or4 orcom orbi2i bitri ) ABECDEEACEZBDEZEJDBEZEABCDFKLJBDGHI $.
|
|
|
|
$( Distribution of disjunction over disjunction. (Contributed by NM,
|
|
25-Feb-1995.) $)
|
|
orordi $p |- ( ( ph \/ ( ps \/ ch ) ) <->
|
|
( ( ph \/ ps ) \/ ( ph \/ ch ) ) ) $=
|
|
( wo oridm orbi1i or4 bitr3i ) ABCDZDAADZIDABDACDDJAIAEFAABCGH $.
|
|
|
|
$( Distribution of disjunction over disjunction. (Contributed by NM,
|
|
25-Feb-1995.) $)
|
|
orordir $p |- ( ( ( ph \/ ps ) \/ ch ) <->
|
|
( ( ph \/ ch ) \/ ( ps \/ ch ) ) ) $=
|
|
( wo oridm orbi2i or4 bitr3i ) ABDZCDICCDZDACDBCDDJCICEFABCCGH $.
|
|
|
|
${
|
|
jca.1 $e |- ( ph -> ps ) $.
|
|
jca.2 $e |- ( ph -> ch ) $.
|
|
$( Deduce conjunction of the consequents of two implications ("join
|
|
consequents with 'and'"). Equivalent to the natural deduction rule
|
|
` /\ ` I ( ` /\ ` introduction), see natded in set.mm. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) $)
|
|
jca $p |- ( ph -> ( ps /\ ch ) ) $=
|
|
( wa pm3.2 sylc ) ABCBCFDEBCGH $.
|
|
$}
|
|
|
|
${
|
|
jcad.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jcad.2 $e |- ( ph -> ( ps -> th ) ) $.
|
|
$( Deduction conjoining the consequents of two implications. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) $)
|
|
jcad $p |- ( ph -> ( ps -> ( ch /\ th ) ) ) $=
|
|
( wa pm3.2 syl6c ) ABCDCDGEFCDHI $.
|
|
$}
|
|
|
|
${
|
|
jca31.1 $e |- ( ph -> ps ) $.
|
|
jca31.2 $e |- ( ph -> ch ) $.
|
|
jca31.3 $e |- ( ph -> th ) $.
|
|
$( Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.) $)
|
|
jca31 $p |- ( ph -> ( ( ps /\ ch ) /\ th ) ) $=
|
|
( wa jca ) ABCHDABCEFIGI $.
|
|
|
|
$( Join three consequents. (Contributed by FL, 1-Aug-2009.) $)
|
|
jca32 $p |- ( ph -> ( ps /\ ( ch /\ th ) ) ) $=
|
|
( wa jca ) ABCDHEACDFGII $.
|
|
$}
|
|
|
|
${
|
|
jcai.1 $e |- ( ph -> ps ) $.
|
|
jcai.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction replacing implication with conjunction. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
jcai $p |- ( ph -> ( ps /\ ch ) ) $=
|
|
( mpd jca ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
jctil.1 $e |- ( ph -> ps ) $.
|
|
jctil.2 $e |- ch $.
|
|
$( Inference conjoining a theorem to left of consequent in an implication.
|
|
(Contributed by NM, 31-Dec-1993.) $)
|
|
jctil $p |- ( ph -> ( ch /\ ps ) ) $=
|
|
( a1i jca ) ACBCAEFDG $.
|
|
|
|
$( Inference conjoining a theorem to right of consequent in an
|
|
implication. (Contributed by NM, 31-Dec-1993.) $)
|
|
jctir $p |- ( ph -> ( ps /\ ch ) ) $=
|
|
( a1i jca ) ABCDCAEFG $.
|
|
$}
|
|
|
|
${
|
|
jctl.1 $e |- ps $.
|
|
$( Inference conjoining a theorem to the left of a consequent.
|
|
(Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen,
|
|
24-Oct-2012.) $)
|
|
jctl $p |- ( ph -> ( ps /\ ph ) ) $=
|
|
( id jctil ) AABADCE $.
|
|
|
|
$( Inference conjoining a theorem to the right of a consequent.
|
|
(Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
24-Oct-2012.) $)
|
|
jctr $p |- ( ph -> ( ph /\ ps ) ) $=
|
|
( id jctir ) AABADCE $.
|
|
$}
|
|
|
|
${
|
|
jctild.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jctild.2 $e |- ( ph -> th ) $.
|
|
$( Deduction conjoining a theorem to left of consequent in an implication.
|
|
(Contributed by NM, 21-Apr-2005.) $)
|
|
jctild $p |- ( ph -> ( ps -> ( th /\ ch ) ) ) $=
|
|
( a1d jcad ) ABDCADBFGEH $.
|
|
$}
|
|
|
|
${
|
|
jctird.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
jctird.2 $e |- ( ph -> th ) $.
|
|
$( Deduction conjoining a theorem to right of consequent in an
|
|
implication. (Contributed by NM, 21-Apr-2005.) $)
|
|
jctird $p |- ( ph -> ( ps -> ( ch /\ th ) ) ) $=
|
|
( a1d jcad ) ABCDEADBFGH $.
|
|
$}
|
|
|
|
$( Conjoin antecedent to left of consequent. (Contributed by NM,
|
|
15-Aug-1994.) $)
|
|
ancl $p |- ( ( ph -> ps ) -> ( ph -> ( ph /\ ps ) ) ) $=
|
|
( wa pm3.2 a2i ) ABABCABDE $.
|
|
|
|
$( Conjoin antecedent to left of consequent. Theorem *4.7 of
|
|
[WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof
|
|
shortened by Wolf Lammen, 24-Mar-2013.) $)
|
|
anclb $p |- ( ( ph -> ps ) <-> ( ph -> ( ph /\ ps ) ) ) $=
|
|
( wa ibar pm5.74i ) ABABCABDE $.
|
|
|
|
$( Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.42 $p |- ( ( ph -> ( ps -> ch ) ) <->
|
|
( ph -> ( ps -> ( ph /\ ch ) ) ) ) $=
|
|
( wi wa ibar imbi2d pm5.74i ) ABCDBACEZDACIBACFGH $.
|
|
|
|
$( Conjoin antecedent to right of consequent. (Contributed by NM,
|
|
15-Aug-1994.) $)
|
|
ancr $p |- ( ( ph -> ps ) -> ( ph -> ( ps /\ ph ) ) ) $=
|
|
( wa pm3.21 a2i ) ABBACABDE $.
|
|
|
|
$( Conjoin antecedent to right of consequent. (Contributed by NM,
|
|
25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) $)
|
|
ancrb $p |- ( ( ph -> ps ) <-> ( ph -> ( ps /\ ph ) ) ) $=
|
|
( wa iba pm5.74i ) ABBACABDE $.
|
|
|
|
${
|
|
ancli.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction conjoining antecedent to left of consequent. (Contributed by
|
|
NM, 12-Aug-1993.) $)
|
|
ancli $p |- ( ph -> ( ph /\ ps ) ) $=
|
|
( id jca ) AABADCE $.
|
|
$}
|
|
|
|
${
|
|
ancri.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction conjoining antecedent to right of consequent. (Contributed by
|
|
NM, 15-Aug-1994.) $)
|
|
ancri $p |- ( ph -> ( ps /\ ph ) ) $=
|
|
( id jca ) ABACADE $.
|
|
$}
|
|
|
|
${
|
|
ancld.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction conjoining antecedent to left of consequent in nested
|
|
implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by
|
|
Wolf Lammen, 1-Nov-2012.) $)
|
|
ancld $p |- ( ph -> ( ps -> ( ps /\ ch ) ) ) $=
|
|
( idd jcad ) ABBCABEDF $.
|
|
$}
|
|
|
|
${
|
|
ancrd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction conjoining antecedent to right of consequent in nested
|
|
implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by
|
|
Wolf Lammen, 1-Nov-2012.) $)
|
|
ancrd $p |- ( ph -> ( ps -> ( ch /\ ps ) ) ) $=
|
|
( idd jcad ) ABCBDABEF $.
|
|
$}
|
|
|
|
$( Conjoin antecedent to left of consequent in nested implication.
|
|
(Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen,
|
|
14-Jul-2013.) $)
|
|
anc2l $p |- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ps -> ( ph /\ ch ) ) ) ) $=
|
|
( wi wa pm5.42 biimpi ) ABCDDABACEDDABCFG $.
|
|
|
|
$( Conjoin antecedent to right of consequent in nested implication.
|
|
(Contributed by NM, 15-Aug-1994.) $)
|
|
anc2r $p |- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ps -> ( ch /\ ph ) ) ) ) $=
|
|
( wi wa pm3.21 imim2d a2i ) ABCDBCAEZDACIBACFGH $.
|
|
|
|
${
|
|
anc2li.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction conjoining antecedent to left of consequent in nested
|
|
implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by
|
|
Wolf Lammen, 7-Dec-2012.) $)
|
|
anc2li $p |- ( ph -> ( ps -> ( ph /\ ch ) ) ) $=
|
|
( id jctild ) ABCADAEF $.
|
|
$}
|
|
|
|
${
|
|
anc2ri.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction conjoining antecedent to right of consequent in nested
|
|
implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by
|
|
Wolf Lammen, 7-Dec-2012.) $)
|
|
anc2ri $p |- ( ph -> ( ps -> ( ch /\ ph ) ) ) $=
|
|
( id jctird ) ABCADAEF $.
|
|
$}
|
|
|
|
$( Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.41 $p |- ( ( ph -> ch ) -> ( ( ph /\ ps ) -> ch ) ) $=
|
|
( wa simpl imim1i ) ABDACABEF $.
|
|
|
|
$( Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.42 $p |- ( ( ps -> ch ) -> ( ( ph /\ ps ) -> ch ) ) $=
|
|
( wa simpr imim1i ) ABDBCABEF $.
|
|
|
|
$( Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell]
|
|
p. 113. (Contributed by NM, 31-Jul-1995.) $)
|
|
pm3.4 $p |- ( ( ph /\ ps ) -> ( ph -> ps ) ) $=
|
|
( wa simpr a1d ) ABCBAABDE $.
|
|
|
|
$( Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell]
|
|
p. 119. (Contributed by NM, 17-May-1998.) $)
|
|
pm4.45im $p |- ( ph <-> ( ph /\ ( ps -> ph ) ) ) $=
|
|
( wi wa ax-1 ancli simpl impbii ) AABACZDAIABEFAIGH $.
|
|
|
|
${
|
|
anim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
anim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( Conjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.) $)
|
|
anim12d $p |- ( ph -> ( ( ps /\ th ) -> ( ch /\ ta ) ) ) $=
|
|
( wa idd syl2and ) ABCDECEHZFGAKIJ $.
|
|
$}
|
|
|
|
${
|
|
anim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Add a conjunct to right of antecedent and consequent in a deduction.
|
|
(Contributed by NM, 3-Apr-1994.) $)
|
|
anim1d $p |- ( ph -> ( ( ps /\ th ) -> ( ch /\ th ) ) ) $=
|
|
( idd anim12d ) ABCDDEADFG $.
|
|
|
|
$( Add a conjunct to left of antecedent and consequent in a deduction.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
anim2d $p |- ( ph -> ( ( th /\ ps ) -> ( th /\ ch ) ) ) $=
|
|
( idd anim12d ) ADDBCADFEG $.
|
|
$}
|
|
|
|
${
|
|
anim12i.1 $e |- ( ph -> ps ) $.
|
|
anim12i.2 $e |- ( ch -> th ) $.
|
|
$( Conjoin antecedents and consequents of two premises. (Contributed by
|
|
NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.) $)
|
|
anim12i $p |- ( ( ph /\ ch ) -> ( ps /\ th ) ) $=
|
|
( wa id syl2an ) ABDBDGZCEFJHI $.
|
|
|
|
$( Variant of ~ anim12i with commutation. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
anim12ci $p |- ( ( ph /\ ch ) -> ( th /\ ps ) ) $=
|
|
( wa anim12i ancoms ) CADBGCDABFEHI $.
|
|
$}
|
|
|
|
${
|
|
anim1i.1 $e |- ( ph -> ps ) $.
|
|
$( Introduce conjunct to both sides of an implication. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
anim1i $p |- ( ( ph /\ ch ) -> ( ps /\ ch ) ) $=
|
|
( id anim12i ) ABCCDCEF $.
|
|
|
|
$( Introduce conjunct to both sides of an implication. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
anim2i $p |- ( ( ch /\ ph ) -> ( ch /\ ps ) ) $=
|
|
( id anim12i ) CCABCEDF $.
|
|
$}
|
|
|
|
${
|
|
anim12ii.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
anim12ii.2 $e |- ( th -> ( ps -> ta ) ) $.
|
|
$( Conjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.) $)
|
|
anim12ii $p |- ( ( ph /\ th ) -> ( ps -> ( ch /\ ta ) ) ) $=
|
|
( wa wi adantr adantl jcad ) ADHBCEABCIDFJDBEIAGKL $.
|
|
$}
|
|
|
|
$( Conjoin antecedents and consequents of two premises. This is the closed
|
|
theorem form of ~ anim12d . Theorem *3.47 of [WhiteheadRussell] p. 113.
|
|
It was proved by Leibniz, and it evidently pleased him enough to call it
|
|
_praeclarum theorema_ (splendid theorem). (Contributed by NM,
|
|
12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
prth $p |- ( ( ( ph -> ps ) /\ ( ch -> th ) )
|
|
-> ( ( ph /\ ch ) -> ( ps /\ th ) ) ) $=
|
|
( wi wa simpl simpr anim12d ) ABEZCDEZFABCDJKGJKHI $.
|
|
|
|
$( Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.3 $p |- ( ( ph \/ ( ps \/ ch ) ) -> ( ph \/ ( ch \/ ps ) ) ) $=
|
|
( wo pm1.4 orim2i ) BCDCBDABCEF $.
|
|
|
|
$( Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.41 $p |- ( ( ps \/ ( ph \/ ps ) ) -> ( ph \/ ps ) ) $=
|
|
( wo olc id jaoi ) BABCZGBADGEF $.
|
|
|
|
$( Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.42 $p |- ( ( -. ph \/ ( ph -> ps ) ) -> ( ph -> ps ) ) $=
|
|
( wn wi pm2.21 id jaoi ) ACABDZHABEHFG $.
|
|
|
|
$( Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.4 $p |- ( ( ph \/ ( ph \/ ps ) ) -> ( ph \/ ps ) ) $=
|
|
( wo orc id jaoi ) AABCZGABDGEF $.
|
|
|
|
${
|
|
pm2.65da.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
pm2.65da.2 $e |- ( ( ph /\ ps ) -> -. ch ) $.
|
|
$( Deduction rule for proof by contradiction. (Contributed by NM,
|
|
12-Jun-2014.) $)
|
|
pm2.65da $p |- ( ph -> -. ps ) $=
|
|
( ex wn pm2.65d ) ABCABCDFABCGEFH $.
|
|
$}
|
|
|
|
$( Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.44 $p |- ( ph <-> ( ph \/ ( ph /\ ps ) ) ) $=
|
|
( wa wo orc id simpl jaoi impbii ) AAABCZDAJEAAJAFABGHI $.
|
|
|
|
$( Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.) $)
|
|
pm4.14 $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) ) $=
|
|
( wi wn wa con34b imbi2i impexp 3bitr4i ) ABCDZDACEZBEZDZDABFCDALFMDKNABCGH
|
|
ABCIALMIJ $.
|
|
|
|
$( Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.) $)
|
|
pm3.37 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ( ph /\ -. ch ) -> -. ps ) ) $=
|
|
( wa wi wn pm4.14 biimpi ) ABDCEACFDBFEABCGH $.
|
|
|
|
$( Theorem to move a conjunct in and out of a negation. (Contributed by NM,
|
|
9-Nov-2003.) $)
|
|
nan $p |- ( ( ph -> -. ( ps /\ ch ) ) <-> ( ( ph /\ ps ) -> -. ch ) ) $=
|
|
( wa wn wi impexp imnan imbi2i bitr2i ) ABDCEZFABKFZFABCDEZFABKGLMABCHIJ $.
|
|
|
|
$( Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) $)
|
|
pm4.15 $p |- ( ( ( ph /\ ps ) -> -. ch ) <-> ( ( ps /\ ch ) -> -. ph ) ) $=
|
|
( wa wn wi con2b nan bitr2i ) BCDZAEFAJEFABDCEFJAGABCHI $.
|
|
|
|
$( Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
|
|
pm4.78 $p |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <->
|
|
( ph -> ( ps \/ ch ) ) ) $=
|
|
( wn wo wi orordi imor orbi12i 3bitr4ri ) ADZBCEZEKBEZKCEZEALFABFZACFZEKBCG
|
|
ALHOMPNABHACHIJ $.
|
|
|
|
$( Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.) $)
|
|
pm4.79 $p |- ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <->
|
|
( ( ps /\ ch ) -> ph ) ) $=
|
|
( wi wo wa id jaoa wn simplim pm3.3 syl5 orrd impbii ) BADZCADZEBCFADZOBAPC
|
|
OGPGHQOPOIBQPBAJBCAKLMN $.
|
|
|
|
$( Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.) $)
|
|
pm4.87 $p |- ( ( ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) /\
|
|
( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) ) ) /\
|
|
( ( ps -> ( ph -> ch ) ) <-> ( ( ps /\ ph ) -> ch ) ) ) $=
|
|
( wa wi wb impexp bi2.04 pm3.2i bicomi ) ABDCEABCEEZFZKBACEEZFZDMBADCEZFLNA
|
|
BCGABCHIOMBACGJI $.
|
|
|
|
$( Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.33 $p |- ( ( ( ph -> ps ) /\ ( ps -> ch ) ) -> ( ph -> ch ) ) $=
|
|
( wi imim1 imp ) ABDBCDACDABCEF $.
|
|
|
|
$( Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.34 $p |- ( ( ( ps -> ch ) /\ ( ph -> ps ) ) -> ( ph -> ch ) ) $=
|
|
( wi imim2 imp ) BCDABDACDBCAEF $.
|
|
|
|
$( Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112.
|
|
(Contributed by NM, 14-Dec-2002.) $)
|
|
pm3.35 $p |- ( ( ph /\ ( ph -> ps ) ) -> ps ) $=
|
|
( wi pm2.27 imp ) AABCBABDE $.
|
|
|
|
$( Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.31 $p |- ( ( ch /\ ( ph -> ps ) ) -> ( ph -> ( ps /\ ch ) ) ) $=
|
|
( wi wa pm3.21 imim2d imp ) CABDABCEZDCBIACBFGH $.
|
|
|
|
${
|
|
imp4.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp4a $p |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) $=
|
|
( wi wa impexp syl6ibr ) ABCDEGGCDHEGFCDEIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp4b $p |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) ) $=
|
|
( wa wi imp4a imp ) ABCDGEHABCDEFIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp4c $p |- ( ph -> ( ( ( ps /\ ch ) /\ th ) -> ta ) ) $=
|
|
( wa wi imp3a ) ABCGDEABCDEHFII $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp4d $p |- ( ph -> ( ( ps /\ ( ch /\ th ) ) -> ta ) ) $=
|
|
( wa imp4a imp3a ) ABCDGEABCDEFHI $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp41 $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $=
|
|
( wa wi imp imp31 ) ABGCDEABCDEHHFIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp42 $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $=
|
|
( wa wi imp32 imp ) ABCGGDEABCDEHFIJ $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp43 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $=
|
|
( wa imp4b imp ) ABGCDGEABCDEFHI $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp44 $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ta ) $=
|
|
( wa imp4c imp ) ABCGDGEABCDEFHI $.
|
|
|
|
$( An importation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
imp45 $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ta ) $=
|
|
( wa imp4d imp ) ABCDGGEABCDEFHI $.
|
|
|
|
$}
|
|
|
|
${
|
|
imp5.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $.
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp5a $p |- ( ph -> ( ps -> ( ch -> ( ( th /\ ta ) -> et ) ) ) ) $=
|
|
( wi wa pm3.31 syl8 ) ABCDEFHHDEIFHGDEFJK $.
|
|
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp5d $p |- ( ( ( ph /\ ps ) /\ ch ) -> ( ( th /\ ta ) -> et ) ) $=
|
|
( wa wi imp31 imp3a ) ABHCHDEFABCDEFIIGJK $.
|
|
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp5g $p |- ( ( ph /\ ps ) -> ( ( ( ch /\ th ) /\ ta ) -> et ) ) $=
|
|
( wa wi imp imp4c ) ABHCDEFABCDEFIIIGJK $.
|
|
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp55 $p |- ( ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) /\ ta ) -> et ) $=
|
|
( wa wi imp4a imp42 ) ABCDHEFABCDEFIGJK $.
|
|
|
|
$( An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
imp511 $p |- ( ( ph /\ ( ( ps /\ ( ch /\ th ) ) /\ ta ) ) -> et ) $=
|
|
( wa wi imp4a imp44 ) ABCDHEFABCDEFIGJK $.
|
|
$}
|
|
|
|
${
|
|
expimpd.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Exportation followed by a deduction version of importation.
|
|
(Contributed by NM, 6-Sep-2008.) $)
|
|
expimpd $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
|
|
( wi ex imp3a ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
exp31.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp31 $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wi wa ex ) ABCDFABGCDEHH $.
|
|
$}
|
|
|
|
${
|
|
exp32.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp32 $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( wa ex exp3a ) ABCDABCFDEGH $.
|
|
$}
|
|
|
|
${
|
|
exp4a.1 $e |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp4a $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa wi impexp syl6ib ) ABCDGEHCDEHHFCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
exp4b.1 $e |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 23-Nov-2012.) $)
|
|
exp4b $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa wi ex exp4a ) ABCDEABCDGEHFIJ $.
|
|
$}
|
|
|
|
${
|
|
exp4c.1 $e |- ( ph -> ( ( ( ps /\ ch ) /\ th ) -> ta ) ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp4c $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi wa exp3a ) ABCDEGABCHDEFII $.
|
|
$}
|
|
|
|
${
|
|
exp4d.1 $e |- ( ph -> ( ( ps /\ ( ch /\ th ) ) -> ta ) ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp4d $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa exp3a exp4a ) ABCDEABCDGEFHI $.
|
|
$}
|
|
|
|
${
|
|
exp41.1 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp41 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi wa ex exp31 ) ABCDEGABHCHDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
exp42.1 $e |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp42 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi wa exp31 exp3a ) ABCDEGABCHDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
exp43.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp43 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa ex exp4b ) ABCDEABGCDGEFHI $.
|
|
$}
|
|
|
|
${
|
|
exp44.1 $e |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp44 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi wa exp32 exp3a ) ABCDEGABCHDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
exp45.1 $e |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ta ) $.
|
|
$( An exportation inference. (Contributed by NM, 26-Apr-1994.) $)
|
|
exp45 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wa exp32 exp4a ) ABCDEABCDGEFHI $.
|
|
$}
|
|
|
|
${
|
|
expr.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Export a wff from a right conjunct. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
expr $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
|
|
( wi exp32 imp ) ABCDFABCDEGH $.
|
|
$}
|
|
|
|
${
|
|
exp5c.1 $e |- ( ph -> ( ( ps /\ ch ) -> ( ( th /\ ta ) -> et ) ) ) $.
|
|
$( An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) $)
|
|
exp5c $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( wi wa exp4a exp3a ) ABCDEFHHABCIDEFGJK $.
|
|
$}
|
|
|
|
${
|
|
exp53.1 $e |- ( ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) /\ ta ) -> et ) $.
|
|
$( An exportation inference. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
exp53 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( wi wa ex exp43 ) ABCDEFHABICDIIEFGJK $.
|
|
$}
|
|
|
|
${
|
|
expl.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Export a wff from a left conjunct. (Contributed by Jeff Hankins,
|
|
28-Aug-2009.) $)
|
|
expl $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
|
|
( exp31 imp3a ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
impr.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Import a wff into a right conjunct. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
impr $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
|
|
( wi ex imp32 ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
impl.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Export a wff from a left conjunct. (Contributed by Mario Carneiro,
|
|
9-Jul-2014.) $)
|
|
impl $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
|
|
( exp3a imp31 ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
impac.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Importation with conjunction in consequent. (Contributed by NM,
|
|
9-Aug-1994.) $)
|
|
impac $p |- ( ( ph /\ ps ) -> ( ch /\ ps ) ) $=
|
|
( wa ancrd imp ) ABCBEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
exbiri.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
|
|
$( Inference form of ~ exbir . This proof is exbiriVD in set.mm
|
|
automatically translated and minimized. (Contributed by Alan Sare,
|
|
31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.) $)
|
|
exbiri $p |- ( ph -> ( ps -> ( th -> ch ) ) ) $=
|
|
( wa biimpar exp31 ) ABDCABFCDEGH $.
|
|
$}
|
|
|
|
${
|
|
pm3.26bda.1 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) $)
|
|
simprbda $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wa biimpa simpld ) ABFCDABCDFEGH $.
|
|
|
|
$( Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) $)
|
|
simplbda $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa biimpa simprd ) ABFCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
pm3.26bi2.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Deduction eliminating a conjunct. Automatically derived from simplbi2VD
|
|
in set.mm. (Contributed by Alan Sare, 31-Dec-2011.) $)
|
|
simplbi2 $p |- ( ps -> ( ch -> ph ) ) $=
|
|
( wa biimpri ex ) BCAABCEDFG $.
|
|
$}
|
|
|
|
$( A theorem similar to the standard definition of the biconditional.
|
|
Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.) $)
|
|
dfbi2 $p |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) $=
|
|
( wb wi wn wa dfbi1 df-an bitr4i ) ABCABDZBADZEDEJKFABGJKHI $.
|
|
|
|
$( Definition ~ df-bi rewritten in an abbreviated form to help intuitive
|
|
understanding of that definition. Note that it is a conjunction of two
|
|
implications; one which asserts properties that follow from the
|
|
biconditional and one which asserts properties that imply the
|
|
biconditional. (Contributed by NM, 15-Aug-2008.) $)
|
|
dfbi $p |- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
|
|
/\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) ) $=
|
|
( wb wi wa dfbi2 biimpi biimpri pm3.2i ) ABCZABDBADEZDKJDJKABFZGJKLHI $.
|
|
|
|
$( Implication in terms of biconditional and conjunction. Theorem *4.71 of
|
|
[WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 2-Dec-2012.) $)
|
|
pm4.71 $p |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) ) $=
|
|
( wa wi wb simpl biantru anclb dfbi2 3bitr4i ) AABCZDZLKADZCABDAKEMLABFGABH
|
|
AKIJ $.
|
|
|
|
$( Implication in terms of biconditional and conjunction. Theorem *4.71 of
|
|
[WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM,
|
|
25-Jul-1999.) $)
|
|
pm4.71r $p |- ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) ) $=
|
|
( wi wa wb pm4.71 ancom bibi2i bitri ) ABCAABDZEABADZEABFJKAABGHI $.
|
|
|
|
${
|
|
pm4.71i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference converting an implication to a biconditional with
|
|
conjunction. Inference from Theorem *4.71 of [WhiteheadRussell]
|
|
p. 120. (Contributed by NM, 4-Jan-2004.) $)
|
|
pm4.71i $p |- ( ph <-> ( ph /\ ps ) ) $=
|
|
( wi wa wb pm4.71 mpbi ) ABDAABEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
pm4.71ri.1 $e |- ( ph -> ps ) $.
|
|
$( Inference converting an implication to a biconditional with
|
|
conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120
|
|
(with conjunct reversed). (Contributed by NM, 1-Dec-2003.) $)
|
|
pm4.71ri $p |- ( ph <-> ( ps /\ ph ) ) $=
|
|
( wi wa wb pm4.71r mpbi ) ABDABAEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
pm4.71rd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction converting an implication to a biconditional with
|
|
conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell]
|
|
p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.) $)
|
|
pm4.71d $p |- ( ph -> ( ps <-> ( ps /\ ch ) ) ) $=
|
|
( wi wa wb pm4.71 sylib ) ABCEBBCFGDBCHI $.
|
|
|
|
$( Deduction converting an implication to a biconditional with
|
|
conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell]
|
|
p. 120. (Contributed by NM, 10-Feb-2005.) $)
|
|
pm4.71rd $p |- ( ph -> ( ps <-> ( ch /\ ps ) ) ) $=
|
|
( wi wa wb pm4.71r sylib ) ABCEBCBFGDBCHI $.
|
|
$}
|
|
|
|
$( Distribution of implication over biconditional. Theorem *5.32 of
|
|
[WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) $)
|
|
pm5.32 $p |- ( ( ph -> ( ps <-> ch ) ) <->
|
|
( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) $=
|
|
( wb wi wn wa notbi imbi2i pm5.74 3bitri df-an bibi12i bitr4i ) ABCDZEZABFZ
|
|
EZFZACFZEZFZDZABGZACGZDPAQTDZERUADUCOUFABCHIAQTJRUAHKUDSUEUBABLACLMN $.
|
|
|
|
${
|
|
pm5.32i.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Distribution of implication over biconditional (inference rule).
|
|
(Contributed by NM, 1-Aug-1994.) $)
|
|
pm5.32i $p |- ( ( ph /\ ps ) <-> ( ph /\ ch ) ) $=
|
|
( wb wi wa pm5.32 mpbi ) ABCEFABGACGEDABCHI $.
|
|
|
|
$( Distribution of implication over biconditional (inference rule).
|
|
(Contributed by NM, 12-Mar-1995.) $)
|
|
pm5.32ri $p |- ( ( ps /\ ph ) <-> ( ch /\ ph ) ) $=
|
|
( wa pm5.32i ancom 3bitr4i ) ABEACEBAECAEABCDFBAGCAGH $.
|
|
$}
|
|
|
|
${
|
|
pm5.32d.1 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 29-Oct-1996.) $)
|
|
pm5.32d $p |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) ) $=
|
|
( wb wi wa pm5.32 sylib ) ABCDFGBCHBDHFEBCDIJ $.
|
|
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 25-Dec-2004.) $)
|
|
pm5.32rd $p |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) ) $=
|
|
( wa pm5.32d ancom 3bitr4g ) ABCFBDFCBFDBFABCDEGCBHDBHI $.
|
|
$}
|
|
|
|
${
|
|
pm5.32da.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 9-Dec-2006.) $)
|
|
pm5.32da $p |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) ) $=
|
|
( wb ex pm5.32d ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
biadan2.1 $e |- ( ph -> ps ) $.
|
|
biadan2.2 $e |- ( ps -> ( ph <-> ch ) ) $.
|
|
$( Add a conjunction to an equivalence. (Contributed by Jeff Madsen,
|
|
20-Jun-2011.) $)
|
|
biadan2 $p |- ( ph <-> ( ps /\ ch ) ) $=
|
|
( wa pm4.71ri pm5.32i bitri ) ABAFBCFABDGBACEHI $.
|
|
$}
|
|
|
|
$( Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.24 $p |- ( ph <-> ( ph /\ ph ) ) $=
|
|
( id pm4.71i ) AAABC $.
|
|
|
|
$( Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 14-Mar-2014.) $)
|
|
anidm $p |- ( ( ph /\ ph ) <-> ph ) $=
|
|
( wa pm4.24 bicomi ) AAABACD $.
|
|
|
|
${
|
|
anidms.1 $e |- ( ( ph /\ ph ) -> ps ) $.
|
|
$( Inference from idempotent law for conjunction. (Contributed by NM,
|
|
15-Jun-1994.) $)
|
|
anidms $p |- ( ph -> ps ) $=
|
|
( ex pm2.43i ) ABAABCDE $.
|
|
$}
|
|
|
|
$( Conjunction idempotence with antecedent. (Contributed by Roy F. Longton,
|
|
8-Aug-2005.) $)
|
|
anidmdbi $p |- ( ( ph -> ( ps /\ ps ) ) <-> ( ph -> ps ) ) $=
|
|
( wa anidm imbi2i ) BBCBABDE $.
|
|
|
|
${
|
|
anasss.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Associative law for conjunction applied to antecedent (eliminates
|
|
syllogism). (Contributed by NM, 15-Nov-2002.) $)
|
|
anasss $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
|
|
( exp31 imp32 ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
anassrs.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Associative law for conjunction applied to antecedent (eliminates
|
|
syllogism). (Contributed by NM, 15-Nov-2002.) $)
|
|
anassrs $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
|
|
( exp32 imp31 ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
$( Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 24-Nov-2012.) $)
|
|
anass $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) ) $=
|
|
( wa id anassrs anasss impbii ) ABDCDZABCDDZABCJJEFABCIIEGH $.
|
|
|
|
${
|
|
sylanl1.1 $e |- ( ph -> ps ) $.
|
|
sylanl1.2 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 10-Mar-2005.) $)
|
|
sylanl1 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
|
|
( wa anim1i sylan ) ACHBCHDEABCFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylanl2.1 $e |- ( ph -> ch ) $.
|
|
sylanl2.2 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 1-Jan-2005.) $)
|
|
sylanl2 $p |- ( ( ( ps /\ ph ) /\ th ) -> ta ) $=
|
|
( wa anim2i sylan ) BAHBCHDEACBFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylanr1.1 $e |- ( ph -> ch ) $.
|
|
sylanr1.2 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 9-Apr-2005.) $)
|
|
sylanr1 $p |- ( ( ps /\ ( ph /\ th ) ) -> ta ) $=
|
|
( wa anim1i sylan2 ) ADHBCDHEACDFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylanr2.1 $e |- ( ph -> th ) $.
|
|
sylanr2.2 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 9-Apr-2005.) $)
|
|
sylanr2 $p |- ( ( ps /\ ( ch /\ ph ) ) -> ta ) $=
|
|
( wa anim2i sylan2 ) CAHBCDHEADCFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylani.1 $e |- ( ph -> ch ) $.
|
|
sylani.2 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( A syllogism inference. (Contributed by NM, 2-May-1996.) $)
|
|
sylani $p |- ( ps -> ( ( ph /\ th ) -> ta ) ) $=
|
|
( wi a1i syland ) BACDEACHBFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylan2i.1 $e |- ( ph -> th ) $.
|
|
sylan2i.2 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( A syllogism inference. (Contributed by NM, 1-Aug-1994.) $)
|
|
sylan2i $p |- ( ps -> ( ( ch /\ ph ) -> ta ) ) $=
|
|
( wi a1i sylan2d ) BADCEADHBFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl2ani.1 $e |- ( ph -> ch ) $.
|
|
syl2ani.2 $e |- ( et -> th ) $.
|
|
syl2ani.3 $e |- ( ps -> ( ( ch /\ th ) -> ta ) ) $.
|
|
$( A syllogism inference. (Contributed by NM, 3-Aug-1999.) $)
|
|
syl2ani $p |- ( ps -> ( ( ph /\ et ) -> ta ) ) $=
|
|
( sylan2i sylani ) ABCFEGFBCDEHIJK $.
|
|
$}
|
|
|
|
${
|
|
sylan9.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylan9.2 $e |- ( th -> ( ch -> ta ) ) $.
|
|
$( Nested syllogism inference conjoining dissimilar antecedents.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
7-May-2011.) $)
|
|
sylan9 $p |- ( ( ph /\ th ) -> ( ps -> ta ) ) $=
|
|
( wi syl9 imp ) ADBEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan9r.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
sylan9r.2 $e |- ( th -> ( ch -> ta ) ) $.
|
|
$( Nested syllogism inference conjoining dissimilar antecedents.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sylan9r $p |- ( ( th /\ ph ) -> ( ps -> ta ) ) $=
|
|
( wi syl9r imp ) DABEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
mtand.1 $e |- ( ph -> -. ch ) $.
|
|
mtand.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( A modus tollens deduction. (Contributed by Jeff Hankins,
|
|
19-Aug-2009.) $)
|
|
mtand $p |- ( ph -> -. ps ) $=
|
|
( ex mtod ) ABCDABCEFG $.
|
|
$}
|
|
|
|
${
|
|
mtord.1 $e |- ( ph -> -. ch ) $.
|
|
mtord.2 $e |- ( ph -> -. th ) $.
|
|
mtord.3 $e |- ( ph -> ( ps -> ( ch \/ th ) ) ) $.
|
|
$( A modus tollens deduction involving disjunction. (Contributed by Jeff
|
|
Hankins, 15-Jul-2009.) $)
|
|
mtord $p |- ( ph -> -. ps ) $=
|
|
( wn wo wi df-or syl6ib mpid mtod ) ABDFABCHZDEABCDIODJGCDKLMN $.
|
|
$}
|
|
|
|
${
|
|
syl2anc.1 $e |- ( ph -> ps ) $.
|
|
syl2anc.2 $e |- ( ph -> ch ) $.
|
|
syl2anc.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( Syllogism inference combined with contraction. (Contributed by NM,
|
|
16-Mar-2012.) $)
|
|
syl2anc $p |- ( ph -> th ) $=
|
|
( ex sylc ) ABCDEFBCDGHI $.
|
|
$}
|
|
|
|
${
|
|
sylancl.1 $e |- ( ph -> ps ) $.
|
|
sylancl.2 $e |- ch $.
|
|
sylancl.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( Syllogism inference combined with modus ponens. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
sylancl $p |- ( ph -> th ) $=
|
|
( a1i syl2anc ) ABCDECAFHGI $.
|
|
$}
|
|
|
|
${
|
|
sylancr.1 $e |- ps $.
|
|
sylancr.2 $e |- ( ph -> ch ) $.
|
|
sylancr.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( Syllogism inference combined with modus ponens. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
sylancr $p |- ( ph -> th ) $=
|
|
( a1i syl2anc ) ABCDBAEHFGI $.
|
|
$}
|
|
|
|
${
|
|
sylanbrc.1 $e |- ( ph -> ps ) $.
|
|
sylanbrc.2 $e |- ( ph -> ch ) $.
|
|
sylanbrc.3 $e |- ( th <-> ( ps /\ ch ) ) $.
|
|
$( Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
sylanbrc $p |- ( ph -> th ) $=
|
|
( wa jca sylibr ) ABCHDABCEFIGJ $.
|
|
$}
|
|
|
|
${
|
|
sylancb.1 $e |- ( ph <-> ps ) $.
|
|
sylancb.2 $e |- ( ph <-> ch ) $.
|
|
sylancb.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by NM,
|
|
3-Sep-2004.) $)
|
|
sylancb $p |- ( ph -> th ) $=
|
|
( syl2anb anidms ) ADABCDAEFGHI $.
|
|
$}
|
|
|
|
${
|
|
sylancbr.1 $e |- ( ps <-> ph ) $.
|
|
sylancbr.2 $e |- ( ch <-> ph ) $.
|
|
sylancbr.3 $e |- ( ( ps /\ ch ) -> th ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by NM,
|
|
3-Sep-2004.) $)
|
|
sylancbr $p |- ( ph -> th ) $=
|
|
( syl2anbr anidms ) ADABCDAEFGHI $.
|
|
$}
|
|
|
|
${
|
|
sylancom.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
sylancom.2 $e |- ( ( ch /\ ps ) -> th ) $.
|
|
$( Syllogism inference with commutation of antecedents. (Contributed by
|
|
NM, 2-Jul-2008.) $)
|
|
sylancom $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa simpr syl2anc ) ABGCBDEABHFI $.
|
|
$}
|
|
|
|
${
|
|
mpdan.1 $e |- ( ph -> ps ) $.
|
|
mpdan.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 23-May-1999.)
|
|
(Proof shortened by Wolf Lammen, 22-Nov-2012.) $)
|
|
mpdan $p |- ( ph -> ch ) $=
|
|
( id syl2anc ) AABCAFDEG $.
|
|
$}
|
|
|
|
${
|
|
mpancom.1 $e |- ( ps -> ph ) $.
|
|
mpancom.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens with commutation of antecedents.
|
|
(Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen,
|
|
7-Apr-2013.) $)
|
|
mpancom $p |- ( ps -> ch ) $=
|
|
( id syl2anc ) BABCDBFEG $.
|
|
$}
|
|
|
|
${
|
|
mpan.1 $e |- ph $.
|
|
mpan.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
mpan $p |- ( ps -> ch ) $=
|
|
( a1i mpancom ) ABCABDFEG $.
|
|
$}
|
|
|
|
${
|
|
mpan2.1 $e |- ps $.
|
|
mpan2.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.)
|
|
(Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
|
|
mpan2 $p |- ( ph -> ch ) $=
|
|
( a1i mpdan ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
mp2an.1 $e |- ph $.
|
|
mp2an.2 $e |- ps $.
|
|
mp2an.3 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
13-Apr-1995.) $)
|
|
mp2an $p |- ch $=
|
|
( mpan ax-mp ) BCEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mp4an.1 $e |- ph $.
|
|
mp4an.2 $e |- ps $.
|
|
mp4an.3 $e |- ch $.
|
|
mp4an.4 $e |- th $.
|
|
mp4an.5 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by Jeff Madsen,
|
|
15-Jun-2010.) $)
|
|
mp4an $p |- ta $=
|
|
( wa pm3.2i mp2an ) ABKCDKEABFGLCDHILJM $.
|
|
$}
|
|
|
|
${
|
|
mpan2d.1 $e |- ( ph -> ch ) $.
|
|
mpan2d.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) $)
|
|
mpan2d $p |- ( ph -> ( ps -> th ) ) $=
|
|
( exp3a mpid ) ABCDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mpand.1 $e |- ( ph -> ps ) $.
|
|
mpand.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
|
|
(Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
mpand $p |- ( ph -> ( ch -> th ) ) $=
|
|
( ancomsd mpan2d ) ACBDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mpani.1 $e |- ps $.
|
|
mpani.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
|
|
mpani $p |- ( ph -> ( ch -> th ) ) $=
|
|
( a1i mpand ) ABCDBAEGFH $.
|
|
$}
|
|
|
|
${
|
|
mpan2i.1 $e |- ch $.
|
|
mpan2i.2 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 19-Nov-2012.) $)
|
|
mpan2i $p |- ( ph -> ( ps -> th ) ) $=
|
|
( a1i mpan2d ) ABCDCAEGFH $.
|
|
$}
|
|
|
|
${
|
|
mp2ani.1 $e |- ps $.
|
|
mp2ani.2 $e |- ch $.
|
|
mp2ani.3 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
12-Dec-2004.) $)
|
|
mp2ani $p |- ( ph -> th ) $=
|
|
( mpani mpi ) ACDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mp2and.1 $e |- ( ph -> ps ) $.
|
|
mp2and.2 $e |- ( ph -> ch ) $.
|
|
mp2and.3 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) $)
|
|
mp2and $p |- ( ph -> th ) $=
|
|
( mpand mpd ) ACDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mpanl1.1 $e |- ph $.
|
|
mpanl1.2 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.)
|
|
(Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
mpanl1 $p |- ( ( ps /\ ch ) -> th ) $=
|
|
( wa jctl sylan ) BABGCDBAEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpanl2.1 $e |- ps $.
|
|
mpanl2.2 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
mpanl2 $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( wa jctr sylan ) AABGCDABEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpanl12.1 $e |- ph $.
|
|
mpanl12.2 $e |- ps $.
|
|
mpanl12.3 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
13-Jul-2005.) $)
|
|
mpanl12 $p |- ( ch -> th ) $=
|
|
( mpanl1 mpan ) BCDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mpanr1.1 $e |- ps $.
|
|
mpanr1.2 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 3-May-1994.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
mpanr1 $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( anassrs mpanl2 ) ABCDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mpanr2.1 $e |- ch $.
|
|
mpanr2.2 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 3-May-1994.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by
|
|
Wolf Lammen, 7-Apr-2013.) $)
|
|
mpanr2 $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa jctr sylan2 ) BABCGDBCEHFI $.
|
|
$}
|
|
|
|
${
|
|
mpanr12.1 $e |- ps $.
|
|
mpanr12.2 $e |- ch $.
|
|
mpanr12.3 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Jul-2009.) $)
|
|
mpanr12 $p |- ( ph -> th ) $=
|
|
( mpanr1 mpan2 ) ACDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mpanlr1.1 $e |- ps $.
|
|
mpanlr1.2 $e |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.)
|
|
(Proof shortened by Wolf Lammen, 7-Apr-2013.) $)
|
|
mpanlr1 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
|
|
( wa jctl sylanl2 ) CABCHDECBFIGJ $.
|
|
$}
|
|
|
|
${
|
|
pm5.74da.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
|
|
$( Distribution of implication over biconditional (deduction rule).
|
|
(Contributed by NM, 4-May-2007.) $)
|
|
pm5.74da $p |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) $=
|
|
( wb ex pm5.74d ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
$( Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.45 $p |- ( ph <-> ( ph /\ ( ph \/ ps ) ) ) $=
|
|
( wo orc pm4.71i ) AABCABDE $.
|
|
|
|
$( Distribution of implication with conjunction. (Contributed by NM,
|
|
31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.) $)
|
|
imdistan $p |- ( ( ph -> ( ps -> ch ) ) <->
|
|
( ( ph /\ ps ) -> ( ph /\ ch ) ) ) $=
|
|
( wi wa pm5.42 impexp bitr4i ) ABCDDABACEZDDABEIDABCFABIGH $.
|
|
|
|
${
|
|
imdistani.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Distribution of implication with conjunction. (Contributed by NM,
|
|
1-Aug-1994.) $)
|
|
imdistani $p |- ( ( ph /\ ps ) -> ( ph /\ ch ) ) $=
|
|
( wa anc2li imp ) ABACEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
imdistanri.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Distribution of implication with conjunction. (Contributed by NM,
|
|
8-Jan-2002.) $)
|
|
imdistanri $p |- ( ( ps /\ ph ) -> ( ch /\ ph ) ) $=
|
|
( com12 impac ) BACABCDEF $.
|
|
$}
|
|
|
|
${
|
|
imdistand.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Distribution of implication with conjunction (deduction rule).
|
|
(Contributed by NM, 27-Aug-2004.) $)
|
|
imdistand $p |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) ) $=
|
|
( wi wa imdistan sylib ) ABCDFFBCGBDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
imdistanda.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Distribution of implication with conjunction (deduction version with
|
|
conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.) $)
|
|
imdistanda $p |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) ) $=
|
|
( wi ex imdistand ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
bi.aa $e |- ( ph <-> ps ) $.
|
|
$( Introduce a left conjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
16-Nov-2013.) $)
|
|
anbi2i $p |- ( ( ch /\ ph ) <-> ( ch /\ ps ) ) $=
|
|
( wb a1i pm5.32i ) CABABECDFG $.
|
|
|
|
$( Introduce a right conjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
16-Nov-2013.) $)
|
|
anbi1i $p |- ( ( ph /\ ch ) <-> ( ps /\ ch ) ) $=
|
|
( wb a1i pm5.32ri ) CABABECDFG $.
|
|
|
|
$( Variant of ~ anbi2i with commutation. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon,
|
|
14-Jun-2011.) $)
|
|
anbi2ci $p |- ( ( ph /\ ch ) <-> ( ch /\ ps ) ) $=
|
|
( wa anbi1i ancom bitri ) ACEBCECBEABCDFBCGH $.
|
|
$}
|
|
|
|
${
|
|
anbi12.1 $e |- ( ph <-> ps ) $.
|
|
anbi12.2 $e |- ( ch <-> th ) $.
|
|
$( Conjoin both sides of two equivalences. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
anbi12i $p |- ( ( ph /\ ch ) <-> ( ps /\ th ) ) $=
|
|
( wa anbi1i anbi2i bitri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
|
|
$( Variant of ~ anbi12i with commutation. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
anbi12ci $p |- ( ( ph /\ ch ) <-> ( th /\ ps ) ) $=
|
|
( wa anbi12i ancom bitri ) ACGBDGDBGABCDEFHBDIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan9bb.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
sylan9bb.2 $e |- ( th -> ( ch <-> ta ) ) $.
|
|
$( Nested syllogism inference conjoining dissimilar antecedents.
|
|
(Contributed by NM, 4-Mar-1995.) $)
|
|
sylan9bb $p |- ( ( ph /\ th ) -> ( ps <-> ta ) ) $=
|
|
( wa wb adantr adantl bitrd ) ADHBCEABCIDFJDCEIAGKL $.
|
|
$}
|
|
|
|
${
|
|
sylan9bbr.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
sylan9bbr.2 $e |- ( th -> ( ch <-> ta ) ) $.
|
|
$( Nested syllogism inference conjoining dissimilar antecedents.
|
|
(Contributed by NM, 4-Mar-1995.) $)
|
|
sylan9bbr $p |- ( ( th /\ ph ) -> ( ps <-> ta ) ) $=
|
|
( wb sylan9bb ancoms ) ADBEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
bid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction adding a left disjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) $)
|
|
orbi2d $p |- ( ph -> ( ( th \/ ps ) <-> ( th \/ ch ) ) ) $=
|
|
( wn wi wo imbi2d df-or 3bitr4g ) ADFZBGLCGDBHDCHABCLEIDBJDCJK $.
|
|
|
|
$( Deduction adding a right disjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) $)
|
|
orbi1d $p |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ th ) ) ) $=
|
|
( wo orbi2d orcom 3bitr4g ) ADBFDCFBDFCDFABCDEGBDHCDHI $.
|
|
|
|
$( Deduction adding a left conjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 16-Nov-2013.) $)
|
|
anbi2d $p |- ( ph -> ( ( th /\ ps ) <-> ( th /\ ch ) ) ) $=
|
|
( wb a1d pm5.32d ) ADBCABCFDEGH $.
|
|
|
|
$( Deduction adding a right conjunct to both sides of a logical
|
|
equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 16-Nov-2013.) $)
|
|
anbi1d $p |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ th ) ) ) $=
|
|
( wb a1d pm5.32rd ) ADBCABCFDEGH $.
|
|
$}
|
|
|
|
$( Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
orbi1 $p |- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) ) $=
|
|
( wb id orbi1d ) ABDZABCGEF $.
|
|
|
|
$( Introduce a right conjunct to both sides of a logical equivalence.
|
|
Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
anbi1 $p |- ( ( ph <-> ps ) -> ( ( ph /\ ch ) <-> ( ps /\ ch ) ) ) $=
|
|
( wb id anbi1d ) ABDZABCGEF $.
|
|
|
|
$( Introduce a left conjunct to both sides of a logical equivalence.
|
|
(Contributed by NM, 16-Nov-2013.) $)
|
|
anbi2 $p |- ( ( ph <-> ps ) -> ( ( ch /\ ph ) <-> ( ch /\ ps ) ) ) $=
|
|
( wb id anbi2d ) ABDZABCGEF $.
|
|
|
|
$( Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
bitr $p |- ( ( ( ph <-> ps ) /\ ( ps <-> ch ) ) -> ( ph <-> ch ) ) $=
|
|
( wb bibi1 biimpar ) ABDACDBCDABCEF $.
|
|
|
|
${
|
|
bi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction joining two equivalences to form equivalence of disjunctions.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
orbi12d $p |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) ) $=
|
|
( wo orbi1d orbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
|
|
$( Deduction joining two equivalences to form equivalence of conjunctions.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
anbi12d $p |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ ta ) ) ) $=
|
|
( wa anbi1d anbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
$( Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
pm5.3 $p |- ( ( ( ph /\ ps ) -> ch ) <->
|
|
( ( ph /\ ps ) -> ( ph /\ ch ) ) ) $=
|
|
( wa wi impexp imdistan bitri ) ABDZCEABCEEIACDEABCFABCGH $.
|
|
|
|
$( Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.) $)
|
|
pm5.61 $p |- ( ( ( ph \/ ps ) /\ -. ps ) <-> ( ph /\ -. ps ) ) $=
|
|
( wn wo biorf orcom syl6rbb pm5.32ri ) BCZABDZAIABADJBAEBAFGH $.
|
|
|
|
${
|
|
adant2.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
adantll $p |- ( ( ( th /\ ph ) /\ ps ) -> ch ) $=
|
|
( wa simpr sylan ) DAFABCDAGEH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
adantlr $p |- ( ( ( ph /\ th ) /\ ps ) -> ch ) $=
|
|
( wa simpl sylan ) ADFABCADGEH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
adantrl $p |- ( ( ph /\ ( th /\ ps ) ) -> ch ) $=
|
|
( wa simpr sylan2 ) DBFABCDBGEH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
adantrr $p |- ( ( ph /\ ( ps /\ th ) ) -> ch ) $=
|
|
( wa simpl sylan2 ) BDFABCBDGEH $.
|
|
$}
|
|
|
|
${
|
|
adantl2.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.) $)
|
|
adantlll $p |- ( ( ( ( ta /\ ph ) /\ ps ) /\ ch ) -> th ) $=
|
|
( wa simpr sylanl1 ) EAGABCDEAHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantllr $p |- ( ( ( ( ph /\ ta ) /\ ps ) /\ ch ) -> th ) $=
|
|
( wa simpl sylanl1 ) AEGABCDAEHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantlrl $p |- ( ( ( ph /\ ( ta /\ ps ) ) /\ ch ) -> th ) $=
|
|
( wa simpr sylanl2 ) EBGABCDEBHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantlrr $p |- ( ( ( ph /\ ( ps /\ ta ) ) /\ ch ) -> th ) $=
|
|
( wa simpl sylanl2 ) BEGABCDBEHFI $.
|
|
$}
|
|
|
|
${
|
|
adantr2.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantrll $p |- ( ( ph /\ ( ( ta /\ ps ) /\ ch ) ) -> th ) $=
|
|
( wa simpr sylanr1 ) EBGABCDEBHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantrlr $p |- ( ( ph /\ ( ( ps /\ ta ) /\ ch ) ) -> th ) $=
|
|
( wa simpl sylanr1 ) BEGABCDBEHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantrrl $p |- ( ( ph /\ ( ps /\ ( ta /\ ch ) ) ) -> th ) $=
|
|
( wa simpr sylanr2 ) ECGABCDECHFI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
adantrrr $p |- ( ( ph /\ ( ps /\ ( ch /\ ta ) ) ) -> th ) $=
|
|
( wa simpl sylanr2 ) CEGABCDCEHFI $.
|
|
$}
|
|
|
|
${
|
|
ad2ant.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.) $)
|
|
ad2antrr $p |- ( ( ( ph /\ ch ) /\ th ) -> ps ) $=
|
|
( adantr adantlr ) ADBCABDEFG $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.) $)
|
|
ad2antlr $p |- ( ( ( ch /\ ph ) /\ th ) -> ps ) $=
|
|
( adantr adantll ) ADBCABDEFG $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
19-Oct-1999.) $)
|
|
ad2antrl $p |- ( ( ch /\ ( ph /\ th ) ) -> ps ) $=
|
|
( wa adantr adantl ) ADFBCABDEGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
19-Oct-1999.) $)
|
|
ad2antll $p |- ( ( ch /\ ( th /\ ph ) ) -> ps ) $=
|
|
( wa adantl ) DAFBCABDEGG $.
|
|
|
|
$( Deduction adding three conjuncts to antecedent. (Contributed by NM,
|
|
28-Jul-2012.) $)
|
|
ad3antrrr $p |- ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) -> ps ) $=
|
|
( wa adantr ad2antrr ) ACGBDEABCFHI $.
|
|
|
|
$( Deduction adding three conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad3antlr $p |- ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) -> ps ) $=
|
|
( wa ad2antlr adantr ) CAGDGBEABCDFHI $.
|
|
|
|
$( Deduction adding 4 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad4antr $p |- ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) -> ps ) $=
|
|
( wa ad3antrrr adantr ) ACHDHEHBFABCDEGIJ $.
|
|
|
|
$( Deduction adding 4 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad4antlr $p |- ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) -> ps ) $=
|
|
( wa ad3antlr adantr ) CAHDHEHBFABCDEGIJ $.
|
|
|
|
$( Deduction adding 5 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad5antr $p |- ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) -> ps ) $=
|
|
( wa ad4antr adantr ) ACIDIEIFIBGABCDEFHJK $.
|
|
|
|
$( Deduction adding 5 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad5antlr $p |- ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) -> ps ) $=
|
|
( wa ad4antlr adantr ) CAIDIEIFIBGABCDEFHJK $.
|
|
|
|
$( Deduction adding 6 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad6antr $p |- ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) -> ps ) $=
|
|
( wa ad5antr adantr ) ACJDJEJFJGJBHABCDEFGIKL $.
|
|
|
|
$( Deduction adding 6 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad6antlr $p |- ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) -> ps ) $=
|
|
( wa ad5antlr adantr ) CAJDJEJFJGJBHABCDEFGIKL $.
|
|
|
|
$( Deduction adding 7 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad7antr $p |- ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) -> ps ) $=
|
|
( wa ad6antr adantr ) ACKDKEKFKGKHKBIABCDEFGHJLM $.
|
|
|
|
$( Deduction adding 7 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad7antlr $p |- ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) -> ps ) $=
|
|
( wa ad6antlr adantr ) CAKDKEKFKGKHKBIABCDEFGHJLM $.
|
|
|
|
$( Deduction adding 8 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad8antr $p |- ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
|
|
( wa ad7antr adantr ) ACLDLELFLGLHLILBJABCDEFGHIKMN $.
|
|
|
|
$( Deduction adding 8 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad8antlr $p |- ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
|
|
( wa ad7antlr adantr ) CALDLELFLGLHLILBJABCDEFGHIKMN $.
|
|
|
|
$( Deduction adding 9 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad9antr $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
|
|
( wa ad8antr adantr ) ACMDMEMFMGMHMIMJMBKABCDEFGHIJLNO $.
|
|
|
|
$( Deduction adding 9 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad9antlr $p |- ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
|
|
( wa ad8antlr adantr ) CAMDMEMFMGMHMIMJMBKABCDEFGHIJLNO $.
|
|
|
|
$( Deduction adding 10 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 4-Jan-2017.) $)
|
|
ad10antr $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
|
|
( wa ad9antr adantr ) ACNDNENFNGNHNINJNKNBLABCDEFGHIJKMOP $.
|
|
|
|
$( Deduction adding 10 conjuncts to antecedent. (Contributed by Mario
|
|
Carneiro, 5-Jan-2017.) $)
|
|
ad10antlr $p |- ( ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
|
|
/\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
|
|
( wa ad9antlr adantr ) CANDNENFNGNHNINJNKNBLABCDEFGHIJKMOP $.
|
|
$}
|
|
|
|
${
|
|
ad2ant2.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
ad2ant2l $p |- ( ( ( th /\ ph ) /\ ( ta /\ ps ) ) -> ch ) $=
|
|
( wa adantrl adantll ) AEBGCDABCEFHI $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
ad2ant2r $p |- ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) -> ch ) $=
|
|
( wa adantrr adantlr ) ABEGCDABCEFHI $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
23-Nov-2007.) $)
|
|
ad2ant2lr $p |- ( ( ( th /\ ph ) /\ ( ps /\ ta ) ) -> ch ) $=
|
|
( wa adantrr adantll ) ABEGCDABCEFHI $.
|
|
|
|
$( Deduction adding two conjuncts to antecedent. (Contributed by NM,
|
|
24-Nov-2007.) $)
|
|
ad2ant2rl $p |- ( ( ( ph /\ th ) /\ ( ta /\ ps ) ) -> ch ) $=
|
|
( wa adantrl adantlr ) AEBGCDABCEFHI $.
|
|
$}
|
|
|
|
$( Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.) $)
|
|
simpll $p |- ( ( ( ph /\ ps ) /\ ch ) -> ph ) $=
|
|
( id ad2antrr ) AABCADE $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.) $)
|
|
simplr $p |- ( ( ( ph /\ ps ) /\ ch ) -> ps ) $=
|
|
( id ad2antlr ) BBACBDE $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.) $)
|
|
simprl $p |- ( ( ph /\ ( ps /\ ch ) ) -> ps ) $=
|
|
( id ad2antrl ) BBACBDE $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.) $)
|
|
simprr $p |- ( ( ph /\ ( ps /\ ch ) ) -> ch ) $=
|
|
( id ad2antll ) CCABCDE $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simplll $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ph ) $=
|
|
( wa simpl ad2antrr ) ABEACDABFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simpllr $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ps ) $=
|
|
( wa simpr ad2antrr ) ABEBCDABFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simplrl $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ps ) $=
|
|
( wa simpl ad2antlr ) BCEBADBCFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simplrr $p |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ch ) $=
|
|
( wa simpr ad2antlr ) BCECADBCFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simprll $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ps ) $=
|
|
( wa simpl ad2antrl ) BCEBADBCFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simprlr $p |- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ch ) $=
|
|
( wa simpr ad2antrl ) BCECADBCFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simprrl $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ch ) $=
|
|
( wa simpl ad2antll ) CDECABCDFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Jeff Hankins,
|
|
28-Jul-2009.) $)
|
|
simprrr $p |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> th ) $=
|
|
( wa simpr ad2antll ) CDEDABCDFG $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-4l $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta ) -> ph ) $=
|
|
( wa simplll adantr ) ABFCFDFAEABCDGH $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-4r $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta ) -> ps ) $=
|
|
( wa simpllr adantr ) ABFCFDFBEABCDGH $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-5l $p |- ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) -> ph ) $=
|
|
( wa simp-4l adantr ) ABGCGDGEGAFABCDEHI $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-5r $p |- ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) -> ps ) $=
|
|
( wa simp-4r adantr ) ABGCGDGEGBFABCDEHI $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-6l $p |- ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) -> ph ) $=
|
|
( wa simp-5l adantr ) ABHCHDHEHFHAGABCDEFIJ $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-6r $p |- ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) -> ps ) $=
|
|
( wa simp-5r adantr ) ABHCHDHEHFHBGABCDEFIJ $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-7l $p |- ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) -> ph ) $=
|
|
( wa simp-6l adantr ) ABICIDIEIFIGIAHABCDEFGJK $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-7r $p |- ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) -> ps ) $=
|
|
( wa simp-6r adantr ) ABICIDIEIFIGIBHABCDEFGJK $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-8l $p |- ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) -> ph ) $=
|
|
( wa simp-7l adantr ) ABJCJDJEJFJGJHJAIABCDEFGHKL $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-8r $p |- ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) -> ps ) $=
|
|
( wa simp-7r adantr ) ABJCJDJEJFJGJHJBIABCDEFGHKL $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-9l $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ph ) $=
|
|
( wa simp-8l adantr ) ABKCKDKEKFKGKHKIKAJABCDEFGHILM $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-9r $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
|
|
( wa simp-8r adantr ) ABKCKDKEKFKGKHKIKBJABCDEFGHILM $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-10l $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ph ) $=
|
|
( wa simp-9l adantr ) ABLCLDLELFLGLHLILJLAKABCDEFGHIJMN $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-10r $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
|
|
( wa simp-9r adantr ) ABLCLDLELFLGLHLILJLBKABCDEFGHIJMN $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-11l $p |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ph ) $=
|
|
( wa simp-10l adantr ) ABMCMDMEMFMGMHMIMJMKMALABCDEFGHIJKNO $.
|
|
|
|
$( Simplification of a conjunction. (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
simp-11r $p |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
|
|
/\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
|
|
( wa simp-10r adantr ) ABMCMDMEMFMGMHMIMJMKMBLABCDEFGHIJKNO $.
|
|
|
|
$( Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell]
|
|
p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf
|
|
Lammen, 9-Dec-2012.) $)
|
|
jaob $p |- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) ) $=
|
|
( wo wi wa pm2.67-2 olc imim1i jca pm3.44 impbii ) ACDZBEZABEZCBEZFNOPABCGC
|
|
MBCAHIJBACKL $.
|
|
|
|
${
|
|
jaoian.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
jaoian.2 $e |- ( ( th /\ ps ) -> ch ) $.
|
|
$( Inference disjoining the antecedents of two implications. (Contributed
|
|
by NM, 23-Oct-2005.) $)
|
|
jaoian $p |- ( ( ( ph \/ th ) /\ ps ) -> ch ) $=
|
|
( wo wi ex jaoi imp ) ADGBCABCHDABCEIDBCFIJK $.
|
|
$}
|
|
|
|
${
|
|
jaodan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
jaodan.2 $e |- ( ( ph /\ th ) -> ch ) $.
|
|
$( Deduction disjoining the antecedents of two implications. (Contributed
|
|
by NM, 14-Oct-2005.) $)
|
|
jaodan $p |- ( ( ph /\ ( ps \/ th ) ) -> ch ) $=
|
|
( wo ex jaod imp ) ABDGCABCDABCEHADCFHIJ $.
|
|
|
|
jaodan.3 $e |- ( ph -> ( ps \/ th ) ) $.
|
|
$( Eliminate a disjunction in a deduction. A translation of natural
|
|
deduction rule ` \/ ` E ( ` \/ ` elimination), see natded in set.mm.
|
|
(Contributed by Mario Carneiro, 29-May-2016.) $)
|
|
mpjaodan $p |- ( ph -> ch ) $=
|
|
( wo jaodan mpdan ) ABDHCGABCDEFIJ $.
|
|
$}
|
|
|
|
$( Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.77 $p |- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) <->
|
|
( ( ps \/ ch ) -> ph ) ) $=
|
|
( wo wi wa jaob bicomi ) BCDAEBAECAEFBACGH $.
|
|
|
|
$( Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.63 $p |- ( ( ph \/ ps ) -> ( ( -. ph \/ ps ) -> ps ) ) $=
|
|
( wo wn pm2.53 idd jaod ) ABCZADBBABEHBFG $.
|
|
|
|
$( Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.64 $p |- ( ( ph \/ ps ) -> ( ( ph \/ -. ps ) -> ph ) ) $=
|
|
( wn wo wi ax-1 orel2 jaoi com12 ) ABCZDABDZAAKAEJAKFBAGHI $.
|
|
|
|
${
|
|
pm2.61ian.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
pm2.61ian.2 $e |- ( ( -. ph /\ ps ) -> ch ) $.
|
|
$( Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.) $)
|
|
pm2.61ian $p |- ( ps -> ch ) $=
|
|
( wi ex wn pm2.61i ) ABCFABCDGAHBCEGI $.
|
|
$}
|
|
|
|
${
|
|
pm2.61dan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
pm2.61dan.2 $e |- ( ( ph /\ -. ps ) -> ch ) $.
|
|
$( Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.) $)
|
|
pm2.61dan $p |- ( ph -> ch ) $=
|
|
( ex wn pm2.61d ) ABCABCDFABGCEFH $.
|
|
$}
|
|
|
|
${
|
|
pm2.61ddan.1 $e |- ( ( ph /\ ps ) -> th ) $.
|
|
pm2.61ddan.2 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
pm2.61ddan.3 $e |- ( ( ph /\ ( -. ps /\ -. ch ) ) -> th ) $.
|
|
$( Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.) $)
|
|
pm2.61ddan $p |- ( ph -> th ) $=
|
|
( wn wa adantlr anassrs pm2.61dan ) ABDEABHZICDACDMFJAMCHDGKLL $.
|
|
$}
|
|
|
|
${
|
|
pm2.61dda.1 $e |- ( ( ph /\ -. ps ) -> th ) $.
|
|
pm2.61dda.2 $e |- ( ( ph /\ -. ch ) -> th ) $.
|
|
pm2.61dda.3 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.) $)
|
|
pm2.61dda $p |- ( ph -> th ) $=
|
|
( wa anassrs wn adantlr pm2.61dan ) ABDABHCDABCDGIACJDBFKLEL $.
|
|
$}
|
|
|
|
${
|
|
condan.1 $e |- ( ( ph /\ -. ps ) -> ch ) $.
|
|
condan.2 $e |- ( ( ph /\ -. ps ) -> -. ch ) $.
|
|
$( Proof by contradiction. (Contributed by NM, 9-Feb-2006.) (Proof
|
|
shortened by Wolf Lammen, 19-Jun-2014.) $)
|
|
condan $p |- ( ph -> ps ) $=
|
|
( wn pm2.65da notnot2 syl ) ABFZFBAJCDEGBHI $.
|
|
$}
|
|
|
|
$( Introduce one conjunct as an antecedent to the other. "abai" stands for
|
|
"and, biconditional, and, implication". (Contributed by NM,
|
|
12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.) $)
|
|
abai $p |- ( ( ph /\ ps ) <-> ( ph /\ ( ph -> ps ) ) ) $=
|
|
( wi biimt pm5.32i ) ABABCABDE $.
|
|
|
|
$( Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.53 $p |- ( ( ( ( ph \/ ps ) \/ ch ) -> th ) <->
|
|
( ( ( ph -> th ) /\ ( ps -> th ) ) /\ ( ch -> th ) ) ) $=
|
|
( wo wi wa jaob anbi1i bitri ) ABEZCEDFKDFZCDFZGADFBDFGZMGKDCHLNMADBHIJ $.
|
|
|
|
$( Swap two conjuncts. Note that the first digit (1) in the label refers to
|
|
the outer conjunct position, and the next digit (2) to the inner conjunct
|
|
position. (Contributed by NM, 12-Mar-1995.) $)
|
|
an12 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
|
|
( wa ancom anbi1i anass 3bitr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH $.
|
|
|
|
$( A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof
|
|
shortened by Wolf Lammen, 25-Dec-2012.) $)
|
|
an32 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ph /\ ch ) /\ ps ) ) $=
|
|
( wa anass an12 ancom 3bitri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.
|
|
|
|
$( A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof
|
|
shortened by Wolf Lammen, 31-Dec-2012.) $)
|
|
an13 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ch /\ ( ps /\ ph ) ) ) $=
|
|
( wa an12 anass ancom 3bitr2i ) ABCDDBACDDBADZCDCIDABCEBACFICGH $.
|
|
|
|
$( A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof
|
|
shortened by Wolf Lammen, 31-Dec-2012.) $)
|
|
an31 $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ch /\ ps ) /\ ph ) ) $=
|
|
( wa an13 anass 3bitr4i ) ABCDDCBADDABDCDCBDADABCEABCFCBAFG $.
|
|
|
|
${
|
|
an12s.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Swap two conjuncts in antecedent. The label suffix "s" means that
|
|
~ an12 is combined with ~ syl (or a variant). (Contributed by NM,
|
|
13-Mar-1996.) $)
|
|
an12s $p |- ( ( ps /\ ( ph /\ ch ) ) -> th ) $=
|
|
( wa an12 sylbi ) BACFFABCFFDBACGEH $.
|
|
|
|
$( Inference commuting a nested conjunction in antecedent. (Contributed by
|
|
NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
ancom2s $p |- ( ( ph /\ ( ch /\ ps ) ) -> th ) $=
|
|
( wa pm3.22 sylan2 ) CBFABCFDCBGEH $.
|
|
|
|
$( Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.) $)
|
|
an13s $p |- ( ( ch /\ ( ps /\ ph ) ) -> th ) $=
|
|
( exp32 com13 imp32 ) CBADABCDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
an32s.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.) $)
|
|
an32s $p |- ( ( ( ph /\ ch ) /\ ps ) -> th ) $=
|
|
( wa an32 sylbi ) ACFBFABFCFDACBGEH $.
|
|
|
|
$( Inference commuting a nested conjunction in antecedent. (Contributed by
|
|
NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
ancom1s $p |- ( ( ( ps /\ ph ) /\ ch ) -> th ) $=
|
|
( wa pm3.22 sylan ) BAFABFCDBAGEH $.
|
|
|
|
$( Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.) $)
|
|
an31s $p |- ( ( ( ch /\ ps ) /\ ph ) -> th ) $=
|
|
( exp31 com13 imp31 ) CBADABCDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
anass1rs.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Commutative-associative law for conjunction in an antecedent.
|
|
(Contributed by Jeff Madsen, 19-Jun-2011.) $)
|
|
anass1rs $p |- ( ( ( ph /\ ch ) /\ ps ) -> th ) $=
|
|
( anassrs an32s ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
$( Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.)
|
|
(Proof shortened by Wolf Lammen, 16-Nov-2013.) $)
|
|
anabs1 $p |- ( ( ( ph /\ ps ) /\ ph ) <-> ( ph /\ ps ) ) $=
|
|
( wa simpl pm4.71i bicomi ) ABCZGACGAABDEF $.
|
|
|
|
$( Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.)
|
|
(Proof shortened by Wolf Lammen, 9-Dec-2012.) $)
|
|
anabs5 $p |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) ) $=
|
|
( wa ibar bicomd pm5.32i ) AABCZBABGABDEF $.
|
|
|
|
$( Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.)
|
|
(Proof shortened by Wolf Lammen, 17-Nov-2013.) $)
|
|
anabs7 $p |- ( ( ps /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) ) $=
|
|
( wa simpr pm4.71ri bicomi ) ABCZBGCGBABDEF $.
|
|
|
|
${
|
|
anabsan.1 $e |- ( ( ( ph /\ ph ) /\ ps ) -> ch ) $.
|
|
$( Absorption of antecedent with conjunction. (Contributed by NM,
|
|
24-Mar-1996.) $)
|
|
anabsan $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wa pm4.24 sylanb ) AAAEBCAFDG $.
|
|
$}
|
|
|
|
${
|
|
anabss1.1 $e |- ( ( ( ph /\ ps ) /\ ph ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.) $)
|
|
anabss1 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( an32s anabsan ) ABCABACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabss4.1 $e |- ( ( ( ps /\ ph ) /\ ps ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) $)
|
|
anabss4 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anabss1 ancoms ) BACBACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabss5.1 $e |- ( ( ph /\ ( ph /\ ps ) ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.) $)
|
|
anabss5 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anassrs anabsan ) ABCAABCDEF $.
|
|
$}
|
|
|
|
${
|
|
anabsi5.1 $e |- ( ph -> ( ( ph /\ ps ) -> ch ) ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.) $)
|
|
anabsi5 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wa imp anabss5 ) ABCAABECDFG $.
|
|
$}
|
|
|
|
${
|
|
anabsi6.1 $e |- ( ph -> ( ( ps /\ ph ) -> ch ) ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
14-Aug-2000.) $)
|
|
anabsi6 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( ancomsd anabsi5 ) ABCABACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabsi7.1 $e |- ( ps -> ( ( ph /\ ps ) -> ch ) ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.) $)
|
|
anabsi7 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anabsi6 ancoms ) BACBACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabsi8.1 $e |- ( ps -> ( ( ps /\ ph ) -> ch ) ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
26-Sep-1999.) $)
|
|
anabsi8 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anabsi5 ancoms ) BACBACDEF $.
|
|
$}
|
|
|
|
${
|
|
anabss7.1 $e |- ( ( ps /\ ( ph /\ ps ) ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.) $)
|
|
anabss7 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anassrs anabss4 ) ABCBABCDEF $.
|
|
$}
|
|
|
|
${
|
|
anabsan2.1 $e |- ( ( ph /\ ( ps /\ ps ) ) -> ch ) $.
|
|
$( Absorption of antecedent with conjunction. (Contributed by NM,
|
|
10-May-2004.) $)
|
|
anabsan2 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( an12s anabss7 ) ABCABBCDEF $.
|
|
$}
|
|
|
|
${
|
|
anabss3.1 $e |- ( ( ( ph /\ ps ) /\ ps ) -> ch ) $.
|
|
$( Absorption of antecedent into conjunction. (Contributed by NM,
|
|
20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.) $)
|
|
anabss3 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( anasss anabsan2 ) ABCABBCDEF $.
|
|
$}
|
|
|
|
$( Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.) $)
|
|
an4 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
|
|
( ( ph /\ ch ) /\ ( ps /\ th ) ) ) $=
|
|
( wa an12 anbi2i anass 3bitr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHAC
|
|
LHI $.
|
|
|
|
$( Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.) $)
|
|
an42 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
|
|
( ( ph /\ ch ) /\ ( th /\ ps ) ) ) $=
|
|
( wa an4 ancom anbi2i bitri ) ABECDEEACEZBDEZEJDBEZEABCDFKLJBDGHI $.
|
|
|
|
${
|
|
an4s.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( Inference rearranging 4 conjuncts in antecedent. (Contributed by NM,
|
|
10-Aug-1995.) $)
|
|
an4s $p |- ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) -> ta ) $=
|
|
( wa an4 sylbi ) ACGBDGGABGCDGGEACBDHFI $.
|
|
$}
|
|
|
|
${
|
|
an41r3s.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( Inference rearranging 4 conjuncts in antecedent. (Contributed by NM,
|
|
10-Aug-1995.) $)
|
|
an42s $p |- ( ( ( ph /\ ch ) /\ ( th /\ ps ) ) -> ta ) $=
|
|
( wa an4s ancom2s ) ACGBDEABCDEFHI $.
|
|
$}
|
|
|
|
$( Distribution of conjunction over conjunction. (Contributed by NM,
|
|
14-Aug-1995.) $)
|
|
anandi $p |- ( ( ph /\ ( ps /\ ch ) ) <->
|
|
( ( ph /\ ps ) /\ ( ph /\ ch ) ) ) $=
|
|
( wa anidm anbi1i an4 bitr3i ) ABCDZDAADZIDABDACDDJAIAEFAABCGH $.
|
|
|
|
$( Distribution of conjunction over conjunction. (Contributed by NM,
|
|
24-Aug-1995.) $)
|
|
anandir $p |- ( ( ( ph /\ ps ) /\ ch ) <->
|
|
( ( ph /\ ch ) /\ ( ps /\ ch ) ) ) $=
|
|
( wa anidm anbi2i an4 bitr3i ) ABDZCDICCDZDACDBCDDJCICEFABCCGH $.
|
|
|
|
${
|
|
anandis.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) -> ta ) $.
|
|
$( Inference that undistributes conjunction in the antecedent.
|
|
(Contributed by NM, 7-Jun-2004.) $)
|
|
anandis $p |- ( ( ph /\ ( ps /\ ch ) ) -> ta ) $=
|
|
( wa an4s anabsan ) ABCFDABACDEGH $.
|
|
$}
|
|
|
|
${
|
|
anandirs.1 $e |- ( ( ( ph /\ ch ) /\ ( ps /\ ch ) ) -> ta ) $.
|
|
$( Inference that undistributes conjunction in the antecedent.
|
|
(Contributed by NM, 7-Jun-2004.) $)
|
|
anandirs $p |- ( ( ( ph /\ ps ) /\ ch ) -> ta ) $=
|
|
( wa an4s anabsan2 ) ABFCDACBCDEGH $.
|
|
$}
|
|
|
|
${
|
|
impbida.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
impbida.2 $e |- ( ( ph /\ ch ) -> ps ) $.
|
|
$( Deduce an equivalence from two implications. (Contributed by NM,
|
|
17-Feb-2007.) $)
|
|
impbida $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( ex impbid ) ABCABCDFACBEFG $.
|
|
$}
|
|
|
|
$( Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM,
|
|
28-Jan-1997.) $)
|
|
pm3.48 $p |- ( ( ( ph -> ps ) /\ ( ch -> th ) )
|
|
-> ( ( ph \/ ch ) -> ( ps \/ th ) ) ) $=
|
|
( wi wo orc imim2i olc jaao ) ABEABDFZCDECBKABDGHDKCDBIHJ $.
|
|
|
|
$( Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.45 $p |- ( ( ph -> ps ) -> ( ( ph /\ ch ) -> ( ps /\ ch ) ) ) $=
|
|
( wi id anim1d ) ABDZABCGEF $.
|
|
|
|
${
|
|
im2an9.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
im2an9.2 $e |- ( th -> ( ta -> et ) ) $.
|
|
$( Deduction joining nested implications to form implication of
|
|
conjunctions. (Contributed by NM, 29-Feb-1996.) $)
|
|
im2anan9 $p |- ( ( ph /\ th ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) ) $=
|
|
( wa wi adantr adantl anim12d ) ADIBCEFABCJDGKDEFJAHLM $.
|
|
|
|
$( Deduction joining nested implications to form implication of
|
|
conjunctions. (Contributed by NM, 29-Feb-1996.) $)
|
|
im2anan9r $p |- ( ( th /\ ph ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) ) $=
|
|
( wa wi im2anan9 ancoms ) ADBEICFIJABCDEFGHKL $.
|
|
$}
|
|
|
|
${
|
|
anim12dan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
anim12dan.2 $e |- ( ( ph /\ th ) -> ta ) $.
|
|
$( Conjoin antecedents and consequents in a deduction. (Contributed by
|
|
Mario Carneiro, 12-May-2014.) $)
|
|
anim12dan $p |- ( ( ph /\ ( ps /\ th ) ) -> ( ch /\ ta ) ) $=
|
|
( wa ex anim12d imp ) ABDHCEHABCDEABCFIADEGIJK $.
|
|
$}
|
|
|
|
${
|
|
orim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
orim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( Disjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
10-May-1994.) $)
|
|
orim12d $p |- ( ph -> ( ( ps \/ th ) -> ( ch \/ ta ) ) ) $=
|
|
( wi wo pm3.48 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
orim1d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Disjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
23-Apr-1995.) $)
|
|
orim1d $p |- ( ph -> ( ( ps \/ th ) -> ( ch \/ th ) ) ) $=
|
|
( idd orim12d ) ABCDDEADFG $.
|
|
|
|
$( Disjoin antecedents and consequents in a deduction. (Contributed by NM,
|
|
23-Apr-1995.) $)
|
|
orim2d $p |- ( ph -> ( ( th \/ ps ) -> ( th \/ ch ) ) ) $=
|
|
( idd orim12d ) ADDBCADFEG $.
|
|
$}
|
|
|
|
$( Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
orim2 $p |- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
|
|
( wi id orim2d ) BCDZBCAGEF $.
|
|
|
|
$( Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM,
|
|
6-Mar-2008.) $)
|
|
pm2.38 $p |- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ch \/ ph ) ) ) $=
|
|
( wi id orim1d ) BCDZBCAGEF $.
|
|
|
|
$( Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM,
|
|
6-Mar-2008.) $)
|
|
pm2.36 $p |- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ch \/ ph ) ) ) $=
|
|
( wo wi pm1.4 pm2.38 syl5 ) ABDBADBCECADABFABCGH $.
|
|
|
|
$( Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM,
|
|
6-Mar-2008.) $)
|
|
pm2.37 $p |- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ph \/ ch ) ) ) $=
|
|
( wi wo pm2.38 pm1.4 syl6 ) BCDBAECAEACEABCFCAGH $.
|
|
|
|
$( Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.73 $p |- ( ( ph -> ps )
|
|
-> ( ( ( ph \/ ps ) \/ ch ) -> ( ps \/ ch ) ) ) $=
|
|
( wi wo pm2.621 orim1d ) ABDABEBCABFG $.
|
|
|
|
$( Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
pm2.74 $p |- ( ( ps -> ph )
|
|
-> ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ch ) ) ) $=
|
|
( wi wo orel2 ax-1 ja orim1d ) BADABEZACBAJADBAFAJGHI $.
|
|
|
|
$( Disjunction distributes over implication. (Contributed by Wolf Lammen,
|
|
5-Jan-2013.) $)
|
|
orimdi $p |- ( ( ph \/ ( ps -> ch ) )
|
|
<-> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
|
|
( wn wi wo imdi df-or imbi12i 3bitr4i ) ADZBCEZEKBEZKCEZEALFABFZACFZEKBCGAL
|
|
HOMPNABHACHIJ $.
|
|
|
|
$( Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.76 $p |- ( ( ph \/ ( ps -> ch ) )
|
|
-> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
|
|
( wi wo orimdi biimpi ) ABCDEABEACEDABCFG $.
|
|
|
|
$( Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.) $)
|
|
pm2.75 $p |- ( ( ph \/ ps )
|
|
-> ( ( ph \/ ( ps -> ch ) ) -> ( ph \/ ch ) ) ) $=
|
|
( wi wo pm2.76 com12 ) ABCDEABEACEABCFG $.
|
|
|
|
$( Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) $)
|
|
pm2.8 $p |- ( ( ph \/ ps ) -> ( ( -. ps \/ ch ) -> ( ph \/ ch ) ) ) $=
|
|
( wo wn pm2.53 con1d orim1d ) ABDZBEACIABABFGH $.
|
|
|
|
$( Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.81 $p |- ( ( ps -> ( ch -> th ) )
|
|
-> ( ( ph \/ ps ) -> ( ( ph \/ ch ) -> ( ph \/ th ) ) ) ) $=
|
|
( wi wo orim2 pm2.76 syl6 ) BCDEZEABFAJFACFADFEABJGACDHI $.
|
|
|
|
$( Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm2.82 $p |- ( ( ( ph \/ ps ) \/ ch ) -> ( ( ( ph \/ -. ch ) \/ th )
|
|
-> ( ( ph \/ ps ) \/ th ) ) ) $=
|
|
( wo wn wi ax-1 pm2.24 orim2d jaoi orim1d ) ABEZCEACFZEZMDMOMGCMOHCNBACBIJK
|
|
L $.
|
|
|
|
$( Theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) $)
|
|
pm2.85 $p |- ( ( ( ph \/ ps ) -> ( ph \/ ch ) )
|
|
-> ( ph \/ ( ps -> ch ) ) ) $=
|
|
( wi wo orimdi biimpri ) ABCDEABEACEDABCFG $.
|
|
|
|
${
|
|
pm3.2ni.1 $e |- -. ph $.
|
|
pm3.2ni.2 $e |- -. ps $.
|
|
$( Infer negated disjunction of negated premises. (Contributed by NM,
|
|
4-Apr-1995.) $)
|
|
pm3.2ni $p |- -. ( ph \/ ps ) $=
|
|
( wo id pm2.21i jaoi mto ) ABEACAABAFBADGHI $.
|
|
$}
|
|
|
|
$( Absorption of redundant internal disjunct. Compare Theorem *4.45 of
|
|
[WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 28-Feb-2014.) $)
|
|
orabs $p |- ( ph <-> ( ( ph \/ ps ) /\ ph ) ) $=
|
|
( wo orc pm4.71ri ) AABCABDE $.
|
|
|
|
$( Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton,
|
|
23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.) $)
|
|
oranabs $p |- ( ( ( ph \/ -. ps ) /\ ps ) <-> ( ph /\ ps ) ) $=
|
|
( wn wo biortn orcom syl6rbb pm5.32ri ) BABCZDZABAIADJBAEIAFGH $.
|
|
|
|
$( Two propositions are equivalent if they are both true. Theorem *5.1 of
|
|
[WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.) $)
|
|
pm5.1 $p |- ( ( ph /\ ps ) -> ( ph <-> ps ) ) $=
|
|
( wb pm5.501 biimpa ) ABABCABDE $.
|
|
|
|
$( Two propositions are equivalent if they are both false. Theorem *5.21 of
|
|
[WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) $)
|
|
pm5.21 $p |- ( ( -. ph /\ -. ps ) -> ( ph <-> ps ) ) $=
|
|
( wn wb pm5.21im imp ) ACBCABDABEF $.
|
|
|
|
$( Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm3.43 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) )
|
|
-> ( ph -> ( ps /\ ch ) ) ) $=
|
|
( wi wa pm3.43i imp ) ABDACDABCEDABCFG $.
|
|
|
|
$( Distributive law for implication over conjunction. Compare Theorem *4.76
|
|
of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof
|
|
shortened by Wolf Lammen, 27-Nov-2013.) $)
|
|
jcab $p |- ( ( ph -> ( ps /\ ch ) )
|
|
<-> ( ( ph -> ps ) /\ ( ph -> ch ) ) ) $=
|
|
( wa wi simpl imim2i simpr jca pm3.43 impbii ) ABCDZEZABEZACEZDMNOLBABCFGLC
|
|
ABCHGIABCJK $.
|
|
|
|
$( Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell]
|
|
p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.) $)
|
|
ordi $p |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) ) $=
|
|
( wn wa wi wo jcab df-or anbi12i 3bitr4i ) ADZBCEZFLBFZLCFZEAMGABGZACGZELBC
|
|
HAMIPNQOABIACIJK $.
|
|
|
|
$( Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) $)
|
|
ordir $p |- ( ( ( ph /\ ps ) \/ ch ) <->
|
|
( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) $=
|
|
( wa wo ordi orcom anbi12i 3bitr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMKNL
|
|
ACGBCGHI $.
|
|
|
|
$( Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.76 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) <->
|
|
( ph -> ( ps /\ ch ) ) ) $=
|
|
( wa wi jcab bicomi ) ABCDEABEACEDABCFG $.
|
|
|
|
$( Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell]
|
|
p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf
|
|
Lammen, 5-Jan-2013.) $)
|
|
andi $p |- ( ( ph /\ ( ps \/ ch ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) ) $=
|
|
( wo wa orc olc jaodan anim2i jaoi impbii ) ABCDZEZABEZACEZDZABPCNOFONGHNMO
|
|
BLABCFICLACBGIJK $.
|
|
|
|
$( Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) $)
|
|
andir $p |- ( ( ( ph \/ ps ) /\ ch ) <->
|
|
( ( ph /\ ch ) \/ ( ps /\ ch ) ) ) $=
|
|
( wo wa andi ancom orbi12i 3bitr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMKNL
|
|
ACGBCGHI $.
|
|
|
|
$( Double distributive law for disjunction. (Contributed by NM,
|
|
12-Aug-1994.) $)
|
|
orddi $p |- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <->
|
|
( ( ( ph \/ ch ) /\ ( ph \/ th ) ) /\
|
|
( ( ps \/ ch ) /\ ( ps \/ th ) ) ) ) $=
|
|
( wa wo ordir ordi anbi12i bitri ) ABECDEZFAKFZBKFZEACFADFEZBCFBDFEZEABKGLN
|
|
MOACDHBCDHIJ $.
|
|
|
|
$( Double distributive law for conjunction. (Contributed by NM,
|
|
12-Aug-1994.) $)
|
|
anddi $p |- ( ( ( ph \/ ps ) /\ ( ch \/ th ) ) <->
|
|
( ( ( ph /\ ch ) \/ ( ph /\ th ) ) \/
|
|
( ( ps /\ ch ) \/ ( ps /\ th ) ) ) ) $=
|
|
( wo wa andir andi orbi12i bitri ) ABECDEZFAKFZBKFZEACFADFEZBCFBDFEZEABKGLN
|
|
MOACDHBCDHIJ $.
|
|
|
|
$( Prove formula-building rules for the biconditional connective. $)
|
|
|
|
$( Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.39 $p |- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) ->
|
|
( ( ph \/ ps ) <-> ( ch \/ th ) ) ) $=
|
|
( wb wa simpl simpr orbi12d ) ACEZBDEZFACBDJKGJKHI $.
|
|
|
|
$( Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.38 $p |- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) ->
|
|
( ( ph /\ ps ) <-> ( ch /\ th ) ) ) $=
|
|
( wb wa simpl simpr anbi12d ) ACEZBDEZFACBDJKGJKHI $.
|
|
|
|
${
|
|
bi2an9.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bi2an9.2 $e |- ( th -> ( ta <-> et ) ) $.
|
|
$( Deduction joining two equivalences to form equivalence of conjunctions.
|
|
(Contributed by NM, 31-Jul-1995.) $)
|
|
bi2anan9 $p |- ( ( ph /\ th ) -> ( ( ps /\ ta ) <-> ( ch /\ et ) ) ) $=
|
|
( wa anbi1d anbi2d sylan9bb ) ABEICEIDCFIABCEGJDEFCHKL $.
|
|
|
|
$( Deduction joining two equivalences to form equivalence of conjunctions.
|
|
(Contributed by NM, 19-Feb-1996.) $)
|
|
bi2anan9r $p |- ( ( th /\ ph ) -> ( ( ps /\ ta ) <-> ( ch /\ et ) ) ) $=
|
|
( wa wb bi2anan9 ancoms ) ADBEICFIJABCDEFGHKL $.
|
|
|
|
$( Deduction joining two biconditionals with different antecedents.
|
|
(Contributed by NM, 12-May-2004.) $)
|
|
bi2bian9 $p |- ( ( ph /\ th ) -> ( ( ps <-> ta ) <-> ( ch <-> et ) ) ) $=
|
|
( wa wb adantr adantl bibi12d ) ADIBCEFABCJDGKDEFJAHLM $.
|
|
$}
|
|
|
|
$( Implication in terms of biconditional and disjunction. Theorem *4.72 of
|
|
[WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 30-Jan-2013.) $)
|
|
pm4.72 $p |- ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) ) $=
|
|
( wi wo wb olc pm2.621 impbid2 orc bi2 syl5 impbii ) ABCZBABDZEZMBNBAFABGHA
|
|
NOBABIBNJKL $.
|
|
|
|
$( Simplify an implication between implications. (Contributed by Paul
|
|
Chapman, 17-Nov-2012.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) $)
|
|
imimorb $p |- ( ( ( ps -> ch ) -> ( ph -> ch ) ) <->
|
|
( ph -> ( ps \/ ch ) ) ) $=
|
|
( wi wo bi2.04 dfor2 imbi2i bitr4i ) BCDZACDDAJCDZDABCEZDJACFLKABCGHI $.
|
|
|
|
$( Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.33 $p |- ( ( ph /\ ( ps -> ch ) ) <->
|
|
( ph /\ ( ( ph /\ ps ) -> ch ) ) ) $=
|
|
( wi wa ibar imbi1d pm5.32i ) ABCDABEZCDABICABFGH $.
|
|
|
|
$( Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.36 $p |- ( ( ph /\ ( ph <-> ps ) ) <-> ( ps /\ ( ph <-> ps ) ) ) $=
|
|
( wb id pm5.32ri ) ABCZABFDE $.
|
|
|
|
${
|
|
bianabs.1 $e |- ( ph -> ( ps <-> ( ph /\ ch ) ) ) $.
|
|
$( Absorb a hypothesis into the second member of a biconditional.
|
|
(Contributed by FL, 15-Feb-2007.) $)
|
|
bianabs $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( wa ibar bitr4d ) ABACECDACFG $.
|
|
$}
|
|
|
|
$( Absorption of disjunction into equivalence. (Contributed by NM,
|
|
6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) $)
|
|
oibabs $p |- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) <-> ( ph <-> ps ) ) $=
|
|
( wo wb wi wn wa ioran pm5.21 sylbi id ja ax-1 impbii ) ABCZABDZEPOPPOFAFBF
|
|
GPABHABIJPKLPOMN $.
|
|
|
|
$( Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who
|
|
call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.)
|
|
(Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
pm3.24 $p |- -. ( ph /\ -. ph ) $=
|
|
( wi wn wa id iman mpbi ) AABAACDCAEAAFG $.
|
|
|
|
$( Theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) $)
|
|
pm2.26 $p |- ( -. ph \/ ( ( ph -> ps ) -> ps ) ) $=
|
|
( wi pm2.27 imori ) AABCBCABDE $.
|
|
|
|
$( Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.11 $p |- ( ( ph -> ps ) \/ ( -. ph -> ps ) ) $=
|
|
( wi wn pm2.5 orri ) ABCADBCABEF $.
|
|
|
|
$( Theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.12 $p |- ( ( ph -> ps ) \/ ( ph -> -. ps ) ) $=
|
|
( wi wn pm2.51 orri ) ABCABDCABEF $.
|
|
|
|
$( Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.14 $p |- ( ( ph -> ps ) \/ ( ps -> ch ) ) $=
|
|
( wi wn ax-1 con3i pm2.21d orri ) ABDZBCDJEBCBJBAFGHI $.
|
|
|
|
$( Theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) $)
|
|
pm5.13 $p |- ( ( ph -> ps ) \/ ( ps -> ph ) ) $=
|
|
( pm5.14 ) ABAC $.
|
|
|
|
$( Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.) $)
|
|
pm5.17 $p |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) $=
|
|
( wn wb wi wa wo bicom dfbi2 orcom df-or bitr2i imnan anbi12i 3bitrri ) ABC
|
|
ZDPADPAEZAPEZFABGZABFCZFAPHPAIQSRTSBAGQABJBAKLABMNO $.
|
|
|
|
$( Theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 15-Oct-2013.) $)
|
|
pm5.15 $p |- ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) $=
|
|
( wb wn xor3 biimpi orri ) ABCZABDCZHDIABEFG $.
|
|
|
|
$( Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 17-Oct-2013.) $)
|
|
pm5.16 $p |- -. ( ( ph <-> ps ) /\ ( ph <-> -. ps ) ) $=
|
|
( wb wn wi wa pm5.18 biimpi imnan mpbi ) ABCZABDCZDZEKLFDKMABGHKLIJ $.
|
|
|
|
$( Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell]
|
|
p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf
|
|
Lammen, 22-Jan-2013.) $)
|
|
xor $p |- ( -. ( ph <-> ps ) <->
|
|
( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) $=
|
|
( wn wa wo wb wi iman anbi12i dfbi2 ioran 3bitr4ri con1bii ) ABCDZBACDZEZAB
|
|
FZABGZBAGZDNCZOCZDQPCRTSUAABHBAHIABJNOKLM $.
|
|
|
|
$( Two ways to express "exclusive or." (Contributed by NM, 3-Jan-2005.)
|
|
(Proof shortened by Wolf Lammen, 24-Jan-2013.) $)
|
|
nbi2 $p |- ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) $=
|
|
( wb wn wo wa xor3 pm5.17 bitr4i ) ABCDABDCABEABFDFABGABHI $.
|
|
|
|
$( An alternate definition of the biconditional. Theorem *5.23 of
|
|
[WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof
|
|
shortened by Wolf Lammen, 3-Nov-2013.) $)
|
|
dfbi3 $p |- ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) $=
|
|
( wn wb wa wo xor pm5.18 notnot anbi2i ancom orbi12i 3bitr4i ) ABCZDCANCZEZ
|
|
NACZEZFABDABEZQNEZFANGABHSPTRBOABIJQNKLM $.
|
|
|
|
$( Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.24 $p |- ( -. ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <->
|
|
( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) $=
|
|
( wb wn wa wo xor dfbi3 xchnxbi ) ABCABDZEBADZEFABEKJEFABGABHI $.
|
|
|
|
$( Conjunction distributes over exclusive-or, using ` -. ( ph <-> ps ) ` to
|
|
express exclusive-or. This is one way to interpret the distributive law
|
|
of multiplication over addition in modulo 2 arithmetic. (Contributed by
|
|
NM, 3-Oct-2008.) $)
|
|
xordi $p |- ( ( ph /\ -. ( ps <-> ch ) ) <->
|
|
-. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) $=
|
|
( wb wn wa wi annim pm5.32 xchbinx ) ABCDZEFAKGABFACFDAKHABCIJ $.
|
|
|
|
$( A wff disjoined with truth is true. (Contributed by NM, 23-May-1999.) $)
|
|
biort $p |- ( ph -> ( ph <-> ( ph \/ ps ) ) ) $=
|
|
( wo orc ax-1 impbid2 ) AAABCZABDAGEF $.
|
|
|
|
$( Theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 20-Jan-2013.) $)
|
|
pm5.55 $p |- ( ( ( ph \/ ps ) <-> ph ) \/ ( ( ph \/ ps ) <-> ps ) ) $=
|
|
( wo wb biort bicomd wn biorf nsyl4 con1i orri ) ABCZADZLBDZNMAMNAALABEFAGB
|
|
LABHFIJK $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Miscellaneous theorems of propositional calculus
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
pm5.21nd.1 $e |- ( ( ph /\ ps ) -> th ) $.
|
|
pm5.21nd.2 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
pm5.21nd.3 $e |- ( th -> ( ps <-> ch ) ) $.
|
|
$( Eliminate an antecedent implied by each side of a biconditional.
|
|
(Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen,
|
|
4-Nov-2013.) $)
|
|
pm5.21nd $p |- ( ph -> ( ps <-> ch ) ) $=
|
|
( ex wb wi a1i pm5.21ndd ) ADBCABDEHACDFHDBCIJAGKL $.
|
|
$}
|
|
|
|
$( Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.35 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) ->
|
|
( ph -> ( ps <-> ch ) ) ) $=
|
|
( wi wa pm5.1 pm5.74rd ) ABDZACDZEABCHIFG $.
|
|
|
|
$( Theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 7-Nov-2013.) $)
|
|
pm5.54 $p |- ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) ) $=
|
|
( wa wb iba bicomd adantl pm5.21ni orri ) ABCZADZJBDJKBBKABAJBAEFZGLHI $.
|
|
|
|
${
|
|
baib.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Move conjunction outside of biconditional. (Contributed by NM,
|
|
13-May-1999.) $)
|
|
baib $p |- ( ps -> ( ph <-> ch ) ) $=
|
|
( wa ibar syl6rbbr ) BCBCEABCFDG $.
|
|
|
|
$( Move conjunction outside of biconditional. (Contributed by NM,
|
|
11-Jul-1994.) $)
|
|
baibr $p |- ( ps -> ( ch <-> ph ) ) $=
|
|
( baib bicomd ) BACABCDEF $.
|
|
|
|
$( Move conjunction outside of biconditional. (Contributed by Mario
|
|
Carneiro, 11-Sep-2015.) $)
|
|
rbaib $p |- ( ch -> ( ph <-> ps ) ) $=
|
|
( wa ancom bitri baib ) ACBABCECBEDBCFGH $.
|
|
|
|
$( Move conjunction outside of biconditional. (Contributed by Mario
|
|
Carneiro, 11-Sep-2015.) $)
|
|
rbaibr $p |- ( ch -> ( ps <-> ph ) ) $=
|
|
( wa ancom bitri baibr ) ACBABCECBEDBCFGH $.
|
|
$}
|
|
|
|
${
|
|
baibd.1 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
|
|
$( Move conjunction outside of biconditional. (Contributed by Mario
|
|
Carneiro, 11-Sep-2015.) $)
|
|
baibd $p |- ( ( ph /\ ch ) -> ( ps <-> th ) ) $=
|
|
( wa ibar bicomd sylan9bb ) ABCDFZCDECDJCDGHI $.
|
|
|
|
$( Move conjunction outside of biconditional. (Contributed by Mario
|
|
Carneiro, 11-Sep-2015.) $)
|
|
rbaibd $p |- ( ( ph /\ th ) -> ( ps <-> ch ) ) $=
|
|
( wa iba bicomd sylan9bb ) ABCDFZDCEDCJDCGHI $.
|
|
$}
|
|
|
|
$( Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm5.44 $p |- ( ( ph -> ps ) -> ( ( ph -> ch ) <->
|
|
( ph -> ( ps /\ ch ) ) ) ) $=
|
|
( wa wi jcab baibr ) ABCDEABEACEABCFG $.
|
|
|
|
$( Conjunction in antecedent versus disjunction in consequent. Theorem *5.6
|
|
of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.) $)
|
|
pm5.6 $p |- ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( ps \/ ch ) ) ) $=
|
|
( wn wa wi wo impexp df-or imbi2i bitr4i ) ABDZECFALCFZFABCGZFALCHNMABCIJK
|
|
$.
|
|
|
|
${
|
|
orcanai.1 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
$( Change disjunction in consequent to conjunction in antecedent.
|
|
(Contributed by NM, 8-Jun-1994.) $)
|
|
orcanai $p |- ( ( ph /\ -. ps ) -> ch ) $=
|
|
( wn ord imp ) ABECABCDFG $.
|
|
$}
|
|
|
|
${
|
|
intnan.1 $e |- -. ph $.
|
|
$( Introduction of conjunct inside of a contradiction. (Contributed by NM,
|
|
16-Sep-1993.) $)
|
|
intnan $p |- -. ( ps /\ ph ) $=
|
|
( wa simpr mto ) BADACBAEF $.
|
|
|
|
$( Introduction of conjunct inside of a contradiction. (Contributed by NM,
|
|
3-Apr-1995.) $)
|
|
intnanr $p |- -. ( ph /\ ps ) $=
|
|
( wa simpl mto ) ABDACABEF $.
|
|
$}
|
|
|
|
${
|
|
intnand.1 $e |- ( ph -> -. ps ) $.
|
|
$( Introduction of conjunct inside of a contradiction. (Contributed by NM,
|
|
10-Jul-2005.) $)
|
|
intnand $p |- ( ph -> -. ( ch /\ ps ) ) $=
|
|
( wa simpr nsyl ) ABCBEDCBFG $.
|
|
|
|
$( Introduction of conjunct inside of a contradiction. (Contributed by NM,
|
|
10-Jul-2005.) $)
|
|
intnanrd $p |- ( ph -> -. ( ps /\ ch ) ) $=
|
|
( wa simpl nsyl ) ABBCEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
mpbiran.1 $e |- ps $.
|
|
mpbiran.2 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Detach truth from conjunction in biconditional. (Contributed by NM,
|
|
27-Feb-1996.) $)
|
|
mpbiran $p |- ( ph <-> ch ) $=
|
|
( wa biantrur bitr4i ) ABCFCEBCDGH $.
|
|
$}
|
|
|
|
${
|
|
mpbiran2.1 $e |- ch $.
|
|
mpbiran2.2 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Detach truth from conjunction in biconditional. (Contributed by NM,
|
|
22-Feb-1996.) $)
|
|
mpbiran2 $p |- ( ph <-> ps ) $=
|
|
( wa biantru bitr4i ) ABCFBECBDGH $.
|
|
$}
|
|
|
|
${
|
|
mpbir2an.1 $e |- ps $.
|
|
mpbir2an.2 $e |- ch $.
|
|
mpbiran2an.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by NM,
|
|
10-May-2005.) $)
|
|
mpbir2an $p |- ph $=
|
|
( mpbiran mpbir ) ACEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mpbi2and.1 $e |- ( ph -> ps ) $.
|
|
mpbi2and.2 $e |- ( ph -> ch ) $.
|
|
mpbi2and.3 $e |- ( ph -> ( ( ps /\ ch ) <-> th ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by NM,
|
|
6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
mpbi2and $p |- ( ph -> th ) $=
|
|
( wa jca mpbid ) ABCHDABCEFIGJ $.
|
|
$}
|
|
|
|
${
|
|
mpbir2and.1 $e |- ( ph -> ch ) $.
|
|
mpbir2and.2 $e |- ( ph -> th ) $.
|
|
mpbir2and.3 $e |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by NM,
|
|
6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
|
|
mpbir2and $p |- ( ph -> ps ) $=
|
|
( wa jca mpbird ) ABCDHACDEFIGJ $.
|
|
$}
|
|
|
|
$( Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F.
|
|
Longton, 21-Jun-2005.) $)
|
|
pm5.62 $p |- ( ( ( ph /\ ps ) \/ -. ps ) <-> ( ph \/ -. ps ) ) $=
|
|
( wa wn wo exmid ordir mpbiran2 ) ABCBDZEAIEBIEBFABIGH $.
|
|
|
|
$( Theorem *5.63 of [WhiteheadRussell] p. 125. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 25-Dec-2012.) $)
|
|
pm5.63 $p |- ( ( ph \/ ps ) <-> ( ph \/ ( -. ph /\ ps ) ) ) $=
|
|
( wn wa wo exmid ordi mpbiran bicomi ) AACZBDEZABEZKAJELAFAJBGHI $.
|
|
|
|
${
|
|
bianfi.1 $e |- -. ph $.
|
|
$( A wff conjoined with falsehood is false. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) $)
|
|
bianfi $p |- ( ph <-> ( ps /\ ph ) ) $=
|
|
( wa intnan 2false ) ABADCABCEF $.
|
|
$}
|
|
|
|
${
|
|
bianfd.1 $e |- ( ph -> -. ps ) $.
|
|
$( A wff conjoined with falsehood is false. (Contributed by NM,
|
|
27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.) $)
|
|
bianfd $p |- ( ph -> ( ps <-> ( ps /\ ch ) ) ) $=
|
|
( wa intnanrd 2falsed ) ABBCEDABCDFG $.
|
|
$}
|
|
|
|
$( Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) $)
|
|
pm4.43 $p |- ( ph <-> ( ( ph \/ ps ) /\ ( ph \/ -. ps ) ) ) $=
|
|
( wn wa wo pm3.24 biorfi ordi bitri ) AABBCZDZEABEAJEDKABFGABJHI $.
|
|
|
|
$( Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.82 $p |- ( ( ( ph -> ps ) /\ ( ph -> -. ps ) ) <-> -. ph ) $=
|
|
( wi wn wa pm2.65 imp pm2.21 jca impbii ) ABCZABDZCZEADZKMNABFGNKMABHALHIJ
|
|
$.
|
|
|
|
$( Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM,
|
|
3-Jan-2005.) $)
|
|
pm4.83 $p |- ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) <-> ps ) $=
|
|
( wn wo wi wa exmid a1bi jaob bitr2i ) BAACZDZBEABEKBEFLBAGHABKIJ $.
|
|
|
|
$( Negation inferred from embedded conjunct. (Contributed by NM,
|
|
20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.) $)
|
|
pclem6 $p |- ( ( ph <-> ( ps /\ -. ph ) ) -> -. ps ) $=
|
|
( wn wa wb ibar nbbn sylib con2i ) BABACZDZEZBJKELCBJFAKGHI $.
|
|
|
|
$( A transitive law of equivalence. Compare Theorem *4.22 of
|
|
[WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.) $)
|
|
biantr $p |- ( ( ( ph <-> ps ) /\ ( ch <-> ps ) ) -> ( ph <-> ch ) ) $=
|
|
( wb id bibi2d biimparc ) CBDZACDABDHCBAHEFG $.
|
|
|
|
$( Disjunction distributes over the biconditional. An axiom of system DS in
|
|
Vladimir Lifschitz, "On calculational proofs" (1998),
|
|
~ http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384 .
|
|
(Contributed by NM, 8-Jan-2005.) (Proof shortened by Wolf Lammen,
|
|
4-Feb-2013.) $)
|
|
orbidi $p |- ( ( ph \/ ( ps <-> ch ) ) <->
|
|
( ( ph \/ ps ) <-> ( ph \/ ch ) ) ) $=
|
|
( wn wb wi wo pm5.74 df-or bibi12i 3bitr4i ) ADZBCEZFLBFZLCFZEAMGABGZACGZEL
|
|
BCHAMIPNQOABIACIJK $.
|
|
|
|
$( Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall,
|
|
ed., _Polish Logic 1920-1939_ (Oxford, 1967), p. 96. (Contributed by NM,
|
|
10-Jan-2005.) $)
|
|
biluk $p |- ( ( ph <-> ps ) <-> ( ( ch <-> ps ) <-> ( ph <-> ch ) ) ) $=
|
|
( wb bicom bibi1i biass bitri mpbi bitr4i ) ABDZCBACDZDZDZCBDLDKCDZMDKNDOBA
|
|
DZCDMKPCABEFBACGHKCMGICBLGJ $.
|
|
|
|
$( Disjunction distributes over the biconditional. Theorem *5.7 of
|
|
[WhiteheadRussell] p. 125. This theorem is similar to ~ orbidi .
|
|
(Contributed by Roy F. Longton, 21-Jun-2005.) $)
|
|
pm5.7 $p |- ( ( ( ph \/ ch ) <-> ( ps \/ ch ) ) <->
|
|
( ch \/ ( ph <-> ps ) ) ) $=
|
|
( wb wo orbidi orcom bibi12i bitr2i ) CABDECAEZCBEZDACEZBCEZDCABFJLKMCAGCBG
|
|
HI $.
|
|
|
|
$( Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by
|
|
NM, 10-Jan-2005.) $)
|
|
bigolden $p |- ( ( ( ph /\ ps ) <-> ph ) <-> ( ps <-> ( ph \/ ps ) ) ) $=
|
|
( wi wa wb wo pm4.71 pm4.72 bicom 3bitr3ri ) ABCAABDZEBABFEKAEABGABHAKIJ $.
|
|
|
|
$( Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F.
|
|
Longton, 23-Jun-2005.) $)
|
|
pm5.71 $p |- ( ( ps -> -. ch ) -> ( ( ( ph \/ ps ) /\ ch ) <->
|
|
( ph /\ ch ) ) ) $=
|
|
( wn wo wa wb orel2 orc impbid1 anbi1d pm2.21 pm5.32rd ja ) BCDZABEZCFACFGB
|
|
DZPACQPABAHABIJKOCPACPAGLMN $.
|
|
|
|
$( Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM,
|
|
3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof
|
|
shortened by Wolf Lammen, 23-Dec-2012.) $)
|
|
pm5.75 $p |- ( ( ( ch -> -. ps ) /\ ( ph <-> ( ps \/ ch ) ) ) ->
|
|
( ( ph /\ -. ps ) <-> ch ) ) $=
|
|
( wo wb wn wa wi anbi1 anbi1i pm5.61 syl6bb pm4.71 biimpi bicomd sylan9bbr
|
|
orcom bitri ) ABCDZEZABFZGZCUAGZCUAHZCTUBSUAGZUCASUAIUECBDZUAGUCSUFUABCQJCB
|
|
KRLUDCUCUDCUCECUAMNOP $.
|
|
|
|
$( Removal of conjunct from one side of an equivalence. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
bimsc1 $p |- ( ( ( ph -> ps ) /\ ( ch <-> ( ps /\ ph ) ) )
|
|
-> ( ch <-> ph ) ) $=
|
|
( wi wa wb simpr ancr impbid2 bibi2d biimpa ) ABDZCBAEZFCAFLMACLMABAGABHIJK
|
|
$.
|
|
|
|
$( The disjunction of the four possible combinations of two wffs and their
|
|
negations is always true. (Contributed by David Abernethy,
|
|
28-Jan-2014.) $)
|
|
4exmid $p |- ( ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) )
|
|
\/ ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) $=
|
|
( wb wn wo wa exmid dfbi3 xor orbi12i mpbi ) ABCZLDZEABFADZBDZFEZAOFBNFEZEL
|
|
GLPMQABHABIJK $.
|
|
|
|
${
|
|
ecase2d.1 $e |- ( ph -> ps ) $.
|
|
ecase2d.2 $e |- ( ph -> -. ( ps /\ ch ) ) $.
|
|
ecase2d.3 $e |- ( ph -> -. ( ps /\ th ) ) $.
|
|
ecase2d.4 $e |- ( ph -> ( ta \/ ( ch \/ th ) ) ) $.
|
|
$( Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.)
|
|
(Proof shortened by Wolf Lammen, 22-Dec-2012.) $)
|
|
ecase2d $p |- ( ph -> ta ) $=
|
|
( wo idd wa pm2.21d mpand jaod mpjaod ) AEECDJAEKACEDABCEFABCLEGMNABDEFAB
|
|
DLEHMNOIP $.
|
|
$}
|
|
|
|
${
|
|
ecase3.1 $e |- ( ph -> ch ) $.
|
|
ecase3.2 $e |- ( ps -> ch ) $.
|
|
ecase3.3 $e |- ( -. ( ph \/ ps ) -> ch ) $.
|
|
$( Inference for elimination by cases. (Contributed by NM, 23-Mar-1995.)
|
|
(Proof shortened by Wolf Lammen, 26-Nov-2012.) $)
|
|
ecase3 $p |- ch $=
|
|
( wo jaoi pm2.61i ) ABGCACBDEHFI $.
|
|
$}
|
|
|
|
${
|
|
ecase.1 $e |- ( -. ph -> ch ) $.
|
|
ecase.2 $e |- ( -. ps -> ch ) $.
|
|
ecase.3 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Inference for elimination by cases. (Contributed by NM,
|
|
13-Jul-2005.) $)
|
|
ecase $p |- ch $=
|
|
( ex pm2.61nii ) ABCABCFGDEH $.
|
|
$}
|
|
|
|
${
|
|
ecase3d.1 $e |- ( ph -> ( ps -> th ) ) $.
|
|
ecase3d.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
ecase3d.3 $e |- ( ph -> ( -. ( ps \/ ch ) -> th ) ) $.
|
|
$( Deduction for elimination by cases. (Contributed by NM, 2-May-1996.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
ecase3d $p |- ( ph -> th ) $=
|
|
( wo jaod pm2.61d ) ABCHDABDCEFIGJ $.
|
|
$}
|
|
|
|
${
|
|
ecased.1 $e |- ( ph -> ( -. ps -> th ) ) $.
|
|
ecased.2 $e |- ( ph -> ( -. ch -> th ) ) $.
|
|
ecased.3 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.) $)
|
|
ecased $p |- ( ph -> th ) $=
|
|
( wn wo wa pm3.11 syl5 ecase3d ) ABHZCHZDEFNOIHBCJADBCKGLM $.
|
|
$}
|
|
|
|
${
|
|
ecase3ad.1 $e |- ( ph -> ( ps -> th ) ) $.
|
|
ecase3ad.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
ecase3ad.3 $e |- ( ph -> ( ( -. ps /\ -. ch ) -> th ) ) $.
|
|
$( Deduction for elimination by cases. (Contributed by NM,
|
|
24-May-2013.) $)
|
|
ecase3ad $p |- ( ph -> th ) $=
|
|
( wn notnot2 syl5 ecased ) ABHZCHZDLHBADBIEJMHCADCIFJGK $.
|
|
$}
|
|
|
|
${
|
|
ccase.1 $e |- ( ( ph /\ ps ) -> ta ) $.
|
|
ccase.2 $e |- ( ( ch /\ ps ) -> ta ) $.
|
|
ccase.3 $e |- ( ( ph /\ th ) -> ta ) $.
|
|
ccase.4 $e |- ( ( ch /\ th ) -> ta ) $.
|
|
$( Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
|
|
(Proof shortened by Wolf Lammen, 6-Jan-2013.) $)
|
|
ccase $p |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta ) $=
|
|
( wo jaoian jaodan ) ACJBEDABECFGKADECHIKL $.
|
|
$}
|
|
|
|
${
|
|
ccased.1 $e |- ( ph -> ( ( ps /\ ch ) -> et ) ) $.
|
|
ccased.2 $e |- ( ph -> ( ( th /\ ch ) -> et ) ) $.
|
|
ccased.3 $e |- ( ph -> ( ( ps /\ ta ) -> et ) ) $.
|
|
ccased.4 $e |- ( ph -> ( ( th /\ ta ) -> et ) ) $.
|
|
$( Deduction for combining cases. (Contributed by NM, 9-May-2004.) $)
|
|
ccased $p |- ( ph -> ( ( ( ps \/ th ) /\ ( ch \/ ta ) ) -> et ) ) $=
|
|
( wo wa wi com12 ccase ) BDKCEKLAFBCDEAFMABCLFGNADCLFHNABELFINADELFJNON
|
|
$.
|
|
$}
|
|
|
|
${
|
|
ccase2.1 $e |- ( ( ph /\ ps ) -> ta ) $.
|
|
ccase2.2 $e |- ( ch -> ta ) $.
|
|
ccase2.3 $e |- ( th -> ta ) $.
|
|
$( Inference for combining cases. (Contributed by NM, 29-Jul-1999.) $)
|
|
ccase2 $p |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta ) $=
|
|
( adantr adantl ccase ) ABCDEFCEBGIDEAHJDECHJK $.
|
|
$}
|
|
|
|
${
|
|
4cases.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
4cases.2 $e |- ( ( ph /\ -. ps ) -> ch ) $.
|
|
4cases.3 $e |- ( ( -. ph /\ ps ) -> ch ) $.
|
|
4cases.4 $e |- ( ( -. ph /\ -. ps ) -> ch ) $.
|
|
$( Inference eliminating two antecedents from the four possible cases that
|
|
result from their true/false combinations. (Contributed by NM,
|
|
25-Oct-2003.) $)
|
|
4cases $p |- ch $=
|
|
( pm2.61ian wn pm2.61i ) BCABCDFHABICEGHJ $.
|
|
$}
|
|
|
|
${
|
|
4casesdan.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
4casesdan.2 $e |- ( ( ph /\ ( ps /\ -. ch ) ) -> th ) $.
|
|
4casesdan.3 $e |- ( ( ph /\ ( -. ps /\ ch ) ) -> th ) $.
|
|
4casesdan.4 $e |- ( ( ph /\ ( -. ps /\ -. ch ) ) -> th ) $.
|
|
$( Deduction eliminating two antecedents from the four possible cases that
|
|
result from their true/false combinations. (Contributed by NM,
|
|
19-Mar-2013.) $)
|
|
4casesdan $p |- ( ph -> th ) $=
|
|
( wi wa expcom wn 4cases ) BCADIABCJDEKABCLZJDFKABLZCJDGKAONJDHKM $.
|
|
$}
|
|
|
|
${
|
|
niabn.1 $e |- ph $.
|
|
$( Miscellaneous inference relating falsehoods. (Contributed by NM,
|
|
31-Mar-1994.) $)
|
|
niabn $p |- ( -. ps -> ( ( ch /\ ps ) <-> -. ph ) ) $=
|
|
( wa wn simpr pm2.24i pm5.21ni ) CBEBAFCBGABDHI $.
|
|
$}
|
|
|
|
$( Lemma for an alternate version of weak deduction theorem. (Contributed by
|
|
NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof
|
|
shortened by Wolf Lammen, 4-Dec-2012.) $)
|
|
dedlem0a $p |- ( ph -> ( ps <-> ( ( ch -> ph ) -> ( ps /\ ph ) ) ) ) $=
|
|
( wa wi iba wb ax-1 biimt syl bitrd ) ABBADZCAEZLEZABFAMLNGACHMLIJK $.
|
|
|
|
$( Lemma for an alternate version of weak deduction theorem. (Contributed by
|
|
NM, 2-Apr-1994.) $)
|
|
dedlem0b $p |- ( -. ph -> ( ps <-> ( ( ps -> ph ) -> ( ch /\ ph ) ) ) ) $=
|
|
( wn wi wa pm2.21 imim2d com23 simpr imim12i con1d com12 impbid ) ADZBBAEZC
|
|
AFZEZOPBQOAQBAQGHIROBRBABDPQABAGCAJKLMN $.
|
|
|
|
$( Lemma for weak deduction theorem. (Contributed by NM, 26-Jun-2002.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
dedlema $p |- ( ph -> ( ps <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) ) $=
|
|
( wa wn wo orc expcom wi simpl a1i pm2.24 adantld jaod impbid ) ABBADZCAEZD
|
|
ZFZBASPRGHAPBRPBIABAJKAQBCABLMNO $.
|
|
|
|
$( Lemma for weak deduction theorem. (Contributed by NM, 15-May-1999.)
|
|
(Proof shortened by Andrew Salmon, 7-May-2011.) $)
|
|
dedlemb $p |- ( -. ph -> ( ch <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) ) $=
|
|
( wn wa wo olc expcom pm2.21 adantld wi simpl a1i jaod impbid ) ADZCBAEZCPE
|
|
ZFZCPSRQGHPQCRPACBACIJRCKPCPLMNO $.
|
|
|
|
${
|
|
elimh.1 $e |- ( ( ph <-> ( ( ph /\ ch ) \/ ( ps /\ -. ch ) ) )
|
|
-> ( ch <-> ta ) ) $.
|
|
elimh.2 $e |- ( ( ps <-> ( ( ph /\ ch ) \/ ( ps /\ -. ch ) ) )
|
|
-> ( th <-> ta ) ) $.
|
|
elimh.3 $e |- th $.
|
|
$( Hypothesis builder for weak deduction theorem. For more information,
|
|
see the Deduction Theorem link on the Metamath Proof Explorer home
|
|
page. (Contributed by NM, 26-Jun-2002.) $)
|
|
elimh $p |- ta $=
|
|
( wa wn wo wb dedlema syl ibi dedlemb mpbii pm2.61i ) CECECAACIBCJZIKZLCE
|
|
LCABMFNOSDEHSBTLDELCABPGNQR $.
|
|
$}
|
|
|
|
${
|
|
dedt.1 $e |- ( ( ph <-> ( ( ph /\ ch ) \/ ( ps /\ -. ch ) ) )
|
|
-> ( th <-> ta ) ) $.
|
|
dedt.2 $e |- ta $.
|
|
$( The weak deduction theorem. For more information, see the Deduction
|
|
Theorem link on the Metamath Proof Explorer home page. (Contributed by
|
|
NM, 26-Jun-2002.) $)
|
|
dedt $p |- ( ch -> th ) $=
|
|
( wa wn wo wb dedlema mpbiri syl ) CAACHBCIHJKZDCABLODEGFMN $.
|
|
$}
|
|
|
|
$( Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version
|
|
of ~ con3 demonstrates the use of the weak deduction theorem ~ dedt to
|
|
derive it from ~ con3i . (Contributed by NM, 27-Jun-2002.)
|
|
(Proof modification is discouraged.) $)
|
|
con3th $p |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) ) $=
|
|
( wi wn wa wo wb id notbid imbi1d imbi2d elimh con3i dedt ) BAABCZBDZADZCBO
|
|
EAODEFZDZQCBRGZPSQTBRTHZIJARBAOAACARCTBRAUAKARGZARAUBHKAHLMN $.
|
|
|
|
$( The consensus theorem. This theorem and its dual (with ` \/ ` and ` /\ `
|
|
interchanged) are commonly used in computer logic design to eliminate
|
|
redundant terms from Boolean expressions. Specifically, we prove that the
|
|
term ` ( ps /\ ch ) ` on the left-hand side is redundant. (Contributed by
|
|
NM, 16-May-2003.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|
|
(Proof shortened by Wolf Lammen, 20-Jan-2013.) $)
|
|
consensus $p |- ( ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) \/ ( ps /\ ch ) ) <->
|
|
( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) ) $=
|
|
( wa wn wo id orc adantrr olc adantrl pm2.61ian jaoi impbii ) ABDZAEZCDZFZB
|
|
CDZFRRRSRGASRABRCOQHIPCRBQOJKLMRSHN $.
|
|
|
|
$( Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Roy F.
|
|
Longton, 21-Jun-2005.) $)
|
|
pm4.42 $p |- ( ph <-> ( ( ph /\ ps ) \/ ( ph /\ -. ps ) ) ) $=
|
|
( wa wn wo wb dedlema dedlemb pm2.61i ) BAABCABDCEFBAAGBAAHI $.
|
|
|
|
${
|
|
ninba.1 $e |- ph $.
|
|
$( Miscellaneous inference relating falsehoods. (Contributed by NM,
|
|
31-Mar-1994.) $)
|
|
ninba $p |- ( -. ps -> ( -. ph <-> ( ch /\ ps ) ) ) $=
|
|
( wn wa niabn bicomd ) BECBFAEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
prlem1.1 $e |- ( ph -> ( et <-> ch ) ) $.
|
|
prlem1.2 $e |- ( ps -> -. th ) $.
|
|
$( A specialized lemma for set theory (to derive the Axiom of Pairing).
|
|
(Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon,
|
|
13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) $)
|
|
prlem1 $p |- ( ph -> ( ps ->
|
|
( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) ) ) $=
|
|
( wa wo wi biimprd adantld pm2.21d adantrd jaao ex ) ABBCIZDEIZJFKARFBSAC
|
|
FBAFCGLMBDFEBDFHNOPQ $.
|
|
$}
|
|
|
|
$( A specialized lemma for set theory (to derive the Axiom of Pairing).
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) $)
|
|
prlem2 $p |- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <->
|
|
( ( ph \/ ch ) /\ ( ( ph /\ ps ) \/ ( ch /\ th ) ) ) ) $=
|
|
( wa wo simpl orim12i pm4.71ri ) ABEZCDEZFACFJAKCABGCDGHI $.
|
|
|
|
${
|
|
oplem1.1 $e |- ( ph -> ( ps \/ ch ) ) $.
|
|
oplem1.2 $e |- ( ph -> ( th \/ ta ) ) $.
|
|
oplem1.3 $e |- ( ps <-> th ) $.
|
|
oplem1.4 $e |- ( ch -> ( th <-> ta ) ) $.
|
|
$( A specialized lemma for set theory (ordered pair theorem). (Contributed
|
|
by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) $)
|
|
oplem1 $p |- ( ph -> ps ) $=
|
|
( wn wa notbii ord syl5bir jcad biimpar syl6 pm2.18d sylibr ) ADBADADJZCE
|
|
KDATCETBJACBDHLABCFMNADEGMOCDEIPQRHS $.
|
|
$}
|
|
|
|
$( Lemma used in construction of real numbers. (Contributed by NM,
|
|
4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
rnlem $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
|
|
( ( ( ph /\ ch ) /\ ( ps /\ th ) ) /\ ( ( ph /\ th ) /\ ( ps /\ ch ) ) ) ) $=
|
|
( wa an4 biimpi an42 biimpri jca adantl impbii ) ABECDEEZACEBDEEZADEBCEEZEM
|
|
NOMNABCDFGOMADBCHZIJOMNOMPGKL $.
|
|
|
|
$( A single axiom for Boolean algebra known as DN_1. See
|
|
~ http://www-unix.mcs.anl.gov/~~mccune/papers/basax/v12.pdf .
|
|
(Contributed by Jeffrey Hankins, 3-Jul-2009.) (Proof shortened by Andrew
|
|
Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.) $)
|
|
dn1 $p |- ( -. ( -. ( -. ( ph \/ ps ) \/ ch ) \/
|
|
-. ( ph \/ -. ( -. ch \/ -. ( ch \/ th ) ) ) ) <-> ch ) $=
|
|
( wo wn wa wi pm2.45 imnan mpbi biorfi orcom ordir bitri pm4.45 anor orbi2i
|
|
anbi2i 3bitrri ) CABEFZCEZACEZGZUBACFCDEZFEFZEZGUBFUGFEFCCUAAGZEZUDUHCUAAFH
|
|
UHFABIUAAJKLUIUHCEUDCUHMUAACNOOUCUGUBCUFACCUEGUFCDPCUEQORSUBUGQT $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Abbreviated conjunction and disjunction of three wff's
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend wff definition to include 3-way disjunction ('or'). $)
|
|
w3o $a wff ( ph \/ ps \/ ch ) $.
|
|
$( Extend wff definition to include 3-way conjunction ('and'). $)
|
|
w3a $a wff ( ph /\ ps /\ ch ) $.
|
|
|
|
$( These abbreviations help eliminate parentheses to aid readability. $)
|
|
|
|
$( Define disjunction ('or') of three wff's. Definition *2.33 of
|
|
[WhiteheadRussell] p. 105. This abbreviation reduces the number of
|
|
parentheses and emphasizes that the order of bracketing is not important
|
|
by virtue of the associative law ~ orass . (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
df-3or $a |- ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) ) $.
|
|
|
|
$( Define conjunction ('and') of three wff's. Definition *4.34 of
|
|
[WhiteheadRussell] p. 118. This abbreviation reduces the number of
|
|
parentheses and emphasizes that the order of bracketing is not important
|
|
by virtue of the associative law ~ anass . (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
df-3an $a |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) ) $.
|
|
|
|
$( Associative law for triple disjunction. (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
3orass $p |- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) ) $=
|
|
( w3o wo df-3or orass bitri ) ABCDABECEABCEEABCFABCGH $.
|
|
|
|
$( Associative law for triple conjunction. (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
3anass $p |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) ) $=
|
|
( w3a wa df-3an anass bitri ) ABCDABECEABCEEABCFABCGH $.
|
|
|
|
$( Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.) $)
|
|
3anrot $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ch /\ ph ) ) $=
|
|
( wa w3a ancom 3anass df-3an 3bitr4i ) ABCDZDJADABCEBCAEAJFABCGBCAHI $.
|
|
|
|
$( Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.) $)
|
|
3orrot $p |- ( ( ph \/ ps \/ ch ) <-> ( ps \/ ch \/ ph ) ) $=
|
|
( wo w3o orcom 3orass df-3or 3bitr4i ) ABCDZDJADABCEBCAEAJFABCGBCAHI $.
|
|
|
|
$( Commutation law for triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3ancoma $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ph /\ ch ) ) $=
|
|
( wa w3a ancom anbi1i df-3an 3bitr4i ) ABDZCDBADZCDABCEBACEJKCABFGABCHBACHI
|
|
$.
|
|
|
|
$( Commutation law for triple disjunction. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
3orcoma $p |- ( ( ph \/ ps \/ ch ) <-> ( ps \/ ph \/ ch ) ) $=
|
|
( wo w3o or12 3orass 3bitr4i ) ABCDDBACDDABCEBACEABCFABCGBACGH $.
|
|
|
|
$( Commutation law for triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3ancomb $p |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ch /\ ps ) ) $=
|
|
( w3a 3ancoma 3anrot bitri ) ABCDBACDACBDABCEBACFG $.
|
|
|
|
$( Commutation law for triple disjunction. (Contributed by Scott Fenton,
|
|
20-Apr-2011.) $)
|
|
3orcomb $p |- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ch \/ ps ) ) $=
|
|
( wo w3o orcom orbi2i 3orass 3bitr4i ) ABCDZDACBDZDABCEACBEJKABCFGABCHACBHI
|
|
$.
|
|
|
|
$( Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.) $)
|
|
3anrev $p |- ( ( ph /\ ps /\ ch ) <-> ( ch /\ ps /\ ph ) ) $=
|
|
( w3a 3ancoma 3anrot bitr4i ) ABCDBACDCBADABCECBAFG $.
|
|
|
|
$( Convert triple conjunction to conjunction, then commute. (Contributed by
|
|
Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
3anan32 $p |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ch ) /\ ps ) ) $=
|
|
( w3a wa df-3an an32 bitri ) ABCDABECEACEBEABCFABCGH $.
|
|
|
|
$( Convert triple conjunction to conjunction, then commute. (Contributed by
|
|
Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon,
|
|
14-Jun-2011.) $)
|
|
3anan12 $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
|
|
( w3a wa 3ancoma 3anass bitri ) ABCDBACDBACEEABCFBACGH $.
|
|
|
|
$( Triple conjunction expressed in terms of triple disjunction. (Contributed
|
|
by Jeff Hankins, 15-Aug-2009.) $)
|
|
3anor $p |- ( ( ph /\ ps /\ ch ) <-> -. ( -. ph \/ -. ps \/ -. ch ) ) $=
|
|
( w3a wa wn w3o df-3an wo anor ianor orbi1i xchbinx df-3or xchbinxr bitri )
|
|
ABCDABEZCEZAFZBFZCFZGZFABCHRSTIZUAIZUBRQFZUAIUDQCJUEUCUAABKLMSTUANOP $.
|
|
|
|
$( Negated triple conjunction expressed in terms of triple disjunction.
|
|
(Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew
|
|
Salmon, 13-May-2011.) $)
|
|
3ianor $p |- ( -. ( ph /\ ps /\ ch ) <-> ( -. ph \/ -. ps \/ -. ch ) ) $=
|
|
( wn w3o w3a 3anor con2bii bicomi ) ADBDCDEZABCFZDKJABCGHI $.
|
|
|
|
$( Negated triple disjunction as triple conjunction. (Contributed by Scott
|
|
Fenton, 19-Apr-2011.) $)
|
|
3ioran $p |- ( -. ( ph \/ ps \/ ch ) <-> ( -. ph /\ -. ps /\ -. ch ) ) $=
|
|
( wo wn wa w3o w3a ioran anbi1i df-3or xchnxbir df-3an 3bitr4i ) ABDZEZCEZF
|
|
ZAEZBEZFZQFABCGZESTQHPUAQABIJOCDRUBOCIABCKLSTQMN $.
|
|
|
|
$( Triple disjunction in terms of triple conjunction. (Contributed by NM,
|
|
8-Oct-2012.) $)
|
|
3oran $p |- ( ( ph \/ ps \/ ch ) <-> -. ( -. ph /\ -. ps /\ -. ch ) ) $=
|
|
( wn w3a w3o 3ioran con1bii bicomi ) ADBDCDEZDABCFZKJABCGHI $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3simpa $p |- ( ( ph /\ ps /\ ch ) -> ( ph /\ ps ) ) $=
|
|
( w3a wa df-3an simplbi ) ABCDABECABCFG $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3simpb $p |- ( ( ph /\ ps /\ ch ) -> ( ph /\ ch ) ) $=
|
|
( w3a wa 3ancomb 3simpa sylbi ) ABCDACBDACEABCFACBGH $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
|
|
(Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
3simpc $p |- ( ( ph /\ ps /\ ch ) -> ( ps /\ ch ) ) $=
|
|
( w3a wa 3anrot 3simpa sylbi ) ABCDBCADBCEABCFBCAGH $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
simp1 $p |- ( ( ph /\ ps /\ ch ) -> ph ) $=
|
|
( w3a 3simpa simpld ) ABCDABABCEF $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
simp2 $p |- ( ( ph /\ ps /\ ch ) -> ps ) $=
|
|
( w3a 3simpa simprd ) ABCDABABCEF $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
simp3 $p |- ( ( ph /\ ps /\ ch ) -> ch ) $=
|
|
( w3a 3simpc simprd ) ABCDBCABCEF $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpl1 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ph ) $=
|
|
( w3a simp1 adantr ) ABCEADABCFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpl2 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ps ) $=
|
|
( w3a simp2 adantr ) ABCEBDABCFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpl3 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ch ) $=
|
|
( w3a simp3 adantr ) ABCECDABCFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpr1 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ps ) $=
|
|
( w3a simp1 adantl ) BCDEBABCDFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpr2 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ch ) $=
|
|
( w3a simp2 adantl ) BCDECABCDFG $.
|
|
|
|
$( Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.) $)
|
|
simpr3 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> th ) $=
|
|
( w3a simp3 adantl ) BCDEDABCDFG $.
|
|
|
|
${
|
|
3simp1i.1 $e |- ( ph /\ ps /\ ch ) $.
|
|
$( Infer a conjunct from a triple conjunction. (Contributed by NM,
|
|
19-Apr-2005.) $)
|
|
simp1i $p |- ph $=
|
|
( w3a simp1 ax-mp ) ABCEADABCFG $.
|
|
|
|
$( Infer a conjunct from a triple conjunction. (Contributed by NM,
|
|
19-Apr-2005.) $)
|
|
simp2i $p |- ps $=
|
|
( w3a simp2 ax-mp ) ABCEBDABCFG $.
|
|
|
|
$( Infer a conjunct from a triple conjunction. (Contributed by NM,
|
|
19-Apr-2005.) $)
|
|
simp3i $p |- ch $=
|
|
( w3a simp3 ax-mp ) ABCECDABCFG $.
|
|
$}
|
|
|
|
${
|
|
3simp1d.1 $e |- ( ph -> ( ps /\ ch /\ th ) ) $.
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by NM,
|
|
4-Sep-2005.) $)
|
|
simp1d $p |- ( ph -> ps ) $=
|
|
( w3a simp1 syl ) ABCDFBEBCDGH $.
|
|
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by NM,
|
|
4-Sep-2005.) $)
|
|
simp2d $p |- ( ph -> ch ) $=
|
|
( w3a simp2 syl ) ABCDFCEBCDGH $.
|
|
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by NM,
|
|
4-Sep-2005.) $)
|
|
simp3d $p |- ( ph -> th ) $=
|
|
( w3a simp3 syl ) ABCDFDEBCDGH $.
|
|
$}
|
|
|
|
${
|
|
3simp1bi.1 $e |- ( ph <-> ( ps /\ ch /\ th ) ) $.
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
simp1bi $p |- ( ph -> ps ) $=
|
|
( w3a biimpi simp1d ) ABCDABCDFEGH $.
|
|
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
simp2bi $p |- ( ph -> ch ) $=
|
|
( w3a biimpi simp2d ) ABCDABCDFEGH $.
|
|
|
|
$( Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
simp3bi $p |- ( ph -> th ) $=
|
|
( w3a biimpi simp3d ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
3adant.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
16-Jul-1995.) $)
|
|
3adant1 $p |- ( ( th /\ ph /\ ps ) -> ch ) $=
|
|
( w3a wa 3simpc syl ) DABFABGCDABHEI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
16-Jul-1995.) $)
|
|
3adant2 $p |- ( ( ph /\ th /\ ps ) -> ch ) $=
|
|
( w3a wa 3simpb syl ) ADBFABGCADBHEI $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
16-Jul-1995.) $)
|
|
3adant3 $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
|
|
( w3a wa 3simpa syl ) ABDFABGCABDHEI $.
|
|
$}
|
|
|
|
${
|
|
3ad2ant.1 $e |- ( ph -> ch ) $.
|
|
$( Deduction adding conjuncts to an antecedent. (Contributed by NM,
|
|
21-Apr-2005.) $)
|
|
3ad2ant1 $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
|
|
( adantr 3adant2 ) ADCBACDEFG $.
|
|
|
|
$( Deduction adding conjuncts to an antecedent. (Contributed by NM,
|
|
21-Apr-2005.) $)
|
|
3ad2ant2 $p |- ( ( ps /\ ph /\ th ) -> ch ) $=
|
|
( adantr 3adant1 ) ADCBACDEFG $.
|
|
|
|
$( Deduction adding conjuncts to an antecedent. (Contributed by NM,
|
|
21-Apr-2005.) $)
|
|
3ad2ant3 $p |- ( ( ps /\ th /\ ph ) -> ch ) $=
|
|
( adantl 3adant1 ) DACBACDEFG $.
|
|
$}
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
|
|
simp1l $p |- ( ( ( ph /\ ps ) /\ ch /\ th ) -> ph ) $=
|
|
( wa simpl 3ad2ant1 ) ABECADABFG $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
|
|
simp1r $p |- ( ( ( ph /\ ps ) /\ ch /\ th ) -> ps ) $=
|
|
( wa simpr 3ad2ant1 ) ABECBDABFG $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
|
|
simp2l $p |- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ps ) $=
|
|
( wa simpl 3ad2ant2 ) BCEABDBCFG $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
|
|
simp2r $p |- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ch ) $=
|
|
( wa simpr 3ad2ant2 ) BCEACDBCFG $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
|
|
simp3l $p |- ( ( ph /\ ps /\ ( ch /\ th ) ) -> ch ) $=
|
|
( wa simpl 3ad2ant3 ) CDEACBCDFG $.
|
|
|
|
$( Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) $)
|
|
simp3r $p |- ( ( ph /\ ps /\ ( ch /\ th ) ) -> th ) $=
|
|
( wa simpr 3ad2ant3 ) CDEADBCDFG $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp11 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ph ) $=
|
|
( w3a simp1 3ad2ant1 ) ABCFDAEABCGH $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp12 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ps ) $=
|
|
( w3a simp2 3ad2ant1 ) ABCFDBEABCGH $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp13 $p |- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ch ) $=
|
|
( w3a simp3 3ad2ant1 ) ABCFDCEABCGH $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp21 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> ps ) $=
|
|
( w3a simp1 3ad2ant2 ) BCDFABEBCDGH $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp22 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> ch ) $=
|
|
( w3a simp2 3ad2ant2 ) BCDFACEBCDGH $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp23 $p |- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> th ) $=
|
|
( w3a simp3 3ad2ant2 ) BCDFADEBCDGH $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp31 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> ch ) $=
|
|
( w3a simp1 3ad2ant3 ) CDEFACBCDEGH $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp32 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> th ) $=
|
|
( w3a simp2 3ad2ant3 ) CDEFADBCDEGH $.
|
|
|
|
$( Simplification of doubly triple conjunction. (Contributed by NM,
|
|
17-Nov-2011.) $)
|
|
simp33 $p |- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> ta ) $=
|
|
( w3a simp3 3ad2ant3 ) CDEFAEBCDEGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpll1 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ph ) $=
|
|
( w3a wa simpl1 adantr ) ABCFDGAEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpll2 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ps ) $=
|
|
( w3a wa simpl2 adantr ) ABCFDGBEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpll3 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ch ) $=
|
|
( w3a wa simpl3 adantr ) ABCFDGCEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simplr1 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ph ) $=
|
|
( w3a wa simpr1 adantr ) DABCFGAEDABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simplr2 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ps ) $=
|
|
( w3a wa simpr2 adantr ) DABCFGBEDABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simplr3 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ch ) $=
|
|
( w3a wa simpr3 adantr ) DABCFGCEDABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simprl1 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph ) $=
|
|
( w3a wa simpl1 adantl ) ABCFDGAEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simprl2 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps ) $=
|
|
( w3a wa simpl2 adantl ) ABCFDGBEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simprl3 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch ) $=
|
|
( w3a wa simpl3 adantl ) ABCFDGCEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simprr1 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
|
|
( w3a wa simpr1 adantl ) DABCFGAEDABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simprr2 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
|
|
( w3a wa simpr2 adantl ) DABCFGBEDABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simprr3 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
|
|
( w3a wa simpr3 adantl ) DABCFGCEDABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl1l $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ph ) $=
|
|
( wa w3a simp1l adantr ) ABFCDGAEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl1r $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ps ) $=
|
|
( wa w3a simp1r adantr ) ABFCDGBEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl2l $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ph ) $=
|
|
( wa w3a simp2l adantr ) CABFDGAECABDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl2r $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ps ) $=
|
|
( wa w3a simp2r adantr ) CABFDGBECABDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl3l $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ph ) $=
|
|
( wa w3a simp3l adantr ) CDABFGAECDABHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl3r $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ps ) $=
|
|
( wa w3a simp3r adantr ) CDABFGBECDABHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr1l $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph ) $=
|
|
( wa w3a simp1l adantl ) ABFCDGAEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr1r $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps ) $=
|
|
( wa w3a simp1r adantl ) ABFCDGBEABCDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr2l $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph ) $=
|
|
( wa w3a simp2l adantl ) CABFDGAECABDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr2r $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps ) $=
|
|
( wa w3a simp2r adantl ) CABFDGBECABDHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr3l $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph ) $=
|
|
( wa w3a simp3l adantl ) CDABFGAECDABHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr3r $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps ) $=
|
|
( wa w3a simp3r adantl ) CDABFGBECDABHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1ll $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th /\ ta ) -> ph ) $=
|
|
( wa simpll 3ad2ant1 ) ABFCFDAEABCGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1lr $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th /\ ta ) -> ps ) $=
|
|
( wa simplr 3ad2ant1 ) ABFCFDBEABCGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1rl $p |- ( ( ( ch /\ ( ph /\ ps ) ) /\ th /\ ta ) -> ph ) $=
|
|
( wa simprl 3ad2ant1 ) CABFFDAECABGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1rr $p |- ( ( ( ch /\ ( ph /\ ps ) ) /\ th /\ ta ) -> ps ) $=
|
|
( wa simprr 3ad2ant1 ) CABFFDBECABGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2ll $p |- ( ( th /\ ( ( ph /\ ps ) /\ ch ) /\ ta ) -> ph ) $=
|
|
( wa simpll 3ad2ant2 ) ABFCFDAEABCGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2lr $p |- ( ( th /\ ( ( ph /\ ps ) /\ ch ) /\ ta ) -> ps ) $=
|
|
( wa simplr 3ad2ant2 ) ABFCFDBEABCGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2rl $p |- ( ( th /\ ( ch /\ ( ph /\ ps ) ) /\ ta ) -> ph ) $=
|
|
( wa simprl 3ad2ant2 ) CABFFDAECABGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2rr $p |- ( ( th /\ ( ch /\ ( ph /\ ps ) ) /\ ta ) -> ps ) $=
|
|
( wa simprr 3ad2ant2 ) CABFFDBECABGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3ll $p |- ( ( th /\ ta /\ ( ( ph /\ ps ) /\ ch ) ) -> ph ) $=
|
|
( wa simpll 3ad2ant3 ) ABFCFDAEABCGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3lr $p |- ( ( th /\ ta /\ ( ( ph /\ ps ) /\ ch ) ) -> ps ) $=
|
|
( wa simplr 3ad2ant3 ) ABFCFDBEABCGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3rl $p |- ( ( th /\ ta /\ ( ch /\ ( ph /\ ps ) ) ) -> ph ) $=
|
|
( wa simprl 3ad2ant3 ) CABFFDAECABGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3rr $p |- ( ( th /\ ta /\ ( ch /\ ( ph /\ ps ) ) ) -> ps ) $=
|
|
( wa simprr 3ad2ant3 ) CABFFDBECABGH $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl11 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ph ) $=
|
|
( w3a simp11 adantr ) ABCGDEGAFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl12 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ps ) $=
|
|
( w3a simp12 adantr ) ABCGDEGBFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl13 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ch ) $=
|
|
( w3a simp13 adantr ) ABCGDEGCFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl21 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ph ) $=
|
|
( w3a simp21 adantr ) DABCGEGAFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl22 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ps ) $=
|
|
( w3a simp22 adantr ) DABCGEGBFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl23 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ch ) $=
|
|
( w3a simp23 adantr ) DABCGEGCFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl31 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ph ) $=
|
|
( w3a simp31 adantr ) DEABCGGAFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl32 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ps ) $=
|
|
( w3a simp32 adantr ) DEABCGGBFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpl33 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ch ) $=
|
|
( w3a simp33 adantr ) DEABCGGCFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr11 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ph ) $=
|
|
( w3a simp11 adantl ) ABCGDEGAFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr12 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ps ) $=
|
|
( w3a simp12 adantl ) ABCGDEGBFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr13 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ch ) $=
|
|
( w3a simp13 adantl ) ABCGDEGCFABCDEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr21 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ph ) $=
|
|
( w3a simp21 adantl ) DABCGEGAFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr22 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ps ) $=
|
|
( w3a simp22 adantl ) DABCGEGBFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr23 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ch ) $=
|
|
( w3a simp23 adantl ) DABCGEGCFDABCEHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr31 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
|
|
( w3a simp31 adantl ) DEABCGGAFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr32 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
|
|
( w3a simp32 adantl ) DEABCGGBFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simpr33 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
|
|
( w3a simp33 adantl ) DEABCGGCFDEABCHI $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1l1 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ph ) $=
|
|
( w3a wa simpl1 3ad2ant1 ) ABCGDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1l2 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ps ) $=
|
|
( w3a wa simpl2 3ad2ant1 ) ABCGDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1l3 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta /\ et ) -> ch ) $=
|
|
( w3a wa simpl3 3ad2ant1 ) ABCGDHECFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1r1 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ph ) $=
|
|
( w3a wa simpr1 3ad2ant1 ) DABCGHEAFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1r2 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ps ) $=
|
|
( w3a wa simpr2 3ad2ant1 ) DABCGHEBFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp1r3 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta /\ et ) -> ch ) $=
|
|
( w3a wa simpr3 3ad2ant1 ) DABCGHECFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2l1 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ph ) $=
|
|
( w3a wa simpl1 3ad2ant2 ) ABCGDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2l2 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ps ) $=
|
|
( w3a wa simpl2 3ad2ant2 ) ABCGDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2l3 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) /\ et ) -> ch ) $=
|
|
( w3a wa simpl3 3ad2ant2 ) ABCGDHECFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2r1 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ph ) $=
|
|
( w3a wa simpr1 3ad2ant2 ) DABCGHEAFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2r2 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ps ) $=
|
|
( w3a wa simpr2 3ad2ant2 ) DABCGHEBFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp2r3 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ch ) $=
|
|
( w3a wa simpr3 3ad2ant2 ) DABCGHECFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3l1 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph ) $=
|
|
( w3a wa simpl1 3ad2ant3 ) ABCGDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3l2 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps ) $=
|
|
( w3a wa simpl2 3ad2ant3 ) ABCGDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3l3 $p |- ( ( ta /\ et /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch ) $=
|
|
( w3a wa simpl3 3ad2ant3 ) ABCGDHECFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3r1 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
|
|
( w3a wa simpr1 3ad2ant3 ) DABCGHEAFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3r2 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
|
|
( w3a wa simpr2 3ad2ant3 ) DABCGHEBFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp3r3 $p |- ( ( ta /\ et /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
|
|
( w3a wa simpr3 3ad2ant3 ) DABCGHECFDABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp11l $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta /\ et ) -> ph ) $=
|
|
( wa w3a simp1l 3ad2ant1 ) ABGCDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp11r $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta /\ et ) -> ps ) $=
|
|
( wa w3a simp1r 3ad2ant1 ) ABGCDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp12l $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta /\ et ) -> ph ) $=
|
|
( wa w3a simp2l 3ad2ant1 ) CABGDHEAFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp12r $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta /\ et ) -> ps ) $=
|
|
( wa w3a simp2r 3ad2ant1 ) CABGDHEBFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp13l $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta /\ et ) -> ph ) $=
|
|
( wa w3a simp3l 3ad2ant1 ) CDABGHEAFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp13r $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta /\ et ) -> ps ) $=
|
|
( wa w3a simp3r 3ad2ant1 ) CDABGHEBFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp21l $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) /\ et ) -> ph ) $=
|
|
( wa w3a simp1l 3ad2ant2 ) ABGCDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp21r $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) /\ et ) -> ps ) $=
|
|
( wa w3a simp1r 3ad2ant2 ) ABGCDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp22l $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) /\ et ) -> ph ) $=
|
|
( wa w3a simp2l 3ad2ant2 ) CABGDHEAFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp22r $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) /\ et ) -> ps ) $=
|
|
( wa w3a simp2r 3ad2ant2 ) CABGDHEBFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp23l $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) /\ et ) -> ph ) $=
|
|
( wa w3a simp3l 3ad2ant2 ) CDABGHEAFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp23r $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) /\ et ) -> ps ) $=
|
|
( wa w3a simp3r 3ad2ant2 ) CDABGHEBFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp31l $p |- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph ) $=
|
|
( wa w3a simp1l 3ad2ant3 ) ABGCDHEAFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp31r $p |- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps ) $=
|
|
( wa w3a simp1r 3ad2ant3 ) ABGCDHEBFABCDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp32l $p |- ( ( ta /\ et /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph ) $=
|
|
( wa w3a simp2l 3ad2ant3 ) CABGDHEAFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp32r $p |- ( ( ta /\ et /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps ) $=
|
|
( wa w3a simp2r 3ad2ant3 ) CABGDHEBFCABDIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp33l $p |- ( ( ta /\ et /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph ) $=
|
|
( wa w3a simp3l 3ad2ant3 ) CDABGHEAFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp33r $p |- ( ( ta /\ et /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps ) $=
|
|
( wa w3a simp3r 3ad2ant3 ) CDABGHEBFCDABIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp111 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ph ) $=
|
|
( w3a simp11 3ad2ant1 ) ABCHDEHFAGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp112 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ps ) $=
|
|
( w3a simp12 3ad2ant1 ) ABCHDEHFBGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp113 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et /\ ze ) -> ch ) $=
|
|
( w3a simp13 3ad2ant1 ) ABCHDEHFCGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp121 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ph ) $=
|
|
( w3a simp21 3ad2ant1 ) DABCHEHFAGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp122 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ps ) $=
|
|
( w3a simp22 3ad2ant1 ) DABCHEHFBGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp123 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et /\ ze ) -> ch ) $=
|
|
( w3a simp23 3ad2ant1 ) DABCHEHFCGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp131 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ph ) $=
|
|
( w3a simp31 3ad2ant1 ) DEABCHHFAGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp132 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ps ) $=
|
|
( w3a simp32 3ad2ant1 ) DEABCHHFBGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp133 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et /\ ze ) -> ch ) $=
|
|
( w3a simp33 3ad2ant1 ) DEABCHHFCGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp211 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ph ) $=
|
|
( w3a simp11 3ad2ant2 ) ABCHDEHFAGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp212 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ps ) $=
|
|
( w3a simp12 3ad2ant2 ) ABCHDEHFBGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp213 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ ze ) -> ch ) $=
|
|
( w3a simp13 3ad2ant2 ) ABCHDEHFCGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp221 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ph ) $=
|
|
( w3a simp21 3ad2ant2 ) DABCHEHFAGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp222 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ps ) $=
|
|
( w3a simp22 3ad2ant2 ) DABCHEHFBGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp223 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ ze ) -> ch ) $=
|
|
( w3a simp23 3ad2ant2 ) DABCHEHFCGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp231 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ph ) $=
|
|
( w3a simp31 3ad2ant2 ) DEABCHHFAGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp232 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ps ) $=
|
|
( w3a simp32 3ad2ant2 ) DEABCHHFBGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp233 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ ze ) -> ch ) $=
|
|
( w3a simp33 3ad2ant2 ) DEABCHHFCGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp311 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ph ) $=
|
|
( w3a simp11 3ad2ant3 ) ABCHDEHFAGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp312 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ps ) $=
|
|
( w3a simp12 3ad2ant3 ) ABCHDEHFBGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp313 $p |- ( ( et /\ ze /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ch ) $=
|
|
( w3a simp13 3ad2ant3 ) ABCHDEHFCGABCDEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp321 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ph ) $=
|
|
( w3a simp21 3ad2ant3 ) DABCHEHFAGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp322 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ps ) $=
|
|
( w3a simp22 3ad2ant3 ) DABCHEHFBGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp323 $p |- ( ( et /\ ze /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ch ) $=
|
|
( w3a simp23 3ad2ant3 ) DABCHEHFCGDABCEIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp331 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
|
|
( w3a simp31 3ad2ant3 ) DEABCHHFAGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp332 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
|
|
( w3a simp32 3ad2ant3 ) DEABCHHFBGDEABCIJ $.
|
|
|
|
$( Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) $)
|
|
simp333 $p |- ( ( et /\ ze /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
|
|
( w3a simp33 3ad2ant3 ) DEABCHHFCGDEABCIJ $.
|
|
|
|
${
|
|
3adantl.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3adantl1 $p |- ( ( ( ta /\ ph /\ ps ) /\ ch ) -> th ) $=
|
|
( w3a wa 3simpc sylan ) EABGABHCDEABIFJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3adantl2 $p |- ( ( ( ph /\ ta /\ ps ) /\ ch ) -> th ) $=
|
|
( w3a wa 3simpb sylan ) AEBGABHCDAEBIFJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3adantl3 $p |- ( ( ( ph /\ ps /\ ta ) /\ ch ) -> th ) $=
|
|
( w3a wa 3simpa sylan ) ABEGABHCDABEIFJ $.
|
|
$}
|
|
|
|
${
|
|
3adantr.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
27-Apr-2005.) $)
|
|
3adantr1 $p |- ( ( ph /\ ( ta /\ ps /\ ch ) ) -> th ) $=
|
|
( w3a wa 3simpc sylan2 ) EBCGABCHDEBCIFJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
27-Apr-2005.) $)
|
|
3adantr2 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
|
|
( w3a wa 3simpb sylan2 ) BECGABCHDBECIFJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
27-Apr-2005.) $)
|
|
3adantr3 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
|
|
( w3a wa 3simpa sylan2 ) BCEGABCHDBCEIFJ $.
|
|
$}
|
|
|
|
${
|
|
3ad2antl.1 $e |- ( ( ph /\ ch ) -> th ) $.
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
4-Aug-2007.) $)
|
|
3ad2antl1 $p |- ( ( ( ph /\ ps /\ ta ) /\ ch ) -> th ) $=
|
|
( adantlr 3adantl2 ) AECDBACDEFGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
4-Aug-2007.) $)
|
|
3ad2antl2 $p |- ( ( ( ps /\ ph /\ ta ) /\ ch ) -> th ) $=
|
|
( adantlr 3adantl1 ) AECDBACDEFGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
4-Aug-2007.) $)
|
|
3ad2antl3 $p |- ( ( ( ps /\ ta /\ ph ) /\ ch ) -> th ) $=
|
|
( adantll 3adantl1 ) EACDBACDEFGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
25-Dec-2007.) $)
|
|
3ad2antr1 $p |- ( ( ph /\ ( ch /\ ps /\ ta ) ) -> th ) $=
|
|
( adantrr 3adantr3 ) ACBDEACDBFGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
27-Dec-2007.) $)
|
|
3ad2antr2 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
|
|
( adantrl 3adantr3 ) ABCDEACDBFGH $.
|
|
|
|
$( Deduction adding conjuncts to antecedent. (Contributed by NM,
|
|
30-Dec-2007.) $)
|
|
3ad2antr3 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
|
|
( adantrl 3adantr1 ) AECDBACDEFGH $.
|
|
$}
|
|
|
|
${
|
|
3anibar.1 $e |- ( ( ph /\ ps /\ ch ) -> ( th <-> ( ch /\ ta ) ) ) $.
|
|
$( Remove a hypothesis from the second member of a biimplication.
|
|
(Contributed by FL, 22-Jul-2008.) $)
|
|
3anibar $p |- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) $=
|
|
( w3a wa simp3 biantrurd bitr4d ) ABCGZDCEHEFLCEABCIJK $.
|
|
$}
|
|
|
|
$( Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) $)
|
|
3mix1 $p |- ( ph -> ( ph \/ ps \/ ch ) ) $=
|
|
( wo w3o orc 3orass sylibr ) AABCDZDABCEAIFABCGH $.
|
|
|
|
$( Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) $)
|
|
3mix2 $p |- ( ph -> ( ps \/ ph \/ ch ) ) $=
|
|
( w3o 3mix1 3orrot sylibr ) AACBDBACDACBEBACFG $.
|
|
|
|
$( Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.) $)
|
|
3mix3 $p |- ( ph -> ( ps \/ ch \/ ph ) ) $=
|
|
( w3o 3mix1 3orrot sylib ) AABCDBCADABCEABCFG $.
|
|
|
|
${
|
|
3mixi.1 $e |- ph $.
|
|
$( Introduction in triple disjunction. (Contributed by Mario Carneiro,
|
|
6-Oct-2014.) $)
|
|
3mix1i $p |- ( ph \/ ps \/ ch ) $=
|
|
( w3o 3mix1 ax-mp ) AABCEDABCFG $.
|
|
|
|
$( Introduction in triple disjunction. (Contributed by Mario Carneiro,
|
|
6-Oct-2014.) $)
|
|
3mix2i $p |- ( ps \/ ph \/ ch ) $=
|
|
( w3o 3mix2 ax-mp ) ABACEDABCFG $.
|
|
|
|
$( Introduction in triple disjunction. (Contributed by Mario Carneiro,
|
|
6-Oct-2014.) $)
|
|
3mix3i $p |- ( ps \/ ch \/ ph ) $=
|
|
( w3o 3mix3 ax-mp ) ABCAEDABCFG $.
|
|
$}
|
|
|
|
${
|
|
3pm3.2i.1 $e |- ph $.
|
|
3pm3.2i.2 $e |- ps $.
|
|
3pm3.2i.3 $e |- ch $.
|
|
$( Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.) $)
|
|
3pm3.2i $p |- ( ph /\ ps /\ ch ) $=
|
|
( w3a wa pm3.2i df-3an mpbir2an ) ABCGABHCABDEIFABCJK $.
|
|
$}
|
|
|
|
$( ~ pm3.2 for a triple conjunction. (Contributed by Alan Sare,
|
|
24-Oct-2011.) $)
|
|
pm3.2an3 $p |- ( ph -> ( ps -> ( ch -> ( ph /\ ps /\ ch ) ) ) ) $=
|
|
( wa w3a wi pm3.2 ex df-3an bicomi syl8ib ) ABCABDZCDZABCEZABCMFLCGHNMABCIJ
|
|
K $.
|
|
|
|
${
|
|
3jca.1 $e |- ( ph -> ps ) $.
|
|
3jca.2 $e |- ( ph -> ch ) $.
|
|
3jca.3 $e |- ( ph -> th ) $.
|
|
$( Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.) $)
|
|
3jca $p |- ( ph -> ( ps /\ ch /\ th ) ) $=
|
|
( wa w3a jca31 df-3an sylibr ) ABCHDHBCDIABCDEFGJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
3jcad.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3jcad.2 $e |- ( ph -> ( ps -> th ) ) $.
|
|
3jcad.3 $e |- ( ph -> ( ps -> ta ) ) $.
|
|
$( Deduction conjoining the consequents of three implications.
|
|
(Contributed by NM, 25-Sep-2005.) $)
|
|
3jcad $p |- ( ph -> ( ps -> ( ch /\ th /\ ta ) ) ) $=
|
|
( w3a wa imp 3jca ex ) ABCDEIABJCDEABCFKABDGKABEHKLM $.
|
|
$}
|
|
|
|
${
|
|
mpbir3an.1 $e |- ps $.
|
|
mpbir3an.2 $e |- ch $.
|
|
mpbir3an.3 $e |- th $.
|
|
mpbir3an.4 $e |- ( ph <-> ( ps /\ ch /\ th ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by NM,
|
|
16-Sep-2011.) $)
|
|
mpbir3an $p |- ph $=
|
|
( w3a 3pm3.2i mpbir ) ABCDIBCDEFGJHK $.
|
|
$}
|
|
|
|
${
|
|
mpbir3and.1 $e |- ( ph -> ch ) $.
|
|
mpbir3and.2 $e |- ( ph -> th ) $.
|
|
mpbir3and.3 $e |- ( ph -> ta ) $.
|
|
mpbir3and.4 $e |- ( ph -> ( ps <-> ( ch /\ th /\ ta ) ) ) $.
|
|
$( Detach a conjunction of truths in a biconditional. (Contributed by
|
|
Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro,
|
|
9-Jan-2015.) $)
|
|
mpbir3and $p |- ( ph -> ps ) $=
|
|
( w3a 3jca mpbird ) ABCDEJACDEFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
syl3anbrc.1 $e |- ( ph -> ps ) $.
|
|
syl3anbrc.2 $e |- ( ph -> ch ) $.
|
|
syl3anbrc.3 $e |- ( ph -> th ) $.
|
|
syl3anbrc.4 $e |- ( ta <-> ( ps /\ ch /\ th ) ) $.
|
|
$( Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.) $)
|
|
syl3anbrc $p |- ( ph -> ta ) $=
|
|
( w3a 3jca sylibr ) ABCDJEABCDFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
3anim123i.1 $e |- ( ph -> ps ) $.
|
|
3anim123i.2 $e |- ( ch -> th ) $.
|
|
3anim123i.3 $e |- ( ta -> et ) $.
|
|
$( Join antecedents and consequents with conjunction. (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
3anim123i $p |- ( ( ph /\ ch /\ ta ) -> ( ps /\ th /\ et ) ) $=
|
|
( w3a 3ad2ant1 3ad2ant2 3ad2ant3 3jca ) ACEJBDFACBEGKCADEHLEAFCIMN $.
|
|
$}
|
|
|
|
${
|
|
3animi.1 $e |- ( ph -> ps ) $.
|
|
$( Add two conjuncts to antecedent and consequent. (Contributed by Jeff
|
|
Hankins, 16-Aug-2009.) $)
|
|
3anim1i $p |- ( ( ph /\ ch /\ th ) -> ( ps /\ ch /\ th ) ) $=
|
|
( id 3anim123i ) ABCCDDECFDFG $.
|
|
|
|
$( Add two conjuncts to antecedent and consequent. (Contributed by Jeff
|
|
Hankins, 19-Aug-2009.) $)
|
|
3anim3i $p |- ( ( ch /\ th /\ ph ) -> ( ch /\ th /\ ps ) ) $=
|
|
( id 3anim123i ) CCDDABCFDFEG $.
|
|
$}
|
|
|
|
${
|
|
bi3.1 $e |- ( ph <-> ps ) $.
|
|
bi3.2 $e |- ( ch <-> th ) $.
|
|
bi3.3 $e |- ( ta <-> et ) $.
|
|
$( Join 3 biconditionals with conjunction. (Contributed by NM,
|
|
21-Apr-1994.) $)
|
|
3anbi123i $p |- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) ) $=
|
|
( wa w3a anbi12i df-3an 3bitr4i ) ACJZEJBDJZFJACEKBDFKOPEFABCDGHLILACEMBD
|
|
FMN $.
|
|
|
|
$( Join 3 biconditionals with disjunction. (Contributed by NM,
|
|
17-May-1994.) $)
|
|
3orbi123i $p |- ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) $=
|
|
( wo w3o orbi12i df-3or 3bitr4i ) ACJZEJBDJZFJACEKBDFKOPEFABCDGHLILACEMBD
|
|
FMN $.
|
|
$}
|
|
|
|
${
|
|
3anbi1i.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding two conjuncts to each side of a biconditional.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi1i $p |- ( ( ph /\ ch /\ th ) <-> ( ps /\ ch /\ th ) ) $=
|
|
( biid 3anbi123i ) ABCCDDECFDFG $.
|
|
|
|
$( Inference adding two conjuncts to each side of a biconditional.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi2i $p |- ( ( ch /\ ph /\ th ) <-> ( ch /\ ps /\ th ) ) $=
|
|
( biid 3anbi123i ) CCABDDCFEDFG $.
|
|
|
|
$( Inference adding two conjuncts to each side of a biconditional.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi3i $p |- ( ( ch /\ th /\ ph ) <-> ( ch /\ th /\ ps ) ) $=
|
|
( biid 3anbi123i ) CCDDABCFDFEG $.
|
|
$}
|
|
|
|
${
|
|
3imp.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
|
|
$( Importation inference. (Contributed by NM, 8-Apr-1994.) $)
|
|
3imp $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( w3a wa df-3an imp31 sylbi ) ABCFABGCGDABCHABCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
3impa.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
|
|
$( Importation from double to triple conjunction. (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3impa $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( exp31 3imp ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3impb.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
|
|
$( Importation from double to triple conjunction. (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3impb $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( exp32 3imp ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3impia.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
|
|
$( Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) $)
|
|
3impia $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( wi ex 3imp ) ABCDABCDFEGH $.
|
|
$}
|
|
|
|
${
|
|
3impib.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) $)
|
|
3impib $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( exp3a 3imp ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3exp.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( Exportation inference. (Contributed by NM, 30-May-1994.) $)
|
|
3exp $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
|
|
( w3a pm3.2an3 syl8 ) ABCABCFDABCGEH $.
|
|
|
|
$( Exportation from triple to double conjunction. (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3expa $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
|
|
( 3exp imp31 ) ABCDABCDEFG $.
|
|
|
|
$( Exportation from triple to double conjunction. (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3expb $p |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $=
|
|
( 3exp imp32 ) ABCDABCDEFG $.
|
|
|
|
$( Exportation from triple conjunction. (Contributed by NM,
|
|
19-May-2007.) $)
|
|
3expia $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
|
|
( wi 3exp imp ) ABCDFABCDEGH $.
|
|
|
|
$( Exportation from triple conjunction. (Contributed by NM,
|
|
19-May-2007.) $)
|
|
3expib $p |- ( ph -> ( ( ps /\ ch ) -> th ) ) $=
|
|
( 3exp imp3a ) ABCDABCDEFG $.
|
|
|
|
$( Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM,
|
|
28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
3com12 $p |- ( ( ps /\ ph /\ ch ) -> th ) $=
|
|
( w3a 3ancoma sylbi ) BACFABCFDBACGEH $.
|
|
|
|
$( Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
3com13 $p |- ( ( ch /\ ps /\ ph ) -> th ) $=
|
|
( w3a 3anrev sylbi ) CBAFABCFDCBAGEH $.
|
|
|
|
$( Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
3com23 $p |- ( ( ph /\ ch /\ ps ) -> th ) $=
|
|
( 3exp com23 3imp ) ACBDABCDABCDEFGH $.
|
|
|
|
$( Commutation in antecedent. Rotate left. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
3coml $p |- ( ( ps /\ ch /\ ph ) -> th ) $=
|
|
( 3com23 3com13 ) ACBDABCDEFG $.
|
|
|
|
$( Commutation in antecedent. Rotate right. (Contributed by NM,
|
|
28-Jan-1996.) $)
|
|
3comr $p |- ( ( ch /\ ph /\ ps ) -> th ) $=
|
|
( 3coml ) BCADABCDEFF $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
16-Feb-2008.) $)
|
|
3adant3r1 $p |- ( ( ph /\ ( ta /\ ps /\ ch ) ) -> th ) $=
|
|
( 3expb 3adantr1 ) ABCDEABCDFGH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
17-Feb-2008.) $)
|
|
3adant3r2 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
|
|
( 3expb 3adantr2 ) ABCDEABCDFGH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
18-Feb-2008.) $)
|
|
3adant3r3 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
|
|
( 3expb 3adantr3 ) ABCDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
3an1rs.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( Swap conjuncts. (Contributed by NM, 16-Dec-2007.) $)
|
|
3an1rs $p |- ( ( ( ph /\ ps /\ th ) /\ ch ) -> ta ) $=
|
|
( w3a wi ex 3exp com34 3imp imp ) ABDGCEABDCEHABCDEABCDEHABCGDEFIJKLM $.
|
|
$}
|
|
|
|
${
|
|
3imp1.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
|
|
$( Importation to left triple conjunction. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3imp1 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $=
|
|
( w3a wi 3imp imp ) ABCGDEABCDEHFIJ $.
|
|
|
|
$( Importation deduction for triple conjunction. (Contributed by NM,
|
|
26-Oct-2006.) $)
|
|
3impd $p |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $=
|
|
( w3a wi com4l 3imp com12 ) BCDGAEBCDAEHABCDEFIJK $.
|
|
|
|
$( Importation to right triple conjunction. (Contributed by NM,
|
|
26-Oct-2006.) $)
|
|
3imp2 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $=
|
|
( w3a 3impd imp ) ABCDGEABCDEFHI $.
|
|
$}
|
|
|
|
${
|
|
3exp1.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( Exportation from left triple conjunction. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
3exp1 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi w3a ex 3exp ) ABCDEGABCHDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
3expd.1 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
|
|
$( Exportation deduction for triple conjunction. (Contributed by NM,
|
|
26-Oct-2006.) $)
|
|
3expd $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( wi w3a com12 3exp com4r ) BCDAEBCDAEGABCDHEFIJK $.
|
|
$}
|
|
|
|
${
|
|
3exp2.1 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( Exportation from right triple conjunction. (Contributed by NM,
|
|
26-Oct-2006.) $)
|
|
3exp2 $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
|
|
( w3a ex 3expd ) ABCDEABCDGEFHI $.
|
|
$}
|
|
|
|
${
|
|
exp5o.1 $e |- ( ( ph /\ ps /\ ch ) -> ( ( th /\ ta ) -> et ) ) $.
|
|
$( A triple exportation inference. (Contributed by Jeff Hankins,
|
|
8-Jul-2009.) $)
|
|
exp5o $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( wi w3a exp3a 3exp ) ABCDEFHHABCIDEFGJK $.
|
|
$}
|
|
|
|
${
|
|
exp516.1 $e |- ( ( ( ph /\ ( ps /\ ch /\ th ) ) /\ ta ) -> et ) $.
|
|
$( A triple exportation inference. (Contributed by Jeff Hankins,
|
|
8-Jul-2009.) $)
|
|
exp516 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( wi w3a exp31 3expd ) ABCDEFHABCDIEFGJK $.
|
|
$}
|
|
|
|
${
|
|
exp520.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta ) ) -> et ) $.
|
|
$( A triple exportation inference. (Contributed by Jeff Hankins,
|
|
8-Jul-2009.) $)
|
|
exp520 $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
|
|
( w3a wa ex exp5o ) ABCDEFABCHDEIFGJK $.
|
|
$}
|
|
|
|
${
|
|
3anassrs.1 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( Associative law for conjunction applied to antecedent (eliminates
|
|
syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
3anassrs $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $=
|
|
( 3exp2 imp41 ) ABCDEABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
3adant1l.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant1l $p |- ( ( ( ta /\ ph ) /\ ps /\ ch ) -> th ) $=
|
|
( wa 3expb adantll 3impb ) EAGBCDABCGDEABCDFHIJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant1r $p |- ( ( ( ph /\ ta ) /\ ps /\ ch ) -> th ) $=
|
|
( wa 3expb adantlr 3impb ) AEGBCDABCGDEABCDFHIJ $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant2l $p |- ( ( ph /\ ( ta /\ ps ) /\ ch ) -> th ) $=
|
|
( wa 3com12 3adant1l ) EBGACDBACDEABCDFHIH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant2r $p |- ( ( ph /\ ( ps /\ ta ) /\ ch ) -> th ) $=
|
|
( wa 3com12 3adant1r ) BEGACDBACDEABCDFHIH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant3l $p |- ( ( ph /\ ps /\ ( ta /\ ch ) ) -> th ) $=
|
|
( wa 3com13 3adant1l ) ECGBADCBADEABCDFHIH $.
|
|
|
|
$( Deduction adding a conjunct to antecedent. (Contributed by NM,
|
|
8-Jan-2006.) $)
|
|
3adant3r $p |- ( ( ph /\ ps /\ ( ch /\ ta ) ) -> th ) $=
|
|
( wa 3com13 3adant1r ) CEGBADCBADEABCDFHIH $.
|
|
$}
|
|
|
|
${
|
|
sylXanc.1 $e |- ( ph -> ps ) $.
|
|
sylXanc.2 $e |- ( ph -> ch ) $.
|
|
sylXanc.3 $e |- ( ph -> th ) $.
|
|
${
|
|
syl12anc.4 $e |- ( ( ps /\ ( ch /\ th ) ) -> ta ) $.
|
|
$( Syllogism combined with contraction. (Contributed by Jeff Hankins,
|
|
1-Aug-2009.) $)
|
|
syl12anc $p |- ( ph -> ta ) $=
|
|
( wa jca32 syl ) ABCDJJEABCDFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
syl21anc.4 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( Syllogism combined with contraction. (Contributed by Jeff Hankins,
|
|
1-Aug-2009.) $)
|
|
syl21anc $p |- ( ph -> ta ) $=
|
|
( wa jca31 syl ) ABCJDJEABCDFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
syl111anc.4 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl3anc $p |- ( ph -> ta ) $=
|
|
( w3a 3jca syl ) ABCDJEABCDFGHKIL $.
|
|
$}
|
|
|
|
sylXanc.4 $e |- ( ph -> ta ) $.
|
|
${
|
|
syl22anc.5 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl22anc $p |- ( ph -> et ) $=
|
|
( wa jca syl12anc ) ABCLDEFABCGHMIJKN $.
|
|
$}
|
|
|
|
${
|
|
syl13anc.5 $e |- ( ( ps /\ ( ch /\ th /\ ta ) ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl13anc $p |- ( ph -> et ) $=
|
|
( w3a 3jca syl2anc ) ABCDELFGACDEHIJMKN $.
|
|
$}
|
|
|
|
${
|
|
syl31anc.5 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl31anc $p |- ( ph -> et ) $=
|
|
( w3a 3jca syl2anc ) ABCDLEFABCDGHIMJKN $.
|
|
$}
|
|
|
|
${
|
|
syl112anc.5 $e |- ( ( ps /\ ch /\ ( th /\ ta ) ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl112anc $p |- ( ph -> et ) $=
|
|
( wa jca syl3anc ) ABCDELFGHADEIJMKN $.
|
|
$}
|
|
|
|
${
|
|
syl121anc.5 $e |- ( ( ps /\ ( ch /\ th ) /\ ta ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl121anc $p |- ( ph -> et ) $=
|
|
( wa jca syl3anc ) ABCDLEFGACDHIMJKN $.
|
|
$}
|
|
|
|
${
|
|
syl211anc.5 $e |- ( ( ( ps /\ ch ) /\ th /\ ta ) -> et ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl211anc $p |- ( ph -> et ) $=
|
|
( wa jca syl3anc ) ABCLDEFABCGHMIJKN $.
|
|
$}
|
|
|
|
sylXanc.5 $e |- ( ph -> et ) $.
|
|
${
|
|
syl23anc.6 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl23anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl13anc ) ABCNDEFGABCHIOJKLMP $.
|
|
$}
|
|
|
|
${
|
|
syl32anc.6 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl32anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl31anc ) ABCDEFNGHIJAEFKLOMP $.
|
|
$}
|
|
|
|
${
|
|
syl122anc.6 $e |- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl122anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl121anc ) ABCDEFNGHIJAEFKLOMP $.
|
|
$}
|
|
|
|
${
|
|
syl212anc.6 $e |- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl212anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl211anc ) ABCDEFNGHIJAEFKLOMP $.
|
|
$}
|
|
|
|
${
|
|
syl221anc.6 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ et ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl221anc $p |- ( ph -> ze ) $=
|
|
( wa jca syl211anc ) ABCDENFGHIADEJKOLMP $.
|
|
$}
|
|
|
|
${
|
|
syl113anc.6 $e |- ( ( ps /\ ch /\ ( th /\ ta /\ et ) ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl113anc $p |- ( ph -> ze ) $=
|
|
( w3a 3jca syl3anc ) ABCDEFNGHIADEFJKLOMP $.
|
|
$}
|
|
|
|
${
|
|
syl131anc.6 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ et ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl131anc $p |- ( ph -> ze ) $=
|
|
( w3a 3jca syl3anc ) ABCDENFGHACDEIJKOLMP $.
|
|
$}
|
|
|
|
${
|
|
syl311anc.6 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ et ) -> ze ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl311anc $p |- ( ph -> ze ) $=
|
|
( w3a 3jca syl3anc ) ABCDNEFGABCDHIJOKLMP $.
|
|
$}
|
|
|
|
sylXanc.6 $e |- ( ph -> ze ) $.
|
|
${
|
|
syl33anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl33anc $p |- ( ph -> si ) $=
|
|
( w3a 3jca syl13anc ) ABCDPEFGHABCDIJKQLMNOR $.
|
|
$}
|
|
|
|
${
|
|
syl222anc.7 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ ( et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl222anc $p |- ( ph -> si ) $=
|
|
( wa jca syl221anc ) ABCDEFGPHIJKLAFGMNQOR $.
|
|
$}
|
|
|
|
${
|
|
syl123anc.7 $e |- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl123anc $p |- ( ph -> si ) $=
|
|
( wa jca syl113anc ) ABCDPEFGHIACDJKQLMNOR $.
|
|
$}
|
|
|
|
${
|
|
syl132anc.7 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ ( et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Jul-2012.) $)
|
|
syl132anc $p |- ( ph -> si ) $=
|
|
( wa jca syl131anc ) ABCDEFGPHIJKLAFGMNQOR $.
|
|
$}
|
|
|
|
${
|
|
syl213anc.7 $e |- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl213anc $p |- ( ph -> si ) $=
|
|
( wa jca syl113anc ) ABCPDEFGHABCIJQKLMNOR $.
|
|
$}
|
|
|
|
${
|
|
syl231anc.7 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ze )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl231anc $p |- ( ph -> si ) $=
|
|
( wa jca syl131anc ) ABCPDEFGHABCIJQKLMNOR $.
|
|
$}
|
|
|
|
${
|
|
syl312anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze ) )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Jul-2012.) $)
|
|
syl312anc $p |- ( ph -> si ) $=
|
|
( wa jca syl311anc ) ABCDEFGPHIJKLAFGMNQOR $.
|
|
$}
|
|
|
|
${
|
|
syl321anc.7 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ze )
|
|
-> si ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Jul-2012.) $)
|
|
syl321anc $p |- ( ph -> si ) $=
|
|
( wa jca syl311anc ) ABCDEFPGHIJKAEFLMQNOR $.
|
|
$}
|
|
|
|
sylXanc.7 $e |- ( ph -> si ) $.
|
|
${
|
|
syl133anc.8 $e |- ( ( ps /\ ( ch /\ th /\ ta ) /\ ( et /\ ze /\ si ) )
|
|
-> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl133anc $p |- ( ph -> rh ) $=
|
|
( w3a 3jca syl131anc ) ABCDEFGHRIJKLMAFGHNOPSQT $.
|
|
$}
|
|
|
|
${
|
|
syl313anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze /\ si ) )
|
|
-> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl313anc $p |- ( ph -> rh ) $=
|
|
( w3a 3jca syl311anc ) ABCDEFGHRIJKLMAFGHNOPSQT $.
|
|
$}
|
|
|
|
${
|
|
syl331anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ si )
|
|
-> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl331anc $p |- ( ph -> rh ) $=
|
|
( w3a 3jca syl311anc ) ABCDEFGRHIJKLAEFGMNOSPQT $.
|
|
$}
|
|
|
|
${
|
|
syl223anc.8 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ ( et /\ ze /\ si )
|
|
) -> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl223anc $p |- ( ph -> rh ) $=
|
|
( wa jca syl213anc ) ABCDERFGHIJKADELMSNOPQT $.
|
|
$}
|
|
|
|
${
|
|
syl232anc.8 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si )
|
|
) -> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl232anc $p |- ( ph -> rh ) $=
|
|
( wa jca syl231anc ) ABCDEFGHRIJKLMNAGHOPSQT $.
|
|
$}
|
|
|
|
${
|
|
syl322anc.8 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ( ze /\ si )
|
|
) -> rh ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl322anc $p |- ( ph -> rh ) $=
|
|
( wa jca syl321anc ) ABCDEFGHRIJKLMNAGHOPSQT $.
|
|
$}
|
|
|
|
sylXanc.8 $e |- ( ph -> rh ) $.
|
|
${
|
|
syl233anc.9 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si /\
|
|
rh ) ) -> mu ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl233anc $p |- ( ph -> mu ) $=
|
|
( wa jca syl133anc ) ABCTDEFGHIJABCKLUAMNOPQRSUB $.
|
|
$}
|
|
|
|
${
|
|
syl323anc.9 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ( ze /\ si /\
|
|
rh ) ) -> mu ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl323anc $p |- ( ph -> mu ) $=
|
|
( wa jca syl313anc ) ABCDEFTGHIJKLMAEFNOUAPQRSUB $.
|
|
$}
|
|
|
|
${
|
|
syl332anc.9 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ ( si /\
|
|
rh ) ) -> mu ) $.
|
|
$( Syllogism combined with contraction. (Contributed by NM,
|
|
11-Mar-2012.) $)
|
|
syl332anc $p |- ( ph -> mu ) $=
|
|
( wa jca syl331anc ) ABCDEFGHITJKLMNOPAHIQRUASUB $.
|
|
$}
|
|
|
|
sylXanc.9 $e |- ( ph -> mu ) $.
|
|
${
|
|
syl333anc.10 $e |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze )
|
|
/\ ( si /\ rh /\ mu ) ) -> la ) $.
|
|
$( A syllogism inference combined with contraction. (Contributed by NM,
|
|
10-Mar-2012.) $)
|
|
syl333anc $p |- ( ph -> la ) $=
|
|
( w3a 3jca syl331anc ) ABCDEFGHIJUBKLMNOPQAHIJRSTUCUAUD $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
syl3an1.1 $e |- ( ph -> ps ) $.
|
|
syl3an1.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an1 $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
|
|
( w3a 3anim1i syl ) ACDHBCDHEABCDFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl3an2.1 $e |- ( ph -> ch ) $.
|
|
syl3an2.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an2 $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
|
|
( wi 3exp syl5 3imp ) BADEACBDEHFBCDEGIJK $.
|
|
$}
|
|
|
|
${
|
|
syl3an3.1 $e |- ( ph -> th ) $.
|
|
syl3an3.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an3 $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
|
|
( 3exp syl7 3imp ) BCAEADBCEFBCDEGHIJ $.
|
|
$}
|
|
|
|
${
|
|
syl3an1b.1 $e |- ( ph <-> ps ) $.
|
|
syl3an1b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an1b $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
|
|
( biimpi syl3an1 ) ABCDEABFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an2b.1 $e |- ( ph <-> ch ) $.
|
|
syl3an2b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an2b $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
|
|
( biimpi syl3an2 ) ABCDEACFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an3b.1 $e |- ( ph <-> th ) $.
|
|
syl3an3b.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an3b $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
|
|
( biimpi syl3an3 ) ABCDEADFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an1br.1 $e |- ( ps <-> ph ) $.
|
|
syl3an1br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an1br $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
|
|
( biimpri syl3an1 ) ABCDEBAFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an2br.1 $e |- ( ch <-> ph ) $.
|
|
syl3an2br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an2br $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
|
|
( biimpri syl3an2 ) ABCDECAFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an3br.1 $e |- ( th <-> ph ) $.
|
|
syl3an3br.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 22-Aug-1995.) $)
|
|
syl3an3br $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
|
|
( biimpri syl3an3 ) ABCDEDAFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl3an.1 $e |- ( ph -> ps ) $.
|
|
syl3an.2 $e |- ( ch -> th ) $.
|
|
syl3an.3 $e |- ( ta -> et ) $.
|
|
syl3an.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
|
|
$( A triple syllogism inference. (Contributed by NM, 13-May-2004.) $)
|
|
syl3an $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
|
|
( w3a 3anim123i syl ) ACELBDFLGABCDEFHIJMKN $.
|
|
$}
|
|
|
|
${
|
|
syl3anb.1 $e |- ( ph <-> ps ) $.
|
|
syl3anb.2 $e |- ( ch <-> th ) $.
|
|
syl3anb.3 $e |- ( ta <-> et ) $.
|
|
syl3anb.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
|
|
$( A triple syllogism inference. (Contributed by NM, 15-Oct-2005.) $)
|
|
syl3anb $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
|
|
( w3a 3anbi123i sylbi ) ACELBDFLGABCDEFHIJMKN $.
|
|
$}
|
|
|
|
${
|
|
syl3anbr.1 $e |- ( ps <-> ph ) $.
|
|
syl3anbr.2 $e |- ( th <-> ch ) $.
|
|
syl3anbr.3 $e |- ( et <-> ta ) $.
|
|
syl3anbr.4 $e |- ( ( ps /\ th /\ et ) -> ze ) $.
|
|
$( A triple syllogism inference. (Contributed by NM, 29-Dec-2011.) $)
|
|
syl3anbr $p |- ( ( ph /\ ch /\ ta ) -> ze ) $=
|
|
( bicomi syl3anb ) ABCDEFGBAHLDCILFEJLKM $.
|
|
$}
|
|
|
|
${
|
|
syld3an3.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
syld3an3.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 20-May-2007.) $)
|
|
syld3an3 $p |- ( ( ph /\ ps /\ ch ) -> ta ) $=
|
|
( w3a simp1 simp2 syl3anc ) ABCHABDEABCIABCJFGK $.
|
|
$}
|
|
|
|
${
|
|
syld3an1.1 $e |- ( ( ch /\ ps /\ th ) -> ph ) $.
|
|
syld3an1.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 7-Jul-2008.) $)
|
|
syld3an1 $p |- ( ( ch /\ ps /\ th ) -> ta ) $=
|
|
( 3com13 syld3an3 ) DBCEDBCAECBDAFHABDEGHIH $.
|
|
$}
|
|
|
|
${
|
|
syld3an2.1 $e |- ( ( ph /\ ch /\ th ) -> ps ) $.
|
|
syld3an2.2 $e |- ( ( ph /\ ps /\ th ) -> ta ) $.
|
|
$( A syllogism inference. (Contributed by NM, 20-May-2007.) $)
|
|
syld3an2 $p |- ( ( ph /\ ch /\ th ) -> ta ) $=
|
|
( 3com23 syld3an3 ) ADCEADCBEACDBFHABDEGHIH $.
|
|
$}
|
|
|
|
${
|
|
syl3anl1.1 $e |- ( ph -> ps ) $.
|
|
syl3anl1.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 24-Feb-2005.) $)
|
|
syl3anl1 $p |- ( ( ( ph /\ ch /\ th ) /\ ta ) -> et ) $=
|
|
( w3a 3anim1i sylan ) ACDIBCDIEFABCDGJHK $.
|
|
$}
|
|
|
|
${
|
|
syl3anl2.1 $e |- ( ph -> ch ) $.
|
|
syl3anl2.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 24-Feb-2005.) $)
|
|
syl3anl2 $p |- ( ( ( ps /\ ph /\ th ) /\ ta ) -> et ) $=
|
|
( w3a wi ex syl3an2 imp ) BADIEFABCDEFJGBCDIEFHKLM $.
|
|
$}
|
|
|
|
${
|
|
syl3anl3.1 $e |- ( ph -> th ) $.
|
|
syl3anl3.2 $e |- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 24-Feb-2005.) $)
|
|
syl3anl3 $p |- ( ( ( ps /\ ch /\ ph ) /\ ta ) -> et ) $=
|
|
( w3a 3anim3i sylan ) BCAIBCDIEFADBCGJHK $.
|
|
$}
|
|
|
|
${
|
|
syl3anl.1 $e |- ( ph -> ps ) $.
|
|
syl3anl.2 $e |- ( ch -> th ) $.
|
|
syl3anl.3 $e |- ( ta -> et ) $.
|
|
syl3anl.4 $e |- ( ( ( ps /\ th /\ et ) /\ ze ) -> si ) $.
|
|
$( A triple syllogism inference. (Contributed by NM, 24-Dec-2006.) $)
|
|
syl3anl $p |- ( ( ( ph /\ ch /\ ta ) /\ ze ) -> si ) $=
|
|
( w3a 3anim123i sylan ) ACEMBDFMGHABCDEFIJKNLO $.
|
|
$}
|
|
|
|
${
|
|
syl3anr1.1 $e |- ( ph -> ps ) $.
|
|
syl3anr1.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 31-Jul-2007.) $)
|
|
syl3anr1 $p |- ( ( ch /\ ( ph /\ th /\ ta ) ) -> et ) $=
|
|
( w3a 3anim1i sylan2 ) ADEICBDEIFABDEGJHK $.
|
|
$}
|
|
|
|
${
|
|
syl3anr2.1 $e |- ( ph -> th ) $.
|
|
syl3anr2.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 1-Aug-2007.) $)
|
|
syl3anr2 $p |- ( ( ch /\ ( ps /\ ph /\ ta ) ) -> et ) $=
|
|
( w3a ancoms syl3anl2 ) BAEICFABDECFGCBDEIFHJKJ $.
|
|
$}
|
|
|
|
${
|
|
syl3anr3.1 $e |- ( ph -> ta ) $.
|
|
syl3anr3.2 $e |- ( ( ch /\ ( ps /\ th /\ ta ) ) -> et ) $.
|
|
$( A syllogism inference. (Contributed by NM, 23-Aug-2007.) $)
|
|
syl3anr3 $p |- ( ( ch /\ ( ps /\ th /\ ph ) ) -> et ) $=
|
|
( w3a 3anim3i sylan2 ) BDAICBDEIFAEBDGJHK $.
|
|
$}
|
|
|
|
${
|
|
3impdi.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) ) -> th ) $.
|
|
$( Importation inference (undistribute conjunction). (Contributed by NM,
|
|
14-Aug-1995.) $)
|
|
3impdi $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( anandis 3impb ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3impdir.1 $e |- ( ( ( ph /\ ps ) /\ ( ch /\ ps ) ) -> th ) $.
|
|
$( Importation inference (undistribute conjunction). (Contributed by NM,
|
|
20-Aug-1995.) $)
|
|
3impdir $p |- ( ( ph /\ ch /\ ps ) -> th ) $=
|
|
( anandirs 3impa ) ACBDACBDEFG $.
|
|
$}
|
|
|
|
${
|
|
3anidm12.1 $e |- ( ( ph /\ ph /\ ps ) -> ch ) $.
|
|
$( Inference from idempotent law for conjunction. (Contributed by NM,
|
|
7-Mar-2008.) $)
|
|
3anidm12 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( 3expib anabsi5 ) ABCAABCDEF $.
|
|
$}
|
|
|
|
${
|
|
3anidm13.1 $e |- ( ( ph /\ ps /\ ph ) -> ch ) $.
|
|
$( Inference from idempotent law for conjunction. (Contributed by NM,
|
|
7-Mar-2008.) $)
|
|
3anidm13 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( 3com23 3anidm12 ) ABCABACDEF $.
|
|
$}
|
|
|
|
${
|
|
3anidm23.1 $e |- ( ( ph /\ ps /\ ps ) -> ch ) $.
|
|
$( Inference from idempotent law for conjunction. (Contributed by NM,
|
|
1-Feb-2007.) $)
|
|
3anidm23 $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( 3expa anabss3 ) ABCABBCDEF $.
|
|
$}
|
|
|
|
${
|
|
3ori.1 $e |- ( ph \/ ps \/ ch ) $.
|
|
$( Infer implication from triple disjunction. (Contributed by NM,
|
|
26-Sep-2006.) $)
|
|
3ori $p |- ( ( -. ph /\ -. ps ) -> ch ) $=
|
|
( wn wa wo ioran w3o df-3or mpbi ori sylbir ) AEBEFABGZECABHNCABCINCGDABC
|
|
JKLM $.
|
|
$}
|
|
|
|
$( Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.) $)
|
|
3jao $p |- ( ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) ->
|
|
( ( ph \/ ch \/ th ) -> ps ) ) $=
|
|
( w3o wo wi w3a df-3or jao syl6 3imp syl5bi ) ACDEACFZDFZABGZCBGZDBGZHBACDI
|
|
PQROBGZPQNBGRSGABCJNBDJKLM $.
|
|
|
|
$( Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.) $)
|
|
3jaob $p |- ( ( ( ph \/ ch \/ th ) -> ps ) <->
|
|
( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) ) $=
|
|
( w3o wi w3a 3mix1 imim1i 3mix2 3mix3 3jca 3jao impbii ) ACDEZBFZABFZCBFZDB
|
|
FZGPQRSAOBACDHICOBCADJIDOBDACKILABCDMN $.
|
|
|
|
${
|
|
3jaoi.1 $e |- ( ph -> ps ) $.
|
|
3jaoi.2 $e |- ( ch -> ps ) $.
|
|
3jaoi.3 $e |- ( th -> ps ) $.
|
|
$( Disjunction of 3 antecedents (inference). (Contributed by NM,
|
|
12-Sep-1995.) $)
|
|
3jaoi $p |- ( ( ph \/ ch \/ th ) -> ps ) $=
|
|
( wi w3a w3o 3pm3.2i 3jao ax-mp ) ABHZCBHZDBHZIACDJBHNOPEFGKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
3jaod.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3jaod.2 $e |- ( ph -> ( th -> ch ) ) $.
|
|
3jaod.3 $e |- ( ph -> ( ta -> ch ) ) $.
|
|
$( Disjunction of 3 antecedents (deduction). (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
3jaod $p |- ( ph -> ( ( ps \/ th \/ ta ) -> ch ) ) $=
|
|
( wi w3o 3jao syl3anc ) ABCIDCIECIBDEJCIFGHBCDEKL $.
|
|
$}
|
|
|
|
${
|
|
3jaoian.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
3jaoian.2 $e |- ( ( th /\ ps ) -> ch ) $.
|
|
3jaoian.3 $e |- ( ( ta /\ ps ) -> ch ) $.
|
|
$( Disjunction of 3 antecedents (inference). (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
3jaoian $p |- ( ( ( ph \/ th \/ ta ) /\ ps ) -> ch ) $=
|
|
( w3o wi ex 3jaoi imp ) ADEIBCABCJDEABCFKDBCGKEBCHKLM $.
|
|
$}
|
|
|
|
${
|
|
3jaodan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
3jaodan.2 $e |- ( ( ph /\ th ) -> ch ) $.
|
|
3jaodan.3 $e |- ( ( ph /\ ta ) -> ch ) $.
|
|
$( Disjunction of 3 antecedents (deduction). (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
3jaodan $p |- ( ( ph /\ ( ps \/ th \/ ta ) ) -> ch ) $=
|
|
( w3o ex 3jaod imp ) ABDEICABCDEABCFJADCGJAECHJKL $.
|
|
$}
|
|
|
|
${
|
|
3jaao.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3jaao.2 $e |- ( th -> ( ta -> ch ) ) $.
|
|
3jaao.3 $e |- ( et -> ( ze -> ch ) ) $.
|
|
$( Inference conjoining and disjoining the antecedents of three
|
|
implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
3jaao $p |- ( ( ph /\ th /\ et ) -> ( ( ps \/ ta \/ ze ) -> ch ) ) $=
|
|
( w3a wi 3ad2ant1 3ad2ant2 3ad2ant3 3jaod ) ADFKBCEGADBCLFHMDAECLFINFAGCL
|
|
DJOP $.
|
|
$}
|
|
|
|
${
|
|
syl3an9b.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
syl3an9b.2 $e |- ( th -> ( ch <-> ta ) ) $.
|
|
syl3an9b.3 $e |- ( et -> ( ta <-> ze ) ) $.
|
|
$( Nested syllogism inference conjoining 3 dissimilar antecedents.
|
|
(Contributed by NM, 1-May-1995.) $)
|
|
syl3an9b $p |- ( ( ph /\ th /\ et ) -> ( ps <-> ze ) ) $=
|
|
( wb wa sylan9bb 3impa ) ADFBGKADLBEFGABCDEHIMJMN $.
|
|
$}
|
|
|
|
${
|
|
bi3d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
bi3d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
bi3d.3 $e |- ( ph -> ( et <-> ze ) ) $.
|
|
$( Deduction joining 3 equivalences to form equivalence of disjunctions.
|
|
(Contributed by NM, 20-Apr-1994.) $)
|
|
3orbi123d $p |- ( ph -> ( ( ps \/ th \/ et ) <-> ( ch \/ ta \/ ze ) ) ) $=
|
|
( wo w3o orbi12d df-3or 3bitr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
|
|
NCEGNO $.
|
|
|
|
$( Deduction joining 3 equivalences to form equivalence of conjunctions.
|
|
(Contributed by NM, 22-Apr-1994.) $)
|
|
3anbi123d $p |- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ ze ) ) ) $=
|
|
( wa w3a anbi12d df-3an 3bitr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
|
|
NCEGNO $.
|
|
$}
|
|
|
|
${
|
|
3anbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
3anbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Deduction conjoining and adding a conjunct to equivalences.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi12d $p |- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ et ) ) ) $=
|
|
( biidd 3anbi123d ) ABCDEFFGHAFIJ $.
|
|
|
|
$( Deduction conjoining and adding a conjunct to equivalences.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi13d $p |- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ ta ) ) ) $=
|
|
( biidd 3anbi123d ) ABCFFDEGAFIHJ $.
|
|
|
|
$( Deduction conjoining and adding a conjunct to equivalences.
|
|
(Contributed by NM, 8-Sep-2006.) $)
|
|
3anbi23d $p |- ( ph -> ( ( et /\ ps /\ th ) <-> ( et /\ ch /\ ta ) ) ) $=
|
|
( biidd 3anbi123d ) AFFBCDEAFIGHJ $.
|
|
$}
|
|
|
|
${
|
|
3anbi1d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction adding conjuncts to an equivalence. (Contributed by NM,
|
|
8-Sep-2006.) $)
|
|
3anbi1d $p |- ( ph -> ( ( ps /\ th /\ ta ) <-> ( ch /\ th /\ ta ) ) ) $=
|
|
( biidd 3anbi12d ) ABCDDEFADGH $.
|
|
|
|
$( Deduction adding conjuncts to an equivalence. (Contributed by NM,
|
|
8-Sep-2006.) $)
|
|
3anbi2d $p |- ( ph -> ( ( th /\ ps /\ ta ) <-> ( th /\ ch /\ ta ) ) ) $=
|
|
( biidd 3anbi12d ) ADDBCEADGFH $.
|
|
|
|
$( Deduction adding conjuncts to an equivalence. (Contributed by NM,
|
|
8-Sep-2006.) $)
|
|
3anbi3d $p |- ( ph -> ( ( th /\ ta /\ ps ) <-> ( th /\ ta /\ ch ) ) ) $=
|
|
( biidd 3anbi13d ) ADDBCEADGFH $.
|
|
$}
|
|
|
|
${
|
|
3anim123d.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
3anim123d.2 $e |- ( ph -> ( th -> ta ) ) $.
|
|
3anim123d.3 $e |- ( ph -> ( et -> ze ) ) $.
|
|
$( Deduction joining 3 implications to form implication of conjunctions.
|
|
(Contributed by NM, 24-Feb-2005.) $)
|
|
3anim123d $p |- ( ph -> ( ( ps /\ th /\ et ) -> ( ch /\ ta /\ ze ) ) ) $=
|
|
( wa w3a anim12d df-3an 3imtr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
|
|
NCEGNO $.
|
|
|
|
$( Deduction joining 3 implications to form implication of disjunctions.
|
|
(Contributed by NM, 4-Apr-1997.) $)
|
|
3orim123d $p |- ( ph -> ( ( ps \/ th \/ et ) -> ( ch \/ ta \/ ze ) ) ) $=
|
|
( wo w3o orim12d df-3or 3imtr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJMBDF
|
|
NCEGNO $.
|
|
$}
|
|
|
|
$( Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) $)
|
|
an6 $p |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <->
|
|
( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) ) $=
|
|
( wa w3a an4 anbi1i bitri df-3an anbi12i 3bitr4i ) ABGZCGZDEGZFGZGZADGZBEGZ
|
|
GZCFGZGZABCHZDEFHZGTUAUCHSOQGZUCGUDOCQFIUGUBUCABDEIJKUEPUFRABCLDEFLMTUAUCLN
|
|
$.
|
|
|
|
$( Analog of ~ an4 for triple conjunction. (Contributed by Scott Fenton,
|
|
16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3an6 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) /\ ( ta /\ et ) ) <->
|
|
( ( ph /\ ch /\ ta ) /\ ( ps /\ th /\ et ) ) ) $=
|
|
( w3a wa an6 bicomi ) ACEGBDFGHABHCDHEFHGACEBDFIJ $.
|
|
|
|
$( Analog of ~ or4 for triple conjunction. (Contributed by Scott Fenton,
|
|
16-Mar-2011.) $)
|
|
3or6 $p |- ( ( ( ph \/ ps ) \/ ( ch \/ th ) \/ ( ta \/ et ) ) <->
|
|
( ( ph \/ ch \/ ta ) \/ ( ps \/ th \/ et ) ) ) $=
|
|
( wo w3o or4 orbi1i bitr2i df-3or orbi12i 3bitr4i ) ABGZCDGZGZEFGZGZACGZEGZ
|
|
BDGZFGZGZOPRHACEHZBDFHZGUDTUBGZRGSTEUBFIUGQRACBDIJKOPRLUEUAUFUCACELBDFLMN
|
|
$.
|
|
|
|
${
|
|
mp3an1.1 $e |- ph $.
|
|
mp3an1.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
21-Nov-1994.) $)
|
|
mp3an1 $p |- ( ( ps /\ ch ) -> th ) $=
|
|
( wa 3expb mpan ) ABCGDEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an2.1 $e |- ps $.
|
|
mp3an2.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
21-Nov-1994.) $)
|
|
mp3an2 $p |- ( ( ph /\ ch ) -> th ) $=
|
|
( 3expa mpanl2 ) ABCDEABCDFGH $.
|
|
$}
|
|
|
|
${
|
|
mp3an3.1 $e |- ch $.
|
|
mp3an3.2 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
21-Nov-1994.) $)
|
|
mp3an3 $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa 3expia mpi ) ABGCDEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an12.1 $e |- ph $.
|
|
mp3an12.2 $e |- ps $.
|
|
mp3an12.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
13-Jul-2005.) $)
|
|
mp3an12 $p |- ( ch -> th ) $=
|
|
( mp3an1 mpan ) BCDFABCDEGHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an13.1 $e |- ph $.
|
|
mp3an13.2 $e |- ch $.
|
|
mp3an13.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
14-Jul-2005.) $)
|
|
mp3an13 $p |- ( ps -> th ) $=
|
|
( mp3an3 mpan ) ABDEABCDFGHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an23.1 $e |- ps $.
|
|
mp3an23.2 $e |- ch $.
|
|
mp3an23.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
14-Jul-2005.) $)
|
|
mp3an23 $p |- ( ph -> th ) $=
|
|
( mp3an3 mpan2 ) ABDEABCDFGHI $.
|
|
$}
|
|
|
|
${
|
|
mp3an1i.1 $e |- ps $.
|
|
mp3an1i.2 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) $)
|
|
mp3an1i $p |- ( ph -> ( ( ch /\ th ) -> ta ) ) $=
|
|
( wa wi w3a com12 mp3an1 ) CDHAEBCDAEIFABCDJEGKLK $.
|
|
$}
|
|
|
|
${
|
|
mp3anl1.1 $e |- ph $.
|
|
mp3anl1.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
mp3anl1 $p |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $=
|
|
( wa wi w3a ex mp3an1 imp ) BCHDEABCDEIFABCJDEGKLM $.
|
|
$}
|
|
|
|
${
|
|
mp3anl2.1 $e |- ps $.
|
|
mp3anl2.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
mp3anl2 $p |- ( ( ( ph /\ ch ) /\ th ) -> ta ) $=
|
|
( wa wi w3a ex mp3an2 imp ) ACHDEABCDEIFABCJDEGKLM $.
|
|
$}
|
|
|
|
${
|
|
mp3anl3.1 $e |- ch $.
|
|
mp3anl3.2 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Feb-2005.) $)
|
|
mp3anl3 $p |- ( ( ( ph /\ ps ) /\ th ) -> ta ) $=
|
|
( wa wi w3a ex mp3an3 imp ) ABHDEABCDEIFABCJDEGKLM $.
|
|
$}
|
|
|
|
${
|
|
mp3anr1.1 $e |- ps $.
|
|
mp3anr1.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) $)
|
|
mp3anr1 $p |- ( ( ph /\ ( ch /\ th ) ) -> ta ) $=
|
|
( wa w3a ancoms mp3anl1 ) CDHAEBCDAEFABCDIEGJKJ $.
|
|
$}
|
|
|
|
${
|
|
mp3anr2.1 $e |- ch $.
|
|
mp3anr2.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
24-Nov-2006.) $)
|
|
mp3anr2 $p |- ( ( ph /\ ( ps /\ th ) ) -> ta ) $=
|
|
( wa w3a ancoms mp3anl2 ) BDHAEBCDAEFABCDIEGJKJ $.
|
|
$}
|
|
|
|
${
|
|
mp3anr3.1 $e |- th $.
|
|
mp3anr3.2 $e |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
19-Oct-2007.) $)
|
|
mp3anr3 $p |- ( ( ph /\ ( ps /\ ch ) ) -> ta ) $=
|
|
( wa w3a ancoms mp3anl3 ) BCHAEBCDAEFABCDIEGJKJ $.
|
|
$}
|
|
|
|
${
|
|
mp3an.1 $e |- ph $.
|
|
mp3an.2 $e |- ps $.
|
|
mp3an.3 $e |- ch $.
|
|
mp3an.4 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM,
|
|
14-May-1999.) $)
|
|
mp3an $p |- th $=
|
|
( mp3an1 mp2an ) BCDFGABCDEHIJ $.
|
|
$}
|
|
|
|
${
|
|
mpd3an3.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
mpd3an3.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) $)
|
|
mpd3an3 $p |- ( ( ph /\ ps ) -> th ) $=
|
|
( wa 3expa mpdan ) ABGCDEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
mpd3an23.1 $e |- ( ph -> ps ) $.
|
|
mpd3an23.2 $e |- ( ph -> ch ) $.
|
|
mpd3an23.3 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) $)
|
|
mpd3an23 $p |- ( ph -> th ) $=
|
|
( id syl3anc ) AABCDAHEFGI $.
|
|
$}
|
|
|
|
${
|
|
mp3and.1 $e |- ( ph -> ps ) $.
|
|
mp3and.2 $e |- ( ph -> ch ) $.
|
|
mp3and.3 $e |- ( ph -> th ) $.
|
|
mp3and.4 $e |- ( ph -> ( ( ps /\ ch /\ th ) -> ta ) ) $.
|
|
$( A deduction based on modus ponens. (Contributed by Mario Carneiro,
|
|
24-Dec-2016.) $)
|
|
mp3and $p |- ( ph -> ta ) $=
|
|
( w3a 3jca mpd ) ABCDJEABCDFGHKIL $.
|
|
$}
|
|
|
|
${
|
|
biimp3a.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
|
|
$( Infer implication from a logical equivalence. Similar to ~ biimpa .
|
|
(Contributed by NM, 4-Sep-2005.) $)
|
|
biimp3a $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( wa biimpa 3impa ) ABCDABFCDEGH $.
|
|
|
|
$( Infer implication from a logical equivalence. Similar to ~ biimpar .
|
|
(Contributed by NM, 2-Jan-2009.) $)
|
|
biimp3ar $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
|
|
( exbiri 3imp ) ABDCABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
3anandis.1 $e |- ( ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) )
|
|
-> ta ) $.
|
|
$( Inference that undistributes a triple conjunction in the antecedent.
|
|
(Contributed by NM, 18-Apr-2007.) $)
|
|
3anandis $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ta ) $=
|
|
( w3a wa simpl simpr1 simpr2 simpr3 syl222anc ) ABCDGZHABACADEANIZABCDJOA
|
|
BCDKOABCDLFM $.
|
|
$}
|
|
|
|
${
|
|
3anandirs.1 $e |- ( ( ( ph /\ th ) /\ ( ps /\ th ) /\ ( ch /\ th ) )
|
|
-> ta ) $.
|
|
$( Inference that undistributes a triple conjunction in the antecedent.
|
|
(Contributed by NM, 25-Jul-2006.) $)
|
|
3anandirs $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $=
|
|
( w3a wa simpl1 simpr simpl2 simpl3 syl222anc ) ABCGZDHADBDCDEABCDINDJZAB
|
|
CDKOABCDLOFM $.
|
|
$}
|
|
|
|
${
|
|
ecase23d.1 $e |- ( ph -> -. ch ) $.
|
|
ecase23d.2 $e |- ( ph -> -. th ) $.
|
|
ecase23d.3 $e |- ( ph -> ( ps \/ ch \/ th ) ) $.
|
|
$( Deduction for elimination by cases. (Contributed by NM,
|
|
22-Apr-1994.) $)
|
|
ecase23d $p |- ( ph -> ps ) $=
|
|
( wo wn ioran sylanbrc w3o 3orass sylib ord mt3d ) ABCDHZACIDIQIEFCDJKABQ
|
|
ABCDLBQHGBCDMNOP $.
|
|
$}
|
|
|
|
${
|
|
3ecase.1 $e |- ( -. ph -> th ) $.
|
|
3ecase.2 $e |- ( -. ps -> th ) $.
|
|
3ecase.3 $e |- ( -. ch -> th ) $.
|
|
3ecase.4 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
|
|
$( Inference for elimination by cases. (Contributed by NM,
|
|
13-Jul-2005.) $)
|
|
3ecase $p |- th $=
|
|
( wi 3exp wn a1d pm2.61i pm2.61nii ) BCDABCDIZIABCDHJAKZOBPDCELLMFGN $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical 'nand' (Sheffer stroke)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare connective for alternative denial ('nand'). $)
|
|
$c -/\ $. $( Overlined 'wedge' (read: 'nand') $)
|
|
|
|
$( Extend wff definition to include alternative denial ('nand'). $)
|
|
wnan $a wff ( ph -/\ ps ) $.
|
|
|
|
$( Define incompatibility, or alternative denial ('not-and' or 'nand'). This
|
|
is also called the Sheffer stroke, represented by a vertical bar, but we
|
|
use a different symbol to avoid ambiguity with other uses of the vertical
|
|
bar. In the second edition of Principia Mathematica (1927), Russell and
|
|
Whitehead used the Sheffer stroke and suggested it as a replacement for
|
|
the "or" and "not" operations of the first edition. However, in practice,
|
|
"or" and "not" are more widely used. After we define the constant true
|
|
` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be
|
|
able to prove these truth table values: ` ( ( T. -/\ T. ) <-> F. ) `
|
|
( ~ trunantru ), ` ( ( T. -/\ F. ) <-> T. ) ` ( ~ trunanfal ),
|
|
` ( ( F. -/\ T. ) <-> T. ) ` ( ~ falnantru ), and
|
|
` ( ( F. -/\ F. ) <-> T. ) ` ( ~ falnanfal ). Contrast with ` /\ `
|
|
( ~ df-an ), ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), and ` \/_ `
|
|
( ~ df-xor ) . (Contributed by Jeff Hoffman, 19-Nov-2007.) $)
|
|
df-nan $a |- ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) ) $.
|
|
|
|
$( Write 'and' in terms of 'nand'. (Contributed by Mario Carneiro,
|
|
9-May-2015.) $)
|
|
nanan $p |- ( ( ph /\ ps ) <-> -. ( ph -/\ ps ) ) $=
|
|
( wnan wa df-nan con2bii ) ABCABDABEF $.
|
|
|
|
$( The 'nand' operator commutes. (Contributed by Mario Carneiro,
|
|
9-May-2015.) $)
|
|
nancom $p |- ( ( ph -/\ ps ) <-> ( ps -/\ ph ) ) $=
|
|
( wa wn wnan ancom notbii df-nan 3bitr4i ) ABCZDBACZDABEBAEJKABFGABHBAHI $.
|
|
|
|
$( Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman,
|
|
19-Nov-2007.) $)
|
|
nannan $p |- ( ( ph -/\ ( ch -/\ ps ) ) <-> ( ph -> ( ch /\ ps ) ) ) $=
|
|
( wnan wa wn wi df-nan anbi2i xchbinx iman bitr4i ) ACBDZDZACBEZFZEZFAOGNAM
|
|
EQAMHMPACBHIJAOKL $.
|
|
|
|
$( Show equivalence between implication and the Nicod version. To derive
|
|
~ nic-dfim , apply ~ nanbi . (Contributed by Jeff Hoffman,
|
|
19-Nov-2007.) $)
|
|
nanim $p |- ( ( ph -> ps ) <-> ( ph -/\ ( ps -/\ ps ) ) ) $=
|
|
( wnan wa wi nannan anidmdbi bitr2i ) ABBCCABBDEABEABBFABGH $.
|
|
|
|
$( Show equivalence between negation and the Nicod version. To derive
|
|
~ nic-dfneg , apply ~ nanbi . (Contributed by Jeff Hoffman,
|
|
19-Nov-2007.) $)
|
|
nannot $p |- ( -. ps <-> ( ps -/\ ps ) ) $=
|
|
( wnan wn wa df-nan anidm xchbinx bicomi ) AABZACIAADAAAEAFGH $.
|
|
|
|
$( Show equivalence between the bidirectional and the Nicod version.
|
|
(Contributed by Jeff Hoffman, 19-Nov-2007.) $)
|
|
nanbi $p |- ( ( ph <-> ps ) <->
|
|
( ( ph -/\ ps ) -/\ ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) ) ) $=
|
|
( wa wn wo wb pm4.57 df-nan nannot anbi12i xchbinxr xchbinx dfbi3 3bitr4ri
|
|
wnan ) ABCZDZADZBDZCZDZCZDPTEABOZAAOZBBOZOZOZABFPTGUGUCUFCUBUCUFHUCQUFUAABH
|
|
UFUDUECTUDUEHRUDSUEAIBIJKJLABMN $.
|
|
|
|
$( Introduce a right anti-conjunct to both sides of a logical equivalence.
|
|
(Contributed by SF, 2-Jan-2018.) $)
|
|
nanbi1 $p |- ( ( ph <-> ps ) -> ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) ) $=
|
|
( wb wa wn wnan anbi1 notbid df-nan 3bitr4g ) ABDZACEZFBCEZFACGBCGLMNABCHIA
|
|
CJBCJK $.
|
|
|
|
$( Introduce a left anti-conjunct to both sides of a logical equivalence.
|
|
(Contributed by SF, 2-Jan-2018.) $)
|
|
nanbi2 $p |- ( ( ph <-> ps ) -> ( ( ch -/\ ph ) <-> ( ch -/\ ps ) ) ) $=
|
|
( wb wnan nanbi1 nancom 3bitr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Join two logical equivalences with anti-conjunction. (Contributed by SF,
|
|
2-Jan-2018.) $)
|
|
nanbi12 $p |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) ->
|
|
( ( ph -/\ ch ) <-> ( ps -/\ th ) ) ) $=
|
|
( wb wnan nanbi1 nanbi2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.
|
|
|
|
${
|
|
nanbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Introduce a right anti-conjunct to both sides of a logical equivalence.
|
|
(Contributed by SF, 2-Jan-2018.) $)
|
|
nanbi1i $p |- ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) $=
|
|
( wb wnan nanbi1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Introduce a left anti-conjunct to both sides of a logical equivalence.
|
|
(Contributed by SF, 2-Jan-2018.) $)
|
|
nanbi2i $p |- ( ( ch -/\ ph ) <-> ( ch -/\ ps ) ) $=
|
|
( wb wnan nanbi2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
nanbi12i.2 $e |- ( ch <-> th ) $.
|
|
$( Join two logical equivalences with anti-conjunction. (Contributed by
|
|
SF, 2-Jan-2018.) $)
|
|
nanbi12i $p |- ( ( ph -/\ ch ) <-> ( ps -/\ th ) ) $=
|
|
( wb wnan nanbi12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
|
|
$}
|
|
|
|
${
|
|
nanbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Introduce a right anti-conjunct to both sides of a logical equivalence.
|
|
(Contributed by SF, 2-Jan-2018.) $)
|
|
nanbi1d $p |- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ th ) ) ) $=
|
|
( wb wnan nanbi1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Introduce a left anti-conjunct to both sides of a logical equivalence.
|
|
(Contributed by SF, 2-Jan-2018.) $)
|
|
nanbi2d $p |- ( ph -> ( ( th -/\ ps ) <-> ( th -/\ ch ) ) ) $=
|
|
( wb wnan nanbi2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
nanbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Join two logical equivalences with anti-conjunction. (Contributed by
|
|
Scott Fenton, 2-Jan-2018.) $)
|
|
nanbi12d $p |- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ ta ) ) ) $=
|
|
( wb wnan nanbi12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical 'xor'
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare connective for exclusive disjunction ('xor'). $)
|
|
$c \/_ $. $( Underlined 'vee' (read: 'xor') $)
|
|
|
|
$( Extend wff definition to include exclusive disjunction ('xor'). $)
|
|
wxo $a wff ( ph \/_ ps ) $.
|
|
|
|
$( Define exclusive disjunction (logical 'xor'). Return true if either the
|
|
left or right, but not both, are true. After we define the constant true
|
|
` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be
|
|
able to prove these truth table values: ` ( ( T. \/_ T. ) <-> F. ) `
|
|
( ~ truxortru ), ` ( ( T. \/_ F. ) <-> T. ) ` ( ~ truxorfal ),
|
|
` ( ( F. \/_ T. ) <-> T. ) ` ( ~ falxortru ), and
|
|
` ( ( F. \/_ F. ) <-> F. ) ` ( ~ falxorfal ). Contrast with ` /\ `
|
|
( ~ df-an ), ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), and ` -/\ `
|
|
( ~ df-nan ) . (Contributed by FL, 22-Nov-2010.) $)
|
|
df-xor $a |- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) $.
|
|
|
|
$( Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.) $)
|
|
xnor $p |- ( ( ph <-> ps ) <-> -. ( ph \/_ ps ) ) $=
|
|
( wxo wb df-xor con2bii ) ABCABDABEF $.
|
|
|
|
$( ` \/_ ` is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) $)
|
|
xorcom $p |- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) ) $=
|
|
( wb wn wxo bicom notbii df-xor 3bitr4i ) ABCZDBACZDABEBAEJKABFGABHBAHI $.
|
|
|
|
$( ` \/_ ` is associative. (Contributed by FL, 22-Nov-2010.) (Proof
|
|
shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
xorass $p |- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) ) $=
|
|
( wxo wb wn biass notbii nbbn pm5.18 con2bii 3bitr4i df-xor bibi1i bibi2i )
|
|
ABDZCEZFABCDZEZFPCDARDQSABEZFZCEZABCEZFZEZQSTCEZFAUCEZFUBUEUFUGABCGHTCIUGUE
|
|
AUCJKLPUACABMNRUDABCMOLHPCMARML $.
|
|
|
|
$( This tautology shows that xor is really exclusive. (Contributed by FL,
|
|
22-Nov-2010.) $)
|
|
excxor $p |- ( ( ph \/_ ps ) <->
|
|
( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) $=
|
|
( wxo wb wn wa wo df-xor xor ancom orbi2i 3bitri ) ABCABDEABEFZBAEZFZGMNBFZ
|
|
GABHABIOPMBNJKL $.
|
|
|
|
$( Two ways to express "exclusive or." (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
xor2 $p |- ( ( ph \/_ ps ) <->
|
|
( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) $=
|
|
( wxo wb wn wo wa df-xor nbi2 bitri ) ABCABDEABFABGEGABHABIJ $.
|
|
|
|
$( ` \/_ ` is negated under negation of one argument. (Contributed by Mario
|
|
Carneiro, 4-Sep-2016.) $)
|
|
xorneg1 $p |- ( ( -. ph \/_ ps ) <-> -. ( ph \/_ ps ) ) $=
|
|
( wn wxo wb df-xor nbbn con2bii xnor 3bitr2i ) ACZBDKBEZCABEZABDCKBFLMABGHA
|
|
BIJ $.
|
|
|
|
$( ` \/_ ` is negated under negation of one argument. (Contributed by Mario
|
|
Carneiro, 4-Sep-2016.) $)
|
|
xorneg2 $p |- ( ( ph \/_ -. ps ) <-> -. ( ph \/_ ps ) ) $=
|
|
( wn wxo xorneg1 xorcom notbii 3bitr4i ) BCZADBADZCAIDABDZCBAEAIFKJABFGH $.
|
|
|
|
$( ` \/_ ` is unchanged under negation of both arguments. (Contributed by
|
|
Mario Carneiro, 4-Sep-2016.) $)
|
|
xorneg $p |- ( ( -. ph \/_ -. ps ) <-> ( ph \/_ ps ) ) $=
|
|
( wn wxo xorneg1 xorneg2 con2bii bitr4i ) ACBCZDAIDZCABDZAIEJKABFGH $.
|
|
|
|
${
|
|
xorbi12.1 $e |- ( ph <-> ps ) $.
|
|
xorbi12.2 $e |- ( ch <-> th ) $.
|
|
$( Equality property for XOR. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
xorbi12i $p |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) ) $=
|
|
( wb wn wxo bibi12i notbii df-xor 3bitr4i ) ACGZHBDGZHACIBDINOABCDEFJKACL
|
|
BDLM $.
|
|
$}
|
|
|
|
${
|
|
xor12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
xor12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
$( Equality property for XOR. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
xorbi12d $p |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) ) $=
|
|
( wb wn wxo bibi12d notbid df-xor 3bitr4g ) ABDHZICEHZIBDJCEJAOPABCDEFGKL
|
|
BDMCEMN $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
True and false constants
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c T. $.
|
|
$c F. $.
|
|
|
|
$( ` T. ` is a wff. $)
|
|
wtru $a wff T. $.
|
|
|
|
$( ` F. ` is a wff. $)
|
|
wfal $a wff F. $.
|
|
|
|
$( Soundness justification theorem for ~ df-tru . (Contributed by Mario
|
|
Carneiro, 17-Nov-2013.) $)
|
|
trujust $p |- ( ( ph <-> ph ) <-> ( ps <-> ps ) ) $=
|
|
( wb biid 2th ) AACBBCADBDE $.
|
|
|
|
$( Definition of ` T. ` , a tautology. ` T. ` is a constant true. In this
|
|
definition ~ biid is used as an antecedent, however, any true wff, such as
|
|
an axiom, can be used in its place. (Contributed by Anthony Hart,
|
|
13-Oct-2010.) $)
|
|
df-tru $a |- ( T. <-> ( ph <-> ph ) ) $.
|
|
|
|
$( Definition of ` F. ` , a contradiction. ` F. ` is a constant false.
|
|
(Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
df-fal $a |- ( F. <-> -. T. ) $.
|
|
|
|
$( ` T. ` is provable. (Contributed by Anthony Hart, 13-Oct-2010.) $)
|
|
tru $p |- T. $=
|
|
( wph wtru wb biid df-tru mpbir ) BAACADAEF $.
|
|
|
|
$( ` F. ` is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Mel L. O'Cat, 11-Mar-2012.) $)
|
|
fal $p |- -. F. $=
|
|
( wfal wtru wn tru notnoti df-fal mtbir ) ABCBDEFG $.
|
|
|
|
${
|
|
trud.1 $e |- ( T. -> ph ) $.
|
|
$( Eliminate ` T. ` as an antecedent. (Contributed by Mario Carneiro,
|
|
13-Mar-2014.) $)
|
|
trud $p |- ph $=
|
|
( wtru tru ax-mp ) CADBE $.
|
|
$}
|
|
|
|
$( If something is true, it outputs ` T. ` . (Contributed by Anthony Hart,
|
|
14-Aug-2011.) $)
|
|
tbtru $p |- ( ph <-> ( ph <-> T. ) ) $=
|
|
( wtru tru tbt ) BACD $.
|
|
|
|
$( If something is not true, it outputs ` F. ` . (Contributed by Anthony
|
|
Hart, 14-Aug-2011.) $)
|
|
nbfal $p |- ( -. ph <-> ( ph <-> F. ) ) $=
|
|
( wfal fal nbn ) BACD $.
|
|
|
|
${
|
|
bitru.1 $e |- ph $.
|
|
$( A theorem is equivalent to truth. (Contributed by Mario Carneiro,
|
|
9-May-2015.) $)
|
|
bitru $p |- ( ph <-> T. ) $=
|
|
( wtru tru 2th ) ACBDE $.
|
|
$}
|
|
|
|
${
|
|
bifal.1 $e |- -. ph $.
|
|
$( A contradiction is equivalent to falsehood. (Contributed by Mario
|
|
Carneiro, 9-May-2015.) $)
|
|
bifal $p |- ( ph <-> F. ) $=
|
|
( wfal fal 2false ) ACBDE $.
|
|
$}
|
|
|
|
$( ` F. ` implies anything. (Contributed by FL, 20-Mar-2011.) (Proof
|
|
shortened by Anthony Hart, 1-Aug-2011.) $)
|
|
falim $p |- ( F. -> ph ) $=
|
|
( wfal fal pm2.21i ) BACD $.
|
|
|
|
$( ` F. ` implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) $)
|
|
falimd $p |- ( ( ph /\ F. ) -> ps ) $=
|
|
( wfal falim adantl ) CBABDE $.
|
|
|
|
$( Anything implies ` T. ` . (Contributed by FL, 20-Mar-2011.) (Proof
|
|
shortened by Anthony Hart, 1-Aug-2011.) $)
|
|
a1tru $p |- ( ph -> T. ) $=
|
|
( wtru tru a1i ) BACD $.
|
|
|
|
$( True can be removed from a conjunction. (Contributed by FL,
|
|
20-Mar-2011.) $)
|
|
truan $p |- ( ( T. /\ ph ) <-> ph ) $=
|
|
( wtru wa simpr a1tru ancri impbii ) BACABADABAEFG $.
|
|
|
|
$( Given falsum, we can define the negation of a wff ` ph ` as the statement
|
|
that a contradiction follows from assuming ` ph ` . (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) $)
|
|
dfnot $p |- ( -. ph <-> ( ph -> F. ) ) $=
|
|
( wn wfal wi pm2.21 id falim ja impbii ) ABZACDACEACJJFJGHI $.
|
|
|
|
${
|
|
inegd.1 $e |- ( ( ph /\ ps ) -> F. ) $.
|
|
$( Negation introduction rule from natural deduction. (Contributed by
|
|
Mario Carneiro, 9-Feb-2017.) $)
|
|
inegd $p |- ( ph -> -. ps ) $=
|
|
( wfal wi wn ex dfnot sylibr ) ABDEBFABDCGBHI $.
|
|
$}
|
|
|
|
${
|
|
efald.1 $e |- ( ( ph /\ -. ps ) -> F. ) $.
|
|
$( Deduction based on reductio ad absurdum. (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) $)
|
|
efald $p |- ( ph -> ps ) $=
|
|
( wn inegd notnotrd ) ABABDCEF $.
|
|
$}
|
|
|
|
${
|
|
pm2.21fal.1 $e |- ( ph -> ps ) $.
|
|
pm2.21fal.2 $e |- ( ph -> -. ps ) $.
|
|
$( If a wff and its negation are provable, then falsum is provable.
|
|
(Contributed by Mario Carneiro, 9-Feb-2017.) $)
|
|
pm2.21fal $p |- ( ph -> F. ) $=
|
|
( wfal pm2.21dd ) ABECDF $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Truth tables
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Some sources define operations on true/false values using truth tables.
|
|
These tables show the results of their operations for all possible
|
|
combinations of true ( ` T. ` ) and false ( ` F. ` ).
|
|
Here we show that our definitions and axioms produce equivalent results for
|
|
` /\ ` (conjunction aka logical 'and') ~ df-an ,
|
|
` \/ ` (disjunction aka logical inclusive 'or') ~ df-or ,
|
|
` -> ` (implies) ~ wi ,
|
|
` -. ` (not) ~ wn ,
|
|
` <-> ` (logical equivalence) ~ df-bi ,
|
|
` -/\ ` (nand aka Sheffer stroke) ~ df-nan , and
|
|
` \/_ ` (exclusive or) ~ df-xor .
|
|
$)
|
|
|
|
$( A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
truantru $p |- ( ( T. /\ T. ) <-> T. ) $=
|
|
( wtru anidm ) AB $.
|
|
|
|
$( A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
truanfal $p |- ( ( T. /\ F. ) <-> F. ) $=
|
|
( wtru wfal wa fal intnan bifal ) ABCBADEF $.
|
|
|
|
$( A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falantru $p |- ( ( F. /\ T. ) <-> F. ) $=
|
|
( wfal wtru wa fal intnanr bifal ) ABCABDEF $.
|
|
|
|
$( A ` /\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falanfal $p |- ( ( F. /\ F. ) <-> F. ) $=
|
|
( wfal anidm ) AB $.
|
|
|
|
$( A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
truortru $p |- ( ( T. \/ T. ) <-> T. ) $=
|
|
( wtru oridm ) AB $.
|
|
|
|
$( A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
truorfal $p |- ( ( T. \/ F. ) <-> T. ) $=
|
|
( wtru wfal wo tru orci bitru ) ABCABDEF $.
|
|
|
|
$( A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falortru $p |- ( ( F. \/ T. ) <-> T. ) $=
|
|
( wfal wtru wo tru olci bitru ) ABCBADEF $.
|
|
|
|
$( A ` \/ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
falorfal $p |- ( ( F. \/ F. ) <-> F. ) $=
|
|
( wfal oridm ) AB $.
|
|
|
|
$( A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
truimtru $p |- ( ( T. -> T. ) <-> T. ) $=
|
|
( wtru wi id bitru ) AABACD $.
|
|
|
|
$( A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
truimfal $p |- ( ( T. -> F. ) <-> F. ) $=
|
|
( wfal wtru wi tru a1bi bicomi ) ABACBADEF $.
|
|
|
|
$( A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falimtru $p |- ( ( F. -> T. ) <-> T. ) $=
|
|
( wfal wtru wi falim bitru ) ABCBDE $.
|
|
|
|
$( A ` -> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
falimfal $p |- ( ( F. -> F. ) <-> T. ) $=
|
|
( wfal wi id bitru ) AABACD $.
|
|
|
|
$( A ` -. ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) $)
|
|
nottru $p |- ( -. T. <-> F. ) $=
|
|
( wfal wtru wn df-fal bicomi ) ABCDE $.
|
|
|
|
$( A ` -. ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
notfal $p |- ( -. F. <-> T. ) $=
|
|
( wfal wn fal bitru ) ABCD $.
|
|
|
|
$( A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
trubitru $p |- ( ( T. <-> T. ) <-> T. ) $=
|
|
( wtru wb biid bitru ) AABACD $.
|
|
|
|
$( A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
trubifal $p |- ( ( T. <-> F. ) <-> F. ) $=
|
|
( wtru wfal wb wn nottru nbbn mpbi bifal ) ABCZADBCIDEABFGH $.
|
|
|
|
$( A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
falbitru $p |- ( ( F. <-> T. ) <-> F. ) $=
|
|
( wfal wtru wb bicom trubifal bitri ) ABCBACAABDEF $.
|
|
|
|
$( A ` <-> ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
falbifal $p |- ( ( F. <-> F. ) <-> T. ) $=
|
|
( wfal wb biid bitru ) AABACD $.
|
|
|
|
$( A ` -/\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
trunantru $p |- ( ( T. -/\ T. ) <-> F. ) $=
|
|
( wtru wnan wn wfal nannot nottru bitr3i ) AABACDAEFG $.
|
|
|
|
$( A ` -/\ ` identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
trunanfal $p |- ( ( T. -/\ F. ) <-> T. ) $=
|
|
( wtru wfal wnan wa wn df-nan truanfal notbii notfal 3bitri ) ABCABDZEBEAAB
|
|
FKBGHIJ $.
|
|
|
|
$( A ` -/\ ` identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
falnantru $p |- ( ( F. -/\ T. ) <-> T. ) $=
|
|
( wfal wtru wnan nancom trunanfal bitri ) ABCBACBABDEF $.
|
|
|
|
$( A ` -/\ ` identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof
|
|
shortened by Andrew Salmon, 13-May-2011.) $)
|
|
falnanfal $p |- ( ( F. -/\ F. ) <-> T. ) $=
|
|
( wfal wnan wn wtru nannot notfal bitr3i ) AABACDAEFG $.
|
|
|
|
$( A ` \/_ ` identity. (Contributed by David A. Wheeler, 8-May-2015.) $)
|
|
truxortru $p |- ( ( T. \/_ T. ) <-> F. ) $=
|
|
( wtru wxo wn wfal wb df-xor trubitru xchbinx nottru bitri ) AABZACDKAAEAAA
|
|
FGHIJ $.
|
|
|
|
$( A ` \/_ ` identity. (Contributed by David A. Wheeler, 8-May-2015.) $)
|
|
truxorfal $p |- ( ( T. \/_ F. ) <-> T. ) $=
|
|
( wtru wfal wxo wn wb df-xor trubifal xchbinx notfal bitri ) ABCZBDAKABEBAB
|
|
FGHIJ $.
|
|
|
|
$( A ` \/_ ` identity. (Contributed by David A. Wheeler, 9-May-2015.) $)
|
|
falxortru $p |- ( ( F. \/_ T. ) <-> T. ) $=
|
|
( wfal wtru wxo wb wn df-xor falbitru notbii notfal 3bitri ) ABCABDZEAEBABF
|
|
KAGHIJ $.
|
|
|
|
$( A ` \/_ ` identity. (Contributed by David A. Wheeler, 9-May-2015.) $)
|
|
falxorfal $p |- ( ( F. \/_ F. ) <-> F. ) $=
|
|
( wfal wxo wtru wn wb df-xor falbifal xchbinx nottru bitri ) AABZCDAKAAECAA
|
|
FGHIJ $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Auxiliary theorems for Alan Sare's virtual deduction tool, part 1
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
ee22.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
ee22.2 $e |- ( ph -> ( ps -> th ) ) $.
|
|
ee22.3 $e |- ( ch -> ( th -> ta ) ) $.
|
|
$( Virtual deduction rule e22 in set.mm without virtual deduction
|
|
connectives. Special theorem needed for Alan Sare's virtual deduction
|
|
translation tool. (Contributed by Alan Sare, 2-May-2011.)
|
|
(New usage is discouraged.) TODO: decide if this is worth keeping. $)
|
|
ee22 $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( syl6c ) ABCDEFGHI $.
|
|
$}
|
|
|
|
${
|
|
ee12an.1 $e |- ( ph -> ps ) $.
|
|
ee12an.2 $e |- ( ph -> ( ch -> th ) ) $.
|
|
ee12an.3 $e |- ( ( ps /\ th ) -> ta ) $.
|
|
$( e12an in set.mm without virtual deduction connectives. Special theorem
|
|
needed for Alan Sare's virtual deduction translation tool. (Contributed
|
|
by Alan Sare, 28-Oct-2011.) TODO: this is frequently used; come up with
|
|
better label. $)
|
|
ee12an $p |- ( ph -> ( ch -> ta ) ) $=
|
|
( wa jctild syl6 ) ACBDIEACDBGFJHK $.
|
|
$}
|
|
|
|
${
|
|
ee23.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
ee23.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.
|
|
ee23.3 $e |- ( ch -> ( ta -> et ) ) $.
|
|
$( e23 in set.mm without virtual deductions. (Contributed by Alan Sare,
|
|
17-Jul-2011.) (New usage is discouraged.) TODO: decide if this is
|
|
worth keeping. $)
|
|
ee23 $p |- ( ph -> ( ps -> ( th -> et ) ) ) $=
|
|
( wi syl6 syldd ) ABDEFHABCEFJGIKL $.
|
|
$}
|
|
|
|
$( Exportation implication also converting head from biconditional to
|
|
conditional. This proof is exbirVD in set.mm automatically translated and
|
|
minimized. (Contributed by Alan Sare, 31-Dec-2011.)
|
|
(New usage is discouraged.) TODO: decide if this is worth keeping. $)
|
|
exbir $p |- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) ->
|
|
( ph -> ( ps -> ( th -> ch ) ) ) ) $=
|
|
( wa wb wi bi2 imim2i exp3a ) ABEZCDFZGABDCGZLMKCDHIJ $.
|
|
|
|
$( ~ impexp with a 3-conjunct antecedent. (Contributed by Alan Sare,
|
|
31-Dec-2011.) $)
|
|
3impexp $p |- ( ( ( ph /\ ps /\ ch ) -> th ) <->
|
|
( ph -> ( ps -> ( ch -> th ) ) ) ) $=
|
|
( w3a wi id 3expd 3impd impbii ) ABCEDFZABCDFFFZKABCDKGHLABCDLGIJ $.
|
|
|
|
$( ~ 3impexp with biconditional consequent of antecedent that is commuted in
|
|
consequent. Derived automatically from 3impexpVD in set.mm. (Contributed
|
|
by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if
|
|
this is worth keeping. $)
|
|
3impexpbicom $p |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <->
|
|
( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) $=
|
|
( w3a wb wi bicom imbi2 biimpcd mpi 3expd 3impexp biimpri syl6ibr impbii )
|
|
ABCFZDEGZHZABCEDGZHHHZTABCUATSUAGZRUAHZDEIZUCTUDSUARJKLMUBRUASUDUBABCUANOUE
|
|
PQ $.
|
|
|
|
${
|
|
3impexpbicomi.1 $e |- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) $.
|
|
$( Deduction form of ~ 3impexpbicom . Derived automatically from
|
|
3impexpbicomiVD in set.mm. (Contributed by Alan Sare, 31-Dec-2011.)
|
|
(New usage is discouraged.) TODO: decide if this is worth keeping. $)
|
|
3impexpbicomi $p |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) $=
|
|
( wb w3a bicomd 3exp ) ABCEDGABCHDEFIJ $.
|
|
$}
|
|
|
|
$( Closed form of ~ ancoms . Derived automatically from ancomsimpVD in
|
|
set.mm. (Contributed by Alan Sare, 31-Dec-2011.) $)
|
|
ancomsimp $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) ) $=
|
|
( wa ancom imbi1i ) ABDBADCABEF $.
|
|
|
|
${
|
|
exp3acom3r.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( Export and commute antecedents. (Contributed by Alan Sare,
|
|
18-Mar-2012.) $)
|
|
exp3acom3r $p |- ( ps -> ( ch -> ( ph -> th ) ) ) $=
|
|
( exp3a com3l ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
$( Implication form of ~ exp3acom23 . (Contributed by Alan Sare,
|
|
22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth
|
|
keeping. $)
|
|
exp3acom23g $p |- ( ( ph -> ( ( ps /\ ch ) -> th ) ) <->
|
|
( ph -> ( ch -> ( ps -> th ) ) ) ) $=
|
|
( wa wi ancomsimp impexp bitri imbi2i ) BCEDFZCBDFFZAKCBEDFLBCDGCBDHIJ $.
|
|
|
|
${
|
|
exp3acom23.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
|
|
$( The exportation deduction ~ exp3a with commutation of the conjoined
|
|
wwfs. (Contributed by Alan Sare, 22-Jul-2012.) $)
|
|
exp3acom23 $p |- ( ph -> ( ch -> ( ps -> th ) ) ) $=
|
|
( exp3a com23 ) ABCDABCDEFG $.
|
|
$}
|
|
|
|
$( Implication form of ~ simplbi2com . (Contributed by Alan Sare,
|
|
22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth
|
|
keeping. $)
|
|
simplbi2comg $p |- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) ) $=
|
|
( wa wb bi2 exp3acom23 ) ABCDZEBCAAHFG $.
|
|
|
|
${
|
|
simplbi2com.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
|
|
$( A deduction eliminating a conjunct, similar to ~ simplbi2 .
|
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf
|
|
Lammen, 10-Nov-2012.) $)
|
|
simplbi2com $p |- ( ch -> ( ps -> ph ) ) $=
|
|
( simplbi2 com12 ) BCAABCDEF $.
|
|
$}
|
|
|
|
${
|
|
ee21.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
ee21.2 $e |- ( ph -> th ) $.
|
|
ee21.3 $e |- ( ch -> ( th -> ta ) ) $.
|
|
$( e21 in set.mm without virtual deductions. (Contributed by Alan Sare,
|
|
18-Mar-2012.) (New usage is discouraged.) TODO: decide if this is
|
|
worth keeping. $)
|
|
ee21 $p |- ( ph -> ( ps -> ta ) ) $=
|
|
( a1d ee22 ) ABCDEFADBGIHJ $.
|
|
$}
|
|
|
|
${
|
|
ee10.1 $e |- ( ph -> ps ) $.
|
|
ee10.2 $e |- ch $.
|
|
ee10.3 $e |- ( ps -> ( ch -> th ) ) $.
|
|
$( e10 in set.mm without virtual deductions. (Contributed by Alan Sare,
|
|
25-Jul-2011.) TODO: this is frequently used; come up with better
|
|
label. $)
|
|
ee10 $p |- ( ph -> th ) $=
|
|
( mpi syl ) ABDEBCDFGHI $.
|
|
$}
|
|
|
|
${
|
|
ee02.1 $e |- ph $.
|
|
ee02.2 $e |- ( ps -> ( ch -> th ) ) $.
|
|
ee02.3 $e |- ( ph -> ( th -> ta ) ) $.
|
|
$( e02 in set.mm without virtual deductions. (Contributed by Alan Sare,
|
|
22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is
|
|
worth keeping. $)
|
|
ee02 $p |- ( ps -> ( ch -> ta ) ) $=
|
|
( a1i sylsyld ) BACDEABFIGHJ $.
|
|
$}
|
|
|
|
$( End of auxiliary theorems for Alan Sare's virtual deduction tool, part 1 $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Half-adders and full adders in propositional calculus
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Propositional calculus deals with truth values, which can be interpreted as
|
|
bits. Using this, we can define the half-adder in pure propositional
|
|
calculus, and show its basic properties.
|
|
|
|
$)
|
|
|
|
$c hadd cadd $.
|
|
$c , $. $( Comma (also used for unordered pair notation later) $)
|
|
|
|
$( Define the half adder (triple XOR). (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
whad $a wff hadd ( ph , ps , ch ) $.
|
|
|
|
$( Define the half adder carry. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
wcad $a wff cadd ( ph , ps , ch ) $.
|
|
|
|
$( Define the half adder (triple XOR). (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
df-had $a |- ( hadd ( ph , ps , ch ) <-> ( ( ph \/_ ps ) \/_ ch ) ) $.
|
|
|
|
$( Define the half adder carry, which is true when at least two arguments are
|
|
true. (Contributed by Mario Carneiro, 4-Sep-2016.) $)
|
|
df-cad $a |- ( cadd ( ph , ps , ch ) <->
|
|
( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) ) $.
|
|
|
|
${
|
|
hadbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
hadbid.2 $e |- ( ph -> ( th <-> ta ) ) $.
|
|
hadbid.3 $e |- ( ph -> ( et <-> ze ) ) $.
|
|
$( Equality theorem for half adder. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
hadbi123d $p |- ( ph ->
|
|
( hadd ( ps , th , et ) <-> hadd ( ch , ta , ze ) ) ) $=
|
|
( wxo whad xorbi12d df-had 3bitr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJM
|
|
BDFNCEGNO $.
|
|
|
|
$( Equality theorem for adder carry. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
cadbi123d $p |- ( ph ->
|
|
( cadd ( ps , th , et ) <-> cadd ( ch , ta , ze ) ) ) $=
|
|
( wa wxo wo wcad anbi12d xorbi12d orbi12d df-cad 3bitr4g ) ABDKZFBDLZKZMC
|
|
EKZGCELZKZMBDFNCEGNATUCUBUEABCDEHIOAFGUAUDJABCDEHIPOQBDFRCEGRS $.
|
|
$}
|
|
|
|
${
|
|
hadbii.1 $e |- ( ph <-> ps ) $.
|
|
hadbii.2 $e |- ( ch <-> th ) $.
|
|
hadbii.3 $e |- ( ta <-> et ) $.
|
|
$( Equality theorem for half adder. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
hadbi123i $p |- ( hadd ( ph , ch , ta ) <-> hadd ( ps , th , et ) ) $=
|
|
( whad wb wtru a1i hadbi123d trud ) ACEJBDFJKLABCDEFABKLGMCDKLHMEFKLIMNO
|
|
$.
|
|
|
|
$( Equality theorem for adder carry. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
cadbi123i $p |- ( cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) ) $=
|
|
( wcad wb wtru a1i cadbi123d trud ) ACEJBDFJKLABCDEFABKLGMCDKLHMEFKLIMNO
|
|
$.
|
|
$}
|
|
|
|
$( Associative law for triple XOR. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
hadass $p |- ( hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ ch ) ) ) $=
|
|
( whad wxo df-had xorass bitri ) ABCDABECEABCEEABCFABCGH $.
|
|
|
|
$( The half adder is the same as the triple biconditional. (Contributed by
|
|
Mario Carneiro, 4-Sep-2016.) $)
|
|
hadbi $p |- ( hadd ( ph , ps , ch ) <-> ( ( ph <-> ps ) <-> ch ) ) $=
|
|
( wxo wb wn whad df-xor df-had xnor bibi1i nbbn bitri 3bitr4i ) ABDZCDOCEFZ
|
|
ABCGABEZCEZOCHABCIROFZCEPQSCABJKOCLMN $.
|
|
|
|
$( Commutative law for triple XOR. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
hadcoma $p |- ( hadd ( ph , ps , ch ) <-> hadd ( ps , ph , ch ) ) $=
|
|
( wxo whad xorcom biid xorbi12i df-had 3bitr4i ) ABDZCDBADZCDABCEBACEKLCCAB
|
|
FCGHABCIBACIJ $.
|
|
|
|
$( Commutative law for triple XOR. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
hadcomb $p |- ( hadd ( ph , ps , ch ) <-> hadd ( ph , ch , ps ) ) $=
|
|
( wxo whad biid xorcom xorbi12i hadass 3bitr4i ) ABCDZDACBDZDABCEACBEAAKLAF
|
|
BCGHABCIACBIJ $.
|
|
|
|
$( Rotation law for triple XOR. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
hadrot $p |- ( hadd ( ph , ps , ch ) <-> hadd ( ps , ch , ph ) ) $=
|
|
( whad hadcoma hadcomb bitri ) ABCDBACDBCADABCEBACFG $.
|
|
|
|
$( Write the adder carry in disjunctive normal form. (Contributed by Mario
|
|
Carneiro, 4-Sep-2016.) $)
|
|
cador $p |- ( cadd ( ph , ps , ch ) <->
|
|
( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ps /\ ch ) ) ) $=
|
|
( wcad wa wxo wo df-cad wn wi xor2 rbaib anbi1d ancom andir 3bitr3g pm5.74i
|
|
w3o df-or bitri 3orass 3bitr4i ) ABCDABEZCABFZEZGZUCACEZBCEZRZABCHUCIZUEJUJ
|
|
UGUHGZJZUFUIUJUEUKUJUDCEABGZCEUEUKUJUDUMCUDUMUJABKLMUDCNABCOPQUCUESUIUCUKGU
|
|
LUCUGUHUAUCUKSTUBT $.
|
|
|
|
$( Write the adder carry in conjunctive normal form. (Contributed by Mario
|
|
Carneiro, 4-Sep-2016.) $)
|
|
cadan $p |- ( cadd ( ph , ps , ch ) <->
|
|
( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) ) $=
|
|
( wa w3o wo wcad w3a ordir wn wi wb simpr con3i biorf pm5.74i df-or 3bitr4i
|
|
syl orcom anbi2i 3bitr3i syl6bb bitr3i anbi12i bitri df-3or anandir df-3an
|
|
ordi cador ) ABDZACDZBCDZEZABFZACFZDBCFZDZABCGUPUQURHULUMFZUNFZUPURDZUQURDZ
|
|
DZUOUSVAUTBFZUTCFZDVDUTBCUJVEVBVFVCUMBFZUPCBFZDVEVBACBIBUMFZBUTFZVGVEBJZUMK
|
|
VKUTKVIVJVKUMUTVKULJUMUTLULBABMNULUMOSPBUMQBUTQRUMBTUTBTRVHURUPCBTUAUBVFULC
|
|
FZVCCULFZCUTFZVLVFCJZULKVOUTKVMVNVOULUTVOUMJZULUTLUMCACMNVPULUMULFUTUMULOUM
|
|
ULTUCSPCULQCUTQRULCTUTCTRABCIUDUEUFULUMUNUGUPUQURUHRABCUKUPUQURUIR $.
|
|
|
|
$( The half adder distributes over negation. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
hadnot $p |- ( -. hadd ( ph , ps , ch ) <->
|
|
hadd ( -. ph , -. ps , -. ch ) ) $=
|
|
( wxo wn whad xorneg biid xorbi12i xorneg2 bitr2i df-had notbii 3bitr4i ) A
|
|
BDZCDZEZAEZBEZDZCEZDZABCFZERSUAFUBOUADQTOUAUAABGUAHIOCJKUCPABCLMRSUALN $.
|
|
|
|
$( The adder carry distributes over negation. (Contributed by Mario
|
|
Carneiro, 4-Sep-2016.) $)
|
|
cadnot $p |- ( -. cadd ( ph , ps , ch ) <->
|
|
cadd ( -. ph , -. ps , -. ch ) ) $=
|
|
( wa w3o wn wo wcad 3ioran ianor 3anbi123i bitri cador notbii cadan 3bitr4i
|
|
w3a ) ABDZACDZBCDZEZFZAFZBFZGZUCCFZGZUDUFGZQZABCHZFUCUDUFHUBRFZSFZTFZQUIRST
|
|
IUKUEULUGUMUHABJACJBCJKLUJUAABCMNUCUDUFOP $.
|
|
|
|
$( Commutative law for adder carry. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
cadcoma $p |- ( cadd ( ph , ps , ch ) <-> cadd ( ps , ph , ch ) ) $=
|
|
( wa wxo wo wcad ancom xorcom anbi2i orbi12i df-cad 3bitr4i ) ABDZCABEZDZFB
|
|
ADZCBAEZDZFABCGBACGNQPSABHORCABIJKABCLBACLM $.
|
|
|
|
$( Commutative law for adder carry. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
cadcomb $p |- ( cadd ( ph , ps , ch ) <-> cadd ( ph , ch , ps ) ) $=
|
|
( wa w3o wcad 3orcoma biid ancom 3orbi123i bitri cador 3bitr4i ) ABDZACDZBC
|
|
DZEZONCBDZEZABCFACBFQONPESNOPGOONNPROHNHBCIJKABCLACBLM $.
|
|
|
|
$( Rotation law for adder carry. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
cadrot $p |- ( cadd ( ph , ps , ch ) <-> cadd ( ps , ch , ph ) ) $=
|
|
( wcad cadcoma cadcomb bitri ) ABCDBACDBCADABCEBACFG $.
|
|
|
|
$( If one parameter is true, the adder carry is true exactly when at least
|
|
one of the other parameters is true. (Contributed by Mario Carneiro,
|
|
8-Sep-2016.) $)
|
|
cad1 $p |- ( ch -> ( cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) ) $=
|
|
( wa wxo wo wcad ibar bicomd orbi2d df-cad wn pm5.63 olc orc adantr id jaoi
|
|
impbii xor2 ancom bitri orbi2i 3bitr4i 3bitr4g ) CABDZCABEZDZFUFUGFZABCGABF
|
|
ZCUHUGUFCUGUHCUGHIJABCKUFUJFZUFUFLZUJDZFUJUIUFUJMUJUKUJUFNUFUJUJAUJBABOPUJQ
|
|
RSUGUMUFUGUJULDUMABTUJULUAUBUCUDUE $.
|
|
|
|
$( If two parameters are true, the adder carry is true. (Contributed by
|
|
Mario Carneiro, 4-Sep-2016.) $)
|
|
cad11 $p |- ( ( ph /\ ps ) -> cadd ( ph , ps , ch ) ) $=
|
|
( wa wxo wo wcad orc df-cad sylibr ) ABDZKCABEDZFABCGKLHABCIJ $.
|
|
|
|
$( If one parameter is false, the adder carry is true exactly when both of
|
|
the other two parameters are true. (Contributed by Mario Carneiro,
|
|
8-Sep-2016.) $)
|
|
cad0 $p |- ( -. ch -> ( cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) ) $=
|
|
( wcad wa wxo wo wn df-cad idd pm2.21 adantrd jaod orc impbid1 syl5bb ) ABC
|
|
DABEZCABFZEZGZCHZQABCIUATQUAQQSUAQJUACQRCQKLMQSNOP $.
|
|
|
|
$( Rotation law for adder carry. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
cadtru $p |- cadd ( T. , T. , ph ) $=
|
|
( wtru wcad tru cad11 mp2an ) BBBBACDDBBAEF $.
|
|
|
|
$( If the first parameter is true, the half adder is equivalent to the
|
|
equality of the other two inputs. (Contributed by Mario Carneiro,
|
|
4-Sep-2016.) $)
|
|
had1 $p |- ( ph -> ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) ) $=
|
|
( whad wb hadbi biass bitri id biidd 2thd sylibr syl5bb ) ABCDZABCEZEZAONAB
|
|
ECEPABCFABCGHAAOOEZEPOEAAQAIAOJKAOOGLM $.
|
|
|
|
$( If the first parameter is false, the half adder is equivalent to the XOR
|
|
of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) $)
|
|
had0 $p |- ( -. ph -> ( hadd ( ph , ps , ch ) <-> ( ps \/_ ch ) ) ) $=
|
|
( wn whad wxo wb had1 hadnot df-xor xorneg bitr3i con1bii 3bitr4g con4bid )
|
|
ADZABCEZBCFZPPBDZCDZESTGZQDRDPSTHABCIUARUADSTFRSTJBCKLMNO $.
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Other axiomatizations of classical propositional calculus
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the Lukasiewicz axioms from Meredith's sole axiom
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Carew Meredith's sole axiom for propositional calculus. This amazing
|
|
formula is thought to be the shortest possible single axiom for
|
|
propositional calculus with inference rule ~ ax-mp , where negation and
|
|
implication are primitive. Here we prove Meredith's axiom from ~ ax-1 ,
|
|
~ ax-2 , and ~ ax-3 . Then from it we derive the Lukasiewicz axioms
|
|
~ luk-1 , ~ luk-2 , and ~ luk-3 . Using these we finally re-derive our
|
|
axioms as ~ ax1 , ~ ax2 , and ~ ax3 , thus proving the equivalence of all
|
|
three systems. C. A. Meredith, "Single Axioms for the Systems (C,N),
|
|
(C,O) and (A,N) of the Two-Valued Propositional Calculus," _The Journal of
|
|
Computing Systems_ vol. 1 (1953), pp. 155-164. Meredith claimed to be
|
|
close to a proof that this axiom is the shortest possible, but the proof
|
|
was apparently never completed.
|
|
|
|
An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic
|
|
somewhat late in life after attending talks by Lukasiewicz and produced
|
|
many remarkable results such as this axiom. From his obituary: "He did
|
|
logic whenever time and opportunity presented themselves, and he did it on
|
|
whatever materials came to hand: in a pub, his favored pint of porter
|
|
within reach, he would use the inside of cigarette packs to write proofs
|
|
for logical colleagues." (Contributed by NM, 14-Dec-2002.) (Proof
|
|
shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Wolf
|
|
Lammen, 28-May-2013.) $)
|
|
meredith $p |- ( ( ( ( ( ph -> ps ) -> ( -. ch -> -. th ) ) -> ch ) ->
|
|
ta ) -> ( ( ta -> ph ) -> ( th -> ph ) ) ) $=
|
|
( wi wn pm2.21 ax-3 imim12i com13 con1d com12 a1d ax-1 imim1d ja ) ABFZCGDG
|
|
FZFZCFZEEAFZDAFZFUAGZUCUBDUDADAUATAGZDCUERSDCFABHCDIJKLMNEDEAEDOPQ $.
|
|
|
|
$( Alias for ~ meredith which "verify markup *" will match to
|
|
~ ax-meredith . (Contributed by NM, 21-Aug-2017.)
|
|
(New usage is discouraged.) $)
|
|
axmeredith $p |- ( ( ( ( ( ph -> ps ) -> ( -. ch -> -. th ) ) -> ch ) ->
|
|
ta ) -> ( ( ta -> ph ) -> ( th -> ph ) ) ) $=
|
|
( meredith ) ABCDEF $.
|
|
|
|
$( Theorem ~ meredith restated as an axiom. This will allow us to ensure
|
|
that the rederivation of ~ ax1 , ~ ax2 , and ~ ax3 below depend only on
|
|
Meredith's sole axiom and not accidentally on a previous theorem above.
|
|
Outside of this section, we will not make use of this axiom. (Contributed
|
|
by NM, 14-Dec-2002.) (New usage is discouraged.) $)
|
|
ax-meredith $a |- ( ( ( ( ( ph -> ps ) -> ( -. ch -> -. th ) ) -> ch ) ->
|
|
ta ) -> ( ( ta -> ph ) -> ( th -> ph ) ) ) $.
|
|
|
|
$( Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(The step numbers refer to Meredith's original paper.) (Contributed by
|
|
NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem1 $p |- ( ( ( ch -> ( -. ph -> ps ) ) -> ta ) -> ( ph -> ta ) ) $=
|
|
( wn wi ax-meredith ax-mp ) DAEZFIBFZEZIFFZJFCJFZFZMDFADFFJDECEFZEKEFZFOFDF
|
|
LFNIBOKDGJPDCLGHDIJAMGH $.
|
|
|
|
$( Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem2 $p |- ( ( ( ph -> ph ) -> ch ) -> ( th -> ch ) ) $=
|
|
( wi wn merlem1 ax-meredith ax-mp ) BBDZAECEZDDADAADZDKBDCBDDAJIAFBBACKGH
|
|
$.
|
|
|
|
$( Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem3 $p |- ( ( ( ps -> ch ) -> ph ) -> ( ch -> ph ) ) $=
|
|
( wi wn merlem2 ax-mp ax-meredith ) AADZCEZJDZDZCDBCDZDZMADCADZDOBEZPDDBDZL
|
|
DZNKKDLDRJKIFKLQFGCABBLHGAACCMHG $.
|
|
|
|
$( Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem4 $p |- ( ta -> ( ( ta -> ph ) -> ( th -> ph ) ) ) $=
|
|
( wi wn ax-meredith merlem3 ax-mp ) AADBEZIDDBDZCDCADBADDZDCKDAABBCFKJCGH
|
|
$.
|
|
|
|
$( Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem5 $p |- ( ( ph -> ps ) -> ( -. -. ph -> ps ) ) $=
|
|
( wi wn ax-meredith merlem1 merlem4 ax-mp ) BBCZBDZJCCBCBCIICCZABCZADZDZBCC
|
|
ZBBBBBEIJNDCCBCZACZOCZKOCZBBBNAEOKDZCMTCCZACQCZRSCUAUBMBLTFAPUAGHOTAKQEHHH
|
|
$.
|
|
|
|
$( Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem6 $p |- ( ch -> ( ( ( ps -> ch ) -> ph ) -> ( th -> ph ) ) ) $=
|
|
( wi merlem4 merlem3 ax-mp ) BCEZIAEDAEEZECJEADIFJBCGH $.
|
|
|
|
$( Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his
|
|
sole axiom. (Contributed by NM, 22-Dec-2002.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merlem7 $p |- ( ph -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) ->
|
|
( -. th -> -. ps ) ) -> th ) ) ) $=
|
|
( wi wn merlem4 merlem6 ax-meredith ax-mp ) BCFZLDFZCEFDGBGFFZDFZFZFZAPFZDN
|
|
LHPAGZFCGZSFFZCFLFZQRFOUAFUBSMOTICEDBUAJKPSCALJKK $.
|
|
|
|
$( Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem8 $p |- ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) ->
|
|
( -. th -> -. ps ) ) -> th ) ) $=
|
|
( wph wi wn ax-meredith merlem7 ax-mp ) EEFZEGZLFFEFEFKKFFZABFCFBDFCGAGFFCF
|
|
FEEEEEHMABCDIJ $.
|
|
|
|
$( Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem9 $p |- ( ( ( ph -> ps ) -> ( ch -> ( th -> ( ps -> ta ) ) ) ) ->
|
|
( et -> ( ch -> ( th -> ( ps -> ta ) ) ) ) ) $=
|
|
( wi wn merlem6 merlem8 ax-mp ax-meredith ) CDBEGZGZGZFHZGBHZPGGZBGABGZGZSO
|
|
GFOGGMRHDHGZHAHGZGUAGRGZTNRGUCPCNQIDMRUBJKBEUAARLKOPBFSLK $.
|
|
|
|
$( Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem10 $p |- ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) $=
|
|
( wi wn ax-meredith merlem9 ax-mp ) AADZAEZJDDADADIIDDZAABDZDZCLDDZAAAAAFLA
|
|
DJCEDDADZADNDKNDLAACAFOAMCBKGHH $.
|
|
|
|
$( Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem11 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
|
|
( wi wn ax-meredith merlem10 ax-mp ) AACZADZICCACACHHCCZAABCZCZKCZAAAAAELMC
|
|
JMCABLFLKJFGG $.
|
|
|
|
$( Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem12 $p |- ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) $=
|
|
( wn wi merlem5 merlem2 ax-mp merlem4 merlem11 ) CBDDBEZEZAEZMAEZEZNLOBBEKE
|
|
LBBFBKCGHAMLIHMAJH $.
|
|
|
|
$( Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|
|
(Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merlem13 $p |- ( ( ph -> ps ) ->
|
|
( ( ( th -> ( -. -. ch -> ch ) ) -> -. -. ph ) -> ps ) ) $=
|
|
( wi wn merlem12 merlem5 ax-mp merlem6 ax-meredith merlem11 ) BBEZAFZDCFFCE
|
|
EZNFZEZFZEZEAEZAEZABEQBEETUAEZUASUBOREZREZSRCDGRBEZRFPEZEREUCEZUDSEUFUGQPEU
|
|
FPCDGQPHIRUEUFOJIRBRNUCKIIAMSTJITALIBBAQAKI $.
|
|
|
|
$( 1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from
|
|
Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
luk-1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( wi wn ax-meredith merlem13 ax-mp ) CCDZAEZEZEJDDKDBDZBCDACDDZDZABDZMDZCCK
|
|
ABFMADZOEZEZERDDSDLDZNPDOLDTABJIGOLRQGHMASOLFHH $.
|
|
|
|
$( 2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from
|
|
Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
luk-2 $p |- ( ( -. ph -> ph ) -> ph ) $=
|
|
( wn wi merlem5 merlem4 ax-mp merlem11 ax-meredith ) ABZACZJACZCZKAJBZCIBMC
|
|
CZICZICZLOPCZPNQAMDIONEFOIGFAMIJIHFJAGF $.
|
|
|
|
$( 3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from
|
|
Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
luk-3 $p |- ( ph -> ( -. ph -> ps ) ) $=
|
|
( wn wi merlem11 merlem1 ax-mp ) ACZHBDZDIDAIDHBEABHIFG $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the standard axioms from the Lukasiewicz axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
luklem1.1 $e |- ( ph -> ps ) $.
|
|
luklem1.2 $e |- ( ps -> ch ) $.
|
|
$( Used to rederive standard propositional axioms from Lukasiewicz'.
|
|
(Contributed by NM, 23-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
luklem1 $p |- ( ph -> ch ) $=
|
|
( wi luk-1 ax-mp ) BCFZACFZEABFIJFDABCGHH $.
|
|
$}
|
|
|
|
$( Used to rederive standard propositional axioms from Lukasiewicz'.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
luklem2 $p |- ( ( ph -> -. ps ) ->
|
|
( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) ) $=
|
|
( wn wi luk-1 luk-3 ax-mp luklem1 ) ABEZFZBACFZFZMDFBDFFLKCFZMFZNAKCGBOFPNF
|
|
BCHBOMGIJBMDGJ $.
|
|
|
|
$( Used to rederive standard propositional axioms from Lukasiewicz'.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
luklem3 $p |- ( ph -> ( ( ( -. ph -> ps ) -> ch ) -> ( th -> ch ) ) ) $=
|
|
( wn wi luk-3 luklem2 luklem1 ) AAEZDEZFJBFCFDCFFAKGJDBCHI $.
|
|
|
|
$( Used to rederive standard propositional axioms from Lukasiewicz'.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
luklem4 $p |- ( ( ( ( -. ph -> ph ) -> ph ) -> ps ) -> ps ) $=
|
|
( wn wi luk-2 luklem3 ax-mp luk-1 luklem1 ) ACADADZBDZBCZBDZBLJDZKMDJCJDJDZ
|
|
NJEJONDAEJJJLFGGLJBHGBEI $.
|
|
|
|
$( Used to rederive standard propositional axioms from Lukasiewicz'.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
luklem5 $p |- ( ph -> ( ps -> ph ) ) $=
|
|
( wn wi luklem3 luklem4 luklem1 ) AACADADBADZDHAAABEAHFG $.
|
|
|
|
$( Used to rederive standard propositional axioms from Lukasiewicz'.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
luklem6 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
|
|
( wi luk-1 wn luklem5 luklem2 luklem4 luklem1 ax-mp ) AABCZCKBCZKCZKAKBDKEZ
|
|
KCZKCMKCZCZPMOCZQNLCRNBEZNCZLNSFTSBCBCLCLSKBBGBLHIINLKDJMOKDJKPHJI $.
|
|
|
|
$( Used to rederive standard propositional axioms from Lukasiewicz'.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
luklem7 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
|
|
( wi luk-1 luklem5 luklem1 luklem6 ax-mp ) ABCDZDJCDZACDZDZBLDZAJCEBKDMNDBJ
|
|
KDZKBJBDOBJFJBCEGJCHGBKLEIG $.
|
|
|
|
$( Used to rederive standard propositional axioms from Lukasiewicz'.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
luklem8 $p |- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) ) $=
|
|
( wi luk-1 luklem7 ax-mp ) CADZABDZCBDZDDIHJDDCABEHIJFG $.
|
|
|
|
$( Standard propositional axiom derived from Lukasiewicz axioms.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax1 $p |- ( ph -> ( ps -> ph ) ) $=
|
|
( luklem5 ) ABC $.
|
|
|
|
$( Standard propositional axiom derived from Lukasiewicz axioms.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax2 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $=
|
|
( wi luklem7 luklem8 luklem6 ax-mp luklem1 ) ABCDDBACDZDZABDZJDZABCEKLAJDZD
|
|
ZMBJAFNJDOMDACGNJLFHII $.
|
|
|
|
$( Standard propositional axiom derived from Lukasiewicz axioms.
|
|
(Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax3 $p |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $=
|
|
( wn wi luklem2 luklem4 luklem1 ) ACZBCDHADADBADZDIHBAAEAIFG $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive Nicod's axiom from the standard axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Prove Nicod's axiom and implication and negation definitions.
|
|
|
|
$)
|
|
|
|
$( Define implication in terms of 'nand'. Analogous to
|
|
` ( ( ph -/\ ( ps -/\ ps ) ) <-> ( ph -> ps ) ) ` . In a pure
|
|
(standalone) treatment of Nicod's axiom, this theorem would be changed to
|
|
a definition ($a statement). (Contributed by NM, 11-Dec-2008.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-dfim $p |- ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -> ps ) ) -/\
|
|
( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -/\ ( ps -/\ ps ) ) )
|
|
-/\
|
|
( ( ph -> ps ) -/\ ( ph -> ps ) ) ) ) $=
|
|
( wnan wi wb nanim bicomi nanbi mpbi ) ABBCCZABDZEJKCJJCKKCCCKJABFGJKHI $.
|
|
|
|
$( Define negation in terms of 'nand'. Analogous to
|
|
` ( ( ph -/\ ph ) <-> -. ph ) ` . In a pure (standalone) treatment of
|
|
Nicod's axiom, this theorem would be changed to a definition ($a
|
|
statement). (Contributed by NM, 11-Dec-2008.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-dfneg $p |- ( ( ( ph -/\ ph ) -/\ -. ph ) -/\
|
|
( ( ( ph -/\ ph ) -/\ ( ph -/\ ph ) ) -/\
|
|
( -. ph -/\ -. ph ) ) ) $=
|
|
( wnan wn wb nannot bicomi nanbi mpbi ) AABZACZDIJBIIBJJBBBJIAEFIJGH $.
|
|
|
|
${
|
|
$( Minor premise. $)
|
|
nic-jmin $e |- ph $.
|
|
$( Major premise. $)
|
|
nic-jmaj $e |- ( ph -/\ ( ch -/\ ps ) ) $.
|
|
$( Derive Nicod's rule of modus ponens using 'nand', from the standard
|
|
one. Although the major and minor premise together also imply ` ch ` ,
|
|
this form is necessary for useful derivations from ~ nic-ax . In a pure
|
|
(standalone) treatment of Nicod's axiom, this theorem would be changed
|
|
to an axiom ($a statement). (Contributed by Jeff Hoffman,
|
|
19-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-mp $p |- ps $=
|
|
( wnan wa wi nannan mpbi simprd ax-mp ) ABDACBACBFFACBGHEABCIJKL $.
|
|
|
|
$( A direct proof of ~ nic-mp . (Contributed by NM, 30-Dec-2008.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-mpALT $p |- ps $=
|
|
( wa wi wn wnan df-nan anbi2i xchbinx mpbi iman mpbir simprd ax-mp ) ABDA
|
|
CBACBFZGARHZFZHZACBIZIZUAEUCAUBFTAUBJUBSACBJKLMARNOPQ $.
|
|
$}
|
|
|
|
$( Nicod's axiom derived from the standard ones. See _Intro. to Math.
|
|
Phil._ by B. Russell, p. 152. Like ~ meredith , the usual axioms can be
|
|
derived from this and vice versa. Unlike ~ meredith , Nicod uses a
|
|
different connective ('nand'), so another form of modus ponens must be
|
|
used in proofs, e.g. ` { ` ~ nic-ax , ~ nic-mp ` } ` is equivalent to
|
|
` { ` ~ luk-1 , ~ luk-2 , ~ luk-3 , ~ ax-mp ` } ` . In a pure
|
|
(standalone) treatment of Nicod's axiom, this theorem would be changed to
|
|
an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-ax $p |- ( ( ph -/\ ( ch -/\ ps ) ) -/\
|
|
( ( ta -/\ ( ta -/\ ta ) ) -/\
|
|
( ( th -/\ ch ) -/\
|
|
( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $=
|
|
( wnan wa wi nannan biimpi simpl imim2i wn imnan df-nan bitr4i imim2d con2b
|
|
con3 mpbir 3bitr4ri syl6ibr syl5bir nanim sylib 3syl pm4.24 jctil ) ACBFFZE
|
|
EEFFZDCFZADFZULFFZFFUIUJUMGHUIUMUJUIACBGZHZACHZUMUIUOABCIJUNCACBKLUPUKULHUM
|
|
UKDCMZHZUPULURDCGMUKDCNDCOPUPURDAMZHZULUPUQUSDACSQADMHADGMUTULADNDARADOUAUB
|
|
UCUKULUDUEUFUJEEEGZHEVAEUGJEEEITUHUIUMUJIT $.
|
|
|
|
$( A direct proof of ~ nic-ax . (Contributed by NM, 11-Dec-2008.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-axALT $p |- ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) )
|
|
-/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $=
|
|
( wnan wa wn anidm df-nan anbi2i notbii iman 3bitr4i bitr4i xchbinx anbi12i
|
|
wi imnan mpbir simpl imim2i con3 imim2d biimpri jctil con2b bitr3i 3bitri
|
|
syl ) ACBFZFZEEEFZFZDCFZADFZUPFZFZFZFULUSGZHZVAACBGZRZEEEGZRZDCHZRZDAHZRZRZ
|
|
GZRZVCVJVEVCACRZVJVBCACBUAUBVMVFVHDACUCUDUJVDEEIUEUFVAVCVKHZGZHVLUTVOULVCUS
|
|
VNAUKGZHAVBHZGZHULVCVPVRUKVQACBJKLAUKJAVBMNUSUNURGVKUNURJUNVEURVJEUMGZHEVDH
|
|
ZGZHUNVEVSWAUMVTEEEJKLEUMJEVDMNUOUQGZHVGVIHZGZHURVJWBWDUOVGUQWCUODCGHVGDCJD
|
|
CSOUQUPUPGZVIUPUPJWEUPADGHZVIUPIADJWFADHRVIADSADUGUHUIPQLUOUQJVGVIMNQPQLVCV
|
|
KMOTULUSJT $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the Lukasiewicz axioms from Nicod's axiom
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$( Minor premise. $)
|
|
nic-imp.1 $e |- ( ph -/\ ( ch -/\ ps ) ) $.
|
|
$( Inference for ~ nic-mp using ~ nic-ax as major premise. (Contributed by
|
|
Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-imp $p |- ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) $=
|
|
( wta wnan nic-ax nic-mp ) ACBGGDCGADGZJGGFFFGGEABCDFHI $.
|
|
$}
|
|
|
|
$( Lemma for ~ nic-id . (Contributed by Jeff Hoffman, 17-Nov-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-idlem1 $p |- ( ( th -/\ ( ta -/\ ( ta -/\ ta ) ) ) -/\
|
|
( ( ( ph -/\ ( ch -/\ ps ) ) -/\ th ) -/\
|
|
( ( ph -/\ ( ch -/\ ps ) ) -/\ th ) ) ) $=
|
|
( wnan nic-ax nic-imp ) ACBFFACFAAFZIFFEEEFFDABCAEGH $.
|
|
|
|
${
|
|
nic-idlem2.1 $e |- ( et -/\ ( ( ph -/\ ( ch -/\ ps ) ) -/\ th ) ) $.
|
|
$( Lemma for ~ nic-id . Inference used by ~ nic-id . (Contributed by Jeff
|
|
Hoffman, 17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-idlem2 $p |- ( ( th -/\ ( ta -/\ ( ta -/\ ta ) ) ) -/\ et ) $=
|
|
( wnan nic-ax nic-imp nic-mp ) FACBHHZDHZHDEEEHHZHZFHZPGOMMFLACHAAHZQHHND
|
|
ABCAEIJJK $.
|
|
$}
|
|
|
|
$( Theorem ~ id expressed with ` -/\ ` . (Contributed by Jeff Hoffman,
|
|
17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-id $p |- ( ta -/\ ( ta -/\ ta ) ) $=
|
|
( wph wps wch wth wnan nic-ax nic-idlem2 nic-idlem1 nic-mp ) BCFZCBFZLFFZDD
|
|
DFZFZFZCCCFFZFAAAFFZOEEEMDQCCCBEGHMNDPCORFKLLOAIHJ $.
|
|
|
|
$( ` -/\ ` is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-swap $p |- ( ( th -/\ ph ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) $=
|
|
( wta wnan nic-id nic-ax nic-mp ) AAADDBADABDZHDDCCCDDAEAAABCFG $.
|
|
|
|
${
|
|
nic-isw1.1 $e |- ( th -/\ ph ) $.
|
|
$( Inference version of ~ nic-swap . (Contributed by Jeff Hoffman,
|
|
17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-isw1 $p |- ( ph -/\ th ) $=
|
|
( wnan nic-swap nic-mp ) BADABDZGCABEF $.
|
|
$}
|
|
|
|
${
|
|
nic-isw2.1 $e |- ( ps -/\ ( th -/\ ph ) ) $.
|
|
$( Inference for swapping nested terms. (Contributed by Jeff Hoffman,
|
|
17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-isw2 $p |- ( ps -/\ ( ph -/\ th ) ) $=
|
|
( wnan nic-swap nic-imp nic-mp nic-isw1 ) BACEZBCAEZEJBEZLDJKKBCAFGHI $.
|
|
$}
|
|
|
|
${
|
|
nic-iimp1.1 $e |- ( ph -/\ ( ch -/\ ps ) ) $.
|
|
nic-iimp1.2 $e |- ( th -/\ ch ) $.
|
|
$( Inference version of ~ nic-imp using right-handed term. (Contributed by
|
|
Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-iimp1 $p |- ( th -/\ ph ) $=
|
|
( wnan nic-imp nic-mp nic-isw1 ) DADCGADGZKFABCDEHIJ $.
|
|
$}
|
|
|
|
${
|
|
nic-iimp2.1 $e |- ( ( ph -/\ ps ) -/\ ( ch -/\ ch ) ) $.
|
|
nic-iimp2.2 $e |- ( th -/\ ph ) $.
|
|
$( Inference version of ~ nic-imp using left-handed term. (Contributed by
|
|
Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-iimp2 $p |- ( th -/\ ( ch -/\ ch ) ) $=
|
|
( wnan nic-isw1 nic-iimp1 ) CCGZBADJABGEHFI $.
|
|
$}
|
|
|
|
${
|
|
nic-idel.1 $e |- ( ph -/\ ( ch -/\ ps ) ) $.
|
|
$( Inference to remove the trailing term. (Contributed by Jeff Hoffman,
|
|
17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-idel $p |- ( ph -/\ ( ch -/\ ch ) ) $=
|
|
( wnan nic-id nic-isw1 nic-imp nic-mp ) CCEZCEAJEZKJCCFGABCJDHI $.
|
|
$}
|
|
|
|
${
|
|
nic-ich.1 $e |- ( ph -/\ ( ps -/\ ps ) ) $.
|
|
nic-ich.2 $e |- ( ps -/\ ( ch -/\ ch ) ) $.
|
|
$( Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-ich $p |- ( ph -/\ ( ch -/\ ch ) ) $=
|
|
( wnan nic-isw1 nic-imp nic-mp ) CCFZBFAJFZKJBEGABBJDHI $.
|
|
$}
|
|
|
|
${
|
|
nic-idbl.1 $e |- ( ph -/\ ( ps -/\ ps ) ) $.
|
|
$( Double the terms. Since doubling is the same as negation, this can be
|
|
viewed as a contraposition inference. (Contributed by Jeff Hoffman,
|
|
17-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-idbl $p |- ( ( ps -/\ ps ) -/\ ( ( ph -/\ ph ) -/\ ( ph -/\ ph ) ) ) $=
|
|
( wnan nic-imp nic-ich ) BBDABDAADABBBCEABBACEF $.
|
|
$}
|
|
|
|
$( (not in Table of Contents)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Biconditional justification from Nicod's axiom
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( For nic-* definitions, the biconditional connective is not used. Instead,
|
|
definitions are made based on this form. ~ nic-bi1 and ~ nic-bi2 are used
|
|
to convert the definitions into usable theorems about one side of the
|
|
implication. (Contributed by Jeff Hoffman, 18-Nov-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-bijust $p |- ( ( ta -/\ ta ) -/\ ( ( ta -/\ ta ) -/\ ( ta -/\ ta ) ) ) $=
|
|
( nic-swap ) AAB $.
|
|
|
|
${
|
|
$( 'Biconditional' premise. $)
|
|
nic-bi1.1 $e |- ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph )
|
|
-/\ ( ps -/\ ps ) ) ) $.
|
|
$( Inference to extract one side of an implication from a definition.
|
|
(Contributed by Jeff Hoffman, 18-Nov-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-bi1 $p |- ( ph -/\ ( ps -/\ ps ) ) $=
|
|
( wnan nic-id nic-iimp1 nic-isw2 nic-idel ) AABBAAABDBBDAADACAEFGH $.
|
|
$}
|
|
|
|
${
|
|
$( 'Biconditional' premise. $)
|
|
nic-bi2.1 $e |- ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph )
|
|
-/\ ( ps -/\ ps ) ) ) $.
|
|
$( Inference to extract the other side of an implication from a
|
|
'biconditional' definition. (Contributed by Jeff Hoffman,
|
|
18-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-bi2 $p |- ( ps -/\ ( ph -/\ ph ) ) $=
|
|
( wnan nic-isw2 nic-id nic-iimp1 nic-idel ) BBAABDZAADZBBDZBKIJCEBFGH $.
|
|
$}
|
|
|
|
$( (not in Table of Contents)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Prove the Lukasiewicz axioms from Nicod's axiom
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$( Minor premise. $)
|
|
nic-smin $e |- ph $.
|
|
$( Major premise. $)
|
|
nic-smaj $e |- ( ph -> ps ) $.
|
|
$( Derive the standard modus ponens from ~ nic-mp . (Contributed by Jeff
|
|
Hoffman, 18-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-stdmp $p |- ps $=
|
|
( wi wnan nic-dfim nic-bi2 nic-mp ) ABBCABEZABBFFZKDKJABGHII $.
|
|
$}
|
|
|
|
$( Proof of ~ luk-1 from ~ nic-ax and ~ nic-mp (and definitions ~ nic-dfim
|
|
and ~ nic-dfneg ). Note that the standard axioms ~ ax-1 , ~ ax-2 , and
|
|
~ ax-3 are proved from the Lukasiewicz axioms by theorems ~ ax1 , ~ ax2 ,
|
|
and ~ ax3 . (Contributed by Jeff Hoffman, 18-Nov-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
nic-luk1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( wta wi nic-dfim nic-bi2 nic-ax nic-isw2 nic-idel nic-bi1 nic-idbl nic-imp
|
|
wnan nic-swap nic-ich nic-mp ) ABEZBCEZACEZEZUANNZRUAEZUCRABBNNZUAUDRABFGUD
|
|
STTNZNZUAUDCCNZBNZAUGNZUINZNZUFUDDDDNNZUKUKUDULABBUGDHIJUKUEUHNUFUEUJUJUHUI
|
|
TUITACFKLMSUHUHUESBUGNZUHUMSBCFGUGBOPMPPUFUASTFKPPUBUCRUAFKQ $.
|
|
|
|
$( Proof of ~ luk-2 from ~ nic-ax and ~ nic-mp . (Contributed by Jeff
|
|
Hoffman, 18-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-luk2 $p |- ( ( -. ph -> ph ) -> ph ) $=
|
|
( wn wi wnan nic-dfim nic-bi2 nic-dfneg nic-iimp1 nic-isw2 nic-isw1 nic-bi1
|
|
nic-id nic-mp ) ABZACZAADZDZOACZROPONPDZSPSONAEFNPPPNDNNDPPDPAGPLHIHJQROAEK
|
|
M $.
|
|
|
|
$( Proof of ~ luk-3 from ~ nic-ax and ~ nic-mp . (Contributed by Jeff
|
|
Hoffman, 18-Nov-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nic-luk3 $p |- ( ph -> ( -. ph -> ps ) ) $=
|
|
( wnan nic-dfim nic-bi1 nic-dfneg nic-bi2 nic-id nic-iimp1 nic-iimp2 nic-mp
|
|
wn wi ) AALZBMZOCCZAOMZQNBBCZOANRCONBDENAACZSASNAFGAHIJPQAODEK $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( This alternative axiom for propositional calculus using the Sheffer Stroke
|
|
was offered by Lukasiewicz in his Selected Works. It improves on Nicod's
|
|
axiom by reducing its number of variables by one.
|
|
|
|
This axiom also uses ~ nic-mp for its constructions.
|
|
|
|
Here, the axiom is proved as a substitution instance of ~ nic-ax .
|
|
(Contributed by Anthony Hart, 31-Jul-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
lukshef-ax1 $p |- ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( th -/\ ( th -/\ th ) )
|
|
-/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $=
|
|
( nic-ax ) ABCDDE $.
|
|
|
|
$( Lemma for ~ renicax . (Contributed by NM, 31-Jul-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
lukshefth1 $p |- ( ( ( ( ta -/\ ps ) -/\ ( ( ph -/\ ta ) -/\ ( ph
|
|
-/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ph -/\ ( ps
|
|
-/\ ch ) ) ) $=
|
|
( wnan lukshef-ax1 nic-mp ) ABCFFZEEEFFZEBFAEFZKFFZFZFZLDDDFFZFZIFZQACBEGPM
|
|
MFFZNQQFFIIIFFJODEFEDFZSFFZFFRLLLFFEEEDGJTOLGHPMMIGHH $.
|
|
|
|
$( Lemma for ~ renicax . (Contributed by NM, 31-Jul-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
lukshefth2 $p |- ( ( ta -/\ th ) -/\ ( ( th -/\ ta ) -/\ ( th
|
|
-/\ ta ) ) ) $=
|
|
( wps wch wph wnan lukshef-ax1 nic-mp lukshefth1 ) AAAFFZBAFABFZKFFBBBFFAJF
|
|
ZCDEFFZAFZNFFZJBEFEBFZPFFZMJADFCAFZRFFZFFOJCEDAGMSJAGHQJFZEEEFFZFZOTFZUCEEE
|
|
ABIOUAENFLEFZUDFFZFFUBUCUCFFTTTFFLNNEGOUEUATGHHHAAABGH $.
|
|
|
|
$( A rederivation of ~ nic-ax from ~ lukshef-ax1 , proving that ~ lukshef-ax1
|
|
with ~ nic-mp can be used as a complete axiomatization of propositional
|
|
calculus. (Contributed by Anthony Hart, 31-Jul-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
renicax $p |- ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) )
|
|
-/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $=
|
|
( wnan lukshefth1 lukshefth2 nic-mp lukshef-ax1 ) EEEFFZDCFADFZLFFZFZACBFFZ
|
|
FZONFZQOMKFZFZPPROFSSACBEDGORHINRRFFSPPFFOOOFFMKHNRROJIIONHI $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Justification for ~ tbw-negdf . (Contributed by Anthony Hart,
|
|
15-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
tbw-bijust $p |- ( ( ph <-> ps ) <-> ( ( ( ph -> ps )
|
|
-> ( ( ps -> ph ) -> F. ) ) -> F. ) ) $=
|
|
( wb wi wn wfal dfbi1 pm2.21 imim2i falim impbii notbii ax-1 pm2.43i 3bitri
|
|
id ja ) ABCABDZBADZEZDZERSFDZDZEZUCFDZABGUAUCUAUCTUBRSFHIUBTRSFTTPTJQIKLUDU
|
|
EUCFHUEUDUCFUEUDDZUDUEMUFJQNKO $.
|
|
|
|
$( The definition of negation, in terms of ` -> ` and ` F. ` . (Contributed
|
|
by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
tbw-negdf $p |- ( ( ( -. ph -> ( ph -> F. ) )
|
|
-> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. ) $=
|
|
( wn wfal wi wb pm2.21 ax-1 falim ja pm2.43i impbii tbw-bijust mpbi ) ABZAC
|
|
DZENODONDZCDDCDNOACFONACPNOGPHIJKNOLM $.
|
|
|
|
$( The first of four axioms in the Tarski-Bernays-Wajsberg system.
|
|
(Contributed by Anthony Hart, 13-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbw-ax1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( imim1 ) ABCD $.
|
|
|
|
$( The second of four axioms in the Tarski-Bernays-Wajsberg system.
|
|
(Contributed by Anthony Hart, 13-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbw-ax2 $p |- ( ph -> ( ps -> ph ) ) $=
|
|
( ax-1 ) ABC $.
|
|
|
|
$( The third of four axioms in the Tarski-Bernays-Wajsberg system.
|
|
(Contributed by Anthony Hart, 13-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbw-ax3 $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
|
|
( peirce ) ABC $.
|
|
|
|
$( The fourth of four axioms in the Tarski-Bernays-Wajsberg system.
|
|
|
|
This axiom was added to the Tarski-Bernays axiom system ( see tb-ax1 ,
|
|
tb-ax2 , and tb-ax3 in set.mm) by Wajsberg for completeness. (Contributed
|
|
by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
tbw-ax4 $p |- ( F. -> ph ) $=
|
|
( falim ) AB $.
|
|
|
|
${
|
|
tbwsyl.1 $e |- ( ph -> ps ) $.
|
|
tbwsyl.2 $e |- ( ps -> ch ) $.
|
|
$( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbwsyl $p |- ( ph -> ch ) $=
|
|
( wi tbw-ax1 ax-mp ) BCFZACFZEABFIJFDABCGHH $.
|
|
$}
|
|
|
|
$( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbwlem1 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
|
|
( wi tbw-ax2 tbw-ax1 tbwsyl tbw-ax3 mpsyl ) BBCDZCDZDAJDKACDZDBLDBJKDZKBJBD
|
|
MBJEJBCFGMKCDKDKJKCFKCHGGAJCFBKLFI $.
|
|
|
|
$( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbwlem2 $p |- ( ( ph -> ( ps -> F. ) ) -> ( ( ( ph -> ch ) -> th )
|
|
-> ( ps -> th ) ) ) $=
|
|
( wfal wi tbw-ax4 tbw-ax1 tbwlem1 ax-mp mpsyl tbwsyl ) ABEFZFZBACFZFZODFBDF
|
|
FBMCFZFZNQOFPMBCFZFZRECFZTCGMUASFFUATFBECHMUASIJJMBCIJAMCHBQOHKBODHL $.
|
|
|
|
$( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbwlem3 $p |- ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ps ) -> ps ) $=
|
|
( wfal wi tbw-ax3 tbw-ax2 tbw-ax1 tbwsyl ax-mp ) ACDADADZBDZKBDZDZLJMACEJKJ
|
|
DMJKFKJBGHIMLBDLDLKLBGLBEHI $.
|
|
|
|
$( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbwlem4 $p |- ( ( ( ph -> F. ) -> ps ) -> ( ( ps -> F. ) -> ph ) ) $=
|
|
( wfal wi tbw-ax4 tbw-ax1 tbwlem1 ax-mp tbwlem2 tbwlem3 tbwsyl ) ACDZBDZLBC
|
|
DZCDZDZNADZBODZMPDZNNDZRCCDZTCENUANDDUATDBCCFNUANGHHNBCGHMRPDDRSDLBOFMRPGHH
|
|
PLADADQDQLNAAIAQJKK $.
|
|
|
|
$( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
tbwlem5 $p |- ( ( ( ph -> ( ps -> F. ) ) -> F. ) -> ph ) $=
|
|
( wfal wi tbw-ax2 tbw-ax1 tbwsyl tbwlem1 ax-mp tbwlem4 ) ACDZABCDZDZDZMCDAD
|
|
AKLDZDNABADOABEBACFGAKLHIAMJI $.
|
|
|
|
$( ~ luk-1 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by
|
|
Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re1luk1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( tbw-ax1 ) ABCD $.
|
|
|
|
$( ~ luk-2 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by
|
|
Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re1luk2 $p |- ( ( -. ph -> ph ) -> ph ) $=
|
|
( wn wi wfal tbw-negdf tbw-ax2 tbwlem4 ax-mp tbw-ax1 tbw-ax3 tbwsyl ) ABZAC
|
|
ZADCZACZANLCZMOCLNCZPDCZCZDCZPAERSCTPCRQFPSGHHNLAIHADJK $.
|
|
|
|
$( ~ luk-3 derived from the Tarski-Bernays-Wajsberg axioms.
|
|
|
|
This theorem, along with ~ re1luk1 and ~ re1luk2 proves that ~ tbw-ax1 ,
|
|
~ tbw-ax2 , ~ tbw-ax3 , and ~ tbw-ax4 , with ~ ax-mp can be used as a
|
|
complete axiom system for all of propositional calculus. (Contributed by
|
|
Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re1luk3 $p |- ( ph -> ( -. ph -> ps ) ) $=
|
|
( wn wfal wi tbw-negdf tbwlem5 ax-mp tbw-ax4 tbw-ax1 tbwlem1 mpsyl ) ACZADE
|
|
ZEZANBEZMBEONMEZDEEDEOAFOQGHNABEZEZAPEDBEZSBINTREETSEADBJNTRKHHNABKHMNBJL
|
|
$.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( A single axiom for propositional calculus offered by Meredith.
|
|
|
|
This axiom is worthy of note, due to it having only 19 symbols, not
|
|
counting parentheses. The more well-known ~ meredith has 21 symbols, sans
|
|
parentheses.
|
|
|
|
See ~ merco2 for another axiom of equal length. (Contributed by Anthony
|
|
Hart, 13-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
merco1 $p |- ( ( ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> th ) -> ta )
|
|
-> ( ( ta -> ph ) -> ( ch -> ph ) ) ) $=
|
|
( wi wfal wn ax-1 falim ja imim2i imim1i meredith syl ) ABFZCGFZFZDFZEFPDHZ
|
|
CHZFZFZDFZEFEAFCAFFUDSERUCDQUBPCGUBUATIUBJKLMMABDCENO $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 17-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem1 $p |- ( ph -> ( F. -> ch ) ) $=
|
|
( wfal wi merco1 ax-mp ) AACADZDZDZACBDZDZHGDZHDZIGCDACDZDZGDHDZMGNDZNDZGDO
|
|
DPCAANGEGNAGOEFGCAGHEFHCDZNDZGDLDZMIDQSDHDTDUACAASHEGNHHTEFHCAGLEFFHJDZKDZI
|
|
KDZJCDNDZGDHDZUCRJDUEDUFCAANJEGNAJUEEFJCAGHEFKCDICDZDZJDUBDZUCUDDJUGDSDKDUH
|
|
DUICBISKEJUGHKUHEFKCIJUBEFFF $.
|
|
|
|
$( ~ tbw-ax4 rederived from ~ merco1 . (Contributed by Anthony Hart,
|
|
17-Sep-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
retbwax4 $p |- ( F. -> ph ) $=
|
|
( wfal wi merco1lem1 ax-mp ) ABACZCZFAADGADE $.
|
|
|
|
$( ~ tbw-ax2 rederived from ~ merco1 . (Contributed by Anthony Hart,
|
|
17-Sep-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
retbwax2 $p |- ( ph -> ( ps -> ph ) ) $=
|
|
( wi wfal merco1lem1 merco1 ax-mp ) AAAACZCZCZABACZCZDACZHCZICZJHACADCZCACZ
|
|
MCOQAEHAAAMFGIPCPCDCNCOJCAHAPDFIPADNFGGMKCZLCZJLCZKACPCACZMCSUAAEKAAAMFGLBD
|
|
CZCJDCZCDCRCSTCAKBUCDFLUBJDRFGGG $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 17-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem2 $p |- ( ( ( ph -> ps ) -> ch ) -> ( ( ( ps -> ta ) -> ( ph ->
|
|
F. ) ) -> ch ) ) $=
|
|
( wi wfal retbwax2 merco1 ax-mp ) CAEZBDEAFEEZFEZEZBEABEZEZNCEKCEELMEOLJGBD
|
|
AFMHICAKBNHI $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 17-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem3 $p |- ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) $=
|
|
( wi wfal merco1lem2 retbwax2 ax-mp ) AAADZAEDDZIDZDZABDCEDDZCADZDZIEDJEDDZ
|
|
LAAEAFKLDPLDKAGJILEFHHNEDMEDDZLODZCAEBFORDQRDOLGMNREFHHH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 17-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem4 $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $=
|
|
( wi wfal merco1lem3 merco1 ax-mp ) CADZBEDZDZBDABDZDZLCDBCDDJAEDZDIEDZDKDM
|
|
JNIFBEAOKGHCABBLGH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 17-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem5 $p |- ( ( ( ( ph -> F. ) -> ch ) -> ta ) -> ( ph -> ta ) ) $=
|
|
( wi wfal merco1lem4 merco1 ax-mp ) CADZAEDZDBDJBDZDKCDACDDIJBFCAABKGH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 17-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem6 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ch -> ( ph -> ps ) ) ) $=
|
|
( wi wfal merco1lem5 merco1lem3 ax-mp merco1 ) ABDZEDCEDZDZEDZADZAJDCJDDJME
|
|
DZDZNLODZPOEDMDQLEEFOELGHJKOFHABMGHJECEAIH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 17-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem7 $p |- ( ph -> ( ( ( ps -> ch ) -> ps ) -> ps ) ) $=
|
|
( wi wfal merco1lem5 merco1 ax-mp merco1lem6 ) BCDZBDZKBDZDZALDBEDKEDZDCDJD
|
|
MBNCFBEKCJGHKBAIH $.
|
|
|
|
$( ~ tbw-ax3 rederived from ~ merco1 . (Contributed by Anthony Hart,
|
|
17-Sep-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
retbwax3 $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
|
|
( wi retbwax2 merco1lem7 ax-mp ) AAACCZABCACACAADGABEF $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 17-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem8 $p |- ( ph -> ( ( ps -> ( ps -> ch ) ) -> ( ps -> ch ) ) ) $=
|
|
( wi merco1lem6 ax-mp ) BBCDZDZHGDZDAIDBCHEHGAEF $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem9 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
|
|
( wfal wi merco1lem8 ax-mp ) CADZAABDZDHDZDZIGABEJABEF $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem10 $p |- ( ( ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) -> ph ) ->
|
|
( th -> ph ) ) $=
|
|
( wi wfal merco1 merco1lem2 ax-mp ) ABFZDGFZFCAFEGFFAFZGFZFKCFECFFZFZOAFDAF
|
|
FMKFOFPCAEAKHMKOLIJABDNOHJ $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem11 $p |- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> F. ) -> ps
|
|
) ) $=
|
|
( wi wfal merco1lem5 merco1lem3 ax-mp merco1lem4 merco1 merco1lem2 ) ADEZBA
|
|
EZCMEZFEZFEZEZFEZFEZEZABEPBEEZOTEZUAQTEZUCRTEZUDTFESEUERFFGTFRHINQTJIOFTGIC
|
|
MTJISAEUBEUAUBEBAPFAKSAUBDLII $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem12 $p |- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ps
|
|
) ) $=
|
|
( wi wfal merco1lem3 merco1 ax-mp merco1lem9 merco1lem11 ) BAEZCADEZEZAEZFE
|
|
ZEFEAEZABEOBEEOAEZQOREZRMPECFEZENESMPCGADOTNHIOAJIOALFKIBAOFAHI $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem13 $p |- ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph ->
|
|
ta ) ) $=
|
|
( wi wfal merco1 merco1lem4 ax-mp merco1lem12 ) DAEZAFEEAEABECBEEZEZLDEADEE
|
|
ALEZMBAECFEEAEZAELENBACAAGOALHIALKFJIDAAALGI $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem14 $p |- ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) ) $=
|
|
( wi wfal merco1lem13 merco1lem8 merco1 ax-mp merco1lem9 merco1lem12 ) CADZ
|
|
AEDDADABDZBDZDZNCDACDDANDZOMNDNDZPDZPABMNFRRPDZDZSPADREDDADZQDTUAMBGPARAQHI
|
|
RPJIIANLEKICAAANHI $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem15 $p |- ( ( ph -> ps ) -> ( ph -> ( ch -> ps ) ) ) $=
|
|
( wi merco1lem14 merco1lem13 ax-mp ) ABDZBDCBDZDAIDZDHJDABIEHBCJFG $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem16 $p |- ( ( ( ph -> ( ps -> ch ) ) -> ta ) -> ( ( ph -> ch ) -> ta
|
|
) ) $=
|
|
( wi wfal merco1lem15 merco1lem11 ax-mp merco1 ) DAEZACEZFEEFEABCEEZEZMDELD
|
|
EELMENACBGLMKFHIDALFMJI $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem17 $p |- ( ( ( ( ( ph -> ps ) -> ph ) -> ch ) -> ta ) -> ( ( ph ->
|
|
ch ) -> ta ) ) $=
|
|
( wfal merco1lem11 merco1lem7 ax-mp merco1lem9 merco1lem4 merco1lem16 mpsyl
|
|
wi merco1 ) DAMZACMZEMZMCMZABMAMZCMZMTDMPDMMQPMZTMZRQCMTPTMZUBCAMZSEMMEMAMZ
|
|
UCSAMZUEMZUESAUDEFUGUGUEMZMZUHUEAMUGEMMAMZUFMUIUJABGUEAUGAUFNHUGUEIHHCASEAN
|
|
HTAMZUAEMMEMPMZUCUBMUAPMZULMZULUAPUKEFUNUNULMZMZUOULAMUNEMMAMZUMMUPUQPEGULA
|
|
UNAUMNHUNULIHHTAUAEPNHHOQCJQACTKLDAPCTNH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
|
|
(Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco1lem18 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ps -> ph ) -> ( ps ->
|
|
ch ) ) ) $=
|
|
( wfal merco1 merco1lem17 ax-mp merco1lem5 merco1lem3 merco1lem4 merco1lem2
|
|
wi merco1lem9 ) BALZABCLZLZNOLZLZLZROBLZALRLZSTNDLZLTLALRLUAOBNTAETUBARFGBC
|
|
ARFGSSRLZLZUCQRDLSDLZLZDLZDLZLZUDRUHLZUIUFUHLZUJUHDLUGLUKUFDDHUHDUFIGRUEUHH
|
|
GPQUHJGUGNLUDLUIUDLRDSDNEUGNUDOKGGSRMGG $.
|
|
|
|
$( ~ tbw-ax1 rederived from ~ merco1 .
|
|
|
|
This theorem, along with ~ retbwax2 , ~ retbwax3 , and ~ retbwax4 , shows
|
|
that ~ merco1 with ~ ax-mp can be used as a complete axiomatization of
|
|
propositional calculus. (Contributed by Anthony Hart, 18-Sep-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
retbwax1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( wi merco1lem18 merco1lem16 ax-mp merco1lem15 merco1lem14 wfal merco1lem10
|
|
merco1 merco1lem9 merco1lem13 ) BCDZABDZACDZDZDZPOQDZDZBQDRDSBACEBACRFGOSUA
|
|
DZDZUBSRDUBDZUCRUBDZUDRUADUEPQOHRUASHGRSUAEGORUBIGUCUBDZJDZUADZUFUGTDZUHUFQ
|
|
DZTDZUIOUBQITADZUGJDZDZQDUJDZUKUIDQADZUGDZUMDUNDZUOUMJDULJDDUGDUQDURUGJJUPU
|
|
LKUMJULUGUQLGQAUFUMUNLGTAUGQUJLGGUGTPHGUHUBDUFDZUFDZUHUFDUSUTDUTUFJUAUSSKUS
|
|
UFMGUHUBUCUFNGGGG $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( A single axiom for propositional calculus offered by Meredith.
|
|
|
|
This axiom has 19 symbols, sans auxiliaries. See notes in ~ merco1 .
|
|
(Contributed by Anthony Hart, 7-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
merco2 $p |- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( th
|
|
-> ph ) -> ( ta -> ( et -> ph ) ) ) ) $=
|
|
( wi wfal falim pm2.04 mpi jarl idd jad looinv 3syl a1dd a1i com4l ) FABGZH
|
|
CGZDGGZDAGZEAUBUCEAGGGFUBUCAEUBTDGZADGDGUCAGUBUAUDCITUADJKUDADDABDLUDDMNADO
|
|
PQRS $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
mercolem1 $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) $=
|
|
( wi wfal merco2 ax-mp ) AAEZFAEZAEEIAIEEEZABEZCEZBDCEZEZEZAAAAAAGZKKPEZQCA
|
|
EZJLEZEZPEZKREZCAALBDGPTEJUAEEZUBUCETOEJPEEZUDOJFEZEFBETEEUEBNAFJAGOUFBTJMG
|
|
HTOAPJSGHPTAUAKKGHHHH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
mercolem2 $p |- ( ( ( ph -> ps ) -> ph ) -> ( ch -> ( th -> ph ) ) ) $=
|
|
( wi wfal merco2 ax-mp ) AAEZFAEZAEEIAIEEEZABEZAEZCDAEEZEZAAAAAAGZKKOEZPIJL
|
|
EZEZOEZKQEZAAALCDGOREJSEEZTUAERNEJOEEZUBNLEJREEZUCLJFEZEJNEEUDABAFCDGLUEANJ
|
|
JGHNLARJMGHRNAOJIGHORASKKGHHHH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
mercolem3 $p |- ( ( ps -> ch ) -> ( ps -> ( ph -> ch ) ) ) $=
|
|
( wi wfal merco2 mercolem2 ax-mp ) AADZEADZADDIAIDDDZBCDZBACDZDZDZAAAAAAFZK
|
|
KODZPCADZJBDZDZODZKQDZCAABBAFOSDJTDDZUAUBDSNDJODDZUCNBDJSDDUDBMJJGNBASJLFHS
|
|
NAOJRFHOSATKKFHHHH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
mercolem4 $p |- ( ( th -> ( et -> ph ) ) -> ( ( ( th -> ch )
|
|
-> ph ) -> ( ta -> ( et -> ph ) ) ) ) $=
|
|
( wi wfal merco2 mercolem1 ax-mp mercolem3 ) AAFZGAFZAFFLALFFFZCEAFZFZCBFZA
|
|
FZDOFFZFZAAAAAAHZNNTFZUAOAFZMCFFZTFZNUBFZOAACRDHTCFZMUDFFZUEUFFUGUDFZUHQMTF
|
|
FZUIMQFZTFZUJLUKFSFULAAAQDEHLUKSPIJMQTMIJCBATUCMHJMUGUDKJTCAUDNNHJJJJ $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
mercolem5 $p |- ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) $=
|
|
( wi wfal merco2 mercolem1 ax-mp mercolem2 ) AAEZFAEZAEEKAKEEEZCCAEDBAEEEZE
|
|
ZAAAAAAGZMMOEZPLCEZOEZMQEZKRENESAAACDBGKRNCHIOCELREESTECNLLJOCARMMGIIII $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
mercolem6 $p |- ( ( ph -> ( ps -> ( ph -> ch ) ) )
|
|
-> ( ps -> ( ph -> ch ) ) ) $=
|
|
( wi wfal merco2 mercolem1 ax-mp mercolem5 mercolem4 ) AADZEADADDKAKDDDZABA
|
|
CDZDZDZNDZAAAAAAFZLLPDZQLLRDZQORDZLSDZLTQMRDZLTDZAODZMDPDUBAOMBGUDMPLGHATDU
|
|
BUCDNOALIRCALOJHHHLTUADZQPUADZLUEDZALDZPDSDUFALPLGUHPSLGHOUEDUFUGDRLOLIUANO
|
|
LTJHHHHHHH $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
mercolem7 $p |- ( ( ph -> ps ) -> ( ( ( ph -> ch )
|
|
-> ( th -> ps ) ) -> ( th -> ps ) ) ) $=
|
|
( wi wfal merco2 mercolem3 mercolem6 ax-mp mercolem5 mercolem4 ) AAEZFAEAEE
|
|
MAMEEEZABEZACEZDBEZEZQEZEZAAAAAAGPSEZNTEZRUAEUARPQHRPQIJATEUAUBEBDARKSCANOL
|
|
JJJ $.
|
|
|
|
$( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
|
|
(Contributed by Anthony Hart, 16-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
mercolem8 $p |- ( ( ph -> ps ) -> ( ( ps -> ( ph -> ch ) )
|
|
-> ( ta -> ( th -> ( ph -> ch ) ) ) ) ) $=
|
|
( wi wfal merco2 mercolem3 ax-mp mercolem7 ) AAFZGAFZAFFLALFFFZABFZBACFZFED
|
|
PFFFZFZAAAAAAHZNNRFZSPMBFZFUAFZRFZNTFZUBQFUCPUAABEDHOUBQIJRMUBFZFUEFZUCUDFO
|
|
UBFUFABCMKOUBQMKJRUEAUBNNHJJJJ $.
|
|
|
|
$( ~ tbw-ax1 rederived from ~ merco2 . (Contributed by Anthony Hart,
|
|
16-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re1tbw1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( wi mercolem8 mercolem3 mercolem6 mpsyl ax-mp ) BCDZABDZJACDZDZDZDNKBLDZND
|
|
DJONABCJKEABCFKOMGHJKLGI $.
|
|
|
|
$( ~ tbw-ax2 rederived from ~ merco2 . (Contributed by Anthony Hart,
|
|
16-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re1tbw2 $p |- ( ph -> ( ps -> ph ) ) $=
|
|
( wi mercolem1 ax-mp mercolem6 ) BABACZCZCZHAICZIAACZACHCJAAABDKAHBDEABGFEB
|
|
AAFE $.
|
|
|
|
$( ~ tbw-ax3 rederived from ~ merco2 . (Contributed by Anthony Hart,
|
|
16-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re1tbw3 $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
|
|
( wi mercolem2 mercolem6 ax-mp ) AACZACAGCCZABCACZACZAAAADIHJCZCKABHIDIHAEF
|
|
F $.
|
|
|
|
$( ~ tbw-ax4 rederived from ~ merco2 .
|
|
|
|
This theorem, along with ~ re1tbw1 , ~ re1tbw2 , and ~ re1tbw3 , shows
|
|
that ~ merco2 , along with ~ ax-mp , can be used as a complete
|
|
axiomatization of propositional calculus. (Contributed by Anthony Hart,
|
|
16-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re1tbw4 $p |- ( F. -> ph ) $=
|
|
( wi wfal re1tbw3 re1tbw2 re1tbw1 ax-mp mercolem3 merco2 ) AABZCABZJABZABZJ
|
|
AADALBMJBAJEALAFGGZJJKBZNKKBZJOBZKABZKBZKBZPKADKSBTPBKREKSKFGGRPBPQBCKAHKAA
|
|
KJJIGGGG $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Justification for ~ rb-imdf . (Contributed by Anthony Hart,
|
|
17-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
rb-bijust $p |- ( ( ph <-> ps ) <-> -. ( -. ( -. ph \/ ps )
|
|
\/ -. ( -. ps \/ ph ) ) ) $=
|
|
( wb wi wn wo dfbi1 imor notbii imbi12i pm4.62 3bitri ) ABCABDZBADZEZDZEAEB
|
|
FZBEAFZEZDZEQESFZEABGPTMQOSABHNRBAHIJITUAQRKIL $.
|
|
|
|
$( The definition of implication, in terms of ` \/ ` and ` -. ` .
|
|
(Contributed by Anthony Hart, 17-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rb-imdf $p |- -. ( -. ( -. ( ph -> ps ) \/ ( -. ph \/ ps ) )
|
|
\/ -. ( -. ( -. ph \/ ps ) \/ ( ph -> ps ) ) ) $=
|
|
( wi wn wo wb imor rb-bijust mpbi ) ABCZADBEZFJDKEDKDJEDEDABGJKHI $.
|
|
|
|
${
|
|
anmp.min $e |- ph $.
|
|
anmp.maj $e |- ( -. ph \/ ps ) $.
|
|
$( Modus ponens for ` \/ ` ` -. ` axiom systems. (Contributed by Anthony
|
|
Hart, 12-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
anmp $p |- ps $=
|
|
( imorri ax-mp ) ABCABDEF $.
|
|
$}
|
|
|
|
$( The first of four axioms in the Russell-Bernays axiom system.
|
|
(Contributed by Anthony Hart, 13-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rb-ax1 $p |- ( -. ( -. ps \/ ch ) \/ ( -. ( ph \/ ps ) \/ ( ph \/ ch ) ) ) $=
|
|
( wn wo wi orim2 imor 3imtr3i imori ) BDCEZABEZDACEZEZBCFLMFKNABCGBCHLMHIJ
|
|
$.
|
|
|
|
$( The second of four axioms in the Russell-Bernays axiom system.
|
|
(Contributed by Anthony Hart, 13-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rb-ax2 $p |- ( -. ( ph \/ ps ) \/ ( ps \/ ph ) ) $=
|
|
( wo wn pm1.4 con3i con1i orri ) ABCZDZBACZKJIKABEFGH $.
|
|
|
|
$( The third of four axioms in the Russell-Bernays axiom system.
|
|
(Contributed by Anthony Hart, 13-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rb-ax3 $p |- ( -. ph \/ ( ps \/ ph ) ) $=
|
|
( wn wo pm2.46 con1i orri ) ACZBADZIHBAEFG $.
|
|
|
|
$( The fourth of four axioms in the Russell-Bernays axiom system.
|
|
(Contributed by Anthony Hart, 13-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rb-ax4 $p |- ( -. ( ph \/ ph ) \/ ph ) $=
|
|
( wo wn pm1.2 con3i con1i orri ) AABZCZAAIHAADEFG $.
|
|
|
|
${
|
|
rbsyl.1 $e |- ( -. ps \/ ch ) $.
|
|
rbsyl.2 $e |- ( ph \/ ps ) $.
|
|
$( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
|
|
(Contributed by Anthony Hart, 18-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rbsyl $p |- ( ph \/ ch ) $=
|
|
( wo wn rb-ax1 anmp ) ABFZACFZEBGCFJGKFDABCHII $.
|
|
$}
|
|
|
|
${
|
|
rblem1.1 $e |- ( -. ph \/ ps ) $.
|
|
rblem1.2 $e |- ( -. ch \/ th ) $.
|
|
$( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
|
|
(Contributed by Anthony Hart, 18-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rblem1 $p |- ( -. ( ph \/ ch ) \/ ( ps \/ th ) ) $=
|
|
( wo wn rb-ax1 anmp rb-ax2 rbsyl ) ACGHZBCGZBDGZCHDGNHOGFBCDIJMCBGZNCBKMC
|
|
AGZPAHBGQHPGECABIJACKLLL $.
|
|
$}
|
|
|
|
$( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
|
|
(Contributed by Anthony Hart, 18-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rblem2 $p |- ( -. ( ch \/ ph ) \/ ( ch \/ ( ph \/ ps ) ) ) $=
|
|
( wn wo rb-ax2 rb-ax3 rbsyl rb-ax1 anmp ) ADZABEZECAEDCLEEKBAELBAFABGHCALIJ
|
|
$.
|
|
|
|
$( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
|
|
(Contributed by Anthony Hart, 18-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rblem3 $p |- ( -. ( ch \/ ph ) \/ ( ( ch \/ ps ) \/ ph ) ) $=
|
|
( wo wn rb-ax2 rblem2 rbsyl ) CADEZACBDZDZJADAJFIACDKCBAGCAFHH $.
|
|
|
|
${
|
|
rblem4.1 $e |- ( -. ph \/ th ) $.
|
|
rblem4.2 $e |- ( -. ps \/ ta ) $.
|
|
rblem4.3 $e |- ( -. ch \/ et ) $.
|
|
$( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
|
|
(Contributed by Anthony Hart, 18-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rblem4 $p |- ( -. ( ( ph \/ ps ) \/ ch ) \/ ( ( et \/ ta ) \/ th ) ) $=
|
|
( wo wn rblem1 rb-ax2 rb-ax1 anmp rbsyl rb-ax4 rblem2 rb-ax3 ) ABJZCJKZCB
|
|
JZAJZFEJZDJUBUDADCFBEIHLGLUABCJZAJZUCUFKZAUBJZUCAUBMUGAUEJZUHUEKUBJUIKUHJ
|
|
BCMAUEUBNOUEAMPPUAUFUFJUFUFQTUFCUFTKUIUFAUEMBCARPCKZUEJUJUFJCBSUEAUJROLPP
|
|
P $.
|
|
$}
|
|
|
|
$( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
|
|
(Contributed by Anthony Hart, 19-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rblem5 $p |- ( -. ( -. -. ph \/ ps ) \/ ( -. -. ps \/ ph ) ) $=
|
|
( wn wo rb-ax2 rb-ax4 rb-ax3 rbsyl anmp rblem1 ) ACZCZBDCABCZCZDNADANELABNK
|
|
ADLCZADKAADAAFAAGHZKOAAOLDLODOLLDLLFLLGHOLEIPJINMDMNDNMMDMMFMMGHNMEIJH $.
|
|
|
|
${
|
|
rblem6.1 $e |- -. ( -. ( -. ph \/ ps ) \/ -. ( -. ps \/ ph ) ) $.
|
|
$( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
|
|
(Contributed by Anthony Hart, 19-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rblem6 $p |- ( -. ph \/ ps ) $=
|
|
( wn wo rb-ax4 rb-ax3 rbsyl rb-ax2 anmp rblem3 rblem5 ) ADBEZDZBDAEDZEZDZ
|
|
MCNDZPEZQDMEPREZSNREZTRNEUARNNENNFNNGHRNIJRONKJPRIJMPLJJ $.
|
|
$}
|
|
|
|
${
|
|
rblem7.1 $e |- -. ( -. ( -. ph \/ ps ) \/ -. ( -. ps \/ ph ) ) $.
|
|
$( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
|
|
(Contributed by Anthony Hart, 19-Aug-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
rblem7 $p |- ( -. ps \/ ph ) $=
|
|
( wn wo rb-ax3 rblem5 anmp ) ADBEDZBDAEZDZEZDZJCKDLEMDJEKIFJLGHH $.
|
|
$}
|
|
|
|
${
|
|
re1axmp.min $e |- ph $.
|
|
re1axmp.maj $e |- ( ph -> ps ) $.
|
|
$( ~ ax-mp derived from Russell-Bernays'. (Contributed by Anthony Hart,
|
|
19-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re1axmp $p |- ps $=
|
|
( wi wn wo rb-imdf rblem6 anmp ) ABCABEZAFBGZDKLABHIJJ $.
|
|
$}
|
|
|
|
$( ~ luk-1 derived from Russell-Bernays'. (Contributed by Anthony Hart,
|
|
19-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re2luk1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
|
|
( wi wn rb-imdf rblem7 rblem6 rb-ax2 rb-ax4 rb-ax3 rbsyl anmp rblem1 rb-ax1
|
|
wo rblem4 ) ABDZEZBCDZACDZDZPZRUBDZSTEZUAPZUBUBUFTUAFGSAEZBPZUFUHEZBECPZEZU
|
|
GCPZPZUFUKUEULUAUEUJPZUKEZUEPZTUJBCFHUNEUEUOPUPUEUOIUEUEUJUOUEEUEUEPUEUEJUE
|
|
UEKLUOUKPUKUOPUOUKUKPUKUKJUKUKKLZUOUKIMNLMUAULACFGNUKUIULPZPZUIUMPZUGBCOUSE
|
|
ZUMUIPZUTUMUIIVAURUKPVBUIULUKUIULUKUIEUIUIPUIUIJUIUIKLULEULULPULULJULULKLUQ
|
|
QUKURILLMLRUHABFHLLUDUCRUBFGM $.
|
|
|
|
$( ~ luk-2 derived from Russell-Bernays'. (Contributed by Anthony Hart,
|
|
19-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re2luk2 $p |- ( ( -. ph -> ph ) -> ph ) $=
|
|
( wn wi wo rb-ax4 rb-ax3 rbsyl rb-ax2 anmp rblem1 rb-imdf rblem6 rblem7 ) A
|
|
BZACZBZADZOACZPNBZADZATBAADZAAEZSAAANADSBZADNUAAUBAAFGZNUCAAUCSDSUCDUCSSDSS
|
|
ESSFGUCSHIUDJIUDJGOTNAKLGRQOAKMI $.
|
|
|
|
$( ~ luk-3 derived from Russell-Bernays'.
|
|
|
|
This theorem, along with ~ re1axmp , ~ re2luk1 , and ~ re2luk2 shows that
|
|
~ rb-ax1 , ~ rb-ax2 , ~ rb-ax3 , and ~ rb-ax4 , along with ~ anmp , can be
|
|
used as a complete axiomatization of propositional calculus. (Contributed
|
|
by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
re2luk3 $p |- ( ph -> ( -. ph -> ps ) ) $=
|
|
( wn wi wo rb-imdf rblem7 rb-ax4 rb-ax3 rbsyl rb-ax2 anmp rblem2 ) ACZNBDZE
|
|
ZAODZNNCZBEZOOSNBFGNREZNSERNETRNNENNHNNIJRNKLRBNMLJQPAOFGL $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Stoic logic indemonstrables (Chrysippus of Soli)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The Greek Stoics developed a system of logic.
|
|
The Stoic Chrysippus, in particular, was often considered one of the greatest
|
|
logicians of antiquity.
|
|
Stoic logic is different from Aristotle's system, since it focuses
|
|
on propositional logic,
|
|
though later thinkers did combine the systems of the Stoics with Aristotle.
|
|
Jan Lukasiewicz reports,
|
|
"For anybody familiar with mathematical logic it is self-evident
|
|
that the Stoic dialectic is the ancient form of modern propositional logic"
|
|
( _On the history of the logic of proposition_ by Jan Lukasiewicz (1934),
|
|
translated in: _Selected Works_ - Edited by Ludwik Borkowski -
|
|
Amsterdam, North-Holland, 1970 pp. 197-217,
|
|
referenced in "History of Logic"
|
|
~ https://www.historyoflogic.com/logic-stoics.htm ).
|
|
For more about Aristotle's system, see ~ barbara and related theorems.
|
|
|
|
A key part of the Stoic logic system is a set of five "indemonstrables"
|
|
assigned to Chrysippus of Soli by Diogenes Laertius, though in
|
|
general it is difficult to assign specific
|
|
ideas to specific thinkers.
|
|
The indemonstrables are described in, for example,
|
|
[Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5.
|
|
These indemonstrables are
|
|
modus ponendo ponens (modus ponens) ~ ax-mp ,
|
|
modus tollendo tollens (modus tollens) ~ mto ,
|
|
modus ponendo tollens I ~ mpto1 ,
|
|
modus ponendo tollens II ~ mpto2 , and
|
|
modus tollendo ponens (exclusive-or version) ~ mtp-xor .
|
|
The first is an axiom, the second is already proved; in this section
|
|
we prove the other three.
|
|
Since we assume or prove all of indemonstrables, the system of logic we use
|
|
here is as at least as strong as the set of Stoic indemonstrables.
|
|
Note that modus tollendo ponens ~ mtp-xor originally used exclusive-or,
|
|
but over time the name modus tollendo ponens has increasingly referred
|
|
to an inclusive-or variation, which is proved in ~ mtp-or .
|
|
This set of indemonstrables is not the entire system of Stoic logic.
|
|
|
|
$)
|
|
|
|
${
|
|
$( Minor premise for modus ponendo tollens 1. $)
|
|
mpto1.1 $e |- ph $.
|
|
$( Major premise for modus ponendo tollens 1. $)
|
|
mpto1.2 $e |- -. ( ph /\ ps ) $.
|
|
$( Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic.
|
|
See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and
|
|
rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after
|
|
~ mpto2 ) as a "safer, and these days much more common" version of modus
|
|
ponendo tollens because it avoids confusion between inclusive-or and
|
|
exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.) $)
|
|
mpto1 $p |- -. ps $=
|
|
( wn imnani ax-mp ) ABECABDFG $.
|
|
$}
|
|
|
|
${
|
|
$( Minor premise for modus ponendo tollens 2. $)
|
|
mpto2.1 $e |- ph $.
|
|
$( Major premise for modus ponendo tollens 2. $)
|
|
mpto2.2 $e |- ( ph \/_ ps ) $.
|
|
$( Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic.
|
|
Note that this uses exclusive-or ` \/_ ` . See rule 2 on
|
|
[Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in
|
|
[Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.)
|
|
(Proof shortened by Wolf Lammen, 12-Nov-2017.) $)
|
|
mpto2 $p |- -. ps $=
|
|
( wn wb wxo df-xor mpbi xor3 ) ABEZCABFEZAKFABGLDABHIABJII $.
|
|
$}
|
|
|
|
${
|
|
$( Minor premise for modus ponendo tollens 2. $)
|
|
mpto2OLD.1 $e |- ph $.
|
|
$( Major premise for modus ponendo tollens 2. $)
|
|
mpto2OLD.2 $e |- ( ph \/_ ps ) $.
|
|
$( Obsolete version of ~ mpto2 as of 12-Nov-2017. (Contributed by David A.
|
|
Wheeler, 3-Jul-2016.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
mpto2OLD $p |- -. ps $=
|
|
( wn wb wxo df-xor mpbi nbbn mpbir con1bii ) BEACABAEBFABFEZABGMDABHIABJK
|
|
LK $.
|
|
$}
|
|
|
|
${
|
|
$( Minor premise for modus tollendo ponens (original exclusive-or version).
|
|
$)
|
|
mtp-xor.1 $e |- -. ph $.
|
|
$( Major premise for modus tollendo ponens (original exclusive-or version).
|
|
$)
|
|
mtp-xor.2 $e |- ( ph \/_ ps ) $.
|
|
$( Modus tollendo ponens (original exclusive-or version), aka disjunctive
|
|
syllogism, one of the five "indemonstrables" in Stoic logic. The rule
|
|
says, "if ` ph ` is not true, and either ` ph ` or ` ps ` (exclusively)
|
|
are true, then ` ps ` must be true." Today the name "modus tollendo
|
|
ponens" often refers to a variant, the inclusive-or version as defined
|
|
in ~ mtp-or . See rule 3 on [Lopez-Astorga] p. 12 (note that the "or"
|
|
is the same as ~ mpto2 , that is, it is exclusive-or ~ df-xor ), rule 3
|
|
of [Sanford] p. 39 (where it is not as clearly stated which kind of "or"
|
|
is used but it appears to be in the same sense as ~ mpto2 ), and rule A5
|
|
in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by
|
|
David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen,
|
|
11-Nov-2017.) $)
|
|
mtp-xor $p |- ps $=
|
|
( wn wxo xorneg mpbir mpto2 notnotri ) BAEZBEZCKLFABFDABGHIJ $.
|
|
|
|
$( Obsolete version of ~ mtp-xor as of 11-Nov-2017. (Contributed by David
|
|
A. Wheeler, 4-Jul-2016.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
mtp-xorOLD $p |- ps $=
|
|
( wn wb wxo df-xor mpbi bicom mtbi xor3 mpbir ) BAEZCBAFZEBNFABFZOABGPEDA
|
|
BHIABJKBALIM $.
|
|
$}
|
|
|
|
${
|
|
$( Minor premise for modus tollendo ponens (inclusive-or version). $)
|
|
mtp-or.1 $e |- -. ph $.
|
|
$( Major premise for modus tollendo ponens (inclusive-or version). $)
|
|
mtp-or.2 $e |- ( ph \/ ps ) $.
|
|
$( Modus tollendo ponens (inclusive-or version), aka disjunctive
|
|
syllogism. This is similar to ~ mtp-xor , one of the five original
|
|
"indemonstrables" in Stoic logic. However, in Stoic logic this rule
|
|
used exclusive-or, while the name modus tollendo ponens often refers to
|
|
a variant of the rule that uses inclusive-or instead. The rule says,
|
|
"if ` ph ` is not true, and ` ph ` or ` ps ` (or both) are true, then
|
|
` ps ` must be true." An alternative phrasing is, "Once you eliminate
|
|
the impossible, whatever remains, no matter how improbable, must be the
|
|
truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of
|
|
the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.)
|
|
(Proof shortened by Wolf Lammen, 11-Nov-2017.) $)
|
|
mtp-or $p |- ps $=
|
|
( wn ori ax-mp ) AEBCABDFG $.
|
|
|
|
$( Obsolete version of ~ mtp-or as of 11-Nov-2017. (Contributed by David
|
|
A. Wheeler, 3-Jul-2016.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
mtp-orOLD $p |- ps $=
|
|
( wn wo wi pm2.53 ax-mp ) AEZBCABFJBGDABHII $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
|
|
Here we extend the language of wffs with predicate calculus, which allows us
|
|
to talk about individual objects in a domain of discussion (which for us will
|
|
be the universe of all sets, so we call them "setvar variables") and make
|
|
true/false statements about predicates, which are relationships between
|
|
objects, such as whether or not two objects are equal. In addition, we
|
|
introduce universal quantification ("for all") in order to make statements
|
|
about whether a wff holds for every object in the domain of discussion.
|
|
Later we introduce existential quantification ("there exists", ~ df-ex )
|
|
which is defined in terms of universal quantification.
|
|
|
|
Our axioms are really axiom _schemes_, and our wff and setvar variables are
|
|
metavariables ranging over expressions in an underlying "object language."
|
|
This is explained here: ~ http://us.metamath.org/mpeuni/mmset.html#axiomnote
|
|
|
|
Our axiom system starts with the predicate calculus axiom schemes system S2
|
|
of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate
|
|
Logic with Identity" [Tarski]. System S2 is defined in the last paragraph on
|
|
p. 77, and repeated on p. 81 of [KalishMontague]. We do not include scheme
|
|
B5 (our ~ sp ) since [KalishMontague] shows it to be logically redundant
|
|
(Lemma 9, p. 87, which we prove as theorem ~ spw below).
|
|
|
|
Theorem ~ spw can be used to prove any instance of ~ sp having no wff
|
|
metavariables and mutually distinct setvar variables. However, it seems that
|
|
~ sp in its general form cannot be derived from only Tarski's schemes. We
|
|
do not include B5 i.e. ~ sp as part of what we call "Tarski's system"
|
|
because we want it to be the smallest set of axioms that is logically
|
|
complete with no redundancies. We later prove ~ sp as theorem ~ ax4
|
|
using the auxiliary axioms that make our system metalogically complete.
|
|
|
|
Our version of Tarski's system S2 consists of propositional calculus plus
|
|
~ ax-gen , ~ ax-5 , ~ ax-17 , ~ ax-9 , ~ ax-8 , ~ ax-13 , and ~ ax-14 . The
|
|
last 3 are equality axioms that represent 3 sub-schemes of Tarski's scheme
|
|
B8. Due to its side-condition ("where ` ph ` is an atomic formula and ` ps `
|
|
is obtained by replacing an occurrence of the variable ` x ` by the variable
|
|
` y ` "), we cannot represent his B8 directly without greatly complicating
|
|
our scheme language, but the simpler schemes ~ ax-8 , ~ ax-13 , and ~ ax-14
|
|
are sufficient for set theory.
|
|
|
|
Tarski's system is exactly equivalent to the traditional axiom system in most
|
|
logic textbooks but has the advantage of being easy to manipulate with a
|
|
computer program, and its simpler metalogic (with no built-in notions of free
|
|
variable and proper substitution) is arguably easier for a non-logician human
|
|
to follow step by step in a proof.
|
|
|
|
However, in our system that derives schemes (rather than object language
|
|
theorems) from other schemes, Tarski's S2 is not complete. For example, we
|
|
cannot derive scheme ~ sp , even though (using ~ spw ) we can derive all
|
|
instances of it that don't involve wff metavariables or bundled setvar
|
|
metavariables. (Two setvar metavariables are "bundled" if they can be
|
|
substituted with the same setvar metavariable i.e. do not have a $d distinct
|
|
variable proviso.) Later we will introduce auxiliary axiom schemes ~ ax-6 ,
|
|
~ ax-7 , ~ ax-12 , and ~ ax-11 that are metatheorems of Tarski's system (i.e.
|
|
are logically redundant) but which give our system the property of
|
|
"metalogical completeness," allowing us to prove directly (instead of, say,
|
|
by induction on formula length) all possible schemes that can be expressed in
|
|
our language.
|
|
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Universal quantifier; define "exists" and "not free"
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new symbols needed for pure predicate calculus. $)
|
|
$c A. $. $( "inverted A" universal quantifier (read: "for all") $)
|
|
$c setvar $. $( Individual variable type (read: "the following is an
|
|
individual (setvar) variable" $)
|
|
|
|
$( Add 'setvar' as a typecode. $)
|
|
$( $j syntax 'setvar'; $)
|
|
|
|
$( Declare some names for individual variables. $)
|
|
$v x $.
|
|
$v y $.
|
|
$v z $.
|
|
$v w $.
|
|
$v v $.
|
|
$v u $.
|
|
$v t $.
|
|
$( Let ` x ` be an individual variable. $)
|
|
vx $f setvar x $.
|
|
$( Let ` y ` be an individual variable. $)
|
|
vy $f setvar y $.
|
|
$( Let ` z ` be an individual variable. $)
|
|
vz $f setvar z $.
|
|
$( Let ` w ` be an individual variable. $)
|
|
vw $f setvar w $.
|
|
$( Let ` v ` be an individual variable. $)
|
|
vv $f setvar v $.
|
|
$( Let ` u ` be an individual variable. $)
|
|
vu $f setvar u $.
|
|
$( Let ` t ` be an individual variable. $)
|
|
vt $f setvar t $.
|
|
|
|
$( Extend wff definition to include the universal quantifier ('for all').
|
|
` A. x ph ` is read " ` ph ` (phi) is true for all ` x ` ." Typically, in
|
|
its final application ` ph ` would be replaced with a wff containing a
|
|
(free) occurrence of the variable ` x ` , for example ` x = y ` . In a
|
|
universe with a finite number of objects, "for all" is equivalent to a big
|
|
conjunction (AND) with one wff for each possible case of ` x ` . When the
|
|
universe is infinite (as with set theory), such a propositional-calculus
|
|
equivalent is not possible because an infinitely long formula has no
|
|
meaning, but conceptually the idea is the same. $)
|
|
wal $a wff A. x ph $.
|
|
|
|
$( Register 'A.' as a primitive expression (lacking a definition). $)
|
|
$( $j primitive 'wal'; $)
|
|
|
|
$( Declare the existential quantifier symbol. $)
|
|
$c E. $. $( Backwards E (read: "there exists") $)
|
|
|
|
$( Extend wff definition to include the existential quantifier ("there
|
|
exists"). $)
|
|
wex $a wff E. x ph $.
|
|
|
|
$( Define existential quantification. ` E. x ph ` means "there exists at
|
|
least one set ` x ` such that ` ph ` is true." Definition of [Margaris]
|
|
p. 49. (Contributed by NM, 5-Aug-1993.) $)
|
|
df-ex $a |- ( E. x ph <-> -. A. x -. ph ) $.
|
|
|
|
$( Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) $)
|
|
alnex $p |- ( A. x -. ph <-> -. E. x ph ) $=
|
|
( wex wn wal df-ex con2bii ) ABCADBEABFG $.
|
|
|
|
$c F/ $. $( The not-free symbol. $)
|
|
|
|
$( Extend wff definition to include the not-free predicate. $)
|
|
wnf $a wff F/ x ph $.
|
|
|
|
$( Define the not-free predicate for wffs. This is read " ` x ` is not free
|
|
in ` ph ` ". Not-free means that the value of ` x ` cannot affect the
|
|
value of ` ph ` , e.g., any occurrence of ` x ` in ` ph ` is effectively
|
|
bound by a "for all" or something that expands to one (such as "there
|
|
exists"). In particular, substitution for a variable not free in a wff
|
|
does not affect its value ( ~ sbf ). An example of where this is used is
|
|
~ stdpc5 . See ~ nf2 for an alternative definition which does not involve
|
|
nested quantifiers on the same variable.
|
|
|
|
Not-free is a commonly used constraint, so it is useful to have a notation
|
|
for it. Surprisingly, there is no common formal notation for it, so here
|
|
we devise one. Our definition lets us work with the not-free notion
|
|
within the logic itself rather than as a metalogical side condition.
|
|
|
|
To be precise, our definition really means "effectively not free," because
|
|
it is slightly less restrictive than the usual textbook definition for
|
|
not-free (which only considers syntactic freedom). For example, ` x ` is
|
|
effectively not free in the bare expression ` x = x ` (see ~ nfequid ),
|
|
even though ` x ` would be considered free in the usual textbook
|
|
definition, because the value of ` x ` in the expression ` x = x ` cannot
|
|
affect the truth of the expression (and thus substitution will not change
|
|
the result).
|
|
|
|
This predicate only applies to wffs. See ~ df-nfc for a not-free
|
|
predicate for class variables. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
df-nf $a |- ( F/ x ph <-> A. x ( ph -> A. x ph ) ) $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Rule scheme ax-gen (Generalization)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
ax-g.1 $e |- ph $.
|
|
$( Rule of Generalization. The postulated inference rule of pure predicate
|
|
calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if
|
|
something is unconditionally true, then it is true for all values of a
|
|
variable. For example, if we have proved ` x = x ` , we can conclude
|
|
` A. x x = x ` or even ` A. y x = x ` . Theorem allt in set.mm shows
|
|
the special case ` A. x T. ` . Theorem ~ spi shows we can go the other
|
|
way also: in other words we can add or remove universal quantifiers from
|
|
the beginning of any theorem as required. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ax-gen $a |- A. x ph $.
|
|
$}
|
|
|
|
${
|
|
gen2.1 $e |- ph $.
|
|
$( Generalization applied twice. (Contributed by NM, 30-Apr-1998.) $)
|
|
gen2 $p |- A. x A. y ph $=
|
|
( wal ax-gen ) ACEBACDFF $.
|
|
$}
|
|
|
|
${
|
|
mpg.1 $e |- ( A. x ph -> ps ) $.
|
|
mpg.2 $e |- ph $.
|
|
$( Modus ponens combined with generalization. (Contributed by NM,
|
|
24-May-1994.) $)
|
|
mpg $p |- ps $=
|
|
( wal ax-gen ax-mp ) ACFBACEGDH $.
|
|
$}
|
|
|
|
${
|
|
mpgbi.1 $e |- ( A. x ph <-> ps ) $.
|
|
mpgbi.2 $e |- ph $.
|
|
$( Modus ponens on biconditional combined with generalization.
|
|
(Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan,
|
|
28-Oct-2008.) $)
|
|
mpgbi $p |- ps $=
|
|
( wal ax-gen mpbi ) ACFBACEGDH $.
|
|
$}
|
|
|
|
${
|
|
mpgbir.1 $e |- ( ph <-> A. x ps ) $.
|
|
mpgbir.2 $e |- ps $.
|
|
$( Modus ponens on biconditional combined with generalization.
|
|
(Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan,
|
|
28-Oct-2008.) $)
|
|
mpgbir $p |- ph $=
|
|
( wal ax-gen mpbir ) ABCFBCEGDH $.
|
|
$}
|
|
|
|
${
|
|
nfi.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Deduce that ` x ` is not free in ` ph ` from the definition.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfi $p |- F/ x ph $=
|
|
( wnf wal wi df-nf mpgbir ) ABDAABEFBABGCH $.
|
|
$}
|
|
|
|
${
|
|
hbth.1 $e |- ph $.
|
|
$( No variable is (effectively) free in a theorem.
|
|
|
|
This and later "hypothesis-building" lemmas, with labels starting
|
|
"hb...", allow us to construct proofs of formulas of the form
|
|
` |- ( ph -> A. x ph ) ` from smaller formulas of this form. These are
|
|
useful for constructing hypotheses that state " ` x ` is (effectively)
|
|
not free in ` ph ` ." (Contributed by NM, 5-Aug-1993.) $)
|
|
hbth $p |- ( ph -> A. x ph ) $=
|
|
( wal ax-gen a1i ) ABDAABCEF $.
|
|
|
|
$( No variable is (effectively) free in a theorem. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
nfth $p |- F/ x ph $=
|
|
( hbth nfi ) ABABCDE $.
|
|
$}
|
|
|
|
$( The true constant has no free variables. (This can also be proven in one
|
|
step with ~ nfv , but this proof does not use ~ ax-17 .) (Contributed by
|
|
Mario Carneiro, 6-Oct-2016.) $)
|
|
nftru $p |- F/ x T. $=
|
|
( wtru tru nfth ) BACD $.
|
|
|
|
${
|
|
nex.1 $e |- -. ph $.
|
|
$( Generalization rule for negated wff. (Contributed by NM,
|
|
18-May-1994.) $)
|
|
nex $p |- -. E. x ph $=
|
|
( wn wex alnex mpgbi ) ADABEDBABFCG $.
|
|
$}
|
|
|
|
${
|
|
nfnth.1 $e |- -. ph $.
|
|
$( No variable is (effectively) free in a non-theorem. (Contributed by
|
|
Mario Carneiro, 6-Dec-2016.) $)
|
|
nfnth $p |- F/ x ph $=
|
|
( wal pm2.21i nfi ) ABAABDCEF $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom scheme ax-5 (Quantified Implication)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ax-5 $a |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $.
|
|
|
|
$( Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by O'Cat, 30-Mar-2008.) $)
|
|
alim $p |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $=
|
|
( ax-5 ) ABCD $.
|
|
|
|
${
|
|
alimi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
alimi $p |- ( A. x ph -> A. x ps ) $=
|
|
( wi wal ax-5 mpg ) ABEACFBCFECABCGDH $.
|
|
|
|
$( Inference doubly quantifying both antecedent and consequent.
|
|
(Contributed by NM, 3-Feb-2005.) $)
|
|
2alimi $p |- ( A. x A. y ph -> A. x A. y ps ) $=
|
|
( wal alimi ) ADFBDFCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
al2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference quantifying antecedent, nested antecedent, and consequent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
al2imi $p |- ( A. x ph -> ( A. x ps -> A. x ch ) ) $=
|
|
( wal wi alimi alim syl ) ADFBCGZDFBDFCDFGAKDEHBCDIJ $.
|
|
$}
|
|
|
|
${
|
|
alanimi.1 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Variant of ~ al2imi with conjunctive antecedent. (Contributed by Andrew
|
|
Salmon, 8-Jun-2011.) $)
|
|
alanimi $p |- ( ( A. x ph /\ A. x ps ) -> A. x ch ) $=
|
|
( wal ex al2imi imp ) ADFBDFCDFABCDABCEGHI $.
|
|
$}
|
|
|
|
${
|
|
alimdh.1 $e |- ( ph -> A. x ph ) $.
|
|
alimdh.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM,
|
|
4-Jan-2002.) $)
|
|
alimdh $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
|
|
( wal wi al2imi syl ) AADGBDGCDGHEABCDFIJ $.
|
|
$}
|
|
|
|
$( Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
albi $p |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) ) $=
|
|
( wb wal bi1 al2imi bi2 impbid ) ABDZCEACEBCEJABCABFGJBACABHGI $.
|
|
|
|
${
|
|
alrimih.1 $e |- ( ph -> A. x ph ) $.
|
|
alrimih.2 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
alrimih $p |- ( ph -> A. x ps ) $=
|
|
( wal alimi syl ) AACFBCFDABCEGH $.
|
|
$}
|
|
|
|
${
|
|
albii.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding universal quantifier to both sides of an equivalence.
|
|
(Contributed by NM, 7-Aug-1994.) $)
|
|
albii $p |- ( A. x ph <-> A. x ps ) $=
|
|
( wb wal albi mpg ) ABEACFBCFECABCGDH $.
|
|
|
|
$( Theorem albii is the congruence law for universal quantification. $)
|
|
$( $j congruence 'albii'; $)
|
|
|
|
$( Inference adding two universal quantifiers to both sides of an
|
|
equivalence. (Contributed by NM, 9-Mar-1997.) $)
|
|
2albii $p |- ( A. x A. y ph <-> A. x A. y ps ) $=
|
|
( wal albii ) ADFBDFCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
hbxfrbi.1 $e |- ( ph <-> ps ) $.
|
|
hbxfrbi.2 $e |- ( ps -> A. x ps ) $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. See ~ hbxfreq for equality version. (Contributed by
|
|
Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
hbxfrbi $p |- ( ph -> A. x ph ) $=
|
|
( wal albii 3imtr4i ) BBCFAACFEDABCDGH $.
|
|
$}
|
|
|
|
${
|
|
nfbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Equality theorem for not-free. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfbii $p |- ( F/ x ph <-> F/ x ps ) $=
|
|
( wal wi wnf albii imbi12i df-nf 3bitr4i ) AACEZFZCEBBCEZFZCEACGBCGMOCABL
|
|
NDABCDHIHACJBCJK $.
|
|
|
|
${
|
|
nfxfr.2 $e |- F/ x ps $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfxfr $p |- F/ x ph $=
|
|
( wnf nfbii mpbir ) ACFBCFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
nfxfrd.2 $e |- ( ch -> F/ x ps ) $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Mario Carneiro, 24-Sep-2016.) $)
|
|
nfxfrd $p |- ( ch -> F/ x ph ) $=
|
|
( wnf nfbii sylibr ) CBDGADGFABDEHI $.
|
|
$}
|
|
$}
|
|
|
|
$( Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) $)
|
|
alex $p |- ( A. x ph <-> -. E. x -. ph ) $=
|
|
( wal wn wex notnot albii alnex bitri ) ABCADZDZBCJBEDAKBAFGJBHI $.
|
|
|
|
$( Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by
|
|
Andrew Salmon, 24-May-2011.) $)
|
|
2nalexn $p |- ( -. A. x A. y ph <-> E. x E. y -. ph ) $=
|
|
( wn wex wal df-ex alex albii xchbinxr bicomi ) ADCEZBEZACFZBFZDMLDZBFOLBGN
|
|
PBACHIJK $.
|
|
|
|
$( Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
exnal $p |- ( E. x -. ph <-> -. A. x ph ) $=
|
|
( wal wn wex alex con2bii ) ABCADBEABFG $.
|
|
|
|
$( Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 4-Jul-2014.) $)
|
|
exim $p |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) ) $=
|
|
( wi wal wex wn con3 al2imi alnex 3imtr3g con4d ) ABDZCEZBCFZACFZNBGZCEAGZC
|
|
EOGPGMQRCABHIBCJACJKL $.
|
|
|
|
${
|
|
eximi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference adding existential quantifier to antecedent and consequent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
eximi $p |- ( E. x ph -> E. x ps ) $=
|
|
( wi wex exim mpg ) ABEACFBCFECABCGDH $.
|
|
|
|
$( Inference adding two existential quantifiers to antecedent and
|
|
consequent. (Contributed by NM, 3-Feb-2005.) $)
|
|
2eximi $p |- ( E. x E. y ph -> E. x E. y ps ) $=
|
|
( wex eximi ) ADFBDFCABDEGG $.
|
|
$}
|
|
|
|
$( A transformation of quantifiers and logical connectives. (Contributed by
|
|
NM, 19-Aug-1993.) $)
|
|
alinexa $p |- ( A. x ( ph -> -. ps ) <-> -. E. x ( ph /\ ps ) ) $=
|
|
( wn wi wal wa wex imnan albii alnex bitri ) ABDEZCFABGZDZCFNCHDMOCABIJNCKL
|
|
$.
|
|
|
|
$( A relationship between two quantifiers and negation. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
alexn $p |- ( A. x E. y -. ph <-> -. E. x A. y ph ) $=
|
|
( wn wex wal exnal albii alnex bitri ) ADCEZBFACFZDZBFLBEDKMBACGHLBIJ $.
|
|
|
|
$( Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew
|
|
Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.) $)
|
|
2exnexn $p |- ( E. x A. y ph <-> -. A. x E. y -. ph ) $=
|
|
( wn wex wal alexn con2bii ) ADCEBFACFBEABCGH $.
|
|
|
|
$( Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
exbi $p |- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) ) $=
|
|
( wb wal wex wi bi1 alimi exim syl bi2 impbid ) ABDZCEZACFZBCFZOABGZCEPQGNR
|
|
CABHIABCJKOBAGZCEQPGNSCABLIBACJKM $.
|
|
|
|
${
|
|
exbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding existential quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 24-May-1994.) $)
|
|
exbii $p |- ( E. x ph <-> E. x ps ) $=
|
|
( wb wex exbi mpg ) ABEACFBCFECABCGDH $.
|
|
$}
|
|
|
|
${
|
|
2exbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding two existential quantifiers to both sides of an
|
|
equivalence. (Contributed by NM, 16-Mar-1995.) $)
|
|
2exbii $p |- ( E. x E. y ph <-> E. x E. y ps ) $=
|
|
( wex exbii ) ADFBDFCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
3exbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding 3 existential quantifiers to both sides of an
|
|
equivalence. (Contributed by NM, 2-May-1995.) $)
|
|
3exbii $p |- ( E. x E. y E. z ph <-> E. x E. y E. z ps ) $=
|
|
( wex exbii 2exbii ) AEGBEGCDABEFHI $.
|
|
$}
|
|
|
|
$( A transformation of quantifiers and logical connectives. (Contributed by
|
|
NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.) $)
|
|
exanali $p |- ( E. x ( ph /\ -. ps ) <-> -. A. x ( ph -> ps ) ) $=
|
|
( wn wa wex wi wal annim exbii exnal bitri ) ABDEZCFABGZDZCFNCHDMOCABIJNCKL
|
|
$.
|
|
|
|
$( Commutation of conjunction inside an existential quantifier. (Contributed
|
|
by NM, 18-Aug-1993.) $)
|
|
exancom $p |- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) ) $=
|
|
( wa ancom exbii ) ABDBADCABEF $.
|
|
|
|
${
|
|
alrimdh.1 $e |- ( ph -> A. x ph ) $.
|
|
alrimdh.2 $e |- ( ps -> A. x ps ) $.
|
|
alrimdh.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
alrimdh $p |- ( ph -> ( ps -> A. x ch ) ) $=
|
|
( wal alimdh syl5 ) BBDHACDHFABCDEGIJ $.
|
|
$}
|
|
|
|
${
|
|
eximdh.1 $e |- ( ph -> A. x ph ) $.
|
|
eximdh.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM,
|
|
20-May-1996.) $)
|
|
eximdh $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
|
|
( wi wal wex alrimih exim syl ) ABCGZDHBDICDIGAMDEFJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
nexdh.1 $e |- ( ph -> A. x ph ) $.
|
|
nexdh.2 $e |- ( ph -> -. ps ) $.
|
|
$( Deduction for generalization rule for negated wff. (Contributed by NM,
|
|
2-Jan-2002.) $)
|
|
nexdh $p |- ( ph -> -. E. x ps ) $=
|
|
( wn wal wex alrimih alnex sylib ) ABFZCGBCHFALCDEIBCJK $.
|
|
$}
|
|
|
|
${
|
|
albidh.1 $e |- ( ph -> A. x ph ) $.
|
|
albidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for universal quantifier (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
albidh $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
|
|
( wb wal alrimih albi syl ) ABCGZDHBDHCDHGALDEFIBCDJK $.
|
|
$}
|
|
|
|
${
|
|
exbidh.1 $e |- ( ph -> A. x ph ) $.
|
|
exbidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for existential quantifier (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
exbidh $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
|
|
( wb wal wex alrimih exbi syl ) ABCGZDHBDICDIGAMDEFJBCDKL $.
|
|
$}
|
|
|
|
$( Simplification of an existentially quantified conjunction. (Contributed
|
|
by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
exsimpl $p |- ( E. x ( ph /\ ps ) -> E. x ph ) $=
|
|
( wa simpl eximi ) ABDACABEF $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of
|
|
[WhiteheadRussell] p. 147. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 4-Jul-2014.) $)
|
|
19.26 $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) ) $=
|
|
( wa wal simpl alimi simpr jca id alanimi impbii ) ABDZCEZACEZBCEZDNOPMACAB
|
|
FGMBCABHGIABMCMJKL $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by
|
|
NM, 3-Feb-2005.) $)
|
|
19.26-2 $p |- ( A. x A. y ( ph /\ ps ) <->
|
|
( A. x A. y ph /\ A. x A. y ps ) ) $=
|
|
( wa wal 19.26 albii bitri ) ABEDFZCFADFZBDFZEZCFKCFLCFEJMCABDGHKLCGI $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed
|
|
by NM, 13-Sep-2011.) $)
|
|
19.26-3an $p |- ( A. x ( ph /\ ps /\ ch )
|
|
<-> ( A. x ph /\ A. x ps /\ A. x ch ) ) $=
|
|
( wa wal w3a 19.26 anbi1i bitri df-3an albii 3bitr4i ) ABEZCEZDFZADFZBDFZEZ
|
|
CDFZEZABCGZDFQRTGPNDFZTEUANCDHUCSTABDHIJUBODABCKLQRTKM $.
|
|
|
|
$( Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 13-May-2011.) $)
|
|
19.29 $p |- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) $=
|
|
( wal wex wa wi pm3.2 alimi exim syl imp ) ACDZBCEZABFZCEZMBOGZCDNPGAQCABHI
|
|
BOCJKL $.
|
|
|
|
$( Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
19.29r $p |- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ ps ) ) $=
|
|
( wex wal wa 19.29 ancoms exancom sylibr ) ACDZBCEZFBAFCDZABFCDLKMBACGHABCI
|
|
J $.
|
|
|
|
$( Variation of Theorem 19.29 of [Margaris] p. 90 with double
|
|
quantification. (Contributed by NM, 3-Feb-2005.) $)
|
|
19.29r2 $p |- ( ( E. x E. y ph /\ A. x A. y ps ) ->
|
|
E. x E. y ( ph /\ ps ) ) $=
|
|
( wex wal wa 19.29r eximi syl ) ADEZCEBDFZCFGKLGZCEABGDEZCEKLCHMNCABDHIJ $.
|
|
|
|
$( Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification.
|
|
(Contributed by NM, 11-Feb-2005.) $)
|
|
19.29x $p |- ( ( E. x A. y ph /\ A. x E. y ps ) ->
|
|
E. x E. y ( ph /\ ps ) ) $=
|
|
( wal wex wa 19.29r 19.29 eximi syl ) ADEZCFBDFZCEGLMGZCFABGDFZCFLMCHNOCABD
|
|
IJK $.
|
|
|
|
$( Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an
|
|
implication (in the form of the right-hand side) into the scope of a
|
|
single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 27-Jun-2014.) $)
|
|
19.35 $p |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) ) $=
|
|
( wi wex wal wn wa 19.26 annim albii alnex anbi2i 3bitr3i con4bii ) ABDZCEZ
|
|
ACFZBCEZDZPGZCFZRSGZHZQGTGABGZHZCFRUECFZHUBUDAUECIUFUACABJKUGUCRBCLMNPCLRSJ
|
|
NO $.
|
|
|
|
${
|
|
19.35i.1 $e |- E. x ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.35i $p |- ( A. x ph -> E. x ps ) $=
|
|
( wi wex wal 19.35 mpbi ) ABECFACGBCFEDABCHI $.
|
|
$}
|
|
|
|
${
|
|
19.35ri.1 $e |- ( A. x ph -> E. x ps ) $.
|
|
$( Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.35ri $p |- E. x ( ph -> ps ) $=
|
|
( wi wex wal 19.35 mpbir ) ABECFACGBCFEDABCHI $.
|
|
$}
|
|
|
|
$( Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.25 $p |- ( A. y E. x ( ph -> ps ) ->
|
|
( E. y A. x ph -> E. y E. x ps ) ) $=
|
|
( wi wex wal 19.35 biimpi alimi exim syl ) ABECFZDGACGZBCFZEZDGNDFODFEMPDMP
|
|
ABCHIJNODKL $.
|
|
|
|
$( Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
19.30 $p |- ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) $=
|
|
( wn wi wal wex wo exnal exim syl5bir df-or albii 3imtr4i ) ADZBEZCFZACFZDZ
|
|
BCGZEABHZCFRTHSOCGQTACIOBCJKUAPCABLMRTLN $.
|
|
|
|
$( Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 27-Jun-2014.) $)
|
|
19.43 $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) ) $=
|
|
( wo wex wn wi wal df-or exbii 19.35 alnex imbi1i 3bitri bitr4i ) ABDZCEZAC
|
|
EZFZBCEZGZRTDQAFZBGZCEUBCHZTGUAPUCCABIJUBBCKUDSTACLMNRTIO $.
|
|
|
|
$( Obsolete proof of ~ 19.43 as of 3-May-2016. Leave this in for the example
|
|
on the mmrecent.html page. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
19.43OLD $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) ) $=
|
|
( wo wn wal wex wa ioran albii 19.26 alnex anbi12i 3bitri notbii df-ex oran
|
|
3bitr4i ) ABDZEZCFZEACGZEZBCGZEZHZESCGUBUDDUAUFUAAEZBEZHZCFUGCFZUHCFZHUFTUI
|
|
CABIJUGUHCKUJUCUKUEACLBCLMNOSCPUBUDQR $.
|
|
|
|
$( Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.33 $p |- ( ( A. x ph \/ A. x ps ) -> A. x ( ph \/ ps ) ) $=
|
|
( wal wo orc alimi olc jaoi ) ACDABEZCDBCDAJCABFGBJCBAHGI $.
|
|
|
|
$( The antecedent provides a condition implying the converse of ~ 19.33 .
|
|
Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM,
|
|
27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof
|
|
shortened by Wolf Lammen, 5-Jul-2014.) $)
|
|
19.33b $p |- ( -. ( E. x ph /\ E. x ps ) ->
|
|
( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) $=
|
|
( wex wa wn wo wal ianor alnex pm2.53 al2imi syl5bir olc syl6com orcomd ord
|
|
wi 19.30 orc jaoi sylbi 19.33 impbid1 ) ACDZBCDZEFZABGZCHZACHZBCHZGZUGUEFZU
|
|
FFZGUIULRZUEUFIUMUOUNUIUMUKULUMAFZCHUIUKACJUHUPBCABKLMUKUJNOUIUNUJULUIUFUJU
|
|
IUJUFABCSPQUJUKTOUAUBABCUCUD $.
|
|
|
|
$( Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.40 $p |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) ) $=
|
|
( wa wex exsimpl simpr eximi jca ) ABDZCEACEBCEABCFJBCABGHI $.
|
|
|
|
$( Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris]
|
|
p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) $)
|
|
19.40-2 $p |- ( E. x E. y ( ph /\ ps ) ->
|
|
( E. x E. y ph /\ E. x E. y ps ) ) $=
|
|
( wa wex 19.40 eximi syl ) ABEDFZCFADFZBDFZEZCFKCFLCFEJMCABDGHKLCGI $.
|
|
|
|
$( Split a biconditional and distribute quantifier. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
albiim $p |- ( A. x ( ph <-> ps ) <->
|
|
( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) ) $=
|
|
( wb wal wi wa dfbi2 albii 19.26 bitri ) ABDZCEABFZBAFZGZCEMCENCEGLOCABHIMN
|
|
CJK $.
|
|
|
|
$( Split a biconditional and distribute 2 quantifiers. (Contributed by NM,
|
|
3-Feb-2005.) $)
|
|
2albiim $p |- ( A. x A. y ( ph <-> ps ) <->
|
|
( A. x A. y ( ph -> ps ) /\ A. x A. y ( ps -> ph ) ) ) $=
|
|
( wb wal wi wa albiim albii 19.26 bitri ) ABEDFZCFABGDFZBAGDFZHZCFNCFOCFHMP
|
|
CABDIJNOCKL $.
|
|
|
|
$( Add/remove a conjunct in the scope of an existential quantifier.
|
|
(Contributed by Raph Levien, 3-Jul-2006.) $)
|
|
exintrbi $p |- ( A. x ( ph -> ps ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) ) $=
|
|
( wi wal wa wb wex pm4.71 albii exbi sylbi ) ABDZCEAABFZGZCEACHNCHGMOCABIJA
|
|
NCKL $.
|
|
|
|
$( Introduce a conjunct in the scope of an existential quantifier.
|
|
(Contributed by NM, 11-Aug-1993.) $)
|
|
exintr $p |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) ) $=
|
|
( wi wal wex wa exintrbi biimpd ) ABDCEACFABGCFABCHI $.
|
|
|
|
$( Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew
|
|
Salmon, 8-Jun-2011.) $)
|
|
alsyl $p |- ( ( A. x ( ph -> ps ) /\ A. x ( ps -> ch ) ) ->
|
|
A. x ( ph -> ch ) ) $=
|
|
( wi pm3.33 alanimi ) ABEBCEACEDABCFG $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom scheme ax-17 (Distinctness) - first use of $d
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Axiom of Distinctness. This axiom quantifies a variable over a formula
|
|
in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the
|
|
preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski]
|
|
p. 77 and Axiom C5-1 of [Monk2] p. 113.
|
|
|
|
(See comments in ~ ax17o about the logical redundancy of ~ ax-17 in the
|
|
presence of our obsolete axioms.)
|
|
|
|
This axiom essentially says that if ` x ` does not occur in ` ph ` ,
|
|
i.e. ` ph ` does not depend on ` x ` in any way, then we can add the
|
|
quantifier ` A. x ` to ` ph ` with no further assumptions. By ~ sp , we
|
|
can also remove the quantifier (unconditionally). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ax-17 $a |- ( ph -> A. x ph ) $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( ~ ax-17 with antecedent. Useful in proofs of deduction versions of
|
|
bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.) $)
|
|
a17d $p |- ( ph -> ( ps -> A. x ps ) ) $=
|
|
( wal wi ax-17 a1i ) BBCDEABCFG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( A rephrasing of ~ ax-17 using the existential quantifier. (Contributed
|
|
by Wolf Lammen, 4-Dec-2017.) $)
|
|
ax17e $p |- ( E. x ph -> ph ) $=
|
|
( wex wn wal df-ex ax-17 con1i sylbi ) ABCADZBEZDAABFAKJBGHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( If ` x ` is not present in ` ph ` , then ` x ` is not free in ` ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfv $p |- F/ x ph $=
|
|
( ax-17 nfi ) ABABCD $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( ~ nfv with antecedent. Useful in proofs of deduction versions of
|
|
bound-variable hypothesis builders such as ~ nfimd . (Contributed by
|
|
Mario Carneiro, 6-Oct-2016.) $)
|
|
nfvd $p |- ( ph -> F/ x ps ) $=
|
|
( wnf nfv a1i ) BCDABCEF $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
alimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM,
|
|
3-Apr-1994.) $)
|
|
alimdv $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
|
|
( ax-17 alimdh ) ABCDADFEG $.
|
|
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM,
|
|
27-Apr-1994.) $)
|
|
eximdv $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
|
|
( ax-17 eximdh ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
2alimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM,
|
|
27-Apr-2004.) $)
|
|
2alimdv $p |- ( ph -> ( A. x A. y ps -> A. x A. y ch ) ) $=
|
|
( wal alimdv ) ABEGCEGDABCEFHH $.
|
|
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM,
|
|
3-Aug-1995.) $)
|
|
2eximdv $p |- ( ph -> ( E. x E. y ps -> E. x E. y ch ) ) $=
|
|
( wex eximdv ) ABEGCEGDABCEFHH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
albidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for universal quantifier (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
albidv $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
|
|
( ax-17 albidh ) ABCDADFEG $.
|
|
|
|
$( Formula-building rule for existential quantifier (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
exbidv $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
|
|
( ax-17 exbidh ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
2albidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for 2 universal quantifiers (deduction rule).
|
|
(Contributed by NM, 4-Mar-1997.) $)
|
|
2albidv $p |- ( ph -> ( A. x A. y ps <-> A. x A. y ch ) ) $=
|
|
( wal albidv ) ABEGCEGDABCEFHH $.
|
|
|
|
$( Formula-building rule for 2 existential quantifiers (deduction rule).
|
|
(Contributed by NM, 1-May-1995.) $)
|
|
2exbidv $p |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) ) $=
|
|
( wex exbidv ) ABEGCEGDABCEFHH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $. $d z ph $.
|
|
3exbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for 3 existential quantifiers (deduction rule).
|
|
(Contributed by NM, 1-May-1995.) $)
|
|
3exbidv $p |- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) ) $=
|
|
( wex exbidv 2exbidv ) ABFHCFHDEABCFGIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $. $d z ph $. $d w ph $.
|
|
4exbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for 4 existential quantifiers (deduction rule).
|
|
(Contributed by NM, 3-Aug-1995.) $)
|
|
4exbidv $p |- ( ph ->
|
|
( E. x E. y E. z E. w ps <-> E. x E. y E. z E. w ch ) ) $=
|
|
( wex 2exbidv ) ABGIFICGIFIDEABCFGHJJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
alrimiv.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
alrimiv $p |- ( ph -> A. x ps ) $=
|
|
( ax-17 alrimih ) ABCACEDF $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
alrimivv.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
31-Jul-1995.) $)
|
|
alrimivv $p |- ( ph -> A. x A. y ps ) $=
|
|
( wal alrimiv ) ABDFCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $.
|
|
alrimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
10-Feb-1997.) $)
|
|
alrimdv $p |- ( ph -> ( ps -> A. x ch ) ) $=
|
|
( ax-17 alrimdh ) ABCDADFBDFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
exlimiv.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90.
|
|
|
|
This inference, along with our many variants such as ~ rexlimdv , is
|
|
used to implement a metatheorem called "Rule C" that is given in many
|
|
logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C
|
|
in [Margaris] p. 40, or Rule C in Hirst and Hirst's _A Primer for Logic
|
|
and Proof_ p. 59 (PDF p. 65) at
|
|
~ http://www.mathsci.appstate.edu/~~hirstjl/primer/hirst.pdf .
|
|
|
|
In informal proofs, the statement "Let ` C ` be an element such that..."
|
|
almost always means an implicit application of Rule C.
|
|
|
|
In essence, Rule C states that if we can prove that some element ` x `
|
|
exists satisfying a wff, i.e. ` E. x ph ( x ) ` where ` ph ( x ) ` has
|
|
` x ` free, then we can use ` ph ( C ) ` as a hypothesis for the proof
|
|
where ` C ` is a new (ficticious) constant not appearing previously in
|
|
the proof, nor in any axioms used, nor in the theorem to be proved. The
|
|
purpose of Rule C is to get rid of the existential quantifier.
|
|
|
|
We cannot do this in Metamath directly. Instead, we use the original
|
|
` ph ` (containing ` x ` ) as an antecedent for the main part of the
|
|
proof. We eventually arrive at ` ( ph -> ps ) ` where ` ps ` is the
|
|
theorem to be proved and does not contain ` x ` . Then we apply
|
|
~ exlimiv to arrive at ` ( E. x ph -> ps ) ` . Finally, we separately
|
|
prove ` E. x ph ` and detach it with modus ponens ~ ax-mp to arrive at
|
|
the final theorem ` ps ` . (Contributed by NM, 5-Aug-1993.) (Revised
|
|
by Wolf Lammen to remove dependency on ax-9 and ax-8, 4-Dec-2017.) $)
|
|
exlimiv $p |- ( E. x ph -> ps ) $=
|
|
( wex eximi ax17e syl ) ACEBCEBABCDFBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ps $.
|
|
exlimivv.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
1-Aug-1995.) $)
|
|
exlimivv $p |- ( E. x E. y ph -> ps ) $=
|
|
( wex exlimiv ) ADFBCABDEGG $.
|
|
$}
|
|
|
|
${
|
|
$d x ch $. $d x ph $.
|
|
exlimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
27-Apr-1994.) (Revised by Wolf Lammen to remove dependency on ax-9 and
|
|
ax-8, 4-Dec-2017.) $)
|
|
exlimdv $p |- ( ph -> ( E. x ps -> ch ) ) $=
|
|
( wex eximdv ax17e syl6 ) ABDFCDFCABCDEGCDHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ch $. $d x ph $. $d y ch $. $d y ph $.
|
|
exlimdvv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
31-Jul-1995.) $)
|
|
exlimdvv $p |- ( ph -> ( E. x E. y ps -> ch ) ) $=
|
|
( wex exlimdv ) ABEGCDABCEFHH $.
|
|
$}
|
|
|
|
${
|
|
$d x ch $. $d x ph $.
|
|
exlimddv.1 $e |- ( ph -> E. x ps ) $.
|
|
exlimddv.2 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Existential elimination rule of natural deduction. (Contributed by
|
|
Mario Carneiro, 15-Jun-2016.) $)
|
|
exlimddv $p |- ( ph -> ch ) $=
|
|
( wex ex exlimdv mpd ) ABDGCEABCDABCFHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
nfdv.1 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Apply the definition of not-free in a context. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
nfdv $p |- ( ph -> F/ x ps ) $=
|
|
( wal wi wnf alrimiv df-nf sylibr ) ABBCEFZCEBCGAKCDHBCIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
$( Quantification of two variables over a formula in which they do not
|
|
occur. (Contributed by Alan Sare, 12-Apr-2011.) $)
|
|
2ax17 $p |- ( ph -> A. x A. y ph ) $=
|
|
( id alrimivv ) AABCADE $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Equality predicate; define substitution
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( --- Start of patch to prevent connective overloading $)
|
|
$c class $.
|
|
|
|
$( Add 'class' as a typecode. $)
|
|
$( $j syntax 'class'; $)
|
|
|
|
$( This syntax construction states that a variable ` x ` , which has been
|
|
declared to be a setvar variable by $f statement vx, is also a class
|
|
expression. This can be justified informally as follows. We know that
|
|
the class builder ` { y | y e. x } ` is a class by ~ cab . Since (when
|
|
` y ` is distinct from ` x ` ) we have ` x = { y | y e. x } ` by
|
|
~ cvjust , we can argue that the syntax " ` class x ` " can be viewed as
|
|
an abbreviation for " ` class { y | y e. x } ` ". See the discussion
|
|
under the definition of class in [Jech] p. 4 showing that "Every set can
|
|
be considered to be a class."
|
|
|
|
While it is tempting and perhaps occasionally useful to view ~ cv as a
|
|
"type conversion" from a setvar variable to a class variable, keep in mind
|
|
that ~ cv is intrinsically no different from any other class-building
|
|
syntax such as ~ cab , ~ cun , or ~ c0 .
|
|
|
|
For a general discussion of the theory of classes and the role of ~ cv ,
|
|
see ~ http://us.metamath.org/mpeuni/mmset.html#class .
|
|
|
|
(The description above applies to set theory, not predicate calculus. The
|
|
purpose of introducing ` class x ` here, and not in set theory where it
|
|
belongs, is to allow us to express i.e. "prove" the ~ weq of predicate
|
|
calculus from the ~ wceq of set theory, so that we don't "overload" the
|
|
` = ` connective with two syntax definitions. This is done to prevent
|
|
ambiguity that would complicate some Metamath parsers.) $)
|
|
cv $a class x $.
|
|
$( --- End of patch to prevent connective overloading $)
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$( (None - the above patch had no old code.) $)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( Declare the equality predicate symbol. $)
|
|
$c = $. $( Equal sign (read: 'is equal to') $)
|
|
|
|
$( --- Start of patch to prevent connective overloading $)
|
|
${
|
|
$v A $.
|
|
$v B $.
|
|
wceq.cA $f class A $.
|
|
wceq.cB $f class B $.
|
|
$( Extend wff definition to include class equality.
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class .
|
|
|
|
(The purpose of introducing ` wff A = B ` here, and not in set theory
|
|
where it belongs, is to allow us to express i.e. "prove" the ~ weq of
|
|
predicate calculus in terms of the ~ wceq of set theory, so that we
|
|
don't "overload" the ` = ` connective with two syntax definitions. This
|
|
is done to prevent ambiguity that would complicate some Metamath
|
|
parsers. For example, some parsers - although not the Metamath program
|
|
- stumble on the fact that the ` = ` in ` x = y ` could be the ` = ` of
|
|
either ~ weq or ~ wceq , although mathematically it makes no
|
|
difference. The class variables ` A ` and ` B ` are introduced
|
|
temporarily for the purpose of this definition but otherwise not used in
|
|
predicate calculus. See ~ df-cleq for more information on the set
|
|
theory usage of ~ wceq .) $)
|
|
wceq $a wff A = B $.
|
|
$}
|
|
|
|
$( Extend wff definition to include atomic formulas using the equality
|
|
predicate.
|
|
|
|
(Instead of introducing ~ weq as an axiomatic statement, as was done in an
|
|
older version of this database, we introduce it by "proving" a special
|
|
case of set theory's more general ~ wceq . This lets us avoid overloading
|
|
the ` = ` connective, thus preventing ambiguity that would complicate
|
|
certain Metamath parsers. However, logically ~ weq is considered to be a
|
|
primitive syntax, even though here it is artificially "derived" from
|
|
~ wceq . Note: To see the proof steps of this syntax proof, type "show
|
|
proof weq /all" in the Metamath program.) (Contributed by NM,
|
|
24-Jan-2006.) $)
|
|
weq $p wff x = y $=
|
|
( cv wceq ) ACBCD $.
|
|
$( --- End of patch to prevent connective overloading $)
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( Extend wff definition to include atomic formulas using the equality
|
|
predicate.
|
|
|
|
After we introduce ~ cv and ~ wceq in set theory, this syntax construction
|
|
becomes redundant, since it can be derived with the proof
|
|
"vx cv vy cv wceq". @)
|
|
weq @a wff x = y @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( Lemma used in proofs of substitution properties. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equs3 $p |- ( E. x ( x = y /\ ph ) <-> -. A. x ( x = y -> -. ph ) ) $=
|
|
( weq wn wi wal wa wex alinexa con2bii ) BCDZAEFBGLAHBILABJK $.
|
|
|
|
${
|
|
speimfw.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Specialization, with additional weakening to allow bundling of ` x ` and
|
|
` y ` . Uses only Tarski's FOL axiom schemes. (Contributed by NM,
|
|
23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) $)
|
|
speimfw $p |- ( -. A. x -. x = y -> ( A. x ph -> E. x ps ) ) $=
|
|
( weq wex wi wn wal eximi df-ex 19.35 3imtr3i ) CDFZCGABHZCGOICJIACJBCGHO
|
|
PCEKOCLABCMN $.
|
|
$}
|
|
|
|
${
|
|
spimfw.1 $e |- ( -. ps -> A. x -. ps ) $.
|
|
spimfw.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Specialization, with additional weakening to allow bundling of ` x ` and
|
|
` y ` . Uses only Tarski's FOL axiom schemes. (Contributed by NM,
|
|
23-Apr-1017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.) $)
|
|
spimfw $p |- ( -. A. x -. x = y -> ( A. x ph -> ps ) ) $=
|
|
( weq wn wal wex speimfw df-ex con1i sylbi syl6 ) CDGHCIHACIBCJZBABCDFKPB
|
|
HCIZHBBCLBQEMNO $.
|
|
$}
|
|
|
|
${
|
|
ax11i.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
ax11i.2 $e |- ( ps -> A. x ps ) $.
|
|
$( Inference that has ~ ax-11 (without ` A. y ` ) as its conclusion. Uses
|
|
only Tarski's FOL axiom schemes. The hypotheses may be eliminable
|
|
without one or more of these axioms in special cases. Proof similar to
|
|
Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.) $)
|
|
ax11i $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wi wal biimprcd alrimih syl6bi ) CDGZABMAHZCIEBNCFMABEJKL $.
|
|
$}
|
|
|
|
$c [ $. $( Left bracket $)
|
|
$c / $. $( Slash. $)
|
|
$c ] $. $( Right bracket $)
|
|
|
|
$( Extend wff definition to include proper substitution (read "the wff that
|
|
results when ` y ` is properly substituted for ` x ` in wff ` ph ` ").
|
|
(Contributed by NM, 24-Jan-2006.) $)
|
|
wsb $a wff [ y / x ] ph $.
|
|
|
|
$( Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the
|
|
preprint). For our notation, we use ` [ y / x ] ph ` to mean "the wff
|
|
that results from the proper substitution of ` y ` for ` x ` in the wff
|
|
` ph ` ." We can also use ` [ y / x ] ph ` in place of the "free for"
|
|
side condition used in traditional predicate calculus; see, for example,
|
|
~ stdpc4 .
|
|
|
|
Our notation was introduced in Haskell B. Curry's _Foundations of
|
|
Mathematical Logic_ (1977), p. 316 and is frequently used in textbooks of
|
|
lambda calculus and combinatory logic. This notation improves the common
|
|
but ambiguous notation, " ` ph ( y ) ` is the wff that results when ` y `
|
|
is properly substituted for ` x ` in ` ph ( x ) ` ." For example, if the
|
|
original ` ph ( x ) ` is ` x = y ` , then ` ph ( y ) ` is ` y = y ` , from
|
|
which we obtain that ` ph ( x ) ` is ` x = x ` . So what exactly does
|
|
` ph ( x ) ` mean? Curry's notation solves this problem.
|
|
|
|
In most books, proper substitution has a somewhat complicated recursive
|
|
definition with multiple cases based on the occurrences of free and bound
|
|
variables in the wff. Instead, we use a single formula that is exactly
|
|
equivalent and gives us a direct definition. We later prove that our
|
|
definition has the properties we expect of proper substitution (see
|
|
theorems ~ sbequ , ~ sbcom2 and ~ sbid2v ).
|
|
|
|
Note that our definition is valid even when ` x ` and ` y ` are replaced
|
|
with the same variable, as ~ sbid shows. We achieve this by having ` x `
|
|
free in the first conjunct and bound in the second. We can also achieve
|
|
this by using a dummy variable, as the alternate definition ~ dfsb7 shows
|
|
(which some logicians may prefer because it doesn't mix free and bound
|
|
variables). Another version that mixes free and bound variables is
|
|
~ dfsb3 . When ` x ` and ` y ` are distinct, we can express proper
|
|
substitution with the simpler expressions of ~ sb5 and ~ sb6 .
|
|
|
|
There are no restrictions on any of the variables, including what
|
|
variables may occur in wff ` ph ` . (Contributed by NM, 5-Aug-1993.) $)
|
|
df-sb $a |- ( [ y / x ] ph <->
|
|
( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ2 $p |- ( x = y -> ( [ y / x ] ph -> ph ) ) $=
|
|
( wsb weq wi wa wex df-sb simpl com12 syl5bi ) ABCDBCEZAFZMAGBHZGZMAABCIPMA
|
|
NOJKL $.
|
|
|
|
$( One direction of a simplified definition of substitution. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sb1 $p |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) $=
|
|
( wsb weq wi wa wex df-sb simprbi ) ABCDBCEZAFKAGBHABCIJ $.
|
|
|
|
${
|
|
sbimi.1 $e |- ( ph -> ps ) $.
|
|
$( Infer substitution into antecedent and consequent of an implication.
|
|
(Contributed by NM, 25-Jun-1998.) $)
|
|
sbimi $p |- ( [ y / x ] ph -> [ y / x ] ps ) $=
|
|
( weq wi wa wex wsb imim2i anim2i eximi anim12i df-sb 3imtr4i ) CDFZAGZQA
|
|
HZCIZHQBGZQBHZCIZHACDJBCDJRUATUCABQEKSUBCABQELMNACDOBCDOP $.
|
|
$}
|
|
|
|
${
|
|
sbbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Infer substitution into both sides of a logical equivalence.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sbbii $p |- ( [ y / x ] ph <-> [ y / x ] ps ) $=
|
|
( wsb biimpi sbimi biimpri impbii ) ACDFBCDFABCDABEGHBACDABEIHJ $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom scheme ax-9 (Existence)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Existence. One of the equality and substitution axioms of
|
|
predicate calculus with equality. This axiom tells us is that at least
|
|
one thing exists. In this form (not requiring that ` x ` and ` y ` be
|
|
distinct) it was used in an axiom system of Tarski (see Axiom B7' in
|
|
footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme
|
|
C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is
|
|
established by ~ ax9o and ~ ax9from9o . A more convenient form of this
|
|
axiom is ~ a9e , which has additional remarks.
|
|
|
|
Raph Levien proved the independence of this axiom from the other logical
|
|
axioms on 12-Apr-2005. See item 16 at
|
|
~ http://us.metamath.org/award2003.html .
|
|
|
|
~ ax-9 can be proved from the weaker version ~ ax9v requiring that the
|
|
variables be distinct; see theorem ~ ax9 .
|
|
|
|
~ ax-9 can also be proved from the Axiom of Separation (in the form that
|
|
we use that axiom, where free variables are not universally quantified).
|
|
See theorem ax9vsep in set.mm.
|
|
|
|
Except by ~ ax9v , this axiom should not be referenced directly. Instead,
|
|
use theorem ~ ax9 . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-9 $a |- -. A. x -. x = y $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Axiom B7 of [Tarski] p. 75, which requires that ` x ` and ` y ` be
|
|
distinct. This trivial proof is intended merely to weaken axiom ~ ax-9
|
|
by adding a distinct variable restriction. From here on, ~ ax-9 should
|
|
not be referenced directly by any other proof, so that theorem ~ ax9
|
|
will show that we can recover ~ ax-9 from this weaker version if it were
|
|
an axiom (as it is in the case of Tarski).
|
|
|
|
Note: Introducing ` x y ` as a distinct variable group "out of the
|
|
blue" with no apparent justification has puzzled some people, but it is
|
|
perfectly sound. All we are doing is adding an additional redundant
|
|
requirement, no different from adding a redundant logical hypothesis,
|
|
that results in a weakening of the theorem. This means that any
|
|
_future_ theorem that references ~ ax9v must have a $d specified for the
|
|
two variables that get substituted for ` x ` and ` y ` . The $d does
|
|
not propagate "backwards" i.e. it does not impose a requirement on
|
|
~ ax-9 .
|
|
|
|
When possible, use of this theorem rather than ~ ax9 is preferred since
|
|
its derivation from axioms is much shorter. (Contributed by NM,
|
|
7-Aug-2015.) $)
|
|
ax9v $p |- -. A. x -. x = y $=
|
|
( ax-9 ) ABC $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( At least one individual exists. Weaker version of ~ a9e . When
|
|
possible, use of this theorem rather than ~ a9e is preferred since its
|
|
derivation from axioms is much shorter. (Contributed by NM,
|
|
3-Aug-2017.) $)
|
|
a9ev $p |- E. x x = y $=
|
|
( weq wex wn wal ax9v df-ex mpbir ) ABCZADJEAFEABGJAHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
exiftru.1 $e |- ph $.
|
|
$( A companion rule to ax-gen, valid only if an individual exists. Unlike
|
|
~ ax-9 , it does not require equality on its interface. Some
|
|
fundamental theorems of predicate logic can be proven from ~ ax-gen ,
|
|
~ ax-5 and this theorem alone, not requiring ~ ax-8 or excessive
|
|
distinct variable conditions. (Contributed by Wolf Lammen,
|
|
12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.) $)
|
|
exiftru $p |- E. x ph $=
|
|
( vy weq wex a9ev a1i eximi ax-mp ) BDEZBFABFBDGKABAKCHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
exiftruOLD.1 $e |- ph $.
|
|
$( Obsolete proof of ~ exiftru as of 9-Dec-2017. (Contributed by Wolf
|
|
Lammen, 12-Nov-2017.) (New usage is discouraged.) $)
|
|
exiftruOLD $p |- E. x ph $=
|
|
( vy wex weq wi wal a9ev a1i 19.35ri id 2th exbii mpbir ) ABEBDFZPGZBEPPB
|
|
PBEPBHBDIJKAQBAQCPLMNO $.
|
|
$}
|
|
|
|
$( Theorem 19.2 of [Margaris] p. 89. Note: This proof is very different
|
|
from Margaris' because we only have Tarski's FOL axiom schemes available
|
|
at this point. See the later ~ 19.2g for a more conventional proof.
|
|
(Contributed by NM, 2-Aug-2017.) (Revised by Wolf Lammen to remove
|
|
dependency on ax-8, 4-Dec-2017.) $)
|
|
19.2 $p |- ( A. x ph -> E. x ph ) $=
|
|
( wi id exiftru 19.35i ) AABAACBADEF $.
|
|
|
|
${
|
|
19.8w.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Weak version of ~ 19.8a . Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen,
|
|
4-Dec-2017.) $)
|
|
19.8w $p |- ( ph -> E. x ph ) $=
|
|
( wal wex 19.2 syl ) AABDABECABFG $.
|
|
$}
|
|
|
|
$( Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.39 $p |- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) ) $=
|
|
( wex wi wal 19.2 imim1i 19.35 sylibr ) ACDZBCDZEACFZLEABECDMKLACGHABCIJ $.
|
|
|
|
$( Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.24 $p |- ( ( A. x ph -> A. x ps ) -> E. x ( ph -> ps ) ) $=
|
|
( wal wi wex 19.2 imim2i 19.35 sylibr ) ACDZBCDZEKBCFZEABECFLMKBCGHABCIJ $.
|
|
|
|
$( Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.34 $p |- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) ) $=
|
|
( wal wex wo 19.2 orim1i 19.43 sylibr ) ACDZBCEZFACEZLFABFCEKMLACGHABCIJ $.
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM,
|
|
28-May-1995.) (Revised by NM, 1-Aug-2017.) (Revised by Wolf Lammen to
|
|
remove dependency on ax-8, 4-Dec-2017.) $)
|
|
19.9v $p |- ( E. x ph <-> ph ) $=
|
|
( wex ax17e ax-17 19.8w impbii ) ABCAABDABABEFG $.
|
|
|
|
$( Special case of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM,
|
|
1-Aug-2017.) (Revised by Wolf Lammen to remove dependency on ax-8,
|
|
4-Dec-2017.) $)
|
|
19.3v $p |- ( A. x ph <-> ph ) $=
|
|
( wal wn wex alex 19.9v con2bii bitr4i ) ABCADZBEZDAABFKAJBGHI $.
|
|
|
|
$( Version of ~ sp when ` x ` does not occur in ` ph ` . This provides the
|
|
other direction of ~ ax-17 . Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen,
|
|
4-Dec-2017.) $)
|
|
spvw $p |- ( A. x ph -> ph ) $=
|
|
( wal 19.3v biimpi ) ABCAABDE $.
|
|
$}
|
|
|
|
${
|
|
$d x z $.
|
|
spimeh.1 $e |- ( ph -> A. x ph ) $.
|
|
spimeh.2 $e |- ( x = z -> ( ph -> ps ) ) $.
|
|
$( Existential introduction, using implicit substitution. Compare Lemma 14
|
|
of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened
|
|
by Wolf Lammen, 10-Dec-2017.) $)
|
|
spimeh $p |- ( ph -> E. x ps ) $=
|
|
( wal wex weq wi a9ev eximi ax-mp 19.35i syl ) AACGBCHEABCCDIZCHABJZCHCDK
|
|
PQCFLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
spimw.1 $e |- ( -. ps -> A. x -. ps ) $.
|
|
spimw.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's
|
|
FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened
|
|
by Wolf Lammen, 7-Aug-2017.) $)
|
|
spimw $p |- ( A. x ph -> ps ) $=
|
|
( weq wn wal wi ax9v spimfw ax-mp ) CDGHCIHACIBJCDKABCDEFLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x ps $.
|
|
spimvw.1 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's
|
|
FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) $)
|
|
spimvw $p |- ( A. x ph -> ps ) $=
|
|
( wn ax-17 spimw ) ABCDBFCGEH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
spnfw.1 $e |- ( -. ph -> A. x -. ph ) $.
|
|
$( Weak version of ~ sp . Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen,
|
|
13-Aug-2017.) $)
|
|
spnfw $p |- ( A. x ph -> ph ) $=
|
|
( vy weq idd spimw ) AABDCBDEAFG $.
|
|
$}
|
|
|
|
${
|
|
sptruw.1 $e |- ph $.
|
|
$( Version of ~ sp when ` ph ` is true. Uses only Tarski's FOL axiom
|
|
schemes. (Contributed by NM, 23-Apr-1017.) $)
|
|
sptruw $p |- ( A. x ph -> ph ) $=
|
|
( wal a1i ) AABDCE $.
|
|
$}
|
|
|
|
${
|
|
spfalw.1 $e |- -. ph $.
|
|
$( Version of ~ sp when ` ph ` is false. Uses only Tarski's FOL axiom
|
|
schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf
|
|
Lammen, 25-Dec-2017.) $)
|
|
spfalw $p |- ( A. x ph -> ph ) $=
|
|
( wn hbth spnfw ) ABADBCEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
cbvaliw.1 $e |- ( A. x ph -> A. y A. x ph ) $.
|
|
cbvaliw.2 $e |- ( -. ps -> A. x -. ps ) $.
|
|
cbvaliw.3 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Change bound variable. Uses only Tarski's FOL axiom schemes. Part of
|
|
Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.) $)
|
|
cbvaliw $p |- ( A. x ph -> A. y ps ) $=
|
|
( wal spimw alrimih ) ACHBDEABCDFGIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x ps $. $d y ph $.
|
|
cbvalivw.1 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Change bound variable. Uses only Tarski's FOL axiom schemes. Part of
|
|
Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.) $)
|
|
cbvalivw $p |- ( A. x ph -> A. y ps ) $=
|
|
( wal spimvw alrimiv ) ACFBDABCDEGH $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom scheme ax-8 (Equality)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Equality. One of the equality and substitution axioms of
|
|
predicate calculus with equality. This is similar to, but not quite, a
|
|
transitive law for equality (proved later as ~ equtr ). This axiom scheme
|
|
is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose
|
|
general form cannot be represented with our notation. Also appears as
|
|
Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16
|
|
of the preprint).
|
|
|
|
The equality symbol was invented in 1527 by Robert Recorde. He chose a
|
|
pair of parallel lines of the same length because "noe .2. thynges, can be
|
|
moare equalle."
|
|
|
|
Note that this axiom is still valid even when any two or all three of
|
|
` x ` , ` y ` , and ` z ` are replaced with the same variable since they
|
|
do not have any distinct variable (Metamath's $d) restrictions. Because
|
|
of this, we say that these three variables are "bundled" (a term coined by
|
|
Raph Levien). (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-8 $a |- ( x = y -> ( x = z -> y = z ) ) $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also
|
|
Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised
|
|
by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.) $)
|
|
equid $p |- x = x $=
|
|
( vy weq wex a9ev ax-8 pm2.43i eximi ax17e mp2b ) BACZBDAACZBDLBAEKLBKLBA
|
|
AFGHLBIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Obsolete proof of ~ equid as of 9-Dec-2017. (Contributed by NM,
|
|
1-Apr-2005.) (New usage is discouraged.) $)
|
|
equidOLD $p |- x = x $=
|
|
( vy weq wn wal ax9v ax-8 pm2.43i con3i alimi mto ax-17 mt3 ) AACZNDZBEZP
|
|
BACZDZBEBAFORBQNQNBAAGHIJKOBLM $.
|
|
$}
|
|
|
|
$( Bound-variable hypothesis builder for ` x = x ` . This theorem tells us
|
|
that any variable, including ` x ` , is effectively not free in
|
|
` x = x ` , even though ` x ` is technically free according to the
|
|
traditional definition of free variable. (Contributed by NM,
|
|
13-Jan-2011.) (Revised by NM, 21-Aug-2017.) $)
|
|
nfequid $p |- F/ y x = x $=
|
|
( weq equid nfth ) AACBADE $.
|
|
|
|
$( Commutative law for equality. Lemma 3 of [KalishMontague] p. 85. See
|
|
also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by NM, 9-Apr-2017.) $)
|
|
equcomi $p |- ( x = y -> y = x ) $=
|
|
( weq equid ax-8 mpi ) ABCAACBACADABAEF $.
|
|
|
|
$( Commutative law for equality. (Contributed by NM, 20-Aug-1993.) $)
|
|
equcom $p |- ( x = y <-> y = x ) $=
|
|
( weq equcomi impbii ) ABCBACABDBADE $.
|
|
|
|
${
|
|
equcoms.1 $e |- ( x = y -> ph ) $.
|
|
$( An inference commuting equality in antecedent. Used to eliminate the
|
|
need for a syllogism. (Contributed by NM, 5-Aug-1993.) $)
|
|
equcoms $p |- ( y = x -> ph ) $=
|
|
( weq equcomi syl ) CBEBCEACBFDG $.
|
|
$}
|
|
|
|
$( A transitive law for equality. (Contributed by NM, 23-Aug-1993.) $)
|
|
equtr $p |- ( x = y -> ( y = z -> x = z ) ) $=
|
|
( weq wi ax-8 equcoms ) BCDACDEBABACFG $.
|
|
|
|
$( A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the
|
|
preprint). (Contributed by NM, 23-Aug-1993.) $)
|
|
equtrr $p |- ( x = y -> ( z = x -> z = y ) ) $=
|
|
( weq equtr com12 ) CADABDCBDCABEF $.
|
|
|
|
$( An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 10-Dec-2017.) $)
|
|
equequ1 $p |- ( x = y -> ( x = z <-> y = z ) ) $=
|
|
( weq ax-8 equtr impbid ) ABDACDBCDABCEABCFG $.
|
|
|
|
$( Obsolete version of ~ equequ1 as of 12-Nov-2017. (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
equequ1OLD $p |- ( x = y -> ( x = z <-> y = z ) ) $=
|
|
( weq ax-8 wi equcomi syl impbid ) ABDZACDZBCDZABCEJBADLKFABGBACEHI $.
|
|
|
|
$( An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 4-Aug-2017.) $)
|
|
equequ2 $p |- ( x = y -> ( z = x <-> z = y ) ) $=
|
|
( weq equequ1 equcom 3bitr3g ) ABDACDBCDCADCBDABCEACFBCFG $.
|
|
|
|
$( One of the two equality axioms of standard predicate calculus, called
|
|
reflexivity of equality. (The other one is ~ stdpc7 .) Axiom 6 of
|
|
[Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant
|
|
quantifier, but it was probably to be compatible with free logic (which is
|
|
valid in the empty domain). (Contributed by NM, 16-Feb-2005.) $)
|
|
stdpc6 $p |- A. x x = x $=
|
|
( weq equid ax-gen ) AABAACD $.
|
|
|
|
$( A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 25-May-2011.) $)
|
|
equtr2 $p |- ( ( x = z /\ y = z ) -> x = y ) $=
|
|
( weq wi equtrr equcoms impcom ) BCDACDZABDZIJECBCBAFGH $.
|
|
|
|
$( Two equivalent ways of expressing ~ ax-12 . See the comment for
|
|
~ ax-12 . (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf
|
|
Lammen, 12-Aug-2017.) $)
|
|
ax12b $p |- ( ( -. x = y -> ( y = z -> A. x y = z ) )
|
|
<-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) ) $=
|
|
( weq wn wal wi a1dd equtrr equcoms con3rr3 com4l com23 mpdd com3r impbii
|
|
id ) ABDZEZBCDZTAFZGZGZSACDZEZUBGGZUCSUBUEUCQHSTUFUASTUEUFUAGZTUDRUDRGCBCBA
|
|
IJKSUETUGUFSUETUAUFQLMNOP $.
|
|
|
|
$( Obsolete version of ~ ax12b as of 12-Aug-2017. (Contributed by NM,
|
|
2-May-2017.) (New usage is discouraged.) $)
|
|
ax12bOLD $p |- ( ( -. x = y -> ( y = z -> A. x y = z ) )
|
|
<-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) ) $=
|
|
( weq wn wal wi wa bi2.04 equtrr equcoms con3d pm4.71d imbi1d pm5.74i bitri
|
|
impexp ) ABDZEZBCDZTAFZGZGZSACDZEZHZUBGZSUEUBGGUCTUFUAGZGZUGUCTSUAGZGUISTUA
|
|
ITUJUHTSUFUATSUETUDRUDRGCBCBAJKLMNOPTUFUAIPSUEUBQP $.
|
|
|
|
${
|
|
$d x y $.
|
|
spfw.1 $e |- ( -. ps -> A. x -. ps ) $.
|
|
spfw.2 $e |- ( A. x ph -> A. y A. x ph ) $.
|
|
spfw.3 $e |- ( -. ph -> A. y -. ph ) $.
|
|
spfw.4 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Weak version of ~ sp . Uses only Tarski's FOL axiom schemes. Lemma 9
|
|
of [KalishMontague] p. 87. This may be the best we can do with minimal
|
|
distinct variable conditions. TO DO: Do we need this theorem? If not,
|
|
maybe it should be deleted. (Contributed by NM, 19-Apr-2017.) $)
|
|
spfw $p |- ( A. x ph -> ph ) $=
|
|
( wal wi ax-5 weq biimprd equcoms spimw syl56 biimpd mpg ) ACIZBJZSAJDSSD
|
|
ITDIBDIAFSBDKBADCGBAJCDCDLZABHMNOPABCDEUAABHQOR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
spnfw.3 $e |- ( -. ph -> A. x -. ph ) $.
|
|
$( Weak version of ~ sp . Uses only Tarski's FOL axiom schemes. Obsolete
|
|
version of ~ spnfw as of 13-Aug-2017. (Contributed by NM, 1-Aug-2017.)
|
|
(New usage is discouraged.) $)
|
|
spnfwOLD $p |- ( A. x ph -> ph ) $=
|
|
( vy wal ax-17 wn weq biidd spfw ) AABDCABEDFAGDFBDHAIJ $.
|
|
$}
|
|
|
|
${
|
|
19.8wOLD.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Obsolete version of ~ 19.8w as of 4-Dec-2017. (Contributed by NM,
|
|
1-Aug-2017.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
19.8wOLD $p |- ( ph -> E. x ph ) $=
|
|
( wn wal wex notnot albii 3imtr3i spnfw con2i df-ex sylibr ) AADZBEZDABFO
|
|
ANBAABENDZPBECAGZAPBQHIJKABLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x ps $. $d y ph $.
|
|
spw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Weak version of specialization scheme ~ sp . Lemma 9 of
|
|
[KalishMontague] p. 87. While it appears that ~ sp in its general form
|
|
does not follow from Tarski's FOL axiom schemes, from this theorem we
|
|
can prove any instance of ~ sp having no wff metavariables and mutually
|
|
distinct setvar variables (see ~ ax11wdemo for an example of the
|
|
procedure to eliminate the hypothesis). Other approximations of ~ sp
|
|
are ~ spfw (minimal distinct variable requirements), ~ spnfw (when ` x `
|
|
is not free in ` -. ph ` ), ~ spvw (when ` x ` does not appear in
|
|
` ph ` ), ~ sptruw (when ` ph ` is true), and ~ spfalw (when ` ph ` is
|
|
false). (Contributed by NM, 9-Apr-2017.) $)
|
|
spw $p |- ( A. x ph -> ph ) $=
|
|
( wal wi ax-17 ax-5 weq biimprd equcoms spimvw syl56 biimpd mpg ) ACFZBGZ
|
|
QAGDQQDFRDFBDFAQDHQBDIBADCBAGCDCDJZABEKLMNABCDSABEOMP $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $.
|
|
$( Obsolete version of ~ spvw as of 4-Dec-2017. (Contributed by NM,
|
|
10-Apr-2017.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
spvwOLD $p |- ( A. x ph -> ph ) $=
|
|
( vy weq biidd spw ) AABCBCDAEF $.
|
|
|
|
$( Obsolete version of ~ 19.3v as of 4-Dec-2017. (Contributed by NM,
|
|
1-Aug-2017.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
19.3vOLD $p |- ( A. x ph <-> ph ) $=
|
|
( wal spvw ax-17 impbii ) ABCAABDABEF $.
|
|
|
|
$( Obsolete version of ~ 19.9v as of 4-Dec-2017. (Contributed by NM,
|
|
28-May-1995.) (Revised by NM, 1-Aug-2017.)
|
|
(New usage is discouraged.) (Proof modification is discouraged.) $)
|
|
19.9vOLD $p |- ( E. x ph <-> ph ) $=
|
|
( wex wn wal df-ex 19.3v con2bii bitr4i ) ABCADZBEZDAABFKAJBGHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
exlimivOLD.1 $e |- ( ph -> ps ) $.
|
|
$( Obsolete version of ~ exlimiv as of 4-Dec-2017. (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
exlimivOLD $p |- ( E. x ph -> ps ) $=
|
|
( wex eximi 19.9v sylib ) ACEBCEBABCDFBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
spfalwOLD.1 $e |- -. ph $.
|
|
$( Obsolete proof of ~ spfalw as of 25-Dec-2017. (Contributed by NM,
|
|
23-Apr-1017.) (New usage is discouraged.) $)
|
|
spfalwOLD $p |- ( A. x ph -> ph ) $=
|
|
( vy wfal wb weq bifal a1i spw ) AEBDAEFBDGACHIJ $.
|
|
$}
|
|
|
|
$( Obsolete version of ~ 19.2 as of 4-Dec-2017. (Contributed by NM,
|
|
2-Aug-2017.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
19.2OLD $p |- ( A. x ph -> E. x ph ) $=
|
|
( weq wn wal wex wi equid notnoti spfalw mt2 idd speimfw ax-mp ) BBCZDZBEZD
|
|
ABEABFGQOBHZPBORIJKAABBOALMN $.
|
|
|
|
${
|
|
$d x y $.
|
|
cbvalw.1 $e |- ( A. x ph -> A. y A. x ph ) $.
|
|
cbvalw.2 $e |- ( -. ps -> A. x -. ps ) $.
|
|
cbvalw.3 $e |- ( A. y ps -> A. x A. y ps ) $.
|
|
cbvalw.4 $e |- ( -. ph -> A. y -. ph ) $.
|
|
cbvalw.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variable. Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 9-Apr-2017.) $)
|
|
cbvalw $p |- ( A. x ph <-> A. y ps ) $=
|
|
( wal weq biimpd cbvaliw wi biimprd equcoms impbii ) ACJBDJABCDEFCDKZABIL
|
|
MBADCGHBANCDRABIOPMQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x ps $. $d y ph $.
|
|
cbvalvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variable. Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 9-Apr-2017.) $)
|
|
cbvalvw $p |- ( A. x ph <-> A. y ps ) $=
|
|
( wal weq biimpd cbvalivw wi biimprd equcoms impbii ) ACFBDFABCDCDGZABEHI
|
|
BADCBAJCDNABEKLIM $.
|
|
|
|
$( Change bound variable. Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 19-Apr-2017.) $)
|
|
cbvexvw $p |- ( E. x ph <-> E. y ps ) $=
|
|
( wn wal wex weq notbid cbvalvw notbii df-ex 3bitr4i ) AFZCGZFBFZDGZFACHB
|
|
DHPROQCDCDIABEJKLACMBDMN $.
|
|
$}
|
|
|
|
${
|
|
$d y z $. $d x y $. $d z ph $. $d y ps $.
|
|
alcomiw.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
|
|
$( Weak version of ~ alcom . Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 10-Apr-2017.) $)
|
|
alcomiw $p |- ( A. x A. y ph -> A. y A. x ph ) $=
|
|
( wal weq biimpd cbvalivw alimi ax-17 wi biimprd equcoms spimvw 3syl ) AD
|
|
GZCGBEGZCGZTDGACGZDGRSCABDEDEHZABFIJKTDLTUADSACBAEDBAMDEUBABFNOPKKQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
hbn1fw.1 $e |- ( A. x ph -> A. y A. x ph ) $.
|
|
hbn1fw.2 $e |- ( -. ps -> A. x -. ps ) $.
|
|
hbn1fw.3 $e |- ( A. y ps -> A. x A. y ps ) $.
|
|
hbn1fw.4 $e |- ( -. ph -> A. y -. ph ) $.
|
|
hbn1fw.5 $e |- ( -. A. y ps -> A. x -. A. y ps ) $.
|
|
hbn1fw.6 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Weak version of ~ ax-6 from which we can prove any ~ ax-6 instance not
|
|
involving wff variables or bundling. Uses only Tarski's FOL axiom
|
|
schemes. (Contributed by NM, 19-Apr-2017.) $)
|
|
hbn1fw $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
|
|
( wal wn cbvalw biimpri con3i biimpi alimi 3syl ) ACKZLZBDKZLZUBCKTCKUASS
|
|
UAABCDEFGHJMZNOIUBTCSUASUAUCPOQR $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $. $d x y $.
|
|
hbn1w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Weak version of ~ hbn1 . Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 9-Apr-2017.) $)
|
|
hbn1w $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
|
|
( wal ax-17 wn hbn1fw ) ABCDACFDGBHCGBDFZCGAHDGJHCGEI $.
|
|
|
|
$( Weak version of ~ hba1 . See comments for ~ ax6w . Uses only Tarski's
|
|
FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) $)
|
|
hba1w $p |- ( A. x ph -> A. x A. x ph ) $=
|
|
( wal wn weq wb cbvalvw a1i notbid spw con2i hbn1w con1i alimi 3syl ) ACF
|
|
ZSGZCFZGZUBCFSCFUASTBDFZGZCDCDHZSUCSUCIUEABCDEJKLZMNTUDCDUFOUBSCSUAABCDEO
|
|
PQR $.
|
|
|
|
$( Weak version of ~ hbe1 . See comments for ~ ax6w . Uses only Tarski's
|
|
FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) $)
|
|
hbe1w $p |- ( E. x ph -> A. x E. x ph ) $=
|
|
( wex wn wal df-ex weq notbid hbn1w hbxfrbi ) ACFAGZCHGCACINBGCDCDJABEKLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d x y $. $d z ph $. $d x ps $.
|
|
hbalw.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
|
|
hbalw.2 $e |- ( ph -> A. x ph ) $.
|
|
$( Weak version of ~ hbal . Uses only Tarski's FOL axiom schemes. Unlike
|
|
~ hbal , this theorem requires that ` x ` and ` y ` be distinct i.e. are
|
|
not bundled. (Contributed by NM, 19-Apr-2017.) $)
|
|
hbalw $p |- ( A. y ph -> A. x A. y ph ) $=
|
|
( wal alimi alcomiw syl ) ADHZACHZDHLCHAMDGIABDCEFJK $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Membership predicate
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the membership predicate symbol. $)
|
|
$c e. $. $( Stylized epsilon $)
|
|
|
|
$( --- Start of patch to prevent connective overloading $)
|
|
${
|
|
$v A $.
|
|
$v B $.
|
|
wcel.cA $f class A $.
|
|
wcel.cB $f class B $.
|
|
$( Extend wff definition to include the membership connective between
|
|
classes.
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class .
|
|
|
|
(The purpose of introducing ` wff A e. B ` here is to allow us to
|
|
express i.e. "prove" the ~ wel of predicate calculus in terms of the
|
|
~ wceq of set theory, so that we don't "overload" the ` e. ` connective
|
|
with two syntax definitions. This is done to prevent ambiguity that
|
|
would complicate some Metamath parsers. The class variables ` A ` and
|
|
` B ` are introduced temporarily for the purpose of this definition but
|
|
otherwise not used in predicate calculus. See ~ df-clab for more
|
|
information on the set theory usage of ~ wcel .) $)
|
|
wcel $a wff A e. B $.
|
|
$}
|
|
|
|
$( Extend wff definition to include atomic formulas with the epsilon
|
|
(membership) predicate. This is read " ` x ` is an element of
|
|
` y ` ," " ` x ` is a member of ` y ` ," " ` x ` belongs to ` y ` ,"
|
|
or " ` y ` contains ` x ` ." Note: The phrase " ` y ` includes
|
|
` x ` " means " ` x ` is a subset of ` y ` ;" to use it also for
|
|
` x e. y ` , as some authors occasionally do, is poor form and causes
|
|
confusion, according to George Boolos (1992 lecture at MIT).
|
|
|
|
This syntactical construction introduces a binary non-logical predicate
|
|
symbol ` e. ` (epsilon) into our predicate calculus. We will eventually
|
|
use it for the membership predicate of set theory, but that is irrelevant
|
|
at this point: the predicate calculus axioms for ` e. ` apply to any
|
|
arbitrary binary predicate symbol. "Non-logical" means that the predicate
|
|
is presumed to have additional properties beyond the realm of predicate
|
|
calculus, although these additional properties are not specified by
|
|
predicate calculus itself but rather by the axioms of a theory (in our
|
|
case set theory) added to predicate calculus. "Binary" means that the
|
|
predicate has two arguments.
|
|
|
|
(Instead of introducing ~ wel as an axiomatic statement, as was done in an
|
|
older version of this database, we introduce it by "proving" a special
|
|
case of set theory's more general ~ wcel . This lets us avoid overloading
|
|
the ` e. ` connective, thus preventing ambiguity that would complicate
|
|
certain Metamath parsers. However, logically ~ wel is considered to be a
|
|
primitive syntax, even though here it is artificially "derived" from
|
|
~ wcel . Note: To see the proof steps of this syntax proof, type "show
|
|
proof wel /all" in the Metamath program.) (Contributed by NM,
|
|
24-Jan-2006.) $)
|
|
wel $p wff x e. y $=
|
|
( cv wcel ) ACBCD $.
|
|
$( --- End of patch to prevent connective overloading $)
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( Extend wff definition to include atomic formulas with the epsilon
|
|
(membership) predicate. This is read " ` x ` is an element of ` y ` ,"
|
|
" ` x ` is a member of ` y ` ," " ` x ` belongs to ` y ` ," or " ` y `
|
|
contains ` x ` ." Note: The phrase " ` y ` includes ` x ` " means
|
|
" ` x ` is a subset of ` y ` "; to use it also for ` x e. y ` (as some
|
|
authors occasionally do) is poor form and causes confusion.
|
|
|
|
After we introduce ~ cv and ~ wcel in set theory, this syntax construction
|
|
becomes redundant, since it can be derived with the proof
|
|
"vx cv vy cv wcel". @)
|
|
wel @a wff x e. y @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( Register class-to-set promotion and class equality and membership as
|
|
primitive expressions. Although these are actually definitions, the above
|
|
ambiguity prevention necessitates our taking class equality as the
|
|
primitive, instead of set equality. $)
|
|
$( $j primitive 'weq' 'wel'; $)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom schemes ax-13 (Left Equality for Binary Predicate)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Left Equality for Binary Predicate. One of the equality and
|
|
substitution axioms for a non-logical predicate in our predicate calculus
|
|
with equality. It substitutes equal variables into the left-hand side of
|
|
an arbitrary binary predicate ` e. ` , which we will use for the set
|
|
membership relation when set theory is introduced. This axiom scheme is a
|
|
sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose
|
|
general form cannot be represented with our notation. Also appears as
|
|
Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint).
|
|
"Non-logical" means that the predicate is not a primitive of predicate
|
|
calculus proper but instead is an extension to it. "Binary" means that
|
|
the predicate has two arguments. In a system of predicate calculus with
|
|
equality, like ours, equality is not usually considered to be a
|
|
non-logical predicate. In systems of predicate calculus without equality,
|
|
it typically would be. (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-13 $a |- ( x = y -> ( x e. z -> y e. z ) ) $.
|
|
|
|
$( An identity law for the non-logical predicate. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
elequ1 $p |- ( x = y -> ( x e. z <-> y e. z ) ) $=
|
|
( weq wel ax-13 wi equcoms impbid ) ABDACEZBCEZABCFKJGBABACFHI $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom schemes ax-14 (Right Equality for Binary Predicate)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Right Equality for Binary Predicate. One of the equality and
|
|
substitution axioms for a non-logical predicate in our predicate calculus
|
|
with equality. It substitutes equal variables into the right-hand side of
|
|
an arbitrary binary predicate ` e. ` , which we will use for the set
|
|
membership relation when set theory is introduced. This axiom scheme is a
|
|
sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose
|
|
general form cannot be represented with our notation. Also appears as
|
|
Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ax-14 $a |- ( x = y -> ( z e. x -> z e. y ) ) $.
|
|
|
|
$( An identity law for the non-logical predicate. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
elequ2 $p |- ( x = y -> ( z e. x <-> z e. y ) ) $=
|
|
( weq wel ax-14 wi equcoms impbid ) ABDCAEZCBEZABCFKJGBABACFHI $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
The orginal axiom schemes of Tarski's predicate calculus are ~ ax-5 ,
|
|
~ ax-17 , ~ ax9v , ~ ax-8 , ~ ax-13 , and ~ ax-14 , together with rule
|
|
~ ax-gen . See ~ http://us.metamath.org/mpeuni/mmset.html#compare . They
|
|
are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These
|
|
are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85.
|
|
|
|
The axiom system of set.mm includes the auxiliary axiom schemes ~ ax-6 ,
|
|
~ ax-7 , ~ ax-12 , and ~ ax-11 , which are not part of Tarski's axiom
|
|
schemes. They are used (and we conjecture are required) to make our system
|
|
"metalogically complete" i.e. able to prove directly all possible schemes
|
|
with wff and setvar metavariables, bundled or not, whose object-language
|
|
instances are valid. ( ~ ax-11 has been proved to be required; see
|
|
~ http://us.metamath.org/award2003.html#9a . Metalogical independence of the
|
|
other three are open problems.)
|
|
|
|
(There are additional predicate calculus axiom schemes included in set.mm
|
|
such as ~ ax-4 , but they can all be proved as theorems from the above.)
|
|
|
|
Terminology: Two setvar (individual) metavariables are "bundled" in an axiom
|
|
or theorem scheme when there is no distinct variable constraint ($d) imposed
|
|
on them. (The term "bundled" is due to Raph Levien.) For example, the ` x `
|
|
and ` y ` in ~ ax9 are bundled, but they are not in ~ ax9v . We also say that
|
|
a scheme is bundled when it has at least one pair of bundled setvar
|
|
metavariables. If distinct variable conditions are added to all setvar
|
|
metavariable pairs in a bundled scheme, we call that the "principal" instance
|
|
of the bundled scheme. For example, ~ ax9v is the principal instance of
|
|
~ ax9 . Whenever a common variable is substituted for two or more bundled
|
|
variables in an axiom or theorem scheme, we call the substitution instance
|
|
"degenerate". For example, the instance ` -. A. x -. x = x ` of ~ ax9 is
|
|
degenerate. An advantage of bundling is ease of use since there are fewer
|
|
distinct variable restrictions ($d) to be concerned with. There is also a
|
|
small economy in being able to state principal and degenerate instances
|
|
simultaneously. A disadvantage is that bundling may present difficulties in
|
|
translations to other proof languages, which typically lack the concept (in
|
|
part because their variables often represent the variables of the object
|
|
language rather than metavariables ranging over them).
|
|
|
|
Because Tarski's axiom schemes are logically complete, they can be used to
|
|
prove any object-language instance of ~ ax-6 , ~ ax-7 , ~ ax-11 , and ~ ax-12
|
|
. "Translating" this to Metamath, it means that Tarski's axioms can prove any
|
|
substitution instance of ~ ax-6 , ~ ax-7 , ~ ax-11 , or ~ ax-12 in which (1)
|
|
there are no wff metavariables and (2) all setvr metavariables are mutually
|
|
distinct i.e. are not bundled. In effect this is mimicking the object
|
|
language by pretending that each setvar metavariable is an object-language
|
|
variable. (There may also be specific instances with wff metavariables
|
|
and/or bundling that are directly provable from Tarski's axiom schemes, but
|
|
it isn't guaranteed. Whether all of them are possible is part of the still
|
|
open metalogical independence problem for our additional axiom schemes.)
|
|
|
|
It can be useful to see how this can be done, both to show that our
|
|
additional schemes are valid metatheorems of Tarski's system and to be able
|
|
to translate object language instances of our proofs into proofs that would
|
|
work with a system using only Tarski's original schemes. In addition, it may
|
|
(or may not) provide insight into the conjectured metalogical independence of
|
|
our additional schemes.
|
|
|
|
The new theorem schemes ~ ax6w , ~ ax7w , ~ ax11w , and ~ ax12w are
|
|
derived using only Tarski's axiom schemes, showing that Tarski's schemes can
|
|
be used to derive all substitution instances of ~ ax-6 , ~ ax-7 , ~ ax-11 ,
|
|
and ~ ax-12 meeting conditions (1) and (2). (The "w" suffix stands for "weak
|
|
version".) Each hypothesis of ~ ax6w , ~ ax7w , and ~ ax11w is of the
|
|
form ` ( x = y -> ( ph <-> ps ) ) ` where ` ps ` is an auxiliary or "dummy"
|
|
wff metavariable in which ` x ` doesn't occur. We can show by induction on
|
|
formula length that the hypotheses can be eliminated in all cases meeting
|
|
conditions (1) and (2). The example ~ ax11wdemo illustrates the techniques
|
|
(equality theorems and bound variable renaming) used to achieve this.
|
|
|
|
We also show the degenerate instances for axioms with bundled variables in
|
|
~ ax7dgen , ~ ax11dgen , ~ ax12dgen1 , ~ ax12dgen2 , ~ ax12dgen3 , and
|
|
~ ax12dgen4 . (Their proofs are trivial, but we include them to be thorough.)
|
|
Combining the principal and degenerate cases _outside_ of Metamath, we show
|
|
that the bundled schemes ~ ax-6 , ~ ax-7 , ~ ax-11 , and ~ ax-12 are schemes
|
|
of Tarski's system, meaning that all object language instances they generate
|
|
are theorems of Tarski's system.
|
|
|
|
It is interesting that Tarski used the bundled scheme ~ ax-9 in an older
|
|
system, so it seems the main purpose of his later ~ ax9v was just to show
|
|
that the weaker unbundled form is sufficient rather than an aesthetic
|
|
objection to bundled free and bound variables. Since we adopt the
|
|
bundled ~ ax-9 as our official axiom, we show that the degenerate
|
|
instance holds in ~ ax9dgen .
|
|
|
|
The case of ~ sp is curious: originally an axiom of Tarski's system, it
|
|
was proved redundant by Lemma 9 of [KalishMontague] p. 86. However, the
|
|
proof is by induction on formula length, and the compact scheme form
|
|
` A. x ph -> ph ` apparently cannot be proved directly from Tarski's other
|
|
axioms. The best we can do seems to be ~ spw , again requiring
|
|
substitution instances of ` ph ` that meet conditions (1) and (2) above.
|
|
Note that our direct proof ~ sp requires ~ ax-11 , which is not part of
|
|
Tarski's system.
|
|
|
|
$)
|
|
|
|
$( Tarski's system uses the weaker ~ ax9v instead of the bundled ~ ax-9 , so
|
|
here we show that the degenerate case of ~ ax-9 can be derived.
|
|
(Contributed by NM, 23-Apr-2017.) $)
|
|
ax9dgen $p |- -. A. x -. x = x $=
|
|
( weq wn wal equid notnoti spfalw mt2 ) AABZCZADIAEZJAIKFGH $.
|
|
|
|
${
|
|
$d y ph $. $d x ps $. $d x y $.
|
|
ax6w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Weak version of ~ ax-6 from which we can prove any ~ ax-6 instance not
|
|
involving wff variables or bundling. Uses only Tarski's FOL axiom
|
|
schemes. (Contributed by NM, 9-Apr-2017.) $)
|
|
ax6w $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
|
|
( hbn1w ) ABCDEF $.
|
|
$}
|
|
|
|
${
|
|
$d y z $. $d x y $. $d z ph $. $d y ps $.
|
|
ax7w.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
|
|
$( Weak version of ~ ax-7 from which we can prove any ~ ax-7 instance not
|
|
involving wff variables or bundling. Uses only Tarski's FOL axiom
|
|
schemes. Unlike ~ ax-7 , this theorem requires that ` x ` and ` y ` be
|
|
distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.) $)
|
|
ax7w $p |- ( A. x A. y ph -> A. y A. x ph ) $=
|
|
( alcomiw ) ABCDEFG $.
|
|
$}
|
|
|
|
$( Degenerate instance of ~ ax-7 where bundled variables ` x ` and ` y ` have
|
|
a common substitution. Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 13-Apr-2017.) $)
|
|
ax7dgen $p |- ( A. x A. x ph -> A. x A. x ph ) $=
|
|
( wal id ) ABCBCD $.
|
|
|
|
${
|
|
$d x ps $.
|
|
ax11wlemw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Lemma for weak version of ~ ax-11 . Uses only Tarski's FOL axiom
|
|
schemes. In some cases, this lemma may lead to shorter proofs than
|
|
~ ax11w . (Contributed by NM, 10-Apr-2017.) $)
|
|
ax11wlem $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( ax-17 ax11i ) ABCDEBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d y z $. $d x ps $. $d z ph $. $d y ch $.
|
|
ax11w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
ax11w.2 $e |- ( y = z -> ( ph <-> ch ) ) $.
|
|
$( Weak version of ~ ax-11 from which we can prove any ~ ax-11 instance not
|
|
involving wff variables or bundling. Uses only Tarski's FOL axiom
|
|
schemes. An instance of the first hypothesis will normally require that
|
|
` x ` and ` y ` be distinct (unless ` x ` does not occur in ` ph ` ).
|
|
(Contributed by NM, 10-Apr-2017.) $)
|
|
ax11w $p |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( wal weq wi spw ax11wlem syl5 ) AEIADEJZOAKDIACEFHLABDEGMN $.
|
|
$}
|
|
|
|
$( Degenerate instance of ~ ax-11 where bundled variables ` x ` and ` y `
|
|
have a common substitution. Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 13-Apr-2017.) $)
|
|
ax11dgen $p |- ( x = x -> ( A. x ph -> A. x ( x = x -> ph ) ) ) $=
|
|
( wal weq wi ax-1 alimi a1i ) ABCBBDZAEZBCEIAJBAIFGH $.
|
|
|
|
${
|
|
$d x y z w v $.
|
|
$( Example of an application of ~ ax11w that results in an instance of
|
|
~ ax-11 for a contrived formula with mixed free and bound variables,
|
|
` ( x e. y /\ A. x z e. x /\ A. y A. z y e. x ) ` , in place of
|
|
` ph ` . The proof illustrates bound variable renaming with ~ cbvalvw
|
|
to obtain fresh variables to avoid distinct variable clashes. Uses only
|
|
Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.) $)
|
|
ax11wdemo $p |- ( x = y
|
|
-> ( A. y ( x e. y /\ A. x z e. x /\ A. y A. z y e. x )
|
|
-> A. x ( x = y -> ( x e. y /\ A. x z e. x /\ A. y A. z y e. x ) ) ) ) $=
|
|
( vw vv wel wal w3a weq elequ1 elequ2 cbvalvw a1i albidv syl5bb 3anbi123d
|
|
wb 3anbi13d ax11w ) ABFZCAFZAGZBAFZCGZBGZHBBFZCDFZDGZEBFZCGZEGZHAEFZUBEAF
|
|
ZCGZEGZHABEABIZTUFUBUHUEUKABBJUBUHQUPUAUGADADCKLMUEUOUPUKUDUNBEBEIZUCUMCB
|
|
EAJNLZUPUNUJEUPUMUICABEKNNOPUQTULUEUOUBBEAKUEUOQUQURMRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x z $.
|
|
$( Weak version (principal instance) of ~ ax-12 . (Because ` y ` and ` z `
|
|
don't need to be distinct, this actually bundles the principal instance
|
|
and the degenerate instance
|
|
` ( -. x = y -> ( y = y -> A. x y = y ) ) ` .) Uses only Tarski's FOL
|
|
axiom schemes. The proof is trivial but is included to complete the set
|
|
~ ax6w , ~ ax7w , and ~ ax11w . (Contributed by NM, 10-Apr-2017.) $)
|
|
ax12w $p |- ( -. x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( weq wn a17d ) ABDEBCDAF $.
|
|
$}
|
|
|
|
$( Degenerate instance of ~ ax-12 where bundled variables ` x ` and ` y `
|
|
have a common substitution. Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 13-Apr-2017.) $)
|
|
ax12dgen1 $p |- ( -. x = x -> ( x = z -> A. x x = z ) ) $=
|
|
( weq wal wi equid pm2.24i ) AACABCZHADEAFG $.
|
|
|
|
$( Degenerate instance of ~ ax-12 where bundled variables ` x ` and ` z `
|
|
have a common substitution. Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 13-Apr-2017.) $)
|
|
ax12dgen2 $p |- ( -. x = y -> ( y = x -> A. x y = x ) ) $=
|
|
( weq wn wal equcomi pm2.21 syl5 ) BACZABCZJDIAEZBAFJKGH $.
|
|
|
|
$( Degenerate instance of ~ ax-12 where bundled variables ` y ` and ` z `
|
|
have a common substitution. Uses only Tarski's FOL axiom schemes.
|
|
(Contributed by NM, 13-Apr-2017.) $)
|
|
ax12dgen3 $p |- ( -. x = y -> ( y = y -> A. x y = y ) ) $=
|
|
( weq wn wal equid ax-gen 2a1i ) ABCDBBCZIAEIABFGH $.
|
|
|
|
$( Degenerate instance of ~ ax-12 where bundled variables ` x ` , ` y ` , and
|
|
` z ` have a common substitution. Uses only Tarski's FOL axiom schemes .
|
|
(Contributed by NM, 13-Apr-2017.) $)
|
|
ax12dgen4 $p |- ( -. x = x -> ( x = x -> A. x x = x ) ) $=
|
|
( ax12dgen1 ) AAB $.
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
|
|
In this section we introduce four additional schemes ~ ax-6 , ~ ax-7 ,
|
|
~ ax-11 , and ~ ax-12 that are not part of Tarski's system but can be proved
|
|
(outside of Metamath) as theorem schemes of Tarski's system. These are
|
|
needed to give our system the property of "metalogical completeness," which
|
|
means that we can prove (with Metamath) all possible schemes expressible in
|
|
our language of wff metavariables ranging over object-language wffs and
|
|
setvar metavariables ranging over object-language individual variables.
|
|
|
|
To show that these schemes are valid metatheorems of Tarski's system S2,
|
|
above we proved from Tarski's system theorems ~ ax6w , ~ ax7w , ~ ax12w ,
|
|
and ~ ax11w , which show that any object-language instance of these schemes
|
|
(emulated by having no wff metavariables and requiring all setvar
|
|
metavariables to be mutually distinct) can be proved using only the schemes
|
|
in Tarski's system S2.
|
|
|
|
An open problem is to show that these four additional schemes are
|
|
metalogically independent from Tarski's. So far, independence of ~ ax-11
|
|
from all others has been shown, and independence of Tarski's ~ ax-9 from all
|
|
others has been shown.
|
|
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom scheme ax-6 (Quantified Negation)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom
|
|
scheme is logically redundant (see ~ ax6w ) but is used as an auxiliary
|
|
axiom to achieve metalogical completeness. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ax-6 $a |- ( -. A. x ph -> A. x -. A. x ph ) $.
|
|
|
|
$( ` x ` is not free in ` -. A. x ph ` . (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Wolf Lammen, 18-Aug-2014.) $)
|
|
hbn1 $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
|
|
( ax-6 ) ABC $.
|
|
|
|
$( ` x ` is not free in ` E. x ph ` . (Contributed by NM, 5-Aug-1993.) $)
|
|
hbe1 $p |- ( E. x ph -> A. x E. x ph ) $=
|
|
( wex wn wal df-ex hbn1 hbxfrbi ) ABCADZBEDBABFIBGH $.
|
|
|
|
$( ` x ` is not free in ` E. x ph ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfe1 $p |- F/ x E. x ph $=
|
|
( wex hbe1 nfi ) ABCBABDE $.
|
|
|
|
$( The analog in our "pure" predicate calculus of axiom 5 of modal logic S5.
|
|
(Contributed by NM, 5-Oct-2005.) $)
|
|
modal-5 $p |- ( -. A. x -. ph -> A. x -. A. x -. ph ) $=
|
|
( wn hbn1 ) ACBD $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom scheme ax-7 (Quantifier Commutation)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Quantifier Commutation. This axiom says universal quantifiers
|
|
can be swapped. One of the 4 axioms of pure predicate calculus. Axiom
|
|
scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as
|
|
Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom
|
|
scheme is logically redundant (see ~ ax7w ) but is used as an auxiliary
|
|
axiom to achieve metalogical completeness. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ax-7 $a |- ( A. x A. y ph -> A. y A. x ph ) $.
|
|
|
|
${
|
|
a7s.1 $e |- ( A. x A. y ph -> ps ) $.
|
|
$( Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) $)
|
|
a7s $p |- ( A. y A. x ph -> ps ) $=
|
|
( wal ax-7 syl ) ACFDFADFCFBADCGEH $.
|
|
$}
|
|
|
|
${
|
|
hbal.1 $e |- ( ph -> A. x ph ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
hbal $p |- ( A. y ph -> A. x A. y ph ) $=
|
|
( wal alimi ax-7 syl ) ACEZABEZCEIBEAJCDFACBGH $.
|
|
$}
|
|
|
|
$( Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) $)
|
|
alcom $p |- ( A. x A. y ph <-> A. y A. x ph ) $=
|
|
( wal ax-7 impbii ) ACDBDABDCDABCEACBEF $.
|
|
|
|
$( Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew
|
|
Salmon, 24-May-2011.) $)
|
|
alrot3 $p |- ( A. x A. y A. z ph <-> A. y A. z A. x ph ) $=
|
|
( wal alcom albii bitri ) ADEZCEBEIBEZCEABEDEZCEIBCFJKCABDFGH $.
|
|
|
|
$( Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.)
|
|
(Proof shortened by Fan Zheng, 6-Jun-2016.) $)
|
|
alrot4 $p |- ( A. x A. y A. z A. w ph <-> A. z A. w A. x A. y ph ) $=
|
|
( wal alrot3 albii bitri ) AEFDFCFZBFACFZEFDFZBFKBFEFDFJLBACDEGHKBDEGI $.
|
|
|
|
${
|
|
hbald.1 $e |- ( ph -> A. y ph ) $.
|
|
hbald.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hbal .
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
hbald $p |- ( ph -> ( A. y ps -> A. x A. y ps ) ) $=
|
|
( wal alimdh ax-7 syl6 ) ABDGZBCGZDGKCGABLDEFHBDCIJ $.
|
|
$}
|
|
|
|
$( Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Wolf Lammen to remove dependency on ~ ax-11 ~ ax-6 ~ ax-9
|
|
~ ax-8 and ~ ax-17 , 8-Jan-2018.) $)
|
|
excom $p |- ( E. x E. y ph <-> E. y E. x ph ) $=
|
|
( wn wal wex alcom notbii exnal 3bitr4i df-ex exbii ) ADZCEZDZBFZMBEZDZCFZA
|
|
CFZBFABFZCFNBEZDQCEZDPSUBUCMBCGHNBIQCIJTOBACKLUARCABKLJ $.
|
|
|
|
$( One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Revised by Wolf
|
|
Lammen to remove dependency on ~ ax-11 ~ ax-6 ~ ax-9 ~ ax-8 and ~ ax-17 ,
|
|
8-Jan-2018.) $)
|
|
excomim $p |- ( E. x E. y ph -> E. y E. x ph ) $=
|
|
( wex excom biimpi ) ACDBDABDCDABCEF $.
|
|
|
|
$( Swap 1st and 3rd existential quantifiers. (Contributed by NM,
|
|
9-Mar-1995.) $)
|
|
excom13 $p |- ( E. x E. y E. z ph <-> E. z E. y E. x ph ) $=
|
|
( wex excom exbii 3bitri ) ADEZCEBEIBEZCEABEZDEZCEKCEDEIBCFJLCABDFGKCDFH $.
|
|
|
|
$( Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) $)
|
|
exrot3 $p |- ( E. x E. y E. z ph <-> E. y E. z E. x ph ) $=
|
|
( wex excom13 excom bitri ) ADECEBEABEZCEDEIDECEABCDFIDCGH $.
|
|
|
|
$( Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) $)
|
|
exrot4 $p |- ( E. x E. y E. z E. w ph <-> E. z E. w E. x E. y ph ) $=
|
|
( wex excom13 exbii bitri ) AEFDFCFZBFACFZDFEFZBFKBFEFDFJLBACDEGHKBEDGI $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom scheme ax-11 (Substitution)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Substitution. One of the 5 equality axioms of predicate
|
|
calculus. The final consequent ` A. x ( x = y -> ph ) ` is a way of
|
|
expressing " ` y ` substituted for ` x ` in wff ` ph ` " (cf. ~ sb6 ). It
|
|
is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105,
|
|
from which it can be proved by cases.
|
|
|
|
The original version of this axiom was ~ ax-11o ("o" for "old") and was
|
|
replaced with this shorter ~ ax-11 in Jan. 2007. The old axiom is proved
|
|
from this one as theorem ~ ax11o . Conversely, this axiom is proved from
|
|
~ ax-11o as theorem ~ ax11 .
|
|
|
|
Juha Arpiainen proved the metalogical independence of this axiom (in the
|
|
form of the older axiom ~ ax-11o ) from the others on 19-Jan-2006. See
|
|
item 9a at ~ http://us.metamath.org/award2003.html .
|
|
|
|
See ~ ax11v and ~ ax11v2 for other equivalents of this axiom that (unlike
|
|
this axiom) have distinct variable restrictions.
|
|
|
|
This axiom scheme is logically redundant (see ~ ax11w ) but is used as an
|
|
auxiliary axiom to achieve metalogical completeness. (Contributed by NM,
|
|
22-Jan-2007.) $)
|
|
ax-11 $a |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $.
|
|
|
|
${
|
|
$d x w $. $d w ph $.
|
|
$( Specialization. A universally quantified wff implies the wff without a
|
|
quantifier Axiom scheme B5 of [Tarski] p. 67 (under his system S2,
|
|
defined in the last paragraph on p. 77). Also appears as Axiom scheme
|
|
C5' in [Megill] p. 448 (p. 16 of the preprint).
|
|
|
|
For the axiom of specialization presented in many logic textbooks, see
|
|
theorem ~ stdpc4 .
|
|
|
|
This theorem shows that our obsolete axiom ~ ax-4 can be derived from
|
|
the others. The proof uses ideas from the proof of Lemma 21 of [Monk2]
|
|
p. 114.
|
|
|
|
It appears that this scheme cannot be derived directly from Tarski's
|
|
axioms without auxiliary axiom scheme ~ ax-11 . It is thought the best
|
|
we can do using only Tarski's axioms is ~ spw . (Contributed by NM,
|
|
21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof
|
|
shortened by Wolf Lammen, 23-Dec-2017.) $)
|
|
sp $p |- ( A. x ph -> ph ) $=
|
|
( vw weq wex wal wi a9ev equcomi ax-17 ax-11 syl2im ax9v con2 al2imi mtoi
|
|
wn syl6 con4d exlimiv ax-mp ) CBDZCEABFZAGZCBHUBUDCUBAUCUBAQZBCDZUEGZBFZU
|
|
CQUBUFUEUECFUHCBIUECJUEBCKLUHUCUFQZBFBCMUGAUIBUFANOPRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x w $. $d w ph $.
|
|
$( Obsolete proof of ~ sp as of 23-Dec-2017. (Contributed by NM,
|
|
21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.)
|
|
(New usage is discouraged.) $)
|
|
spOLD $p |- ( A. x ph -> ph ) $=
|
|
( vw wal wi weq wn ax9v equcomi ax-17 ax-11 syl2im con2 al2imi mtoi con4d
|
|
syl6 con3i alrimiv mt3 ) ABDZAEZCBFZGZCDCBHUBGUDCUCUBUCAUAUCAGZBCFZUEEZBD
|
|
ZUAGUCUFUEUECDUHCBIUECJUEBCKLUHUAUFGZBDBCHUGAUIBUFAMNOQPRST $.
|
|
$}
|
|
|
|
$( Show that the original axiom ~ ax-5o can be derived from ~ ax-5 and
|
|
others. See ~ ax5 for the rederivation of ~ ax-5 from ~ ax-5o .
|
|
|
|
Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114.
|
|
(Contributed by NM, 21-May-2008.) (Proof modification is discouraged.) $)
|
|
ax5o $p |- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) ) $=
|
|
( wal wi wn sp con2i hbn1 con1i alimi 3syl ax-5 syl5 ) ACDZOCDZOBECDBCDOOFZ
|
|
CDZFZSCDPROQCGHQCISOCORACIJKLOBCMN $.
|
|
|
|
$( Show that the original axiom ~ ax-6o can be derived from ~ ax-6 and
|
|
others. See ~ ax6 for the rederivation of ~ ax-6 from ~ ax-6o .
|
|
|
|
Normally, ~ ax6o should be used rather than ~ ax-6o , except by theorems
|
|
specifically studying the latter's properties. (Contributed by NM,
|
|
21-May-2008.) $)
|
|
ax6o $p |- ( -. A. x -. A. x ph -> ph ) $=
|
|
( wal wn sp ax-6 nsyl4 ) ABCZAHDBCABEABFG $.
|
|
|
|
$( Abbreviated version of ~ ax6o . (Contributed by NM, 5-Aug-1993.) $)
|
|
a6e $p |- ( E. x A. x ph -> ph ) $=
|
|
( wal wex wn df-ex ax6o sylbi ) ABCZBDIEBCEAIBFABGH $.
|
|
|
|
$( The analog in our "pure" predicate calculus of the Brouwer axiom (B) of
|
|
modal logic S5. (Contributed by NM, 5-Oct-2005.) $)
|
|
modal-b $p |- ( ph -> A. x -. A. x -. ph ) $=
|
|
( wn wal ax6o con4i ) ACZBDCBDAGBEF $.
|
|
|
|
${
|
|
spi.1 $e |- A. x ph $.
|
|
$( Inference rule reversing generalization. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
spi $p |- ph $=
|
|
( wal sp ax-mp ) ABDACABEF $.
|
|
$}
|
|
|
|
${
|
|
sps.1 $e |- ( ph -> ps ) $.
|
|
$( Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) $)
|
|
sps $p |- ( A. x ph -> ps ) $=
|
|
( wal sp syl ) ACEABACFDG $.
|
|
$}
|
|
|
|
${
|
|
spsd.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) $)
|
|
spsd $p |- ( ph -> ( A. x ps -> ch ) ) $=
|
|
( wal sp syl5 ) BDFBACBDGEH $.
|
|
$}
|
|
|
|
$( If a wff is true, it is true for at least one instance. Special case of
|
|
Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.8a $p |- ( ph -> E. x ph ) $=
|
|
( wn wal wex sp con2i df-ex sylibr ) AACZBDZCABEKAJBFGABHI $.
|
|
|
|
$( Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar
|
|
variables. (Contributed by O'Cat, 31-Mar-2008.) $)
|
|
19.2g $p |- ( A. x ph -> E. y ph ) $=
|
|
( wex 19.8a sps ) AACDBACEF $.
|
|
|
|
${
|
|
19.21bi.1 $e |- ( ph -> A. x ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.21bi $p |- ( ph -> ps ) $=
|
|
( wal sp syl ) ABCEBDBCFG $.
|
|
$}
|
|
|
|
${
|
|
19.23bi.1 $e |- ( E. x ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.23bi $p |- ( ph -> ps ) $=
|
|
( wex 19.8a syl ) AACEBACFDG $.
|
|
$}
|
|
|
|
${
|
|
nexr.1 $e |- -. E. x ph $.
|
|
$( Inference from ~ 19.8a . (Contributed by Jeff Hankins, 26-Jul-2009.) $)
|
|
nexr $p |- -. ph $=
|
|
( wex 19.8a mto ) AABDCABEF $.
|
|
$}
|
|
|
|
$( Consequence of the definition of not-free. (Contributed by Mario
|
|
Carneiro, 26-Sep-2016.) $)
|
|
nfr $p |- ( F/ x ph -> ( ph -> A. x ph ) ) $=
|
|
( wnf wal wi df-nf sp sylbi ) ABCAABDEZBDIABFIBGH $.
|
|
|
|
${
|
|
nfri.1 $e |- F/ x ph $.
|
|
$( Consequence of the definition of not-free. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
nfri $p |- ( ph -> A. x ph ) $=
|
|
( wnf wal wi nfr ax-mp ) ABDAABEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
nfrd.1 $e |- ( ph -> F/ x ps ) $.
|
|
$( Consequence of the definition of not-free in a context. (Contributed by
|
|
Mario Carneiro, 11-Aug-2016.) $)
|
|
nfrd $p |- ( ph -> ( ps -> A. x ps ) ) $=
|
|
( wnf wal wi nfr syl ) ABCEBBCFGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
alimd.1 $e |- F/ x ph $.
|
|
alimd.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
alimd $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
|
|
( nfri alimdh ) ABCDADEGFH $.
|
|
$}
|
|
|
|
${
|
|
alrimi.1 $e |- F/ x ph $.
|
|
alrimi.2 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
alrimi $p |- ( ph -> A. x ps ) $=
|
|
( nfri alrimih ) ABCACDFEG $.
|
|
$}
|
|
|
|
${
|
|
nfd.1 $e |- F/ x ph $.
|
|
nfd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduce that ` x ` is not free in ` ps ` in a context. (Contributed by
|
|
Mario Carneiro, 24-Sep-2016.) $)
|
|
nfd $p |- ( ph -> F/ x ps ) $=
|
|
( wal wi wnf alrimi df-nf sylibr ) ABBCFGZCFBCHALCDEIBCJK $.
|
|
$}
|
|
|
|
${
|
|
nfdh.1 $e |- ( ph -> A. x ph ) $.
|
|
nfdh.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduce that ` x ` is not free in ` ps ` in a context. (Contributed by
|
|
Mario Carneiro, 24-Sep-2016.) $)
|
|
nfdh $p |- ( ph -> F/ x ps ) $=
|
|
( nfi nfd ) ABCACDFEG $.
|
|
$}
|
|
|
|
${
|
|
alrimdd.1 $e |- F/ x ph $.
|
|
alrimdd.2 $e |- ( ph -> F/ x ps ) $.
|
|
alrimdd.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
alrimdd $p |- ( ph -> ( ps -> A. x ch ) ) $=
|
|
( wal nfrd alimd syld ) ABBDHCDHABDFIABCDEGJK $.
|
|
$}
|
|
|
|
${
|
|
alrimd.1 $e |- F/ x ph $.
|
|
alrimd.2 $e |- F/ x ps $.
|
|
alrimd.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
alrimd $p |- ( ph -> ( ps -> A. x ch ) ) $=
|
|
( wnf a1i alrimdd ) ABCDEBDHAFIGJ $.
|
|
$}
|
|
|
|
${
|
|
eximd.1 $e |- F/ x ph $.
|
|
eximd.2 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
eximd $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
|
|
( nfri eximdh ) ABCDADEGFH $.
|
|
$}
|
|
|
|
${
|
|
nexd.1 $e |- F/ x ph $.
|
|
nexd.2 $e |- ( ph -> -. ps ) $.
|
|
$( Deduction for generalization rule for negated wff. (Contributed by
|
|
Mario Carneiro, 24-Sep-2016.) $)
|
|
nexd $p |- ( ph -> -. E. x ps ) $=
|
|
( nfri nexdh ) ABCACDFEG $.
|
|
$}
|
|
|
|
${
|
|
albid.1 $e |- F/ x ph $.
|
|
albid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for universal quantifier (deduction rule).
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) $)
|
|
albid $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
|
|
( nfri albidh ) ABCDADEGFH $.
|
|
$}
|
|
|
|
${
|
|
exbid.1 $e |- F/ x ph $.
|
|
exbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for existential quantifier (deduction rule).
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) $)
|
|
exbid $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
|
|
( nfri exbidh ) ABCDADEGFH $.
|
|
$}
|
|
|
|
${
|
|
nfbidf.1 $e |- F/ x ph $.
|
|
nfbidf.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( An equality theorem for effectively not free. (Contributed by Mario
|
|
Carneiro, 4-Oct-2016.) $)
|
|
nfbidf $p |- ( ph -> ( F/ x ps <-> F/ x ch ) ) $=
|
|
( wal wi wnf albid imbi12d df-nf 3bitr4g ) ABBDGZHZDGCCDGZHZDGBDICDIAOQDE
|
|
ABCNPFABCDEFJKJBDLCDLM $.
|
|
$}
|
|
|
|
$( Closed theorem version of bound-variable hypothesis builder ~ hbn .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
hbnt $p |- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) ) $=
|
|
( wn wal wi ax6o con1i con3 al2imi syl5 ) ACZABDZCZBDZALEZBDKBDNAABFGOMKBAL
|
|
HIJ $.
|
|
|
|
${
|
|
hbn.1 $e |- ( ph -> A. x ph ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` -. ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen,
|
|
17-Dec-2017.) $)
|
|
hbn $p |- ( -. ph -> A. x -. ph ) $=
|
|
( wal wi wn hbnt mpg ) AABDEAFZIBDEBABGCH $.
|
|
|
|
$( Obsolete proof of ~ hbn as of 16-Dec-2017. (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.) $)
|
|
hbnOLD $p |- ( -. ph -> A. x -. ph ) $=
|
|
( wn wal sp con3i hbn1 alrimih syl ) ADZABEZDZKBELAABFGMKBABHALCGIJ $.
|
|
$}
|
|
|
|
$( A closed version of ~ 19.9 . (Contributed by NM, 5-Aug-1993.) $)
|
|
19.9ht $p |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) ) $=
|
|
( wex wn wal wi df-ex hbnt con1d syl5bi ) ABCADBEZDAABEFBEZAABGLAKABHIJ $.
|
|
|
|
$( A closed version of ~ 19.9 . (Contributed by NM, 5-Aug-1993.) (Revised
|
|
by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen,
|
|
30-Dec-2017.) $)
|
|
19.9t $p |- ( F/ x ph -> ( E. x ph <-> ph ) ) $=
|
|
( wnf wex wal wi df-nf 19.9ht sylbi 19.8a impbid1 ) ABCZABDZALAABEFBEMAFABG
|
|
ABHIABJK $.
|
|
|
|
${
|
|
19.9h.1 $e |- ( ph -> A. x ph ) $.
|
|
$( A wff may be existentially quantified with a variable not free in it.
|
|
Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)
|
|
(Proof shortened by Wolf Lammen, 5-Jan-2018.) $)
|
|
19.9h $p |- ( E. x ph <-> ph ) $=
|
|
( wnf wex wb nfi 19.9t ax-mp ) ABDABEAFABCGABHI $.
|
|
$( Obsolete proof of ~ 19.9h as of 5-Jan-2018. (Contributed by FL,
|
|
24-Mar-2007.) (New usage is discouraged.) $)
|
|
19.9hOLD $p |- ( E. x ph <-> ph ) $=
|
|
( wex wal wi 19.9ht mpg 19.8a impbii ) ABDZAAABEFKAFBABGCHABIJ $.
|
|
$}
|
|
|
|
${
|
|
19.9d.1 $e |- ( ps -> F/ x ph ) $.
|
|
$( A deduction version of one direction of ~ 19.9 . (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) $)
|
|
19.9d $p |- ( ps -> ( E. x ph -> ph ) ) $=
|
|
( wex wnf wb 19.9t syl biimpd ) BACEZABACFKAGDACHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.9.1 $e |- F/ x ph $.
|
|
$( A wff may be existentially quantified with a variable not free in it.
|
|
Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)
|
|
(Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf
|
|
Lammen, 30-Dec-2017.) $)
|
|
19.9 $p |- ( E. x ph <-> ph ) $=
|
|
( nfri 19.9h ) ABABCDE $.
|
|
|
|
$( Obsolete proof of ~ 19.9 as of 30-Dec-2017. (Contributed by FL,
|
|
24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.)
|
|
(New usage is discouraged.) $)
|
|
19.9OLD $p |- ( E. x ph <-> ph ) $=
|
|
( wnf wex wb 19.9t ax-mp ) ABDABEAFCABGH $.
|
|
$}
|
|
|
|
${
|
|
19.3.1 $e |- F/ x ph $.
|
|
$( A wff may be quantified with a variable not free in it. Theorem 19.3 of
|
|
[Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
19.3 $p |- ( A. x ph <-> ph ) $=
|
|
( wal sp nfri impbii ) ABDAABEABCFG $.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` A. x ph ` . Example in Appendix in [Megill] p. 450
|
|
(p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed
|
|
by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Dec-2017.) $)
|
|
hba1 $p |- ( A. x ph -> A. x A. x ph ) $=
|
|
( wn wex wal hbe1 hbn alex albii 3imtr4i ) ACZBDZCZMBEABEZNBELBKBFGABHZNMBO
|
|
IJ $.
|
|
|
|
$( Obsolete proof of ~ hba1 as of 15-Dec-2017 (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.) $)
|
|
hba1OLD $p |- ( A. x ph -> A. x A. x ph ) $=
|
|
( wal wn sp con2i hbn1 con1i alimi 3syl ) ABCZKDZBCZDZNBCKBCMKLBEFLBGNKBKMA
|
|
BGHIJ $.
|
|
|
|
$( ` x ` is not free in ` A. x ph ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfa1 $p |- F/ x A. x ph $=
|
|
( wal hba1 nfi ) ABCBABDE $.
|
|
|
|
${
|
|
a5i.1 $e |- ( A. x ph -> ps ) $.
|
|
$( Inference version of ~ ax5o . (Contributed by NM, 5-Aug-1993.) $)
|
|
a5i $p |- ( A. x ph -> A. x ps ) $=
|
|
( wal nfa1 alrimi ) ACEBCACFDG $.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` F/ x ph ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfnf1 $p |- F/ x F/ x ph $=
|
|
( wnf wal wi df-nf nfa1 nfxfr ) ABCAABDEZBDBABFIBGH $.
|
|
|
|
${
|
|
nfnd.1 $e |- ( ph -> F/ x ps ) $.
|
|
$( If in a context ` x ` is not free in ` ps ` , it is not free in
|
|
` -. ps ` . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof
|
|
shortened by Wolf Lammen, 28-Dec-2017.) $)
|
|
nfnd $p |- ( ph -> F/ x -. ps ) $=
|
|
( wnf wn nfnf1 wal wi df-nf hbnt sylbi nfd syl ) ABCEZBFZCEDOPCBCGOBBCHIC
|
|
HPPCHIBCJBCKLMN $.
|
|
|
|
$( Obsolete proof of ~ nfnd as of 28-Dec-2017. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) (New usage is discouraged.) $)
|
|
nfndOLD $p |- ( ph -> F/ x -. ps ) $=
|
|
( wnf wn nfnf1 wal ax6o con1i wi df-nf con3 al2imi sylbi syl5 nfd syl ) A
|
|
BCEZBFZCEDSTCBCGTBCHZFZCHZSTCHZUCBBCIJSBUAKZCHUCUDKBCLUEUBTCBUAMNOPQR $.
|
|
|
|
$}
|
|
|
|
${
|
|
nfn.1 $e |- F/ x ph $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` -. ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfn $p |- F/ x -. ph $=
|
|
( wn wnf wtru a1i nfnd trud ) ADBEFABABEFCGHI $.
|
|
$}
|
|
|
|
$( Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Wolf Lammen, 2-Jan-2018.) $)
|
|
19.38 $p |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) ) $=
|
|
( wex wal wi wn alnex pm2.21 alimi sylbir ax-1 ja ) ACDZBCEABFZCEZNGAGZCEPA
|
|
CHQOCABIJKBOCBALJM $.
|
|
|
|
$( Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM,
|
|
27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened
|
|
by Wolf Lammen, 3-Jan-2018.) $)
|
|
19.21t $p |- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) ) $=
|
|
( wnf wi wal nfr ax-5 syl9 wex 19.9t imbi1d 19.38 syl6bir impbid ) ACDZABEC
|
|
FZABCFZEZPAACFQRACGABCHIPSACJZREQPTARACKLABCMNO $.
|
|
|
|
${
|
|
19.21.1 $e |- F/ x ph $.
|
|
$( Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of
|
|
as " ` x ` is not free in ` ph ` ." (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 24-Sep-2016.) $)
|
|
19.21 $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
|
|
( wnf wi wal wb 19.21t ax-mp ) ACEABFCGABCGFHDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
19.21h.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of
|
|
as " ` x ` is not free in ` ph ` ." (Contributed by NM, 1-Aug-2017.)
|
|
(Proof shortened by Wolf Lammen, 1-Jan-2018.) $)
|
|
19.21h $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
|
|
( nfi 19.21 ) ABCACDEF $.
|
|
$}
|
|
|
|
${
|
|
stdpc5.1 $e |- F/ x ph $.
|
|
$( An axiom scheme of standard predicate calculus that emulates Axiom 5 of
|
|
[Mendelson] p. 69. The hypothesis ` F/ x ph ` can be thought of as
|
|
emulating " ` x ` is not free in ` ph ` ." With this definition, the
|
|
meaning of "not free" is less restrictive than the usual textbook
|
|
definition; for example ` x ` would not (for us) be free in ` x = x ` by
|
|
~ nfequid . This theorem scheme can be proved as a metatheorem of
|
|
Mendelson's axiom system, even though it is slightly stronger than his
|
|
Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro,
|
|
12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) $)
|
|
stdpc5 $p |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) $=
|
|
( wi wal 19.21 biimpi ) ABECFABCFEABCDGH $.
|
|
|
|
$( Obsolete proof of ~ stdpc5 as of 1-Jan-2018. (Contributed by NM,
|
|
22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
|
|
(New usage is discouraged.) $)
|
|
stdpc5OLD $p |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) $=
|
|
( wal wi nfri alim syl5 ) AACEABFCEBCEACDGABCHI $.
|
|
$}
|
|
|
|
$( Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
|
|
19.23t $p |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
|
|
( wnf wi wal wex exim 19.9t biimpd syl9r nfr imim2d 19.38 syl6 impbid ) BCD
|
|
ZABECFZACGZBEZRSBCGZQBABCHQUABBCIJKQTSBCFZERQBUBSBCLMABCNOP $.
|
|
|
|
${
|
|
19.23.1 $e |- F/ x ps $.
|
|
$( Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 24-Sep-2016.) $)
|
|
19.23 $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
|
|
( wnf wi wal wex wb 19.23t ax-mp ) BCEABFCGACHBFIDABCJK $.
|
|
$}
|
|
|
|
${
|
|
19.23h.1 $e |- ( ps -> A. x ps ) $.
|
|
$( Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf
|
|
Lammen, 1-Jan-2018.) $)
|
|
19.23h $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
|
|
( nfi 19.23 ) ABCBCDEF $.
|
|
$}
|
|
|
|
${
|
|
exlimi.1 $e |- F/ x ps $.
|
|
exlimi.2 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
exlimi $p |- ( E. x ph -> ps ) $=
|
|
( wi wex 19.23 mpgbi ) ABFACGBFCABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
exlimih.1 $e |- ( ps -> A. x ps ) $.
|
|
exlimih.2 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof
|
|
shortened by Wolf Lammen, 1-Jan-2018.) $)
|
|
exlimih $p |- ( E. x ph -> ps ) $=
|
|
( nfi exlimi ) ABCBCDFEG $.
|
|
|
|
$( Obsolete proof of ~ exlimih as of 1-Jan-2018. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|
|
(New usage is discouraged.) $)
|
|
exlimihOLD $p |- ( E. x ph -> ps ) $=
|
|
( wi wex 19.23h mpgbi ) ABFACGBFCABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
exlimd.1 $e |- F/ x ph $.
|
|
exlimd.2 $e |- F/ x ch $.
|
|
exlimd.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) $)
|
|
exlimd $p |- ( ph -> ( E. x ps -> ch ) ) $=
|
|
( wi wal wex alrimi 19.23 sylib ) ABCHZDIBDJCHANDEGKBCDFLM $.
|
|
$}
|
|
|
|
${
|
|
exlimdh.1 $e |- ( ph -> A. x ph ) $.
|
|
exlimdh.2 $e |- ( ch -> A. x ch ) $.
|
|
exlimdh.3 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
28-Jan-1997.) $)
|
|
exlimdh $p |- ( ph -> ( E. x ps -> ch ) ) $=
|
|
( nfi exlimd ) ABCDADEHCDFHGI $.
|
|
$}
|
|
|
|
${
|
|
nfimd.1 $e |- ( ph -> F/ x ps ) $.
|
|
nfimd.2 $e |- ( ph -> F/ x ch ) $.
|
|
$( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
|
|
in ` ( ps -> ch ) ` . (Contributed by Mario Carneiro, 24-Sep-2016.)
|
|
(Proof shortened by Wolf Lammen, 30-Dec-2017.) $)
|
|
nfimd $p |- ( ph -> F/ x ( ps -> ch ) ) $=
|
|
( wnf wal nfnf1 nfr imim2d 19.21t biimprd syl9r alrimd df-nf syl6ibr sylc
|
|
wi ) ABDGZCDGZBCSZDGZEFTUAUBUBDHZSZDHUCTUAUEDBDICDIUAUBBCDHZSZTUDUACUFBCD
|
|
JKTUDUGBCDLMNOUBDPQR $.
|
|
|
|
$( Obsolete proof of ~ nfimd as of 29-Dec-2017. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) (New usage is discouraged.) $)
|
|
nfimdOLD $p |- ( ph -> F/ x ( ps -> ch ) ) $=
|
|
( wnf wi wal wa nfa1 wn hbnt pm2.21 alimi imim2i adantr ax-1 adantl df-nf
|
|
jad ex syl alimd imp anbi12i 3imtr4i syl2anc ) ABDGZCDGZBCHZDGZEFBBDIHZDI
|
|
ZCCDIZHZDIZJUKUKDIZHZDIZUIUJJULUNUQUTUNUPUSDUMDKUNBLZVADIZHZUPUSHBDMVCUPU
|
|
SVCUPJBCURVCVAURHUPVBURVAVAUKDBCNOPQUPCURHVCUOURCCUKDCBROPSUAUBUCUDUEUIUN
|
|
UJUQBDTCDTUFUKDTUGUH $.
|
|
$}
|
|
|
|
${
|
|
hbim1.1 $e |- ( ph -> A. x ph ) $.
|
|
hbim1.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( A closed form of ~ hbim . (Contributed by NM, 5-Aug-1993.) $)
|
|
hbim1 $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
|
|
( wi wal a2i 19.21h sylibr ) ABFZABCGZFKCGABLEHABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
nfim1.1 $e |- F/ x ph $.
|
|
nfim1.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( A closed form of ~ nfim . (Contributed by NM, 5-Aug-1993.) (Revised by
|
|
Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen,
|
|
2-Jan-2018.) $)
|
|
nfim1 $p |- F/ x ( ph -> ps ) $=
|
|
( wi nfri nfrd hbim1 nfi ) ABFCABCACDGABCEHIJ $.
|
|
|
|
$( A closed form of ~ nfim . (Contributed by NM, 5-Aug-1993.) (Revised by
|
|
Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) $)
|
|
nfim1OLD $p |- F/ x ( ph -> ps ) $=
|
|
( wi wal nfrd a2i 19.21 sylibr nfi ) ABFZCMABCGZFMCGABNABCEHIABCDJKL $.
|
|
$}
|
|
|
|
${
|
|
nfim.1 $e |- F/ x ph $.
|
|
nfim.2 $e |- F/ x ps $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph -> ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.)
|
|
(Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
|
|
nfim $p |- F/ x ( ph -> ps ) $=
|
|
( wnf a1i nfim1 ) ABCDBCFAEGH $.
|
|
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph -> ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.)
|
|
(New usage is discouraged.) $)
|
|
nfimOLD $p |- F/ x ( ph -> ps ) $=
|
|
( wi wnf wtru a1i nfimd trud ) ABFCGHABCACGHDIBCGHEIJK $.
|
|
$}
|
|
|
|
${
|
|
hbimd.1 $e |- ( ph -> A. x ph ) $.
|
|
hbimd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
hbimd.3 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hbim .
|
|
(Contributed by NM, 1-Jan-2002.) (Proof shortened by Wolf Lammen,
|
|
3-Jan-2018.) $)
|
|
hbimd $p |- ( ph -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) ) $=
|
|
( wi nfdh nfimd nfrd ) ABCHDABCDABDEFIACDEGIJK $.
|
|
|
|
$( Obsolete proof of ~ hbimd as of 16-Dec-2017. (Contributed by NM,
|
|
1-Jan-2002.) (New usage is discouraged.) $)
|
|
hbimdOLD $p |- ( ph -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) ) $=
|
|
( wi wal wn alrimih sp hbn1 nsyl4 con1i con3 al2imi syl2im alimi syl6 jad
|
|
pm2.21 ax-1 ) ABCBCHZDIZABJZUFDIZUEABBDIZHZDIUFUHJZDIZUGAUIDEFKUKBUHBUKBD
|
|
LBDMNOUIUJUFDBUHPQRUFUDDBCUBSTACCDIUEGCUDDCBUCSTUA $.
|
|
$}
|
|
|
|
${
|
|
hbim.1 $e |- ( ph -> A. x ph ) $.
|
|
hbim.2 $e |- ( ps -> A. x ps ) $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph -> ps ) ` . (Contributed by NM, 5-Aug-1993.) (Proof shortened
|
|
by O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) $)
|
|
hbim $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
|
|
( wal wi a1i hbim1 ) ABCDBBCFGAEHI $.
|
|
|
|
$( Obsolete proof of ~ hbim as of 1-Jan-2018. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.)
|
|
(New usage is discouraged.) $)
|
|
hbimOLD $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
|
|
( wi wal wn hbn pm2.21 alrimih ax-1 ja ) ABABFZCGAHNCACDIABJKBNCEBALKM $.
|
|
$}
|
|
|
|
$( Obsolete proof of ~ 19.23t as of 1-Jan-2018. (Contributed by NM,
|
|
7-Nov-2005.) (New usage is discouraged.) $)
|
|
19.23tOLD $p |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
|
|
( wnf wi wal wex exim 19.9t imbi2d syl5ib nfnf1 nfe1 a1i nfimd 19.8a imim1d
|
|
id alrimdd impbid ) BCDZABEZCFZACGZBEZUCUDBCGZEUAUEABCHUAUFBUDBCIJKUAUEUBCB
|
|
CLUAUDBCUDCDUAACMNUAROUAAUDBAUDEUAACPNQST $.
|
|
|
|
${
|
|
19.23hOLD.1 $e |- ( ps -> A. x ps ) $.
|
|
$( Obsolete proof of ~ 19.23h as of 1-Jan-2018. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|
|
(New usage is discouraged.) $)
|
|
19.23hOLD $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
|
|
( wi wal wex exim 19.9h syl6ib hbe1 hbim 19.8a imim1i alrimih impbii ) AB
|
|
EZCFZACGZBEZRSBCGBABCHBCDIJTQCSBCACKDLASBACMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x z $.
|
|
spimehOLD.1 $e |- ( ph -> A. x ph ) $.
|
|
spimehOLD.2 $e |- ( x = z -> ( ph -> ps ) ) $.
|
|
$( Obsolete proof of ~ spimeh as of 10-Dec-2017. (Contributed by NM,
|
|
7-Aug-1994.) (New usage is discouraged.) $)
|
|
spimehOLD $p |- ( ph -> E. x ps ) $=
|
|
( wn wal wex wi weq ax9v id hbth hba1 a1i hbn hbimd ax-mp sp nsyli sylibr
|
|
con3i alrimih mt3 con2i df-ex ) ABGZCHZGBCIUIAUIAGZJZCDKZGZCHCDLUKGUMCUKC
|
|
AAJZUKUKCHJAMZUNUIUJCUNCUONUIUICHJUNUHCOPUJUJCHJUNACEQPRSQULUKULABUIFUHCT
|
|
UAUCUDUEUFBCUGUB $.
|
|
$}
|
|
|
|
${
|
|
nfand.1 $e |- ( ph -> F/ x ps ) $.
|
|
nfand.2 $e |- ( ph -> F/ x ch ) $.
|
|
$( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
|
|
in ` ( ps /\ ch ) ` . (Contributed by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfand $p |- ( ph -> F/ x ( ps /\ ch ) ) $=
|
|
( wa wn wi df-an nfnd nfimd nfxfrd ) BCGBCHZIZHADBCJAODABNDEACDFKLKM $.
|
|
|
|
nfand.3 $e |- ( ph -> F/ x th ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ nf3an .
|
|
(Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro,
|
|
16-Oct-2016.) $)
|
|
nf3and $p |- ( ph -> F/ x ( ps /\ ch /\ th ) ) $=
|
|
( w3a wa df-3an nfand nfxfrd ) BCDIBCJZDJAEBCDKANDEABCEFGLHLM $.
|
|
$}
|
|
|
|
${
|
|
nfan.1 $e |- F/ x ph $.
|
|
nfan.2 $e |- F/ x ps $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph /\ ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.)
|
|
(Proof shortned by Wolf Lammen, 2-Jan-2018.) $)
|
|
nfan $p |- F/ x ( ph /\ ps ) $=
|
|
( wa wnf wtru a1i nfand trud ) ABFCGHABCACGHDIBCGHEIJK $.
|
|
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , then it is not free in
|
|
` ( ph -/\ ps ) ` . (Contributed by Scott Fenton, 2-Jan-2018.) $)
|
|
nfnan $p |- F/ x ( ph -/\ ps ) $=
|
|
( wnan wa wn df-nan nfan nfn nfxfr ) ABFABGZHCABIMCABCDEJKL $.
|
|
|
|
$( Obsolete proof of ~ nfan as of 2-Jan-2018. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) (New usage is discouraged.) $)
|
|
nfanOLD $p |- F/ x ( ph /\ ps ) $=
|
|
( wa wn wi df-an nfn nfim nfxfr ) ABFABGZHZGCABINCAMCDBCEJKJL $.
|
|
|
|
nfan.3 $e |- F/ x ch $.
|
|
$( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
|
|
` ( ph /\ ps /\ ch ) ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nf3an $p |- F/ x ( ph /\ ps /\ ch ) $=
|
|
( w3a wa df-3an nfan nfxfr ) ABCHABIZCIDABCJMCDABDEFKGKL $.
|
|
$}
|
|
|
|
${
|
|
hb.1 $e |- ( ph -> A. x ph ) $.
|
|
hb.2 $e |- ( ps -> A. x ps ) $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph /\ ps ) ` . (Contributed by NM, 5-Aug-1993.) (Proof shortened
|
|
by Wolf Lammen, 2-Jan-2018.) $)
|
|
hban $p |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) ) $=
|
|
( wa nfi nfan nfri ) ABFCABCACDGBCEGHI $.
|
|
$( Obsolete proof of ~ hban as of 2-Jan-2018. (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.) $)
|
|
hbanOLD $p |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) ) $=
|
|
( wa wn wi df-an hbn hbim hbxfrbi ) ABFABGZHZGCABINCAMCDBCEJKJL $.
|
|
hb.3 $e |- ( ch -> A. x ch ) $.
|
|
$( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
|
|
` ( ph /\ ps /\ ch ) ` . (Contributed by NM, 14-Sep-2003.) (Proof
|
|
shortened by Wolf Lammen, 2-Jan-2018.) $)
|
|
hb3an $p |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) ) $=
|
|
( w3a nfi nf3an nfri ) ABCHDABCDADEIBDFICDGIJK $.
|
|
|
|
$( Obsolete proof of ~ hb3an as of 2-Jan-2018. (Contributed by NM,
|
|
14-Sep-2003.) (New usage is discouraged.) $)
|
|
hb3anOLD $p |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) ) $=
|
|
( w3a wa df-3an hban hbxfrbi ) ABCHABIZCIDABCJMCDABDEFKGKL $.
|
|
$}
|
|
|
|
${
|
|
nfbid.1 $e |- ( ph -> F/ x ps ) $.
|
|
nfbid.2 $e |- ( ph -> F/ x ch ) $.
|
|
$( If in a context ` x ` is not free in ` ps ` and ` ch ` , it is not free
|
|
in ` ( ps <-> ch ) ` . (Contributed by Mario Carneiro, 24-Sep-2016.)
|
|
(Proof shortened by Wolf Lammen, 29-Dec-2017.) $)
|
|
nfbid $p |- ( ph -> F/ x ( ps <-> ch ) ) $=
|
|
( wb wi wa dfbi2 nfimd nfand nfxfrd ) BCGBCHZCBHZIADBCJANODABCDEFKACBDFEK
|
|
LM $.
|
|
|
|
$( Obsolete proof of ~ nfbid as of 29-Dec-2017. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) (New usage is discouraged.) $)
|
|
nfbidOLD $p |- ( ph -> F/ x ( ps <-> ch ) ) $=
|
|
( wb wi wn dfbi1 nfimd nfnd nfxfrd ) BCGBCHZCBHZIZHZIADBCJAQDANPDABCDEFKA
|
|
ODACBDFEKLKLM $.
|
|
$}
|
|
|
|
${
|
|
nf.1 $e |- F/ x ph $.
|
|
nf.2 $e |- F/ x ps $.
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph <-> ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.)
|
|
(Proof shortened by Wolf Lammen, 2-Jan-2018.) $)
|
|
nfbi $p |- F/ x ( ph <-> ps ) $=
|
|
( wb wnf wtru a1i nfbid trud ) ABFCGHABCACGHDIBCGHEIJK $.
|
|
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph <-> ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.)
|
|
(New usage is discouraged.) $)
|
|
nfbiOLD $p |- F/ x ( ph <-> ps ) $=
|
|
( wb wi wa dfbi2 nfim nfan nfxfr ) ABFABGZBAGZHCABIMNCABCDEJBACEDJKL $.
|
|
|
|
$( If ` x ` is not free in ` ph ` and ` ps ` , it is not free in
|
|
` ( ph \/ ps ) ` . (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfor $p |- F/ x ( ph \/ ps ) $=
|
|
( wo wn wi df-or nfn nfim nfxfr ) ABFAGZBHCABIMBCACDJEKL $.
|
|
|
|
nf.3 $e |- F/ x ch $.
|
|
$( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , it is not free in
|
|
` ( ph \/ ps \/ ch ) ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nf3or $p |- F/ x ( ph \/ ps \/ ch ) $=
|
|
( w3o wo df-3or nfor nfxfr ) ABCHABIZCIDABCJMCDABDEFKGKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
equsalhw.1 $e |- ( ps -> A. x ps ) $.
|
|
equsalhw.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Weaker version of ~ equsalh (requiring distinct variables) without using
|
|
~ ax-12 . (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf
|
|
Lammen, 28-Dec-2017.) $)
|
|
equsalhw $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
|
|
( weq wi wal wex 19.23h pm5.74i albii a9ev a1bi 3bitr4i ) CDGZBHZCIQCJZBH
|
|
QAHZCIBQBCEKTRCQABFLMSBCDNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
equsalhwOLD.1 $e |- ( ps -> A. x ps ) $.
|
|
equsalhwOLD.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Obsolete proof of ~ equsalhw as of 28-Dec-2017. (Contributed by NM,
|
|
29-Nov-2015.) (New usage is discouraged.) $)
|
|
equsalhwOLD $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
|
|
( weq wi wal sp impbii syl6bbr pm5.74i albii a1d alrimih ax9v con3 al2imi
|
|
wn mtoi ax6o syl bitr4i ) CDGZAHZCIUEBCIZHZCIZBUFUHCUEAUGUEABUGFUGBBCJEKL
|
|
MNBUIBUHCEBUGUEEOPUIUGTZCIZTBUIUKUETZCICDQUHUJULCUEUGRSUABCUBUCKUD $.
|
|
$}
|
|
|
|
${
|
|
19.21hOLD.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Obsolete proof of ~ 19.21h as of 1-Jan-2018. (Contributed by NM,
|
|
1-Aug-2017.) (New usage is discouraged.) $)
|
|
19.21hOLD $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
|
|
( wi wal alim syl5 hba1 hbim sp imim2i alrimih impbii ) ABEZCFZABCFZEZAAC
|
|
FPQDABCGHROCAQCDBCIJQBABCKLMN $.
|
|
$}
|
|
|
|
${
|
|
hbex.1 $e |- ( ph -> A. x ph ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
hbex $p |- ( E. y ph -> A. x E. y ph ) $=
|
|
( wex wn wal df-ex hbn hbal hbxfrbi ) ACEAFZCGZFBACHMBLBCABDIJIK $.
|
|
$}
|
|
|
|
${
|
|
nfal.1 $e |- F/ x ph $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfal $p |- F/ x A. y ph $=
|
|
( wal nfri hbal nfi ) ACEBABCABDFGH $.
|
|
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf
|
|
Lammen, 30-Dec-2017.) $)
|
|
nfex $p |- F/ x E. y ph $=
|
|
( wex nfri hbex nfi ) ACEBABCABDFGH $.
|
|
|
|
$( Obsolete proof of ~ nfex as of 30-Dec-2017. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) (New usage is discouraged.) $)
|
|
nfexOLD $p |- F/ x E. y ph $=
|
|
( wex wn wal df-ex nfn nfal nfxfr ) ACEAFZCGZFBACHMBLBCABDIJIK $.
|
|
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` F/ y ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf
|
|
Lammen, 30-Dec-2017.) $)
|
|
nfnf $p |- F/ x F/ y ph $=
|
|
( wnf wal wi df-nf nfal nfim nfxfr ) ACEAACFZGZCFBACHMBCALBDABCDIJIK $.
|
|
|
|
$( Obsolete proof of ~ nfnf as of 30-Dec-2017. (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) (New usage is discouraged.) $)
|
|
nfnfOLD $p |- F/ x F/ y ph $=
|
|
( wnf wal wi df-nf wtru a1i nfal nfimd trud nfxfr ) ACEAACFZGZCFBACHPBCPB
|
|
EIAOBABEIDJOBEIABCDKJLMKN $.
|
|
$}
|
|
|
|
$( Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake
|
|
sometimes made by beginners! But sometimes the converse does hold, as in
|
|
~ 19.12vv and ~ r19.12sn . (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Wolf Lammen, 3-Jan-2018.) $)
|
|
19.12 $p |- ( E. x A. y ph -> A. y E. x ph ) $=
|
|
( wal wex nfa1 nfex sp eximi alrimi ) ACDZBEABECKCBACFGKABACHIJ $.
|
|
|
|
$( Obsolete proof of ~ 19.12 as of 3-Jan-2018. (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.) $)
|
|
19.12OLD $p |- ( E. x A. y ph -> A. y E. x ph ) $=
|
|
( wal wex hba1 hbex sp eximi alrimih ) ACDZBEABECKCBACFGKABACHIJ $.
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
dvelimhw.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelimhw.2 $e |- ( ps -> A. z ps ) $.
|
|
dvelimhw.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( dvelimhw.4 $e |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $. $)
|
|
dvelimhw.4 $e |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $.
|
|
$( Proof of ~ dvelimh without using ~ ax-12 but with additional distinct
|
|
variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.)
|
|
(Revised by NM, 1-Aug-2017.) $)
|
|
dvelimhw $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( weq wal wn wi ax-17 hbn1 equcomi alimi syl56 a1i hbimd equsalhw 3imtr3g
|
|
hbald albii ) CDJZCKLZEDJZAMZEKZUICKBBCKUFUHCEUFENUFUGACUECOUGDEJZUFUJCKU
|
|
GCKEDPIUJUGCDEPQRAACKMUFFSTUCABEDGHUAZUIBCUKUDUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
cbv3hv.1 $e |- ( ph -> A. y ph ) $.
|
|
cbv3hv.2 $e |- ( ps -> A. x ps ) $.
|
|
cbv3hv.3 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Lemma for ~ ax10 . Similar to ~ cbv3h . Requires distinct variables
|
|
but avoids ~ ax-12 . (Contributed by NM, 25-Jul-2015.) (Proof
|
|
shortened by Wolf Lammen, 29-Dec-2017.) $)
|
|
cbv3hv $p |- ( A. x ph -> A. y ps ) $=
|
|
( wal alimi wex weq wi a9ev eximi ax-mp 19.35i 19.9h sylib a7s syl ) ACHZ
|
|
ADHZCHBDHZAUBCEIAUCDCUABDUABCJBABCCDKZCJABLZCJCDMUDUECGNOPBCFQRIST $.
|
|
|
|
$( Obsolete proof of ~ cbv3hv as of 29-Dec-2017. (Contributed by NM,
|
|
25-Jul-2015.) (New usage is discouraged.) $)
|
|
cbv3hvOLD $p |- ( A. x ph -> A. y ps ) $=
|
|
( wal alimi wi weq wn ax9v hba1 hbim hbn sp syl5 con3i alrimih mt3 a7s
|
|
syl ) ACHZADHZCHBDHZAUECEIAUFDCUDBDUDBJZCDKZLZCHCDMUGLUICUGCUDBCACNFOPUHU
|
|
GUDAUHBACQGRSTUAIUBUC $.
|
|
$}
|
|
|
|
${
|
|
nfald.1 $e |- F/ y ph $.
|
|
nfald.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` A. y ph ` .
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf
|
|
Lammen, 6-Jan-2018.) $)
|
|
nfald $p |- ( ph -> F/ x A. y ps ) $=
|
|
( wnf wal alrimi nfnf1 nfal hba1 sp nfrd hbald nfd syl ) ABCGZDHZBDHZCGAR
|
|
DEFISTCRCDBCJKSBCDRDLSBCRDMNOPQ $.
|
|
|
|
$( Obsolete proof of ~ nfald as of 6-Jan-2018. (Contributed by Mario
|
|
Carneiro, 24-Sep-2016.) (New usage is discouraged.) $)
|
|
nfaldOLD $p |- ( ph -> F/ x A. y ps ) $=
|
|
( wnf wal alrimi nfnf1 nfal nfr al2imi ax-7 syl6 nfd syl ) ABCGZDHZBDHZCG
|
|
ARDEFISTCRCDBCJKSTBCHZDHTCHRBUADBCLMBDCNOPQ $.
|
|
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` E. y ph ` .
|
|
(Contributed by Mario Carneiro, 24-Sep-2016.) $)
|
|
nfexd $p |- ( ph -> F/ x E. y ps ) $=
|
|
( wex wn wal df-ex nfnd nfald nfxfrd ) BDGBHZDIZHACBDJAOCANCDEABCFKLKM $.
|
|
$}
|
|
|
|
$( Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nfa2 $p |- F/ x A. y A. x ph $=
|
|
( wal nfa1 nfal ) ABDBCABEF $.
|
|
|
|
$( Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nfia1 $p |- F/ x ( A. x ph -> A. x ps ) $=
|
|
( wal nfa1 nfim ) ACDBCDCACEBCEF $.
|
|
|
|
$( Obsolete proof of ~ 19.9t as of 30-Dec-2017. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|
|
(New usage is discouraged.) $)
|
|
19.9tOLD $p |- ( F/ x ph -> ( E. x ph <-> ph ) ) $=
|
|
( wnf wex wn wal df-ex id nfnd nfrd con1d syl5bi 19.8a impbid1 ) ABCZABDZAP
|
|
AEZBFZEOAABGOAROQBOABOHIJKLABMN $.
|
|
|
|
$( Obsolete proof of ~ excomim as of 8-Jan-2018. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
|
|
(New usage is discouraged.) $)
|
|
excomimOLD $p |- ( E. x E. y ph -> E. y E. x ph ) $=
|
|
( wex 19.8a 2eximi nfe1 nfex 19.9 sylib ) ACDBDABDZCDZBDLAKBCABEFLBKBCABGHI
|
|
J $.
|
|
|
|
$( Obsolete proof of ~ excom as of 8-Jan-2018. (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.) $)
|
|
excomOLD $p |- ( E. x E. y ph <-> E. y E. x ph ) $=
|
|
( wex excomim impbii ) ACDBDABDCDABCEACBEF $.
|
|
|
|
${
|
|
19.16.1 $e |- F/ x ph $.
|
|
$( Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.16 $p |- ( A. x ( ph <-> ps ) -> ( ph <-> A. x ps ) ) $=
|
|
( wal wb 19.3 albi syl5bbr ) AACEABFCEBCEACDGABCHI $.
|
|
$}
|
|
|
|
${
|
|
19.17.1 $e |- F/ x ps $.
|
|
$( Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.17 $p |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> ps ) ) $=
|
|
( wb wal albi 19.3 syl6bb ) ABECFACFBCFBABCGBCDHI $.
|
|
$}
|
|
|
|
${
|
|
19.19.1 $e |- F/ x ph $.
|
|
$( Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.19 $p |- ( A. x ( ph <-> ps ) -> ( ph <-> E. x ps ) ) $=
|
|
( wex wb wal 19.9 exbi syl5bbr ) AACEABFCGBCEACDHABCIJ $.
|
|
$}
|
|
|
|
$( Obsolete proof of ~ 19.21t as of 30-Dec-2017. (Contributed by NM,
|
|
27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.)
|
|
(New usage is discouraged.) $)
|
|
19.21tOLD $p |- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) ) $=
|
|
( wnf wi wal id nfrd alim syl9 nfa1 a1i nfimd sp imim2i alimi syl6 impbid )
|
|
ACDZABEZCFZABCFZEZSAACFUAUBSACSGZHABCIJSUCUCCFUASUCCSAUBCUDUBCDSBCKLMHUCTCU
|
|
BBABCNOPQR $.
|
|
|
|
${
|
|
19.21-2.1 $e |- F/ x ph $.
|
|
19.21-2.2 $e |- F/ y ph $.
|
|
$( Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed
|
|
by NM, 4-Feb-2005.) $)
|
|
19.21-2 $p |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) ) $=
|
|
( wi wal 19.21 albii bitri ) ABGDHZCHABDHZGZCHAMCHGLNCABDFIJAMCEIK $.
|
|
$}
|
|
|
|
${
|
|
19.21bbi.1 $e |- ( ph -> A. x A. y ps ) $.
|
|
$( Inference removing double quantifier. (Contributed by NM,
|
|
20-Apr-1994.) $)
|
|
19.21bbi $p |- ( ph -> ps ) $=
|
|
( wal 19.21bi ) ABDABDFCEGG $.
|
|
$}
|
|
|
|
$( An alternative definition of ~ df-nf , which does not involve nested
|
|
quantifiers on the same variable. (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nf2 $p |- ( F/ x ph <-> ( E. x ph -> A. x ph ) ) $=
|
|
( wnf wal wi wex df-nf nfa1 19.23 bitri ) ABCAABDZEBDABFKEABGAKBABHIJ $.
|
|
|
|
$( An alternative definition of ~ df-nf . (Contributed by Mario Carneiro,
|
|
24-Sep-2016.) $)
|
|
nf3 $p |- ( F/ x ph <-> A. x ( E. x ph -> ph ) ) $=
|
|
( wnf wex wal wi nf2 nfe1 19.21 bitr4i ) ABCABDZABEFKAFBEABGKABABHIJ $.
|
|
|
|
$( Variable ` x ` is effectively not free in ` ph ` iff ` ph ` is always true
|
|
or always false. (Contributed by Mario Carneiro, 24-Sep-2016.) $)
|
|
nf4 $p |- ( F/ x ph <-> ( A. x ph \/ A. x -. ph ) ) $=
|
|
( wnf wex wal wi wn wo nf2 imor orcom alnex orbi2i bitr4i 3bitri ) ABCABDZA
|
|
BEZFPGZQHZQAGBEZHZABIPQJSQRHUARQKTRQABLMNO $.
|
|
|
|
${
|
|
19.27.1 $e |- F/ x ps $.
|
|
$( Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.27 $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
|
|
( wa wal 19.26 19.3 anbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.28.1 $e |- F/ x ph $.
|
|
$( Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.28 $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
|
|
( wa wal 19.26 19.3 anbi1i bitri ) ABECFACFZBCFZEALEABCGKALACDHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.36.1 $e |- F/ x ps $.
|
|
$( Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.36 $p |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) ) $=
|
|
( wi wex wal 19.35 19.9 imbi2i bitri ) ABECFACGZBCFZELBEABCHMBLBCDIJK $.
|
|
|
|
19.36i.2 $e |- E. x ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.36i $p |- ( A. x ph -> ps ) $=
|
|
( wi wex wal 19.36 mpbi ) ABFCGACHBFEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
19.37.1 $e |- F/ x ph $.
|
|
$( Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.37 $p |- ( E. x ( ph -> ps ) <-> ( ph -> E. x ps ) ) $=
|
|
( wi wex wal 19.35 19.3 imbi1i bitri ) ABECFACGZBCFZEAMEABCHLAMACDIJK $.
|
|
$}
|
|
|
|
$( Obsolete proof of 19.38 as of 2-Jan-2018. (Contributed by NM,
|
|
5-Aug-1993.) (New usage is discouraged.) $)
|
|
19.38OLD $p |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) ) $=
|
|
( wex wal wi nfe1 nfa1 nfim 19.8a sp imim12i alrimi ) ACDZBCEZFABFCNOCACGBC
|
|
HIANOBACJBCKLM $.
|
|
|
|
${
|
|
19.32.1 $e |- F/ x ph $.
|
|
$( Theorem 19.32 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 24-Sep-2016.) $)
|
|
19.32 $p |- ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) $=
|
|
( wn wi wal wo nfn 19.21 df-or albii 3bitr4i ) AEZBFZCGNBCGZFABHZCGAPHNBC
|
|
ACDIJQOCABKLAPKM $.
|
|
$}
|
|
|
|
${
|
|
19.31.1 $e |- F/ x ps $.
|
|
$( Theorem 19.31 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.31 $p |- ( A. x ( ph \/ ps ) <-> ( A. x ph \/ ps ) ) $=
|
|
( wo wal 19.32 orcom albii 3bitr4i ) BAEZCFBACFZEABEZCFLBEBACDGMKCABHILBH
|
|
J $.
|
|
$}
|
|
|
|
${
|
|
19.44.1 $e |- F/ x ps $.
|
|
$( Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.44 $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ ps ) ) $=
|
|
( wo wex 19.43 19.9 orbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.45.1 $e |- F/ x ph $.
|
|
$( Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) $)
|
|
19.45 $p |- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) ) $=
|
|
( wo wex 19.43 19.9 orbi1i bitri ) ABECFACFZBCFZEALEABCGKALACDHIJ $.
|
|
$}
|
|
|
|
${
|
|
19.41.1 $e |- F/ x ps $.
|
|
$( Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
19.41 $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
|
|
( wa wex 19.40 id exlimi anim2i syl pm3.21 eximd impcom impbii ) ABEZCFZA
|
|
CFZBEZQRBCFZESABCGTBRBBCDBHIJKBRQBAPCDBALMNO $.
|
|
$}
|
|
|
|
${
|
|
19.42.1 $e |- F/ x ph $.
|
|
$( Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) $)
|
|
19.42 $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
|
|
( wa wex 19.41 exancom ancom 3bitr4i ) BAECFBCFZAEABECFAKEBACDGABCHAKIJ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nfan1.1 $e |- F/ x ph $.
|
|
nfan1.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( A closed form of ~ nfan . (Contributed by Mario Carneiro,
|
|
3-Oct-2016.) $)
|
|
nfan1 $p |- F/ x ( ph /\ ps ) $=
|
|
( wa wal nfrd imdistani 19.28 sylibr nfi ) ABFZCMABCGZFMCGABNABCEHIABCDJK
|
|
L $.
|
|
$}
|
|
|
|
${
|
|
exan.1 $e |- ( E. x ph /\ ps ) $.
|
|
$( Place a conjunct in the scope of an existential quantifier.
|
|
(Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
exan $p |- E. x ( ph /\ ps ) $=
|
|
( wex wal wa nfe1 19.28 mpgbi 19.29r ax-mp ) ACEZBCFGZABGCEMBGNCMBCACHIDJ
|
|
ABCKL $.
|
|
$}
|
|
|
|
${
|
|
hbnd.1 $e |- ( ph -> A. x ph ) $.
|
|
hbnd.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
|
|
$( Deduction form of bound-variable hypothesis builder ~ hbn .
|
|
(Contributed by NM, 3-Jan-2002.) $)
|
|
hbnd $p |- ( ph -> ( -. ps -> A. x -. ps ) ) $=
|
|
( wal wi wn alrimih hbnt syl ) ABBCFGZCFBHZMCFGALCDEIBCJK $.
|
|
$}
|
|
|
|
${
|
|
aaan.1 $e |- F/ y ph $.
|
|
aaan.2 $e |- F/ x ps $.
|
|
$( Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) $)
|
|
aaan $p |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) ) $=
|
|
( wa wal 19.28 albii nfal 19.27 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
|
|
CBCDFKLM $.
|
|
$}
|
|
|
|
${
|
|
eeor.1 $e |- F/ y ph $.
|
|
eeor.2 $e |- F/ x ps $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.) $)
|
|
eeor $p |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) ) $=
|
|
( wo wex 19.45 exbii nfex 19.44 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
|
|
CBCDFKLM $.
|
|
$}
|
|
|
|
$( Quantified "excluded middle." Exercise 9.2a of Boolos, p. 111,
|
|
_Computability and Logic_. (Contributed by NM, 10-Dec-2000.) $)
|
|
qexmid $p |- E. x ( ph -> A. x ph ) $=
|
|
( wal 19.8a 19.35ri ) AABCZBFBDE $.
|
|
|
|
$( A property related to substitution that unlike ~ equs5 doesn't require a
|
|
distinctor antecedent. (Contributed by NM, 2-Feb-2007.) $)
|
|
equs5a $p |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) ) $=
|
|
( weq wal wa wi nfa1 ax-11 imp exlimi ) BCDZACEZFLAGZBEZBNBHLMOABCIJK $.
|
|
|
|
$( A property related to substitution that unlike ~ equs5 doesn't require a
|
|
distinctor antecedent. (Contributed by NM, 2-Feb-2007.) $)
|
|
equs5e $p |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) ) $=
|
|
( weq wa wex wi nfe1 wn wal equs3 ax-11 con3rr3 df-ex syl6ibr sylbi alrimi
|
|
) BCDZAEZBFZRACFZGZBSBHTRAIZGBJZIZUBABCKUERUCCJZIUARUFUDUCBCLMACNOPQ $.
|
|
|
|
${
|
|
exlimdd.1 $e |- F/ x ph $.
|
|
exlimdd.2 $e |- F/ x ch $.
|
|
exlimdd.3 $e |- ( ph -> E. x ps ) $.
|
|
exlimdd.4 $e |- ( ( ph /\ ps ) -> ch ) $.
|
|
$( Existential elimination rule of natural deduction. (Contributed by
|
|
Mario Carneiro, 9-Feb-2017.) $)
|
|
exlimdd $p |- ( ph -> ch ) $=
|
|
( wex ex exlimd mpd ) ABDICGABCDEFABCHJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Special case of Theorem 19.21 of [Margaris] p. 90. _Notational
|
|
convention_: We sometimes suffix with "v" the label of a theorem
|
|
eliminating a hypothesis such as ` F/ x ph ` in ~ 19.21 via the use of
|
|
distinct variable conditions combined with ~ nfv . Conversely, we
|
|
sometimes suffix with "f" the label of a theorem introducing such a
|
|
hypothesis to eliminate the need for the distinct variable condition;
|
|
e.g. ~ euf derived from ~ df-eu . The "f" stands for "not free in"
|
|
which is less restrictive than "does not occur in." (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.21v $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
|
|
( nfv 19.21 ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM,
|
|
28-Jun-1998.) $)
|
|
19.23v $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
|
|
( nfv 19.23 ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ps $.
|
|
$( Theorem 19.23 of [Margaris] p. 90 extended to two variables.
|
|
(Contributed by NM, 10-Aug-2004.) $)
|
|
19.23vv $p |- ( A. x A. y ( ph -> ps ) <-> ( E. x E. y ph -> ps ) ) $=
|
|
( wi wal wex 19.23v albii bitri ) ABEDFZCFADGZBEZCFLCGBEKMCABDHILBCHJ $.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d ps x $.
|
|
$( Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew
|
|
Salmon, 24-May-2011.) $)
|
|
pm11.53 $p |- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) ) $=
|
|
( wi wal wex 19.21v albii nfv nfal 19.23 bitri ) ABEDFZCFABDFZEZCFACGOENP
|
|
CABDHIAOCBCDBCJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.) $)
|
|
19.27v $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
|
|
( nfv 19.27 ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.) $)
|
|
19.28v $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
|
|
( nfv 19.28 ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Special case of Theorem 19.36 of [Margaris] p. 90. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
19.36v $p |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) ) $=
|
|
( nfv 19.36 ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
19.36aiv.1 $e |- E. x ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.36aiv $p |- ( A. x ph -> ps ) $=
|
|
( nfv 19.36i ) ABCBCEDF $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $.
|
|
$( Special case of ~ 19.12 where its converse holds. (Contributed by NM,
|
|
18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.) $)
|
|
19.12vv $p |- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) ) $=
|
|
( wi wal wex 19.21v exbii nfv nfal 19.36 19.36v albii 19.21 bitr2i 3bitri
|
|
) ABEZDFZCGABDFZEZCGACFZTEZRCGZDFZSUACABDHIATCBCDBCJKLUEUBBEZDFUCUDUFDABC
|
|
MNUBBDADCADJKOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Special case of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.37v $p |- ( E. x ( ph -> ps ) <-> ( ph -> E. x ps ) ) $=
|
|
( nfv 19.37 ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
19.37aiv.1 $e |- E. x ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.37aiv $p |- ( ph -> E. x ps ) $=
|
|
( wi wex 19.37v mpbi ) ABECFABCFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.41v $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
|
|
( nfv 19.41 ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ps $.
|
|
$( Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by
|
|
NM, 30-Apr-1995.) $)
|
|
19.41vv $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x E. y ph /\ ps ) ) $=
|
|
( wa wex 19.41v exbii bitri ) ABEDFZCFADFZBEZCFKCFBEJLCABDGHKBCGI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ps $. $d z ps $.
|
|
$( Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by
|
|
NM, 30-Apr-1995.) $)
|
|
19.41vvv $p |- ( E. x E. y E. z ( ph /\ ps ) <->
|
|
( E. x E. y E. z ph /\ ps ) ) $=
|
|
( wa wex 19.41vv exbii 19.41v bitri ) ABFEGDGZCGAEGDGZBFZCGMCGBFLNCABDEHI
|
|
MBCJK $.
|
|
$}
|
|
|
|
${
|
|
$d w ps $. $d x ps $. $d y ps $. $d z ps $.
|
|
$( Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by
|
|
FL, 14-Jul-2007.) $)
|
|
19.41vvvv $p |- ( E. w E. x E. y E. z ( ph /\ ps ) <->
|
|
( E. w E. x E. y E. z ph /\ ps ) ) $=
|
|
( wa wex 19.41vvv exbii 19.41v bitri ) ABGEHDHCHZFHAEHDHCHZBGZFHNFHBGMOFA
|
|
BCDEIJNBFKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
19.42v $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
|
|
( nfv 19.42 ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $.
|
|
$( Distribution of existential quantifiers. (Contributed by NM,
|
|
9-Mar-1995.) $)
|
|
exdistr $p |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) ) $=
|
|
( wa wex 19.42v exbii ) ABEDFABDFECABDGH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
$( Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by
|
|
NM, 16-Mar-1995.) $)
|
|
19.42vv $p |- ( E. x E. y ( ph /\ ps ) <-> ( ph /\ E. x E. y ps ) ) $=
|
|
( wa wex exdistr 19.42v bitri ) ABEDFCFABDFZECFAJCFEABCDGAJCHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $. $d z ph $.
|
|
$( Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by
|
|
NM, 21-Sep-2011.) $)
|
|
19.42vvv $p |- ( E. x E. y E. z ( ph /\ ps )
|
|
<-> ( ph /\ E. x E. y E. z ps ) ) $=
|
|
( wa wex 19.42vv exbii 19.42v bitri ) ABFEGDGZCGABEGDGZFZCGAMCGFLNCABDEHI
|
|
AMCJK $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d z ph $.
|
|
$( Distribution of existential quantifiers. (Contributed by NM,
|
|
17-Mar-1995.) $)
|
|
exdistr2 $p |- ( E. x E. y E. z ( ph /\ ps ) <->
|
|
E. x ( ph /\ E. y E. z ps ) ) $=
|
|
( wa wex 19.42vv exbii ) ABFEGDGABEGDGFCABDEHI $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d z ph $. $d z ps $.
|
|
$( Distribution of existential quantifiers. (Contributed by NM,
|
|
9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3exdistr $p |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <->
|
|
E. x ( ph /\ E. y ( ps /\ E. z ch ) ) ) $=
|
|
( w3a wex wa 3anass 2exbii 19.42vv exdistr anbi2i 3bitri exbii ) ABCGZFHE
|
|
HZABCFHIEHZIZDRABCIZIZFHEHAUAFHEHZITQUBEFABCJKAUAEFLUCSABCEFMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d z ph $. $d w ph $. $d z ps $. $d w ps $. $d w ch $.
|
|
$( Distribution of existential quantifiers. (Contributed by NM,
|
|
9-Mar-1995.) $)
|
|
4exdistr $p |- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
|
|
E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) ) $=
|
|
( wa wex anass exbii 19.42v anbi2i 3bitri bitri ) ABICDIZIZHJZGJZFJZABCDH
|
|
JIZGJIZFJIZEUAAUCIZFJUDTUEFTABUBIZIZGJAUFGJZIUESUGGSABQIZIZHJZUGRUJHABQKL
|
|
UKAUIHJZIABQHJZIZIUGAUIHMULUNABQHMNUNUFAUMUBBCDHMNNOPLAUFGMUHUCABUBGMNOLA
|
|
UCFMPL $.
|
|
$}
|
|
|
|
${
|
|
eean.1 $e |- F/ y ph $.
|
|
eean.2 $e |- F/ x ps $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
eean $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) ) $=
|
|
( wa wex 19.42 exbii nfex 19.41 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
|
|
CBCDFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) $)
|
|
eeanv $p |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) ) $=
|
|
( nfv eean ) ABCDADEBCEF $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d z ph $. $d x z ps $. $d x y ch $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
eeeanv $p |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <->
|
|
( E. x ph /\ E. y ps /\ E. z ch ) ) $=
|
|
( w3a wex wa df-3an 3exbii eeanv exbii anbi1i 19.41v 3bitr4i 3bitri ) ABC
|
|
GZFHEHDHABIZCIZFHEHZDHSEHZCFHZIZDHZADHZBEHZUCGZRTDEFABCJKUAUDDSCEFLMUBDHZ
|
|
UCIUFUGIZUCIUEUHUIUJUCABDELNUBUCDOUFUGUCJPQ $.
|
|
$}
|
|
|
|
${
|
|
$d z ph $. $d w ph $. $d x ps $. $d y ps $. $d y z $. $d w x $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.) $)
|
|
ee4anv $p |- ( E. x E. y E. z E. w ( ph /\ ps ) <->
|
|
( E. x E. y ph /\ E. z E. w ps ) ) $=
|
|
( wa wex excom exbii eeanv 2exbii 3bitri ) ABGFHZEHDHZCHNDHZEHZCHADHZBFHZ
|
|
GZEHCHRCHSEHGOQCNDEIJPTCEABDFKLRSCEKM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
nexdv.1 $e |- ( ph -> -. ps ) $.
|
|
$( Deduction for generalization rule for negated wff. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
nexdv $p |- ( ph -> -. E. x ps ) $=
|
|
( nfv nexd ) ABCACEDF $.
|
|
$}
|
|
|
|
$( One of the two equality axioms of standard predicate calculus, called
|
|
substitutivity of equality. (The other one is ~ stdpc6 .) Translated to
|
|
traditional notation, it can be
|
|
read: " ` x = y -> ( ph ( x , x ) -> ph ( x , y ) ) ` , provided that
|
|
` y ` is free for ` x ` in ` ph ( x , x ) ` ." Axiom 7 of [Mendelson]
|
|
p. 95. (Contributed by NM, 15-Feb-2005.) $)
|
|
stdpc7 $p |- ( x = y -> ( [ x / y ] ph -> ph ) ) $=
|
|
( wsb wi sbequ2 equcoms ) ACBDAECBACBFG $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ1 $p |- ( x = y -> ( ph -> [ y / x ] ph ) ) $=
|
|
( weq wsb wa wi wex pm3.4 19.8a df-sb sylanbrc ex ) BCDZAABCEZNAFZNAGPBHONA
|
|
IPBJABCKLM $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ12 $p |- ( x = y -> ( ph <-> [ y / x ] ph ) ) $=
|
|
( weq wsb sbequ1 sbequ2 impbid ) BCDAABCEABCFABCGH $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.)
|
|
(Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
|
|
sbequ12r $p |- ( x = y -> ( [ x / y ] ph <-> ph ) ) $=
|
|
( wsb wb weq sbequ12 bicomd equcoms ) ACBDZAECBCBFAJACBGHI $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ12a $p |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) ) $=
|
|
( weq wsb sbequ12 wb equcoms bitr3d ) BCDAABCEACBEZABCFAJGCBACBFHI $.
|
|
|
|
$( An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p.
|
|
15 of the preprint). (Contributed by NM, 5-Aug-1993.) $)
|
|
sbid $p |- ( [ x / x ] ph <-> ph ) $=
|
|
( wsb weq wb equid sbequ12 ax-mp bicomi ) AABBCZBBDAJEBFABBGHI $.
|
|
|
|
$( A version of ~ sb4 that doesn't require a distinctor antecedent.
|
|
(Contributed by NM, 2-Feb-2007.) $)
|
|
sb4a $p |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) ) $=
|
|
( wal wsb weq wa wex wi sb1 equs5a syl ) ACDZBCEBCFZMGBHNAIBDMBCJABCKL $.
|
|
|
|
$( One direction of a simplified definition of substitution that unlike ~ sb4
|
|
doesn't require a distinctor antecedent. (Contributed by NM,
|
|
2-Feb-2007.) $)
|
|
sb4e $p |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) ) $=
|
|
( wsb weq wa wex wi wal sb1 equs5e syl ) ABCDBCEZAFBGMACGHBIABCJABCKL $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Axiom scheme ax-12 (Quantified Equality)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Axiom of Quantified Equality. One of the equality and substitution axioms
|
|
of predicate calculus with equality.
|
|
|
|
An equivalent way to express this axiom that may be easier to understand
|
|
is ` ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) ` (see
|
|
~ ax12b ). Recall that in the intended interpretation, our variables are
|
|
metavariables ranging over the variables of predicate calculus (the object
|
|
language). In order for the first antecedent ` -. x = y ` to hold, ` x `
|
|
and ` y ` must have different values and thus cannot be the same
|
|
object-language variable. Similarly, ` x ` and ` z ` cannot be the same
|
|
object-language variable. Therefore, ` x ` will not occur in the wff
|
|
` y = z ` when the first two antecedents hold, so analogous to ~ ax-17 ,
|
|
the conclusion ` ( y = z -> A. x y = z ) ` follows.
|
|
|
|
The original version of this axiom was ~ ax-12o and was replaced with this
|
|
shorter ~ ax-12 in December 2015. The old axiom is proved from this one
|
|
as theorem ~ ax12o . Conversely, this axiom is proved from ~ ax-12o as
|
|
theorem ~ ax12 .
|
|
|
|
The primary purpose of this axiom is to provide a way to introduce the
|
|
quantifier ` A. x ` on ` y = z ` even when ` x ` and ` y ` are substituted
|
|
with the same variable. In this case, the first antecedent becomes
|
|
` -. x = x ` and the axiom still holds.
|
|
|
|
Although this version is shorter, the original version ~ ax12o may be more
|
|
practical to work with because of the "distinctor" form of its
|
|
antecedents. A typical application of ~ ax12o is in ~ dvelimh which
|
|
converts a distinct variable pair to the distinctor antecendent
|
|
` -. A. x x = y ` .
|
|
|
|
This axiom can be weakened if desired by adding distinct variable
|
|
restrictions on pairs ` x , z ` and ` y , z ` . To show that, we add
|
|
these restrictions to theorem ~ ax12v and use only ~ ax12v for further
|
|
derivations. Thus, ~ ax12v should be the only theorem referencing this
|
|
axiom. Other theorems can reference either ~ ax12v or ~ ax12o .
|
|
|
|
This axiom scheme is logically redundant (see ~ ax12w ) but is used as an
|
|
auxiliary axiom to achieve metalogical completeness. (Contributed by NM,
|
|
21-Dec-2015.) (New usage is discouraged.) $)
|
|
ax-12 $a |- ( -. x = y -> ( y = z -> A. x y = z ) ) $.
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( A weaker version of ~ ax-12 with distinct variable restrictions on pairs
|
|
` x , z ` and ` y , z ` . In order to show that this weakening is
|
|
adequate, this should be the only theorem referencing ~ ax-12 directly.
|
|
(Contributed by NM, 30-Jun-2016.) $)
|
|
ax12v $p |- ( -. x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( ax-12 ) ABCD $.
|
|
$}
|
|
|
|
${
|
|
$d w y $. $d w z $.
|
|
$( Lemma for ~ ax12o . Similar to ~ equvin but with a negated equality.
|
|
(Contributed by NM, 24-Dec-2015.) $)
|
|
ax12olem1 $p |- ( E. w ( y = w /\ -. z = w ) <-> -. y = z ) $=
|
|
( weq wn wa wex ax-8 equcomi con3and exlimiv ax-17 wi equcoms com12 con3d
|
|
syl6 jctild spimeh impbii ) ACDZBCDZEZFZCGABDZEZUDUFCUAUEUBUAUECBDZUBACBH
|
|
CBIQJKUFUDCAUFCLCADZUFUCUAUHUBUEUBUHUEUHUEMCBUGUHBADUECBAHBAIQNOPCAIRST
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d w x z $. $d w y $.
|
|
ax12olem2.1 $e |- ( -. x = y -> ( y = w -> A. x y = w ) ) $.
|
|
$( Lemma for ~ ax12o . Negate the equalities in ~ ax-12 , shown as the
|
|
hypothesis. (Contributed by NM, 24-Dec-2015.) $)
|
|
ax12olem2 $p |- ( -. x = y -> ( -. y = z -> A. x -. y = z ) ) $=
|
|
( weq wn wa wex wal anim1d ax-17 anim2i 19.26 sylibr syl6 19.12 ax12olem1
|
|
eximdv albii 3imtr3g ) ABFGZBDFZCDFGZHZDIZUFAJZBCFGZUHAJUBUFUEAJZDIUGUBUE
|
|
UIDUBUEUCAJZUDHZUIUBUCUJUDEKUKUJUDAJZHUIUDULUJUDALMUCUDANOPSUEDAQPBCDRZUF
|
|
UHAUMTUA $.
|
|
$}
|
|
|
|
$( Lemma for ~ ax12o . Show the equivalence of an intermediate equivalent to
|
|
~ ax12o with the conjunction of ~ ax-12 and a variant with negated
|
|
equalities. (Contributed by NM, 24-Dec-2015.) $)
|
|
ax12olem3 $p |- ( ( -. x = y -> ( -. A. x -. y = z -> A. x y = z ) )
|
|
<-> ( ( -. x = y -> ( y = z -> A. x y = z ) )
|
|
/\ ( -. x = y -> ( -. y = z -> A. x -. y = z ) ) ) ) $=
|
|
( weq wn wal wi wa sp con2i imim1i imim2i con1d jca imim1d com12 imim3i imp
|
|
con1 impbii ) ABDEZBCDZEZAFZEZUBAFZGZGZUAUBUFGZGZUAUCUDGZGZHUHUJULUGUIUAUBU
|
|
EUFUDUBUCAIJKLUGUKUAUGUDUBUFUBUEUBAILMLNUJULUHUIUKUGUAUKUIUGUKUEUBUFUBUDSOP
|
|
QRT $.
|
|
|
|
${
|
|
$d w x z $. $d w y z $.
|
|
ax12olem4.1 $e |- ( -. x = y -> ( y = z -> A. x y = z ) ) $.
|
|
ax12olem4.2 $e |- ( -. x = y -> ( y = w -> A. x y = w ) ) $.
|
|
$( Lemma for ~ ax12o . Construct an intermediate equivalent to ~ ax-12
|
|
from two instances of ~ ax-12 . (Contributed by NM, 24-Dec-2015.) $)
|
|
ax12olem4 $p |- ( -. x = y -> ( -. A. x -. y = z -> A. x y = z ) ) $=
|
|
( weq wn wal wi ax12olem2 ax12olem3 mpbir2an ) ABGHZBCGZHZAIZHOAIZJJNORJJ
|
|
NPQJJEABCDFKABCLM $.
|
|
$}
|
|
|
|
${
|
|
ax12olem5.1 $e |- ( -. x = y -> ( -. A. x -. y = z -> A. x y = z ) ) $.
|
|
$( Lemma for ~ ax12o . See ~ ax12olem6 for derivation of ~ ax12o from the
|
|
conclusion. (Contributed by NM, 24-Dec-2015.) $)
|
|
ax12olem5 $p |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( weq wal wn wex wi exnal 19.8a hbe1 hba1 hbim syl5bi exlimih syl5 sylbir
|
|
df-ex ) ABEZAFGTGZAHZBCEZUCAFZITAJUCUCAHZUBUDUCAKUAUEUDIAUEUDAUCALUCAMNUE
|
|
UCGAFGUAUDUCASDOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $.
|
|
ax12olem6.1 $e |- ( -. A. x x = z -> ( z = w -> A. x z = w ) ) $.
|
|
ax12olem6.2 $e |- ( -. A. x x = y -> ( y = w -> A. x y = w ) ) $.
|
|
$( Lemma for ~ ax12o . Derivation of ~ ax12o from the hypotheses, without
|
|
using ~ ax12o . (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised
|
|
by NM, 24-Dec-2015.) $)
|
|
ax12olem6 $p |- ( -. A. x x = y
|
|
-> ( -. A. x x = z -> ( y = z -> A. x y = z ) ) ) $=
|
|
( weq wn wi hbn1 hbim1 ax-17 equcom equequ1 syl5bb imbi2d dvelimhw 19.21h
|
|
wal syl6ib pm2.86d ) ABGASHZACGZASHZBCGZUEASZUBUDUEIZUGASUDUFIUDCDGZIUGAB
|
|
DUDUHAUCAJZEKUGDLDBGZUHUEUDUHDCGUJUECDMDBCNOPFQUDUEAUIRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $.
|
|
ax12olem7.1 $e |- ( -. x = z -> ( -. A. x -. z = w -> A. x z = w ) ) $.
|
|
ax12olem7.2 $e |- ( -. x = y -> ( -. A. x -. y = w -> A. x y = w ) ) $.
|
|
$( Lemma for ~ ax12o . Derivation of ~ ax12o from the hypotheses, without
|
|
using ~ ax12o . (Contributed by NM, 24-Dec-2015.) $)
|
|
ax12olem7 $p |- ( -. A. x x = y
|
|
-> ( -. A. x x = z -> ( y = z -> A. x y = z ) ) ) $=
|
|
( ax12olem5 ax12olem6 ) ABCDACDEGABDFGH $.
|
|
$}
|
|
|
|
${
|
|
$d x w v $. $d y w v $. $d z w v $.
|
|
$( Derive set.mm's original ~ ax-12o from the shorter ~ ax-12 .
|
|
(Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) $)
|
|
ax12o $p |- ( -. A. z z = x -> ( -. A. z z = y
|
|
-> ( x = y -> A. z x = y ) ) ) $=
|
|
( vw vv ax12v ax12olem4 ax12olem7 ) CABDCBDECBDFCBEFGCADECADFCAEFGH $.
|
|
$}
|
|
|
|
$( Derive ~ ax-12 from ~ ax12v via ~ ax12o . This shows that the weakening
|
|
in ~ ax12v is still sufficient for a complete system. (Contributed by NM,
|
|
21-Dec-2015.) $)
|
|
ax12 $p |- ( -. x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( weq wn wal wi wa sp con3i adantr equtrr equcoms con3rr3 imp nsyl ax12o ex
|
|
sylc pm2.43d ) ABDZEZBCDZUCAFZUBUCUCUDGZUBUCHZUAAFZEZACDZAFZEUEUBUHUCUGUAUA
|
|
AIJKUFUIUJUBUCUIEUCUIUAUIUAGCBCBALMNOUIAIPBCAQSRT $.
|
|
|
|
${
|
|
$d x v w $. $d y v w $.
|
|
$( Lemma for ~ ax10 . Change bound variable. (Contributed by NM,
|
|
22-Jul-2015.) $)
|
|
ax10lem1 $p |- ( A. x x = w -> A. y y = w ) $=
|
|
( vv weq wal ax-8 cbvalivw syl ) ACEZAFDCEZDFBCEZBFJKADADCGHKLDBDBCGHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x z $.
|
|
$( Lemma for ~ ax10 . Change free variable. (Contributed by NM,
|
|
25-Jul-2015.) $)
|
|
ax10lem2 $p |- ( A. x x = y -> A. x x = z ) $=
|
|
( weq wal wn hbe1 equequ2 biimprd con3rr3 19.8a syl6 ax-17 equequ1 notbid
|
|
wex spimeh pm2.61d1 exlimih exnal 3imtr3i con4i ) ACDZAEZABDZAEZUCFZAPUEF
|
|
ZAPZUDFUFFUGUIAUHAGUGCBDZUIUGUJUHUIUJUEUCUJUCUECBAHIJUHAKLUJFZUHACUKAMUCU
|
|
HUKUCUEUJACBNOIQRSUCATUEATUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d w x y $. $d w x z $.
|
|
$( Lemma for ~ ax10 . Similar to ~ ax-10 but with distinct variables.
|
|
(Contributed by NM, 25-Jul-2015.) $)
|
|
ax10lem3 $p |- ( A. x x = y -> A. y y = x ) $=
|
|
( vz vw weq wal ax10lem2 ax10lem1 syl ) ABEAFACEAFZBAEBFZABCGJDAEDFZKJDCE
|
|
DFLADCHDCAGIDBAHII $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ps $. $d x ph $.
|
|
dvelimv.1 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Similar to ~ dvelim with first hypothesis replaced by distinct variable
|
|
condition. (Contributed by NM, 25-Jul-2015.) $)
|
|
dvelimv $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( weq wal wn wi ax-17 alrimih sp a2i alimi syl ax10lem3 con3i hbn1 hban
|
|
a1d syl5ibr ax12o imp a17d hbimd hbald biimpd ax9v con3 al2imi mtoi nsyl2
|
|
wa syl56 expcom ax-11 syl2im pm2.27 syld pm2.61d2 ) CDGZCHIZCEGZCHZBBCHZJ
|
|
ZVEIZVCVGBEDGZAJZEHZVHVCUNZVKCHVFBVIBEHZJZEHVKBVNEBEKZBVMVIVOUALVNVJEVIVM
|
|
AVMAVIBBEMFUBNOPVLVJCEVHVCEVHECGZEHZIZVHEHVQVEECQRVRVHEVPESVEVQCEQRLPVCEK
|
|
TVLVIACVHVCCVDCSVBCSTVHVCVIVICHJEDCUCUDVLACUEUFUGVKBCVKBIZEHZBVKVIBJZEHZV
|
|
TIVJWAEVIABVIABFUHNOWBVTVIIZEHEDUIWAVSWCEVIBUJUKULPVSEKUMOUOUPVEBVDBJZCHZ
|
|
VFVEVDBVMWEVDCMVOBCEUQURVDWDBCVDBUSUKUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w y $.
|
|
$( Quantifier introduction when one pair of variables is distinct.
|
|
(Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) $)
|
|
dveeq2 $p |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $=
|
|
( vw weq equequ2 dvelimv ) CDECBEABDDBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w z y $.
|
|
$( Lemma for ~ ax10 . Change bound variable. (Contributed by NM,
|
|
8-Jul-2016.) $)
|
|
ax10lem4 $p |- ( A. x x = w -> A. y y = x ) $=
|
|
( vz weq wal wn wi ax10lem1 equequ1 dvelimv wb equequ2 sps albidh biimprd
|
|
hba1 syl6 syl7 spsd pm2.43d com12 pm2.18d ) ACEZAFZBAEZBFZUGGZUEUGUHUEUGU
|
|
HUDUEUGHAUEBCEZBFZUHUDUGABCIUHUDUDBFZUJUGHDCEUDBADDACJKUKUGUJUKUFUIBUDBQU
|
|
DUFUILBACBMNOPRSTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d w z $. $d u v w $. $d v x $. $d v y $.
|
|
$( Lemma for ~ ax10 . Change free and bound variables. (Contributed by
|
|
NM, 22-Jul-2015.) $)
|
|
ax10lem5 $p |- ( A. z z = w -> A. y y = x ) $=
|
|
( vv vu weq wal ax10lem1 ax10lem4 syl ) CDGCHZAEGAHZBAGBHLFEGFHZMLEDGEHNC
|
|
EDIEFDJKFAEIKABEJK $.
|
|
$}
|
|
|
|
$( Lemma for ~ ax10 . Similar to ~ ax10o but with reversed antecedent.
|
|
(Contributed by NM, 25-Jul-2015.) $)
|
|
ax10lem6 $p |- ( A. y y = x -> ( A. x ph -> A. y ph ) ) $=
|
|
( weq wal wi ax-11 sps pm2.27 al2imi syld ) CBDZCEABEZLAFZCEZACELMOFCACBGHL
|
|
NACLAIJK $.
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( Derive set.mm's original ~ ax-10 from others. (Contributed by NM,
|
|
25-Jul-2015.) (Revised by NM, 7-Nov-2015.) $)
|
|
ax10 $p |- ( A. x x = y -> A. y y = x ) $=
|
|
( vz weq wal wn ax9v wex df-ex wi wa dveeq2 imp ax10lem6 equcomi ax10lem5
|
|
alimi syl6 syl56 exp3acom23 pm2.18 exlimdv syl5bir mpi ) ABDAEZCADZFCEFZB
|
|
ADBEZCAGUGUFCHUEUHUFCIUEUFUHCUEUFUHFZUHJUHUEUIUFUHUIUFKUFBEZUEACDZAEZUHUI
|
|
UFUJBACLMUEUJUFAEULUFBANUFUKACAOQRABACPSTUHUARUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d w ph $. $d w z $.
|
|
$( Generalization of ~ ax16 . (Contributed by NM, 25-Jul-2015.) $)
|
|
a16g $p |- ( A. x x = y -> ( ph -> A. z ph ) ) $=
|
|
( vw weq wex wal wi a9ev ax10lem5 wn hbn1 pm2.21 alrimih ax-17 ja equcomi
|
|
ax-1 ax-11 syl2im ax-5 syl6 com23 syl5 exlimih mpsyl ) EDFZEGBCFBHUHEHZAA
|
|
DHZIZEDJDEBCKUHUIUKIZEUIUKULEHUILULEUHEMUIUKNOUKULEUKEPUKUISOQUIDEFZDHZUH
|
|
UKEDEDKUHAUNUJUHAUMAIDHZUNUJIUHUMAAEHUOEDRAEPADETUAUMADUBUCUDUEUFUG $.
|
|
$}
|
|
|
|
$( Commutation law for identical variable specifiers. The antecedent and
|
|
consequent are true when ` x ` and ` y ` are substituted with the same
|
|
variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
aecom $p |- ( A. x x = y -> A. y y = x ) $=
|
|
( ax10 ) ABC $.
|
|
|
|
${
|
|
alequcoms.1 $e |- ( A. x x = y -> ph ) $.
|
|
$( A commutation rule for identical variable specifiers. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
aecoms $p |- ( A. y y = x -> ph ) $=
|
|
( weq wal aecom syl ) CBECFBCEBFACBGDH $.
|
|
$}
|
|
|
|
${
|
|
nalequcoms.1 $e |- ( -. A. x x = y -> ph ) $.
|
|
$( A commutation rule for distinct variable specifiers. (Contributed by
|
|
NM, 2-Jan-2002.) $)
|
|
naecoms $p |- ( -. A. y y = x -> ph ) $=
|
|
( weq wal aecom nsyl4 con1i ) ACBECFZBCEBFJABCGDHI $.
|
|
$}
|
|
|
|
${
|
|
$d x v $. $d y v $.
|
|
$( Theorem showing that ~ ax-9 follows from the weaker version ~ ax9v .
|
|
(Even though this theorem depends on ~ ax-9 , all references of ~ ax-9
|
|
are made via ~ ax9v . An earlier version stated ~ ax9v as a separate
|
|
axiom, but having two axioms caused some confusion.)
|
|
|
|
This theorem should be referenced in place of ~ ax-9 so that all proofs
|
|
can be traced back to ~ ax9v . (Contributed by NM, 12-Nov-2013.)
|
|
(Revised by NM, 25-Jul-2015.) $)
|
|
ax9 $p |- -. A. x -. x = y $=
|
|
( vv weq wal wn sp nsyl3 wi ax9v dveeq2 hba1 wb equequ2 syl notbid albidh
|
|
mtbii syl6com con3i alrimiv mt3 pm2.61i ) ABDZAEZUDFZAEZFZUGUDUEUFAGUDAGH
|
|
UEFZUHIZCBDZFZCECBJUJFULCUKUJUIUKUKAEZUHABCKUMACDZFZAEUGACJUMUOUFAUKALUMU
|
|
NUDUMUKUNUDMUKAGCBANOPQRSTUAUBUC $.
|
|
$}
|
|
|
|
$( Show that the original axiom ~ ax-9o can be derived from ~ ax9 and
|
|
others. See ~ ax9from9o for the rederivation of ~ ax9 from ~ ax-9o .
|
|
|
|
Normally, ~ ax9o should be used rather than ~ ax-9o , except by theorems
|
|
specifically studying the latter's properties. (Contributed by NM,
|
|
5-Aug-1993.) (Proof modification is discouraged.) $)
|
|
ax9o $p |- ( A. x ( x = y -> A. x ph ) -> ph ) $=
|
|
( weq wal wi wn ax9 con3 al2imi mtoi ax6o syl ) BCDZABEZFZBEZOGZBEZGAQSNGZB
|
|
EBCHPRTBNOIJKABLM $.
|
|
|
|
$( At least one individual exists. This is not a theorem of free logic,
|
|
which is sound in empty domains. For such a logic, we would add this
|
|
theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the
|
|
system consisting of ~ ax-5 through ~ ax-14 and ~ ax-17 , all axioms other
|
|
than ~ ax9 are believed to be theorems of free logic, although the system
|
|
without ~ ax9 is probably not complete in free logic. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
a9e $p |- E. x x = y $=
|
|
( weq wex wn wal ax9 df-ex mpbir ) ABCZADJEAFEABGJAHI $.
|
|
|
|
$( Show that ~ ax-10o can be derived from ~ ax-10 in the form of ~ ax10 .
|
|
Normally, ~ ax10o should be used rather than ~ ax-10o , except by theorems
|
|
specifically studying the latter's properties. (Contributed by NM,
|
|
16-May-2008.) (Proof modification is discouraged.) $)
|
|
ax10o $p |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $=
|
|
( weq wal wi ax10 ax-11 equcoms sps pm2.27 al2imi sylsyld ) BCDZBECBDZCEABE
|
|
ZOAFZCEZACEBCGNPRFZBSCBACBHIJOQACOAKLM $.
|
|
|
|
$( All variables are effectively bound in an identical variable specifier.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
hbae $p |- ( A. x x = y -> A. z A. x x = y ) $=
|
|
( weq wal wi wn ax12o syl7 ax10o aecoms pm2.43i syl5 pm2.61ii a5i ax-7 syl
|
|
sp ) ABDZAEZSCEZAETCESUAACADCEZCBDCEZTUAFZTSUBGUCGUASARABCHIUDACSACJKUDBCTS
|
|
BEZBCDBEUATUESABJLSBCJMKNOSACPQ $.
|
|
|
|
$( All variables are effectively bound in an identical variable specifier.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfae $p |- F/ z A. x x = y $=
|
|
( weq wal hbae nfi ) ABDAECABCFG $.
|
|
|
|
$( All variables are effectively bound in a distinct variable specifier.
|
|
Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
hbnae $p |- ( -. A. x x = y -> A. z -. A. x x = y ) $=
|
|
( weq wal hbae hbn ) ABDAECABCFG $.
|
|
|
|
$( All variables are effectively bound in a distinct variable specifier.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfnae $p |- F/ z -. A. x x = y $=
|
|
( weq wal nfae nfn ) ABDAECABCFG $.
|
|
|
|
${
|
|
hbnalequs.1 $e |- ( A. z -. A. x x = y -> ph ) $.
|
|
$( Rule that applies ~ hbnae to antecedent. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
hbnaes $p |- ( -. A. x x = y -> ph ) $=
|
|
( weq wal wn hbnae syl ) BCFBGHZKDGABCDIEJ $.
|
|
$}
|
|
|
|
$( A variable is effectively not free in an equality if it is not either of
|
|
the involved variables. ` F/ ` version of ~ ax-12o . (Contributed by
|
|
Mario Carneiro, 6-Oct-2016.) $)
|
|
nfeqf $p |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y ) $=
|
|
( weq wal wn wa nfnae nfan wi ax12o imp nfd ) CADCEFZCBDCEFZGABDZCNOCCACHCB
|
|
CHINOPPCEJABCKLM $.
|
|
|
|
$( Lemma used in proofs of substitution properties. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) $)
|
|
equs4 $p |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) ) $=
|
|
( weq wi wal wa wex a9e 19.29 mpan2 ancl imp eximi syl ) BCDZAEZBFZQPGZBHZP
|
|
AGZBHRPBHTBCIQPBJKSUABQPUAPALMNO $.
|
|
|
|
${
|
|
equsal.1 $e |- F/ x ps $.
|
|
equsal.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A useful equivalence related to substitution. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised
|
|
by Mario Carneiro, 3-Oct-2016.) $)
|
|
equsal $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
|
|
( weq wal 19.3 syl6bbr pm5.74i albii nfri a1d alrimi ax9o impbii bitr4i
|
|
wi ) CDGZASZCHTBCHZSZCHZBUAUCCTAUBTABUBFBCEIJKLBUDBUCCEBUBTBCEMNOBCDPQR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
equsalh.1 $e |- ( ps -> A. x ps ) $.
|
|
equsalh.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A useful equivalence related to substitution. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equsalh $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
|
|
( nfi equsal ) ABCDBCEGFH $.
|
|
$}
|
|
|
|
${
|
|
equsex.1 $e |- F/ x ps $.
|
|
equsex.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A useful equivalence related to substitution. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) $)
|
|
equsex $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
|
|
( weq wn wi wex wal exnal df-an exbii nfn notbid equsal con2bii 3bitr4i
|
|
wa ) CDGZAHZIZHZCJUCCKZHUAATZCJBUCCLUFUDCUAAMNUEBUBBHCDBCEOUAABFPQRS $.
|
|
$}
|
|
|
|
${
|
|
equsexh.1 $e |- ( ps -> A. x ps ) $.
|
|
equsexh.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A useful equivalence related to substitution. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equsexh $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
|
|
( nfi equsex ) ABCDBCEGFH $.
|
|
$}
|
|
|
|
${
|
|
dvelimh.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelimh.2 $e |- ( ps -> A. z ps ) $.
|
|
dvelimh.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Version of ~ dvelim without any variable restrictions. (Contributed by
|
|
NM, 1-Oct-2002.) $)
|
|
dvelimh $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( weq wal wn wi hba1 ax10o aecoms syl5 a1d wa hbnae hban ax12o imp a1i ex
|
|
hbimd hbald pm2.61i equsalh albii 3imtr3g ) CDICJKZEDIZALZEJZUNCJZBBCJCEI
|
|
CJZUKUNUOLZLUPUQUKUNUNEJZUPUOUMEMURUOLECUNECNOPQUPKZUKUQUSUKRZUMCEUSUKECE
|
|
ESCDESTUTULACUSUKCCECSCDCSTUSUKULULCJLEDCUAUBAACJLUTFUCUEUFUDUGABEDGHUHZU
|
|
NBCVAUIUJ $.
|
|
$}
|
|
|
|
${
|
|
dral1.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|
|
(Contributed by NM, 24-Nov-1994.) $)
|
|
dral1 $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $=
|
|
( weq wal hbae biimpd alimdh ax10o syld biimprd wi aecoms impbid ) CDFCGZ
|
|
ACGZBDGZQRBCGSQABCCDCHQABEIJBCDKLQSADGZRQBADCDDHQABEMJTRNDCADCKOLP $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|
|
(Contributed by NM, 27-Feb-2005.) $)
|
|
dral2 $p |- ( A. x x = y -> ( A. z ph <-> A. z ps ) ) $=
|
|
( weq wal hbae albidh ) CDGCHABECDEIFJ $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|
|
(Contributed by NM, 27-Feb-2005.) $)
|
|
drex1 $p |- ( A. x x = y -> ( E. x ph <-> E. y ps ) ) $=
|
|
( weq wal wn wex notbid dral1 df-ex 3bitr4g ) CDFCGZAHZCGZHBHZDGZHACIBDIN
|
|
PROQCDNABEJKJACLBDLM $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|
|
(Contributed by NM, 27-Feb-2005.) $)
|
|
drex2 $p |- ( A. x x = y -> ( E. z ph <-> E. z ps ) ) $=
|
|
( weq wal hbae exbidh ) CDGCHABECDEIFJ $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
(Contributed by Mario Carneiro, 4-Oct-2016.) $)
|
|
drnf1 $p |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) $=
|
|
( weq wal wi wnf dral1 imbi12d df-nf 3bitr4g ) CDFCGZAACGZHZCGBBDGZHZDGAC
|
|
IBDIPRCDNABOQEABCDEJKJACLBDLM $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
(Contributed by Mario Carneiro, 4-Oct-2016.) $)
|
|
drnf2 $p |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) ) $=
|
|
( weq wal wi wnf dral2 imbi12d df-nf 3bitr4g ) CDGCHZAAEHZIZEHBBEHZIZEHAE
|
|
JBEJQSCDEOABPRFABCDEFKLKAEMBEMN $.
|
|
$}
|
|
|
|
${
|
|
exdistrf.1 $e |- ( -. A. x x = y -> F/ y ph ) $.
|
|
$( Distribution of existential quantifiers, with a bound-variable
|
|
hypothesis saying that ` y ` is not free in ` ph ` , but ` x ` can be
|
|
free in ` ph ` (and there is no distinct variable condition on ` x ` and
|
|
` y ` ). (Contributed by Mario Carneiro, 20-Mar-2013.) $)
|
|
exdistrf $p |- ( E. x E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) ) $=
|
|
( weq wal wa wex wi biidd drex1 drex2 nfe1 19.9 19.8a eximi sylbi syl6bir
|
|
anim2i wn nfnae 19.40 19.9d anim1d syl5 eximd pm2.61i ) CDFCGZABHZDIZCIZA
|
|
BDIZHZCIZJUIULUJCIZCIZUOUPUKCDCUJUJCDUIUJKLMUQUPUOUPCUJCNOUJUNCBUMABDPTQR
|
|
SUIUAZUKUNCCDCUBUKADIZUMHURUNABDUCURUSAUMAURDEUDUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
nfald2.1 $e |- F/ y ph $.
|
|
nfald2.2 $e |- ( ( ph /\ -. A. x x = y ) -> F/ x ps ) $.
|
|
$( Variation on ~ nfald which adds the hypothesis that ` x ` and ` y ` are
|
|
distinct in the inner subproof. (Contributed by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
nfald2 $p |- ( ph -> F/ x A. y ps ) $=
|
|
( weq wal wnf wn wa nfnae nfan nfald ex nfa1 biidd drnf1 mpbiri pm2.61d2
|
|
) ACDGCHZBDHZCIZAUAJZUCAUDKBCDAUDDECDDLMFNOUAUCUBDIBDPUBUBCDUAUBQRST $.
|
|
|
|
$( Variation on ~ nfexd which adds the hypothesis that ` x ` and ` y ` are
|
|
distinct in the inner subproof. (Contributed by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
nfexd2 $p |- ( ph -> F/ x E. y ps ) $=
|
|
( wex wn wal df-ex weq wa nfnd nfald2 nfxfrd ) BDGBHZDIZHACBDJAQCAPCDEACD
|
|
KCIHLBCFMNMO $.
|
|
$}
|
|
|
|
$( Closed theorem form of ~ spim . (Contributed by NM, 15-Jan-2008.)
|
|
(Revised by Mario Carneiro, 17-Oct-2016.) $)
|
|
spimt $p |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) ->
|
|
( A. x ph -> ps ) ) $=
|
|
( wnf weq wi wal wa nfnf1 nfa1 sp adantl nfr adantr embantd imim2d impancom
|
|
nfan alimd ax9o syl6 ) BCEZCDFZABGZGZCHZIACHZUDBCHZGZCHZBUCUHUGUKUCUHIZUFUJ
|
|
CUCUHCBCJACKSULUEUIUDULABUIUHAUCACLMUCBUIGUHBCNOPQTRBCDUAUB $.
|
|
|
|
${
|
|
spim.1 $e |- F/ x ps $.
|
|
spim.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Specialization, using implicit substitution. Compare Lemma 14 of
|
|
[Tarski] p. 70. The ~ spim series of theorems requires that only one
|
|
direction of the substitution hypothesis hold. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) $)
|
|
spim $p |- ( A. x ph -> ps ) $=
|
|
( wnf weq wi wal ax-gen spimt mp2an ) BCGCDHABIIZCJACJBIENCFKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
spime.1 $e |- F/ x ph $.
|
|
spime.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Existential introduction, using implicit substitution. Compare Lemma 14
|
|
of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario
|
|
Carneiro, 3-Oct-2016.) $)
|
|
spime $p |- ( ph -> E. x ps ) $=
|
|
( wn wal wex nfn weq con3d spim con2i df-ex sylibr ) ABGZCHZGBCIRAQAGCDAC
|
|
EJCDKABFLMNBCOP $.
|
|
$}
|
|
|
|
${
|
|
spimed.1 $e |- ( ch -> F/ x ph ) $.
|
|
spimed.2 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Deduction version of ~ spime . (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 3-Oct-2016.) $)
|
|
spimed $p |- ( ch -> ( ph -> E. x ps ) ) $=
|
|
( wnf wex wi wa nfnf1 id nfan1 weq adantld spime ex syl ) CADHZABDIZJFTAU
|
|
ATAKBDETADADLTMNDEOABTGPQRS $.
|
|
$}
|
|
|
|
${
|
|
cbv1h.1 $e |- ( ph -> ( ps -> A. y ps ) ) $.
|
|
cbv1h.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
cbv1h.3 $e |- ( ph -> ( x = y -> ( ps -> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbv1h $p |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) ) $=
|
|
( wal wi sps al2imi ax-7 syl6 weq com23 syl6d ax9o a7s syld ) AEIZDIZBDIZ
|
|
UCEIZCEIZUBUCBEIZDIUDUABUFDABUFJEFKLBDEMNAUDUEJEDADIZUCCEUGUCDEOZCDIZJZDI
|
|
CABUJDABUHCUIAUHBCHPGQLCDERNLST $.
|
|
$}
|
|
|
|
${
|
|
cbv1.1 $e |- ( ph -> F/ y ps ) $.
|
|
cbv1.2 $e |- ( ph -> F/ x ch ) $.
|
|
cbv1.3 $e |- ( ph -> ( x = y -> ( ps -> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
3-Oct-2016.) $)
|
|
cbv1 $p |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) ) $=
|
|
( nfrd cbv1h ) ABCDEABEFIACDGIHJ $.
|
|
$}
|
|
|
|
${
|
|
cbv2h.1 $e |- ( ph -> ( ps -> A. y ps ) ) $.
|
|
cbv2h.2 $e |- ( ph -> ( ch -> A. x ch ) ) $.
|
|
cbv2h.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbv2h $p |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) ) $=
|
|
( wal weq wb wi bi1 syl6 cbv1h equcomi bi2 syl56 a7s impbid ) AEIDIBDIZCE
|
|
IZABCDEFGADEJZBCKZBCLHBCMNOAUBUALEDACBEDGFEDJUCAUDCBLEDPHBCQROST $.
|
|
$}
|
|
|
|
${
|
|
cbv2.1 $e |- ( ph -> F/ y ps ) $.
|
|
cbv2.2 $e |- ( ph -> F/ x ch ) $.
|
|
cbv2.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
3-Oct-2016.) $)
|
|
cbv2 $p |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) ) $=
|
|
( nfrd cbv2h ) ABCDEABEFIACDGIHJ $.
|
|
$}
|
|
|
|
${
|
|
cbv3.1 $e |- F/ y ph $.
|
|
cbv3.2 $e |- F/ x ps $.
|
|
cbv3.3 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution, that
|
|
does not use ~ ax-12o . (Contributed by NM, 5-Aug-1993.) $)
|
|
cbv3 $p |- ( A. x ph -> A. y ps ) $=
|
|
( wtru wal wi wnf a1i weq cbv1 tru ax-gen mpg ) HDIACIBDIJCHABCDADKHELBCK
|
|
HFLCDMABJJHGLNHDOPQ $.
|
|
$}
|
|
|
|
${
|
|
cbv3h.1 $e |- ( ph -> A. y ph ) $.
|
|
cbv3h.2 $e |- ( ps -> A. x ps ) $.
|
|
cbv3h.3 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) (Proof modification is discouraged.) $)
|
|
cbv3h $p |- ( A. x ph -> A. y ps ) $=
|
|
( weq wal wi a1i cbv1h stdpc6 mpg ) DDHZDIACIBDIJCOABCDAADIJOEKBBCIJOFKCD
|
|
HABJJOGKLDMN $.
|
|
$}
|
|
|
|
${
|
|
cbval.1 $e |- F/ y ph $.
|
|
cbval.2 $e |- F/ x ps $.
|
|
cbval.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
3-Oct-2016.) $)
|
|
cbval $p |- ( A. x ph <-> A. y ps ) $=
|
|
( wal weq biimpd cbv3 wi biimprd equcoms impbii ) ACHBDHABCDEFCDIZABGJKBA
|
|
DCFEBALCDPABGMNKO $.
|
|
$}
|
|
|
|
${
|
|
cbvex.1 $e |- F/ y ph $.
|
|
cbvex.2 $e |- F/ x ps $.
|
|
cbvex.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbvex $p |- ( E. x ph <-> E. y ps ) $=
|
|
( wn wal wex nfn weq notbid cbval notbii df-ex 3bitr4i ) AHZCIZHBHZDIZHAC
|
|
JBDJSUARTCDADEKBCFKCDLABGMNOACPBDPQ $.
|
|
$}
|
|
|
|
${
|
|
chvar.1 $e |- F/ x ps $.
|
|
chvar.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
chvar.3 $e |- ph $.
|
|
$( Implicit substitution of ` y ` for ` x ` into a theorem. (Contributed
|
|
by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro,
|
|
3-Oct-2016.) $)
|
|
chvar $p |- ps $=
|
|
( weq biimpd spim mpg ) ABCABCDECDHABFIJGK $.
|
|
$}
|
|
|
|
$( A variable introduction law for equality. Lemma 15 of [Monk2] p. 109,
|
|
however we do not require ` z ` to be distinct from ` x ` and ` y `
|
|
(making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 25-May-2011.) $)
|
|
equvini $p |- ( x = y -> E. z ( x = z /\ z = y ) ) $=
|
|
( weq wal wa wex wi equcomi alimi a9e jctir a1d 19.29 syl6 eximi ax-mp 2a1i
|
|
anc2ri wn 19.29r wo ioran nfeqf ax-8 anc2li equcoms spimed sylbi ecase3 ) C
|
|
ADZCEZCBDZCEZABDZACDZUMFZCGZHZULUOUPCEZUMCGZFZURULVBUOULUTVAUKUPCCAIZJCBKLM
|
|
UPUMCNOUNUOUPCGZUNFURUNUOVDUNUOVDUKCGVDCAKUKUPCVCPQRSUPUMCUAOULUNUBTULTUNTF
|
|
ZUSULUNUCUOUQVECAABCUDUOUQHACUPUOUMACBUEUFUGUHUIUJ $.
|
|
|
|
$( A variable elimination law for equality with no distinct variable
|
|
requirements. (Compare ~ equvini .) (Contributed by NM, 1-Mar-2013.)
|
|
(Proof shortened by Mario Carneiro, 17-Oct-2016.) $)
|
|
equveli $p |- ( A. z ( z = x <-> z = y ) -> x = y ) $=
|
|
( weq wb wal wi wa albiim equequ1 imbi12d sps dral1 equid sp equcomi syl6bi
|
|
mpi wn pm2.61i syl adantld dral2 a1bi biimpri a1d wnf nfeqf equtr ax-8 mpii
|
|
imim12d ax-gen spimt sylancl ex adantrd sylbi ) CADZCBDZECFUSUTGZCFZUTUSGZC
|
|
FZHZABDZUSUTCIUTCFZVEVFGVGVDVFVBVGVDBBDZBADZGZBFZVFVCVJCBUTVCVJECUTUTVHUSVI
|
|
CBBJCBAJKLMVKVIVFVKVHVIBNVJBORBAPUAQUBVGSZVBVFVDUSCFZVLVBVFGZGVMVNVLVMVBAAD
|
|
ZVFGZCFVFVAVPCACUSVAVPECUSUSVOUTVFCAAJCABJKLUCVPVFCVFVPVOVFANZUDUELQUFVMSZV
|
|
LVNVRVLHVFCUGUSVAVFGGZCFVNABCUHVSCUSVAVOVFVQUSVOUSUTVFCAAUICABUJULUKUMVAVFC
|
|
AUNUOUPTUQTUR $.
|
|
|
|
${
|
|
equs45f.1 $e |- F/ y ph $.
|
|
$( Two ways of expressing substitution when ` y ` is not free in ` ph ` .
|
|
(Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) $)
|
|
equs45f $p |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) $=
|
|
( weq wa wex wi wal nfri anim2i eximi equs5a syl equs4 impbii ) BCEZAFZBG
|
|
ZQAHBIZSQACIZFZBGTRUBBAUAQACDJKLABCMNABCOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
spimv.1 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( A version of ~ spim with a distinct variable requirement instead of a
|
|
bound variable hypothesis. (Contributed by NM, 5-Aug-1993.) $)
|
|
spimv $p |- ( A. x ph -> ps ) $=
|
|
( nfv spim ) ABCDBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d u v $. $d x y $. $d u w $.
|
|
$( A "distinctor elimination" lemma with no restrictions on variables in
|
|
the consequent. (Contributed by NM, 8-Nov-2006.) $)
|
|
aev $p |- ( A. x x = y -> A. z w = v ) $=
|
|
( vu weq wal hbae ax10lem5 ax-8 spimv syl alrimih ) ABGAHZDEGZCABCIOFEGZF
|
|
HPEFABJQPFDFDEKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
ax11v2.1 $e |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) $.
|
|
$( Recovery of ~ ax-11o from ~ ax11v . This proof uses ~ ax-10 and
|
|
~ ax-11 . TODO: figure out if this is useful, or if it should be
|
|
simplified or eliminated. (Contributed by NM, 2-Feb-2007.) $)
|
|
ax11v2 $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( weq wal wn wex wi a9ev wa wb equequ2 adantl dveeq2 imp nfa1 imbi1d sps
|
|
albid syl imbi2d imbi12d mpbii ex exlimdv mpi ) BCFZBGHZDCFZDIUIAUIAJZBGZ
|
|
JZJZDCKUJUKUODUJUKUOUJUKLZBDFZAUQAJZBGZJZJUOEUPUQUIUTUNUKUQUIMUJDCBNZOUPU
|
|
SUMAUPUKBGZUSUMMUJUKVBBCDPQVBURULBUKBRUKURULMBUKUQUIAVASTUAUBUCUDUEUFUGUH
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
ax11a2.1 $e |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) ) $.
|
|
$( Derive ~ ax-11o from a hypothesis in the form of ~ ax-11 . ~ ax-10 and
|
|
~ ax-11 are used by the proof, but not ~ ax-10o or ~ ax-11o . TODO:
|
|
figure out if this is useful, or if it should be simplified or
|
|
eliminated. (Contributed by NM, 2-Feb-2007.) $)
|
|
ax11a2 $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( wal weq wi ax-17 syl5 ax11v2 ) ABCDAADFBDGZLAHBFADIEJK $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( Derivation of set.mm's original ~ ax-11o from ~ ax-10 and the shorter
|
|
~ ax-11 that has replaced it.
|
|
|
|
An open problem is whether this theorem can be proved without relying on
|
|
~ ax-16 or ~ ax-17 (given all of the original and new versions of ~ sp
|
|
through ~ ax-15 ).
|
|
|
|
Another open problem is whether this theorem can be proved without
|
|
relying on ~ ax12o .
|
|
|
|
Theorem ~ ax11 shows the reverse derivation of ~ ax-11 from ~ ax-11o .
|
|
|
|
Normally, ~ ax11o should be used rather than ~ ax-11o , except by
|
|
theorems specifically studying the latter's properties. (Contributed by
|
|
NM, 3-Feb-2007.) $)
|
|
ax11o $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( vz ax-11 ax11a2 ) ABCDABDEF $.
|
|
$}
|
|
|
|
$( A bidirectional version of ~ ax11o . (Contributed by NM, 30-Jun-2006.) $)
|
|
ax11b $p |- ( ( -. A. x x = y /\ x = y ) ->
|
|
( ph <-> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wn wa wi ax11o imp sp com12 adantl impbid ) BCDZBEFZOGAOAHZBEZPOA
|
|
RHABCIJORAHPROAQBKLMN $.
|
|
|
|
$( Lemma used in proofs of substitution properties. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equs5 $p |- ( -. A. x x = y ->
|
|
( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wn wa wi nfnae nfa1 ax11o imp3a exlimd ) BCDZBEFZNAGNAHZBEZBBCBIP
|
|
BJONAQABCKLM $.
|
|
|
|
${
|
|
dvelimf.1 $e |- F/ x ph $.
|
|
dvelimf.2 $e |- F/ z ps $.
|
|
dvelimf.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Version of ~ dvelimv without any variable restrictions. (Contributed by
|
|
NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
dvelimf $p |- ( -. A. x x = y -> F/ x ps ) $=
|
|
( weq wi wal wn equsal bicomi nfnae wa nfan ax12o impcom nfd nfimd nfald2
|
|
wnf a1i nfxfrd ) BEDIZAJZEKZCDICKLZCUHBABEDGHMNUIUGCECDEOUICEICKLZPZUFACU
|
|
KUFCUIUJCCDCOCECOQUJUIUFUFCKJEDCRSTACUCUKFUDUAUBUE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
spv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Specialization, using implicit substitution. (Contributed by NM,
|
|
30-Aug-1993.) $)
|
|
spv $p |- ( A. x ph -> ps ) $=
|
|
( weq biimpd spimv ) ABCDCDFABEGH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
spimev.1 $e |- ( x = y -> ( ph -> ps ) ) $.
|
|
$( Distinct-variable version of ~ spime . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
spimev $p |- ( ph -> E. x ps ) $=
|
|
( nfv spime ) ABCDACFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
speiv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
speiv.2 $e |- ps $.
|
|
$( Inference from existential specialization, using implicit substitution.
|
|
(Contributed by NM, 19-Aug-1993.) $)
|
|
speiv $p |- E. x ph $=
|
|
( wex weq biimprd spimev ax-mp ) BACGFBACDCDHABEIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( A variable introduction law for equality. Lemma 15 of [Monk2] p. 109.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
equvin $p |- ( x = y <-> E. z ( x = z /\ z = y ) ) $=
|
|
( weq wa wex equvini equtr imp exlimiv impbii ) ABDZACDZCBDZEZCFABCGOLCMN
|
|
LACBHIJK $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $.
|
|
cbvalv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbvalv $p |- ( A. x ph <-> A. y ps ) $=
|
|
( nfv cbval ) ABCDADFBCFEG $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
cbvexv $p |- ( E. x ph <-> E. y ps ) $=
|
|
( nfv cbvex ) ABCDADFBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y z $. $d w x $. $d w z $.
|
|
cbval2.1 $e |- F/ z ph $.
|
|
cbval2.2 $e |- F/ w ph $.
|
|
cbval2.3 $e |- F/ x ps $.
|
|
cbval2.4 $e |- F/ y ps $.
|
|
cbval2.5 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro,
|
|
6-Oct-2016.) $)
|
|
cbval2 $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
|
|
( wal nfal weq wb wa nfv nfan cbval 19.28v wi expcom pm5.32d pm5.32 mpbir
|
|
3bitr3i ) ADLZBFLZCEAEDGMBCFIMCENZUGUHOUAUIUGPZUIUHPZOUIAPZDLUIBPZFLUJUKU
|
|
LUMDFUIAFUIFQHRUIBDUIDQJRDFNZUIABUIUNABOKUBUCSUIADTUIBFTUFUIUGUHUDUES $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro,
|
|
6-Oct-2016.) $)
|
|
cbvex2 $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
|
|
( wex nfex weq wb wa nfv nfan cbvex 19.42v wi expcom pm5.32d pm5.32 mpbir
|
|
3bitr3i ) ADLZBFLZCEAEDGMBCFIMCENZUGUHOUAUIUGPZUIUHPZOUIAPZDLUIBPZFLUJUKU
|
|
LUMDFUIAFUIFQHRUIBDUIDQJRDFNZUIABUIUNABOKUBUCSUIADTUIBFTUFUIUGUHUDUES $.
|
|
$}
|
|
|
|
${
|
|
$d z w ph $. $d x y ps $. $d x w $. $d z y $.
|
|
cbval2v.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 4-Feb-2005.) $)
|
|
cbval2v $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
|
|
( nfv cbval2 ) ABCDEFAEHAFHBCHBDHGI $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 26-Jul-1995.) $)
|
|
cbvex2v $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
|
|
( nfv cbvex2 ) ABCDEFAEHAFHBCHBDHGI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
cbvald.1 $e |- F/ y ph $.
|
|
cbvald.2 $e |- ( ph -> F/ y ps ) $.
|
|
cbvald.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction used to change bound variables, using implicit substitution,
|
|
particularly useful in conjunction with ~ dvelim . (Contributed by NM,
|
|
2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
cbvald $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
|
|
( wal wb nfri alrimiv nfvd cbv2 syl ) AAEIZDIBDICEIJAPDAEFKLABCDEGACDMHNO
|
|
$.
|
|
|
|
$( Deduction used to change bound variables, using implicit substitution,
|
|
particularly useful in conjunction with ~ dvelim . (Contributed by NM,
|
|
2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
cbvexd $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
|
|
( wn wal wex nfnd weq wb notbi syl6ib cbvald notbid df-ex 3bitr4g ) ABIZD
|
|
JZICIZEJZIBDKCEKAUBUDAUAUCDEFABEGLADEMBCNUAUCNHBCOPQRBDSCEST $.
|
|
$}
|
|
|
|
${
|
|
$d ps y $. $d ch x $. $d ph x $. $d ph y $.
|
|
cbvaldva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
|
|
$( Rule used to change the bound variable in a universal quantifier with
|
|
implicit substitution. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
cbvaldva $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
|
|
( nfv nfvd weq wb ex cbvald ) ABCDEAEGABEHADEIBCJFKL $.
|
|
|
|
$( Rule used to change the bound variable in an existential quantifier with
|
|
implicit substitution. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
cbvexdva $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
|
|
( nfv nfvd weq wb ex cbvexd ) ABCDEAEGABEHADEIBCJFKL $.
|
|
$}
|
|
|
|
${
|
|
$v f $.
|
|
$v g $.
|
|
$( Define temporary individual variables. $)
|
|
cbvex4v.vf $f setvar f $.
|
|
cbvex4v.vg $f setvar g $.
|
|
$d w z ch $. $d u v ph $. $d x y ps $. $d f g ps $. $d f w $.
|
|
$d g z $. $d u v w x y z $.
|
|
cbvex4v.1 $e |- ( ( x = v /\ y = u ) -> ( ph <-> ps ) ) $.
|
|
cbvex4v.2 $e |- ( ( z = f /\ w = g ) -> ( ps <-> ch ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 26-Jul-1995.) $)
|
|
cbvex4v $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $=
|
|
( wex weq wa 2exbidv cbvex2v 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZINH
|
|
NUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
chv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
chv.2 $e |- ph $.
|
|
$( Implicit substitution of ` y ` for ` x ` into a theorem. (Contributed
|
|
by NM, 20-Apr-1994.) $)
|
|
chvarv $p |- ps $=
|
|
( spv mpg ) ABCABCDEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( When the class variables in definition ~ df-clel are replaced with
|
|
setvar variables, this theorem of predicate calculus is the result.
|
|
This theorem provides part of the justification for the consistency of
|
|
that definition, which "overloads" the setvar variables in ~ wel with
|
|
the class variables in ~ wcel . Note: This proof is referenced on the
|
|
Metamath Proof Explorer Home Page and shouldn't be changed.
|
|
(Contributed by NM, 28-Jan-2004.)
|
|
(Proof modification is discouraged.) $)
|
|
cleljust $p |- ( x e. y <-> E. z ( z = x /\ z e. y ) ) $=
|
|
( weq wel wa wex ax-17 elequ1 equsexh bicomi ) CADCBEZFCGABEZLMCAMCHCABIJ
|
|
K $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( When the class variables in definition ~ df-clel are replaced with
|
|
setvar variables, this theorem of predicate calculus is the result.
|
|
This theorem provides part of the justification for the consistency of
|
|
that definition, which "overloads" the setvar variables in ~ wel with
|
|
the class variables in ~ wcel . (Contributed by NM, 28-Jan-2004.)
|
|
(Revised by Mario Carneiro, 21-Dec-2016.) $)
|
|
cleljustALT $p |- ( x e. y <-> E. z ( z = x /\ z e. y ) ) $=
|
|
( weq wel wa wex nfv elequ1 equsex bicomi ) CADCBEZFCGABEZLMCAMCHCABIJK
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d z ps $.
|
|
dvelim.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelim.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( This theorem can be used to eliminate a distinct variable restriction on
|
|
` x ` and ` z ` and replace it with the "distinctor" ` -. A. x x = y `
|
|
as an antecedent. ` ph ` normally has ` z ` free and can be read
|
|
` ph ( z ) ` , and ` ps ` substitutes ` y ` for ` z ` and can be read
|
|
` ph ( y ) ` . We don't require that ` x ` and ` y ` be distinct: if
|
|
they aren't, the distinctor will become false (in multiple-element
|
|
domains of discourse) and "protect" the consequent.
|
|
|
|
To obtain a closed-theorem form of this inference, prefix the hypotheses
|
|
with ` A. x A. z ` , conjoin them, and apply ~ dvelimdf .
|
|
|
|
Other variants of this theorem are ~ dvelimh (with no distinct variable
|
|
restrictions), ~ dvelimhw (that avoids ~ ax-12 ), and ~ dvelimALT (that
|
|
avoids ~ ax-10 ). (Contributed by NM, 23-Nov-1994.) $)
|
|
dvelim $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( ax-17 dvelimh ) ABCDEFBEHGI $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $.
|
|
dvelimnf.1 $e |- F/ x ph $.
|
|
dvelimnf.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Version of ~ dvelim using "not free" notation. (Contributed by Mario
|
|
Carneiro, 9-Oct-2016.) $)
|
|
dvelimnf $p |- ( -. A. x x = y -> F/ x ps ) $=
|
|
( nfv dvelimf ) ABCDEFBEHGI $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w y $.
|
|
$( Quantifier introduction when one pair of variables is distinct.
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
dveeq1 $p |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( vw weq equequ1 dvelimv ) DCEBCEABDDBCFG $.
|
|
|
|
$( Quantifier introduction when one pair of variables is distinct.
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
dveel1 $p |- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) ) $=
|
|
( vw wel elequ1 dvelimv ) DCEBCEABDDBCFG $.
|
|
|
|
$( Quantifier introduction when one pair of variables is distinct.
|
|
(Contributed by NM, 2-Jan-2002.) $)
|
|
dveel2 $p |- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) ) $=
|
|
( vw wel elequ2 dvelimv ) CDECBEABDDBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d w y $. $d w z $. $d w x $. $( ` w ` is dummy. $)
|
|
$( Axiom ~ ax-15 is redundant if we assume ~ ax-17 . Remark 9.6 in
|
|
[Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.
|
|
|
|
Note that ` w ` is a dummy variable introduced in the proof. On the web
|
|
page, it is implicitly assumed to be distinct from all other variables.
|
|
(This is made explicit in the database file set.mm). Its purpose is to
|
|
satisfy the distinct variable requirements of ~ dveel2 and ~ ax-17 . By
|
|
the end of the proof it has vanished, and the final theorem has no
|
|
distinct variable requirements. (Contributed by NM, 29-Jun-1995.)
|
|
(Proof modification is discouraged.) $)
|
|
ax15 $p |- ( -. A. z z = x -> ( -. A. z z = y ->
|
|
( x e. y -> A. z x e. y ) ) ) $=
|
|
( vw weq wal wn wel hbn1 dveel2 hbim1 elequ1 imbi2d dvelim nfa1 nfn 19.21
|
|
wi syl6ib pm2.86d ) CAECFGZCBEZCFZGZABHZUECFZUAUDUERZUGCFUDUFRUDDBHZRUGCA
|
|
DUDUHCUBCICBDJKDAEUHUEUDDABLMNUDUECUCCUBCOPQST $.
|
|
$}
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
drsb1 $p |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) ) $=
|
|
( weq wal wi wa wex wsb wb equequ1 sps imbi1d anbi1d drex1 anbi12d 3bitr4g
|
|
df-sb ) BCEZBFZBDEZAGZUBAHZBIZHCDEZAGZUFAHZCIZHABDJACDJUAUCUGUEUIUAUBUFATUB
|
|
UFKBBCDLMZNUDUHBCUAUBUFAUJOPQABDSACDSR $.
|
|
|
|
$( One direction of a simplified definition of substitution. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sb2 $p |- ( A. x ( x = y -> ph ) -> [ y / x ] ph ) $=
|
|
( weq wi wal wa wex wsb sp equs4 df-sb sylanbrc ) BCDZAEZBFONAGBHABCIOBJABC
|
|
KABCLM $.
|
|
|
|
$( The specialization axiom of standard predicate calculus. It states that
|
|
if a statement ` ph ` holds for all ` x ` , then it also holds for the
|
|
specific case of ` y ` (properly) substituted for ` x ` . Translated to
|
|
traditional notation, it can be read: " ` A. x ph ( x ) -> ph ( y ) ` ,
|
|
provided that ` y ` is free for ` x ` in ` ph ( x ) ` ." Axiom 4 of
|
|
[Mendelson] p. 69. See also ~ spsbc and ~ rspsbc . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
stdpc4 $p |- ( A. x ph -> [ y / x ] ph ) $=
|
|
( wal weq wi wsb ax-1 alimi sb2 syl ) ABDBCEZAFZBDABCGAMBALHIABCJK $.
|
|
|
|
$( Substitution has no effect on a non-free variable. (Contributed by NM,
|
|
30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
sbft $p |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) ) $=
|
|
( wnf wsb weq wa wex sb1 wal simpr ax-gen 19.23t mpbii syl5 nfr stdpc4 syl6
|
|
wi impbid ) ABDZABCEZAUBBCFZAGZBHZUAAABCIUAUDASZBJUEASUFBUCAKLUDABMNOUAAABJ
|
|
UBABPABCQRT $.
|
|
|
|
${
|
|
sbf.1 $e |- F/ x ph $.
|
|
$( Substitution for a variable not free in a wff does not affect it.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) $)
|
|
sbf $p |- ( [ y / x ] ph <-> ph ) $=
|
|
( wnf wsb wb sbft ax-mp ) ABEABCFAGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
sbh.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Substitution for a variable not free in a wff does not affect it.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sbh $p |- ( [ y / x ] ph <-> ph ) $=
|
|
( nfi sbf ) ABCABDEF $.
|
|
$}
|
|
|
|
$( Substitution has no effect on a bound variable. (Contributed by NM,
|
|
1-Jul-2005.) $)
|
|
sbf2 $p |- ( [ y / x ] A. x ph <-> A. x ph ) $=
|
|
( wal nfa1 sbf ) ABDBCABEF $.
|
|
|
|
${
|
|
sb6x.1 $e |- F/ x ph $.
|
|
$( Equivalence involving substitution for a variable not free.
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) $)
|
|
sb6x $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
|
|
( wsb weq wi wal sbf biidd equsal bitr4i ) ABCEABCFZAGBHABCDIAABCDMAJKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nfs1f.1 $e |- F/ x ph $.
|
|
$( If ` x ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfs1f $p |- F/ x [ y / x ] ph $=
|
|
( wsb sbf nfxfr ) ABCEABABCDFDG $.
|
|
$}
|
|
|
|
$( Substitution does not change an identical variable specifier.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sbequ5 $p |- ( [ w / z ] A. x x = y <-> A. x x = y ) $=
|
|
( weq wal nfae sbf ) ABEAFCDABCGH $.
|
|
|
|
$( Substitution does not change a distinctor. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sbequ6 $p |- ( [ w / z ] -. A. x x = y <-> -. A. x x = y ) $=
|
|
( weq wal wn nfnae sbf ) ABEAFGCDABCHI $.
|
|
|
|
${
|
|
sbt.1 $e |- ph $.
|
|
$( A substitution into a theorem remains true. (See ~ chvar and ~ chvarv
|
|
for versions using implicit substitution.) (Contributed by NM,
|
|
21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sbt $p |- [ y / x ] ph $=
|
|
( wsb nfth sbf mpbir ) ABCEADABCABDFGH $.
|
|
$}
|
|
|
|
$( Substitution applied to an atomic wff. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equsb1 $p |- [ y / x ] x = y $=
|
|
( weq wi wsb sb2 id mpg ) ABCZIDIABEAIABFIGH $.
|
|
|
|
$( Substitution applied to an atomic wff. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
equsb2 $p |- [ y / x ] y = x $=
|
|
( weq wi wsb sb2 equcomi mpg ) ABCBACZDIABEAIABFABGH $.
|
|
|
|
${
|
|
sbied.1 $e |- F/ x ph $.
|
|
sbied.2 $e |- ( ph -> F/ x ch ) $.
|
|
sbied.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
|
|
$( Conversion of implicit substitution to explicit substitution (deduction
|
|
version of ~ sbie ). (Contributed by NM, 30-Jun-1994.) (Revised by
|
|
Mario Carneiro, 4-Oct-2016.) $)
|
|
sbied $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
|
|
( wsb wex weq wa sb1 wb wi bi1 syl6 imp3a syld wal eximd syl5 19.9d com23
|
|
nfrd bi2 alimd sb2 impbid ) ABDEIZCAUJCDJZCUJDEKZBLZDJAUKBDEMAUMCDFAULBCA
|
|
ULBCNZBCOHBCPQRUAUBCADGUCSACCDTZUJACDGUEAUOULBOZDTUJACUPDFAULCBAULUNCBOHB
|
|
CUFQUDUGBDEUHQSUI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
sbiedv.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
|
|
$( Conversion of implicit substitution to explicit substitution (deduction
|
|
version of ~ sbie ). (Contributed by NM, 7-Jan-2017.) $)
|
|
sbiedv $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
|
|
( nfv nfvd weq wb ex sbied ) ABCDEADGACDHADEIBCJFKL $.
|
|
$}
|
|
|
|
${
|
|
sbie.1 $e |- F/ x ps $.
|
|
sbie.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit substitution.
|
|
(Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) $)
|
|
sbie $p |- ( [ y / x ] ph <-> ps ) $=
|
|
( wsb wb wtru nftru wnf a1i weq wi sbied trud ) ACDGBHIABCDCJBCKIELCDMABH
|
|
NIFLOP $.
|
|
$}
|
|
|
|
${
|
|
sb6f.1 $e |- F/ y ph $.
|
|
$( Equivalence for substitution when ` y ` is not free in ` ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) $)
|
|
sb6f $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
|
|
( wsb weq wi wal nfri sbimi sb4a syl sb2 impbii ) ABCEZBCFAGBHZOACHZBCEPA
|
|
QBCACDIJABCKLABCMN $.
|
|
|
|
$( Equivalence for substitution when ` y ` is not free in ` ph ` .
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) $)
|
|
sb5f $p |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) $=
|
|
( wsb weq wi wal wa wex sb6f equs45f bitr4i ) ABCEBCFZAGBHNAIBJABCDKABCDL
|
|
M $.
|
|
$}
|
|
|
|
$( Special case of a bound-variable hypothesis builder for substitution.
|
|
(Contributed by NM, 2-Feb-2007.) $)
|
|
hbsb2a $p |- ( [ y / x ] A. y ph -> A. x [ y / x ] ph ) $=
|
|
( wal wsb weq wi sb4a sb2 a5i syl ) ACDBCEBCFAGZBDABCEZBDABCHLMBABCIJK $.
|
|
|
|
$( Special case of a bound-variable hypothesis builder for substitution.
|
|
(Contributed by NM, 2-Feb-2007.) $)
|
|
hbsb2e $p |- ( [ y / x ] ph -> A. x [ y / x ] E. y ph ) $=
|
|
( wsb weq wex wi wal sb4e sb2 a5i syl ) ABCDBCEACFZGZBHMBCDZBHABCINOBMBCJKL
|
|
$.
|
|
|
|
${
|
|
hbsb3.1 $e |- ( ph -> A. y ph ) $.
|
|
$( If ` y ` is not free in ` ph ` , ` x ` is not free in
|
|
` [ y / x ] ph ` . (Contributed by NM, 5-Aug-1993.) $)
|
|
hbsb3 $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
|
|
( wsb wal sbimi hbsb2a syl ) ABCEZACFZBCEJBFAKBCDGABCHI $.
|
|
$}
|
|
|
|
${
|
|
nfs1.1 $e |- F/ y ph $.
|
|
$( If ` y ` is not free in ` ph ` , ` x ` is not free in
|
|
` [ y / x ] ph ` . (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfs1 $p |- F/ x [ y / x ] ph $=
|
|
( wsb nfri hbsb3 nfi ) ABCEBABCACDFGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Proof of older axiom ~ ax-16 . (Contributed by NM, 8-Nov-2006.)
|
|
(Revised by NM, 22-Sep-2017.) $)
|
|
ax16 $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
|
|
( a16g ) ABCBD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
ax16i.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
|
|
ax16i.2 $e |- ( ps -> A. x ps ) $.
|
|
$( Inference with ~ ax16 as its conclusion. (Contributed by NM,
|
|
20-May-2008.) (Proof modification is discouraged.) $)
|
|
ax16i $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
|
|
( weq wal wi nfv ax-8 cbv3 spimv equcomi syl syl5com alimdv mpcom alimi
|
|
biimpcd nfi biimprd syl6com 3syl ) CDHZCIEDHZEIZCEHZEIZAACIZJUFUGCEUFEKUG
|
|
CKCEDLMUHECHZEIZUJUFUHUMUGUFECECDLNUFUGULEUFDCHZUGULCDOUGDEHUNULJEDODECLP
|
|
QRSULUIEECOZTPAUJBEIUKAUIBEUIABFUARBAECBCGUBAEKULUIBAJUOUIABFUCPMUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
$( Alternate proof of ~ ax16 . (Contributed by NM, 17-May-2008.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax16ALT $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
|
|
( vz wsb sbequ12 ax-17 hbsb3 ax16i ) AABDEBCDABDFABDADGHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d z ph $.
|
|
$( Alternate proof of ~ ax16 . (Contributed by NM, 8-Nov-2006.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax16ALT2 $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
|
|
( weq wal aev wsb sbequ12 biimpcd alimdv nfv nfs1 stdpc7 cbv3 syl6com syl
|
|
vz wi ) BCDBEBQDZQEZAABEZRBCQBQFATABQGZQEUAASUBQSAUBABQHIJUBAQBABQAQKZLUC
|
|
AQBMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( A generalization of axiom ~ ax-16 . Alternate proof of ~ a16g that uses
|
|
~ df-sb . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 25-May-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
a16gALT $p |- ( A. x x = y -> ( ph -> A. z ph ) ) $=
|
|
( weq wal aev ax16ALT2 biidd dral1 biimprd sylsyld ) BCEBFDBEDFZAABFZADFZ
|
|
BCDDBGABCHMONAADBMAIJKL $.
|
|
|
|
$( A generalization of axiom ~ ax-16 . (Contributed by NM, 5-Aug-1993.) $)
|
|
a16gb $p |- ( A. x x = y -> ( ph <-> A. z ph ) ) $=
|
|
( weq wal a16g sp impbid1 ) BCEBFAADFABCDGADHI $.
|
|
|
|
$( If dtru in set.mm is false, then there is only one element in the
|
|
universe, so everything satisfies ` F/ ` . (Contributed by Mario
|
|
Carneiro, 7-Oct-2016.) $)
|
|
a16nf $p |- ( A. x x = y -> F/ z ph ) $=
|
|
( weq wal nfae a16g nfd ) BCEBFADBCDGABCDHI $.
|
|
$}
|
|
|
|
$( One direction of a simplified definition of substitution when variables
|
|
are distinct. (Contributed by NM, 5-Aug-1993.) $)
|
|
sb3 $p |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) ) $=
|
|
( weq wal wn wa wex wi wsb equs5 sb2 syl6 ) BCDZBEFNAGBHNAIBEABCJABCKABCLM
|
|
$.
|
|
|
|
$( One direction of a simplified definition of substitution when variables
|
|
are distinct. (Contributed by NM, 5-Aug-1993.) $)
|
|
sb4 $p |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( wsb weq wa wex wal wn wi sb1 equs5 syl5 ) ABCDBCEZAFBGNBHINAJBHABCKABCLM
|
|
$.
|
|
|
|
$( Simplified definition of substitution when variables are distinct.
|
|
(Contributed by NM, 27-May-1997.) $)
|
|
sb4b $p |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wn wsb wi sb4 sb2 impbid1 ) BCDZBEFABCGLAHBEABCIABCJK $.
|
|
|
|
$( An alternate definition of proper substitution that, like ~ df-sb , mixes
|
|
free and bound variables to avoid distinct variable requirements.
|
|
(Contributed by NM, 17-Feb-2005.) $)
|
|
dfsb2 $p |- ( [ y / x ] ph <->
|
|
( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) $=
|
|
( wsb weq wa wi wal wo sp sbequ2 sps orc ee12an sb4 olc syl6 pm2.61i sbequ1
|
|
wn imp sb2 jaoi impbii ) ABCDZBCEZAFZUFAGBHZIZUFBHZUEUIGUJUFUEAUIUFBJUFUEAG
|
|
BABCKLUGUHMNUJTUEUHUIABCOUHUGPQRUGUEUHUFAUEABCSUAABCUBUCUD $.
|
|
|
|
$( An alternate definition of proper substitution ~ df-sb that uses only
|
|
primitive connectives (no defined terms) on the right-hand side.
|
|
(Contributed by NM, 6-Mar-2007.) $)
|
|
dfsb3 $p |- ( [ y / x ] ph <->
|
|
( ( x = y -> -. ph ) -> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wa wi wal wo wn wsb df-or dfsb2 imnan imbi1i 3bitr4i ) BCDZAEZPAFBGZH
|
|
QIZRFABCJPAIFZRFQRKABCLTSRPAMNO $.
|
|
|
|
$( Bound-variable hypothesis builder for substitution. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
hbsb2 $p |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) ) $=
|
|
( weq wal wn wsb wi sb4 sb2 a5i syl6 ) BCDZBEFABCGZMAHZBENBEABCIONBABCJKL
|
|
$.
|
|
|
|
$( Bound-variable hypothesis builder for substitution. (Contributed by Mario
|
|
Carneiro, 4-Oct-2016.) $)
|
|
nfsb2 $p |- ( -. A. x x = y -> F/ x [ y / x ] ph ) $=
|
|
( weq wal wn wsb nfnae hbsb2 nfd ) BCDBEFABCGBBCBHABCIJ $.
|
|
|
|
$( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbequi $p |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) $=
|
|
( weq wal wsb wi wn wa wex hbsb2 stdpc7 sbequ1 sylan9 sps adantr drsb1 syld
|
|
ex equvini eximi 19.35 sylib nfsb2 19.9d syl9 sbequ2 biimprd sylan9r biimpd
|
|
syl com23 pm2.61ii ) DBEZDFZDCEZDFZBCEZADBGZADCGZHZHUPIZUSURIZVBVCUSVDVBHVC
|
|
USJUTVADKZVDVAVCUTUTDFZUSVEADBLUSVBDKZVFVEHUSBDEZUQJZDKVGBCDUAVIVBDVHUTAUQV
|
|
AABDMADCNZOUBULUTVADUCUDOVAVDDADCUEUFUGTUMUPUSVBUPUSJUTAVAUPUTAHZUSUOVKDADB
|
|
UHPQUSAABCGZUPVAABCNUPVAVLADBCRUIUJSTURUSVBURUSJUTAVAURUTACBGZUSAURUTVMADCB
|
|
RUKABCMOURAVAHZUSUQVNDVJPQSTUN $.
|
|
|
|
$( An equality theorem for substitution. Used in proof of Theorem 9.7 in
|
|
[Megill] p. 449 (p. 16 of the preprint). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sbequ $p |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) $=
|
|
( weq wsb sbequi wi equcoms impbid ) BCEADBFZADCFZABCDGLKHCBACBDGIJ $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed
|
|
by NM, 27-Feb-2005.) $)
|
|
drsb2 $p |- ( A. x x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) $=
|
|
( weq wsb wb sbequ sps ) BCEADBFADCFGBABCDHI $.
|
|
|
|
$( Negation inside and outside of substitution are equivalent. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
sbn $p |- ( [ y / x ] -. ph <-> -. [ y / x ] ph ) $=
|
|
( wn wsb weq wal wi sbequ2 nsyld sps sb4 wa wex sb1 equs3 sylib syl6 sylibr
|
|
con2i pm2.61i sbequ1 con3rr3 sb2 notnot sbbii con3i df-sb sylanbrc impbii )
|
|
ADZBCEZABCEZDZBCFZBGZULUNHZUOUQBUOULAUMUKBCIABCIJKUPDULUOUKHZBGZUNUKBCLUMUS
|
|
UMUOAMBNUSDABCOABCPQTRUAUNURUOUKMBNZULUOAUMABCUBUCUNUOUKDZHBGZDUTVBUMVBVABC
|
|
EUMVABCUDAVABCAUEUFSUGUKBCPSUKBCUHUIUJ $.
|
|
|
|
$( Removal of implication from substitution. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sbi1 $p |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
|
|
( weq wal wi wsb sbequ2 syl5d sbequ1 syl6d sps sb4 ax-2 al2imi syl6 pm2.61i
|
|
wn sb2 ) CDEZCFZABGZCDHZACDHZBCDHZGGZUAUGCUAUDUEBUFUAUEAUDBACDIUCCDIJBCDKLM
|
|
UBSZUEUAAGZCFZUDUFACDNUHUDUAUCGZCFZUJUFGUCCDNULUJUABGZCFUFUKUIUMCUAABOPBCDT
|
|
QQJR $.
|
|
|
|
$( Introduction of implication into substitution. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sbi2 $p |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) ) $=
|
|
( wsb wi wn sbn pm2.21 sbimi sylbir ax-1 ja ) ACDEZBCDEABFZCDEZNGAGZCDEPACD
|
|
HQOCDABIJKBOCDBALJM $.
|
|
|
|
$( Implication inside and outside of substitution are equivalent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sbim $p |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
|
|
( wi wsb sbi1 sbi2 impbii ) ABECDFACDFBCDFEABCDGABCDHI $.
|
|
|
|
$( Logical OR inside and outside of substitution are equivalent.
|
|
(Contributed by NM, 29-Sep-2002.) $)
|
|
sbor $p |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) ) $=
|
|
( wn wi wsb wo sbim sbn imbi1i bitri df-or sbbii 3bitr4i ) AEZBFZCDGZACDGZE
|
|
ZBCDGZFZABHZCDGSUAHRPCDGZUAFUBPBCDIUDTUAACDJKLUCQCDABMNSUAMO $.
|
|
|
|
${
|
|
sbrim.1 $e |- F/ x ph $.
|
|
$( Substitution with a variable not free in antecedent affects only the
|
|
consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario
|
|
Carneiro, 4-Oct-2016.) $)
|
|
sbrim $p |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) $=
|
|
( wi wsb sbim sbf imbi1i bitri ) ABFCDGACDGZBCDGZFAMFABCDHLAMACDEIJK $.
|
|
$}
|
|
|
|
${
|
|
sblim.1 $e |- F/ x ps $.
|
|
$( Substitution with a variable not free in consequent affects only the
|
|
antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario
|
|
Carneiro, 4-Oct-2016.) $)
|
|
sblim $p |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) ) $=
|
|
( wi wsb sbim sbf imbi2i bitri ) ABFCDGACDGZBCDGZFLBFABCDHMBLBCDEIJK $.
|
|
$}
|
|
|
|
$( Conjunction inside and outside of a substitution are equivalent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sban $p |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) ) $=
|
|
( wn wi wsb wa sbn sbim imbi2i bitri xchbinx df-an sbbii 3bitr4i ) ABEZFZEZ
|
|
CDGZACDGZBCDGZEZFZEABHZCDGUAUBHTRCDGZUDRCDIUFUAQCDGZFUDAQCDJUGUCUABCDIKLMUE
|
|
SCDABNOUAUBNP $.
|
|
|
|
$( Conjunction inside and outside of a substitution are equivalent.
|
|
(Contributed by NM, 14-Dec-2006.) $)
|
|
sb3an $p |- ( [ y / x ] ( ph /\ ps /\ ch ) <->
|
|
( [ y / x ] ph /\ [ y / x ] ps /\ [ y / x ] ch ) ) $=
|
|
( w3a wsb wa df-3an sbbii sban anbi1i bitr4i 3bitri ) ABCFZDEGABHZCHZDEGPDE
|
|
GZCDEGZHZADEGZBDEGZSFZOQDEABCIJPCDEKTUAUBHZSHUCRUDSABDEKLUAUBSIMN $.
|
|
|
|
$( Equivalence inside and outside of a substitution are equivalent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sbbi $p |- ( [ y / x ] ( ph <-> ps )
|
|
<-> ( [ y / x ] ph <-> [ y / x ] ps ) ) $=
|
|
( wb wsb wi wa dfbi2 sbbii sbim anbi12i sban 3bitr4i bitri ) ABEZCDFABGZBAG
|
|
ZHZCDFZACDFZBCDFZEZPSCDABIJQCDFZRCDFZHUAUBGZUBUAGZHTUCUDUFUEUGABCDKBACDKLQR
|
|
CDMUAUBINO $.
|
|
|
|
${
|
|
sblbis.1 $e |- ( [ y / x ] ph <-> ps ) $.
|
|
$( Introduce left biconditional inside of a substitution. (Contributed by
|
|
NM, 19-Aug-1993.) $)
|
|
sblbis $p |- ( [ y / x ] ( ch <-> ph ) <-> ( [ y / x ] ch <-> ps ) ) $=
|
|
( wb wsb sbbi bibi2i bitri ) CAGDEHCDEHZADEHZGLBGCADEIMBLFJK $.
|
|
$}
|
|
|
|
${
|
|
sbrbis.1 $e |- ( [ y / x ] ph <-> ps ) $.
|
|
$( Introduce right biconditional inside of a substitution. (Contributed by
|
|
NM, 18-Aug-1993.) $)
|
|
sbrbis $p |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> [ y / x ] ch ) ) $=
|
|
( wb wsb sbbi bibi1i bitri ) ACGDEHADEHZCDEHZGBMGACDEILBMFJK $.
|
|
$}
|
|
|
|
${
|
|
sbrbif.1 $e |- F/ x ch $.
|
|
sbrbif.2 $e |- ( [ y / x ] ph <-> ps ) $.
|
|
$( Introduce right biconditional inside of a substitution. (Contributed by
|
|
NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) $)
|
|
sbrbif $p |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> ch ) ) $=
|
|
( wb wsb sbrbis sbf bibi2i bitri ) ACHDEIBCDEIZHBCHABCDEGJNCBCDEFKLM $.
|
|
$}
|
|
|
|
$( A specialization theorem. (Contributed by NM, 5-Aug-1993.) $)
|
|
spsbe $p |- ( [ y / x ] ph -> E. x ph ) $=
|
|
( wsb wn wal wex stdpc4 sbn sylib con2i df-ex sylibr ) ABCDZAEZBFZEABGPNPOB
|
|
CDNEOBCHABCIJKABLM $.
|
|
|
|
$( Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 25-May-2011.) $)
|
|
spsbim $p |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
|
|
( wi wal wsb stdpc4 sbi1 syl ) ABEZCFKCDGACDGBCDGEKCDHABCDIJ $.
|
|
|
|
$( Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) $)
|
|
spsbbi $p |- ( A. x ( ph <-> ps ) -> ( [ y / x ] ph <-> [ y / x ] ps ) ) $=
|
|
( wb wal wsb stdpc4 sbbi sylib ) ABEZCFKCDGACDGBCDGEKCDHABCDIJ $.
|
|
|
|
${
|
|
sbbid.1 $e |- F/ x ph $.
|
|
sbbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Deduction substituting both sides of a biconditional. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sbbid $p |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) ) $=
|
|
( wb wal wsb alrimi spsbbi syl ) ABCHZDIBDEJCDEJHANDFGKBCDELM $.
|
|
$}
|
|
|
|
$( Elimination of equality from antecedent after substitution. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
sbequ8 $p |- ( [ y / x ] ph <-> [ y / x ] ( x = y -> ph ) ) $=
|
|
( wsb weq wi equsb1 a1bi sbim bitr4i ) ABCDZBCEZBCDZKFLAFBCDMKBCGHLABCIJ $.
|
|
|
|
$( A variable not free remains so after substitution with a distinct variable
|
|
(closed form of ~ nfsb4 ). (Contributed by NM, 7-Apr-2004.) (Revised by
|
|
Mario Carneiro, 4-Oct-2016.) $)
|
|
nfsb4t $p |- ( A. x F/ z ph ->
|
|
( -. A. z z = y -> F/ z [ y / x ] ph ) ) $=
|
|
( wnf wal weq wn wsb wi wa sbequ12 sps drnf2 biimpcd a1dd nfa1 nfnae nfan
|
|
wb nfeqf adantl sp adantr nfimd nfald sb4b nfbidf imbi2d syl5ibrcom pm2.61d
|
|
ex exp3a nfsb2 drsb1 syl5ib pm2.61d2 ) ADEZBFZDBGDFZDCGDFHZABCIZDEZJUSUTHZV
|
|
AVCUSBCGZBFZVDVAKZVCJZUSVFVCVGURVFVCJBVFURVCAVBBCDVEAVBTBABCLMNOMPUSVHVFHZV
|
|
GVEAJZBFZDEZJUSVGVLUSVGKZVJDBUSVGBURBQVDVABDBBRDCBRSSVMVEADVGVEDEUSBCDUAUBU
|
|
SURVGURBUCUDUEUFULVIVCVLVGVIVBVKDBCDRABCUGUHUIUJUKUMVAADCIZDEUTVCADCUNVNVBD
|
|
BDADBCUONUPUQ $.
|
|
|
|
${
|
|
nfsb4.1 $e |- F/ z ph $.
|
|
$( A variable not free remains so after substitution with a distinct
|
|
variable. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
4-Oct-2016.) $)
|
|
nfsb4 $p |- ( -. A. z z = y -> F/ z [ y / x ] ph ) $=
|
|
( wnf weq wal wn wsb wi nfsb4t mpg ) ADFDCGDHIABCJDFKBABCDLEM $.
|
|
$}
|
|
|
|
${
|
|
dvelimdf.1 $e |- F/ x ph $.
|
|
dvelimdf.2 $e |- F/ z ph $.
|
|
dvelimdf.3 $e |- ( ph -> F/ x ps ) $.
|
|
dvelimdf.4 $e |- ( ph -> F/ z ch ) $.
|
|
dvelimdf.5 $e |- ( ph -> ( z = y -> ( ps <-> ch ) ) ) $.
|
|
$( Deduction form of ~ dvelimf . This version may be useful if we want to
|
|
avoid ~ ax-17 and use ~ ax-16 instead. (Contributed by NM,
|
|
7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
dvelimdf $p |- ( ph -> ( -. A. x x = y -> F/ x ch ) ) $=
|
|
( weq wal wn wnf wa wsb wi alrimi nfsb4t syl imp nfnae nfan adantr nfbidf
|
|
wb sbied mpbid ex ) ADELDMNZCDOZAUKPZBFEQZDOZULAUKUOABDOZFMUKUORAUPFHISBF
|
|
EDTUAUBUMUNCDAUKDGDEDUCUDAUNCUGUKABCFEHJKUHUEUFUIUJ $.
|
|
$}
|
|
|
|
$( A composition law for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbco $p |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) $=
|
|
( wsb wb weq equsb2 sbequ12 bicomd sbimi ax-mp sbbi mpbi ) ACBDZAEZBCDZNBCD
|
|
ABCDECBFZBCDPBCGQOBCQANACBHIJKNABCLM $.
|
|
|
|
${
|
|
sbid2.1 $e |- F/ x ph $.
|
|
$( An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sbid2 $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
|
|
( wsb sbco sbf bitri ) ACBEBCEABCEAABCFABCDGH $.
|
|
$}
|
|
|
|
$( An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sbidm $p |- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph ) $=
|
|
( wsb wb weq equsb2 sbequ12r sbimi ax-mp sbbi mpbi ) ABCDZAEZBCDZMBCDMECBFZ
|
|
BCDOBCGPNBCACBHIJMABCKL $.
|
|
|
|
${
|
|
sbco2.1 $e |- F/ z ph $.
|
|
$( A composition law for substitution. (Contributed by NM, 30-Jun-1994.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sbco2 $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
|
|
( weq wal wsb wb sbid2 sbequ syl5bbr sbequ12 bitr3d sps wn nfnae nfsb4 wi
|
|
nfs1 a1i sbied bicomd pm2.61i ) BCFZBGZABDHZDCHZABCHZIZUEUJBUEAUHUIAUGDBH
|
|
UEUHADBEJUGBCDKLZABCMNOUFPZUIUHULAUHBCBCBQUGDCBABDETRUEAUHISULUKUAUBUCUD
|
|
$.
|
|
$}
|
|
|
|
${
|
|
sbco2d.1 $e |- F/ x ph $.
|
|
sbco2d.2 $e |- F/ z ph $.
|
|
sbco2d.3 $e |- ( ph -> F/ z ps ) $.
|
|
$( A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sbco2d $p |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) ) $=
|
|
( wsb wi nfim1 sbco2 sbrim sbbii bitri 3bitr3i pm5.74ri ) ABCEIZEDIZBCDIZ
|
|
ABJZCEIZEDIZUACDIASJZATJUACDEABEGHKLUCARJZEDIUDUBUEEDABCEFMNAREDGMOABCDFM
|
|
PQ $.
|
|
$}
|
|
|
|
$( A composition law for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbco3 $p |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) $=
|
|
( weq wal wsb wb drsb1 sbequ12a alimi spsbbi syl bitr3d wn sbco sbbii nfnae
|
|
nfsb2 sbco2d syl5rbbr pm2.61i ) BCEZBFZABCGZCDGZACBGZBDGZHUDUEBDGZUFUHUEBCD
|
|
IUDUEUGHZBFUIUHHUCUJBABCJKUEUGBDLMNUHUECBGZBDGUDOZUFUKUGBDACBPQULUECDBBCCRB
|
|
CBRABCSTUAUB $.
|
|
|
|
$( A commutativity law for substitution. (Contributed by NM,
|
|
27-May-1997.) $)
|
|
sbcom $p |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph ) $=
|
|
( weq wal wsb wb wn wa drsb1 nfae sbbid bitr3d nfnae albid sb4b sbequ12 sps
|
|
wi adantr wnf nfeqf 19.21t syl adantrr alcom bi2.04 albii aecom con3i sylan
|
|
nfan syl5bb adantrl imbi2d sylan9bbr sylan9bb 3bitr4d pm2.61ian ex pm2.61ii
|
|
adantl ) BCEZBFZDCEZDFZABCGZDCGZADCGZBCGZHZVEIZVGIZVLBDEBFZVMVNJZVLVOVLVPVO
|
|
VHBCGVIVKVHBDCKVOVHVJBCBDBLABDCKMNUAVOIZVPJZVFVDATZBFZTZDFZVDVFATZDFZTZBFZV
|
|
IVKVRVFVSTZBFZDFZWBWFVQVMWIWBHVNVQVMJZWHWADVQVMDBDDOBCDOZUMWJVFBUBWHWAHDCBU
|
|
CVFVSBUDUEPUFVQVNWIWFHVMWIWGDFZBFVQVNJZWFWGDBUGWMWLWEBVQVNBBDBODCBOZUMWLVDW
|
|
CTZDFZWMWEWGWODVFVDAUHUIWMVDDUBZWPWEHVQDBEDFZIVNWQWRVODBUJUKBCDUCULVDWCDUDU
|
|
EUNPUNUONVPVIWBHVQVNVIVFVHTZDFVMWBVHDCQVMWSWADWKVMVHVTVFABCQUPPUQVCVPVKWFHV
|
|
QVMVKVDVJTZBFVNWFVJBCQVNWTWEBWNVNVJWDVDADCQUPPURVCUSUTVAVEVJVIVKVEAVHDCBCDL
|
|
VDAVHHBABCRSMVDVJVKHBVJBCRSNVGVHVIVKVFVHVIHDVHDCRSVGAVJBCDCBLVFAVJHDADCRSMN
|
|
VB $.
|
|
|
|
${
|
|
sb5rf.1 $e |- F/ y ph $.
|
|
$( Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by
|
|
Mario Carneiro, 6-Oct-2016.) $)
|
|
sb5rf $p |- ( ph <-> E. y ( y = x /\ [ y / x ] ph ) ) $=
|
|
( weq wsb wa wex sbid2 sb1 sylbir stdpc7 imp exlimi impbii ) ACBEZABCFZGZ
|
|
CHZAQCBFSACBDIQCBJKRACDPQAACBLMNO $.
|
|
|
|
$( Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by
|
|
Mario Carneiro, 6-Oct-2016.) $)
|
|
sb6rf $p |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) ) $=
|
|
( weq wsb wi wal sbequ1 equcoms com12 alrimi sb2 sbid2 sylib impbii ) ACB
|
|
EZABCFZGZCHZASCDQARARGBCABCIJKLTRCBFARCBMACBDNOP $.
|
|
|
|
$( Substitution of variable in universal quantifier. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sb8 $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
|
|
( wal wsb nfal stdpc4 alrimi nfs1 stdpc7 cbv3 impbii ) ABEZABCFZCENOCACBD
|
|
GABCHIOACBABCDJDACBKLM $.
|
|
|
|
$( Substitution of variable in existential quantifier. (Contributed by NM,
|
|
12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sb8e $p |- ( E. x ph <-> E. y [ y / x ] ph ) $=
|
|
( wn wal wsb wex nfn sb8 sbn albii bitri notbii df-ex 3bitr4i ) AEZBFZEAB
|
|
CGZEZCFZEABHSCHRUARQBCGZCFUAQBCACDIJUBTCABCKLMNABOSCOP $.
|
|
$}
|
|
|
|
$( Commutation of quantification and substitution variables. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sb9i $p |- ( A. x [ x / y ] ph -> A. y [ y / x ] ph ) $=
|
|
( weq wal wsb wi drsb1 drsb2 bitr3d dral1 biimprd wn nfnae hbsb2 alimd sbco
|
|
stdpc4 sylib alimi a7s syl6 pm2.61i ) CBDCEZACBFZBEZABCFZCEZGUDUHUFUGUECBUD
|
|
ACCFUGUEACBCHACBCIJKLUDMZUFUECEZBEUHUIUEUJBCBBNACBOPUEUHCBUFUGCUFUEBCFUGUEB
|
|
CRABCQSTUAUBUC $.
|
|
|
|
$( Commutation of quantification and substitution variables. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sb9 $p |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) $=
|
|
( wsb wal sb9i impbii ) ACBDBEABCDCEABCFACBFG $.
|
|
|
|
${
|
|
$d x y $. $d x z $. $d y z $. $d ph z $.
|
|
$( This is a version of ~ ax-11o when the variables are distinct. Axiom
|
|
(C8) of [Monk2] p. 105. See theorem ~ ax11v2 for the rederivation of
|
|
~ ax-11o from this theorem. (Contributed by NM, 5-Aug-1993.) $)
|
|
ax11v $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wi ax-1 ax16 syl5 a1d ax11o pm2.61i ) BCDZBEZMAMAFZBEZFZFNQMAON
|
|
PAMGOBCHIJABCKL $.
|
|
|
|
$( Alternate proof of ~ ax11v that avoids theorem ~ ax16 and is proved
|
|
directly from ~ ax-11 rather than via ~ ax11o . (Contributed by Jim
|
|
Kingdon, 15-Dec-2017.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
ax11vALT $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( vz cv wceq wex wi wal a9e ax-17 ax-11 syl5 equequ2 imbi1d albidv imbi2d
|
|
imbi12d mpbii exlimiv ax-mp ) DEZCEZFZDGBEZUCFZAUFAHZBIZHZHZDCJUDUJDUDUEU
|
|
BFZAUKAHZBIZHZHUJAADIUKUMADKABDLMUDUKUFUNUIDCBNZUDUMUHAUDULUGBUDUKUFAUOOP
|
|
QRSTUA $.
|
|
|
|
$( Two equivalent ways of expressing the proper substitution of ` y ` for
|
|
` x ` in ` ph ` , when ` x ` and ` y ` are distinct. Theorem 6.2 of
|
|
[Quine] p. 40. The proof does not involve ~ df-sb . (Contributed by
|
|
NM, 14-Apr-2008.) $)
|
|
sb56 $p |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) $=
|
|
( weq wi wal nfa1 ax11v sp com12 impbid equsex ) ABCDZAEZBFZBCNBGMAOABCHO
|
|
MANBIJKL $.
|
|
|
|
$( Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40.
|
|
Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
sb6 $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
|
|
( weq wi wa wex wal wsb sb56 anbi2i df-sb sp pm4.71ri 3bitr4i ) BCDZAEZPA
|
|
FBGZFQQBHZFABCISRSQABCJKABCLSQQBMNO $.
|
|
|
|
$( Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40.
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
sb5 $p |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) $=
|
|
( wsb weq wi wal wa wex sb6 sb56 bitr4i ) ABCDBCEZAFBGMAHBIABCJABCKL $.
|
|
$}
|
|
|
|
${
|
|
$d y z $. $d x y $.
|
|
$( Lemma for ~ equsb3 . (Contributed by Raph Levien and FL, 4-Dec-2005.)
|
|
(Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
equsb3lem $p |- ( [ x / y ] y = z <-> x = z ) $=
|
|
( weq nfv equequ1 sbie ) BCDACDZBAHBEBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d w x $.
|
|
$( Substitution applied to an atomic wff. (Contributed by Raph Levien and
|
|
FL, 4-Dec-2005.) $)
|
|
equsb3 $p |- ( [ x / y ] y = z <-> x = z ) $=
|
|
( vw weq wsb equsb3lem sbbii nfv sbco2 3bitr3i ) BCEZBDFZDAFDCEZDAFLBAFAC
|
|
EMNDADBCGHLBADLDIJADCGK $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d w x $.
|
|
$( Substitution applied to an atomic membership wff. (Contributed by NM,
|
|
7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
elsb3 $p |- ( [ x / y ] y e. z <-> x e. z ) $=
|
|
( vw wel wsb nfv sbco2 elequ1 sbie sbbii 3bitr3i ) DCEZDBFZBAFMDAFBCEZBAF
|
|
ACEZMDABMBGHNOBAMODBODGDBCIJKMPDAPDGDACIJL $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d w x $.
|
|
$( Substitution applied to an atomic membership wff. (Contributed by
|
|
Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
14-Jun-2011.) $)
|
|
elsb4 $p |- ( [ x / y ] z e. y <-> z e. x ) $=
|
|
( vw wel wsb nfv sbco2 elequ2 sbie sbbii 3bitr3i ) CDEZDBFZBAFMDAFCBEZBAF
|
|
CAEZMDABMBGHNOBAMODBODGDBCIJKMPDAPDGDACIJL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( ` x ` is not free in ` [ y / x ] ph ` when ` x ` and ` y ` are
|
|
distinct. (Contributed by NM, 5-Aug-1993.) $)
|
|
hbs1 $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
|
|
( weq wal wsb wi ax16 hbsb2 pm2.61i ) BCDBEABCFZKBEGKBCHABCIJ $.
|
|
|
|
$( ` x ` is not free in ` [ y / x ] ph ` when ` x ` and ` y ` are
|
|
distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfs1v $p |- F/ x [ y / x ] ph $=
|
|
( wsb hbs1 nfi ) ABCDBABCEF $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $.
|
|
$( Two ways of expressing " ` x ` is (effectively) not free in ` ph ` ."
|
|
(Contributed by NM, 29-May-2009.) $)
|
|
sbhb $p |- ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) $=
|
|
( wal wi wsb nfv sb8 imbi2i 19.21v bitr4i ) AABDZEAABCFZCDZEAMECDLNAABCAC
|
|
GHIAMCJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z ph $.
|
|
$( Two ways of expressing " ` x ` is (effectively) not free in ` ph ` ."
|
|
(Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario
|
|
Carneiro, 6-Oct-2016.) $)
|
|
sbnf2 $p |- ( F/ x ph
|
|
<-> A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) ) $=
|
|
( wsb wb wal wi wnf 2albiim sbhb albii alcom 3bitri nfv nfs1v sblim bitri
|
|
wa sb8 df-nf anbi12i anidm 3bitr2ri ) ABCEZABDEZFDGCGUEUFHZDGCGZUFUEHZDGZ
|
|
CGZSABIZULSULUEUFCDJULUHULUKULAUFHZBGZDGZUGCGZDGUHULAABGHZBGZUMDGZBGUOABU
|
|
AZUQUSBABDKLUMBDMNUNUPDUNUMBCEZCGUPUMBCUMCOTVAUGCAUFBCABDPQLRLUGDCMNULAUE
|
|
HZBGZCGZUKULURVBCGZBGVDUTUQVEBABCKLVBBCMNVCUJCVCVBBDEZDGUJVBBDVBDOTVFUIDA
|
|
UEBDABCPQLRLRUBULUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d y z $.
|
|
nfsb.1 $e |- F/ z ph $.
|
|
$( If ` z ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` when
|
|
` y ` and ` z ` are distinct. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfsb $p |- F/ z [ y / x ] ph $=
|
|
( weq wal wsb wnf a16nf nfsb4 pm2.61i ) DCFDGABCHZDIMDCDJABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d y z $.
|
|
hbsb.1 $e |- ( ph -> A. z ph ) $.
|
|
$( If ` z ` is not free in ` ph ` , it is not free in ` [ y / x ] ph ` when
|
|
` y ` and ` z ` are distinct. (Contributed by NM, 12-Aug-1993.) $)
|
|
hbsb $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $=
|
|
( wsb nfi nfsb nfri ) ABCFDABCDADEGHI $.
|
|
$}
|
|
|
|
${
|
|
$d y z $.
|
|
nfsbd.1 $e |- F/ x ph $.
|
|
nfsbd.2 $e |- ( ph -> F/ z ps ) $.
|
|
$( Deduction version of ~ nfsb . (Contributed by NM, 15-Feb-2013.) $)
|
|
nfsbd $p |- ( ph -> F/ z [ y / x ] ps ) $=
|
|
( weq wal wsb wnf wn wi alrimi nfsb4t syl a16nf pm2.61d2 ) AEDHEIZBCDJZEK
|
|
ZABEKZCISLUAMAUBCFGNBCDEOPTEDEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d w y $.
|
|
$( Equivalence for double substitution. (Contributed by NM,
|
|
3-Feb-2005.) $)
|
|
2sb5 $p |- ( [ z / x ] [ w / y ] ph <->
|
|
E. x E. y ( ( x = z /\ y = w ) /\ ph ) ) $=
|
|
( wsb weq wa wex sb5 19.42v anass exbii anbi2i 3bitr4ri bitri ) ACEFZBDFB
|
|
DGZQHZBIRCEGZHAHZCIZBIQBDJSUBBRTAHZHZCIRUCCIZHUBSRUCCKUAUDCRTALMQUERACEJN
|
|
OMP $.
|
|
|
|
$( Equivalence for double substitution. (Contributed by NM,
|
|
3-Feb-2005.) $)
|
|
2sb6 $p |- ( [ z / x ] [ w / y ] ph <->
|
|
A. x A. y ( ( x = z /\ y = w ) -> ph ) ) $=
|
|
( wsb weq wi wal wa sb6 19.21v impexp albii imbi2i 3bitr4ri bitri ) ACEFZ
|
|
BDFBDGZRHZBISCEGZJAHZCIZBIRBDKTUCBSUAAHZHZCISUDCIZHUCTSUDCLUBUECSUAAMNRUF
|
|
SACEKOPNQ $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d x w $. $d y z $.
|
|
$( Commutativity law for substitution. Used in proof of Theorem 9.7 of
|
|
[Megill] p. 449 (p. 16 of the preprint). (Contributed by NM,
|
|
27-May-1997.) $)
|
|
sbcom2 $p |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) $=
|
|
( weq wal wsb wb wn wi albii 19.21v sb4b imbi2d albidv nfae sbequ12 sbbid
|
|
sps wa alcom bi2.04 bitri 3bitr3i a1i sylan9bbr sylan9bb 3bitr4d pm2.61ii
|
|
ex bitr3d ) BCFZBGZDEFZDGZABCHZDEHZADEHZBCHZIZUNJZUPJZVAVBVCUAZUOUMAKZBGZ
|
|
KZDGZUMUOAKZDGZKZBGZURUTVHVLIVDUMVIKZBGZDGVMDGZBGVHVLVMDBUBVNVGDVNUOVEKZB
|
|
GVGVMVPBUMUOAUCLUOVEBMUDLVOVKBUMVIDMLUEUFVCURUOUQKZDGVBVHUQDENVBVQVGDVBUQ
|
|
VFUOABCNOPUGVBUTUMUSKZBGVCVLUSBCNVCVRVKBVCUSVJUMADENOPUHUIUKUNUSURUTUNAUQ
|
|
DEBCDQUMAUQIBABCRTSUMUSUTIBUSBCRTULUPUQURUTUOUQURIDUQDERTUPAUSBCDEBQUOAUS
|
|
IDADERTSULUJ $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $. $d w x z $.
|
|
$( Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew
|
|
Salmon, 17-Jun-2011.) $)
|
|
pm11.07 $p |- ( [ w / x ] [ y / z ] ph <-> [ y / x ] [ w / z ] ph ) $=
|
|
( weq wa wex wsb a9ev pm3.2i 2th eeanv 3bitr4i anbi1i 19.41vv 2sb5 ) BEFZ
|
|
DCFZGZAGDHBHZBCFZDEFZGZAGDHBHZADCIBEIADEIBCITDHBHZAGUDDHBHZAGUAUEUFUGARBH
|
|
ZSDHZGZUBBHZUCDHZGZUFUGUJUMUHUIBEJDCJKUKULBCJDEJKLRSBDMUBUCBDMNOTABDPUDAB
|
|
DPNABDECQABDCEQN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Equivalence for substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sb6a $p |- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) ) $=
|
|
( wsb weq wi wal sb6 wb sbequ12 equcoms pm5.74i albii bitri ) ABCDBCEZAFZ
|
|
BGOACBDZFZBGABCHPRBOAQAQICBACBJKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x w $. $d y z $. $d z w $.
|
|
2sb5rf.1 $e |- F/ z ph $.
|
|
2sb5rf.2 $e |- F/ w ph $.
|
|
$( Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
2sb5rf $p |- ( ph <->
|
|
E. z E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) ) $=
|
|
( weq wsb wex sb5rf 19.42v sbcom2 anbi2i anass bitri exbii nfsb 3bitr4ri
|
|
wa ) ADBHZABDIZTZDJUAECHZTZACEIBDIZTZEJZDJABDFKUCUHDUAUDUBCEIZTZTZEJUAUJE
|
|
JZTUHUCUAUJELUGUKEUGUEUITUKUFUIUEACEBDMNUAUDUIOPQUBULUAUBCEABDEGRKNSQP $.
|
|
|
|
$( Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
2sb6rf $p |- ( ph <->
|
|
A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) ) $=
|
|
( weq wsb wi wal wa sb6rf 19.21v sbcom2 imbi2i impexp bitri albii nfsb
|
|
3bitr4ri ) ADBHZABDIZJZDKUBECHZLZACEIBDIZJZEKZDKABDFMUDUIDUBUEUCCEIZJZJZE
|
|
KUBUKEKZJUIUDUBUKENUHULEUHUFUJJULUGUJUFACEBDOPUBUEUJQRSUCUMUBUCCEABDEGTMP
|
|
UASR $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( An alternate definition of proper substitution ~ df-sb . By introducing
|
|
a dummy variable ` z ` in the definiens, we are able to eliminate any
|
|
distinct variable restrictions among the variables ` x ` , ` y ` , and
|
|
` ph ` of the definiendum. No distinct variable conflicts arise because
|
|
` z ` effectively insulates ` x ` from ` y ` . To achieve this, we use
|
|
a chain of two substitutions in the form of ~ sb5 , first ` z ` for
|
|
` x ` then ` y ` for ` z ` . Compare Definition 2.1'' of [Quine] p. 17,
|
|
which is obtained from this theorem by applying ~ df-clab . Theorem
|
|
~ sb7h provides a version where ` ph ` and ` z ` don't have to be
|
|
distinct. (Contributed by NM, 28-Jan-2004.) $)
|
|
dfsb7 $p |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
|
|
( wsb weq wa wex sb5 sbbii nfv sbco2 3bitr3i ) ABDEZDCEBDFAGBHZDCEABCEDCF
|
|
OGDHNODCABDIJABCDADKLODCIM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
sb7f.1 $e |- F/ z ph $.
|
|
$( This version of ~ dfsb7 does not require that ` ph ` and ` z ` be
|
|
distinct. This permits it to be used as a definition for substitution
|
|
in a formalization that omits the logically redundant axiom ~ ax-17 i.e.
|
|
that doesn't have the concept of a variable not occurring in a wff.
|
|
( ~ df-sb is also suitable, but its mixing of free and bound variables
|
|
is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.)
|
|
(Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sb7f $p |- ( [ y / x ] ph <->
|
|
E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
|
|
( wsb weq wa wex sb5 sbbii sbco2 3bitr3i ) ABDFZDCFBDGAHBIZDCFABCFDCGOHDI
|
|
NODCABDJKABCDELODCJM $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
sb7h.1 $e |- ( ph -> A. z ph ) $.
|
|
$( This version of ~ dfsb7 does not require that ` ph ` and ` z ` be
|
|
distinct. This permits it to be used as a definition for substitution
|
|
in a formalization that omits the logically redundant axiom ~ ax-17 i.e.
|
|
that doesn't have the concept of a variable not occurring in a wff.
|
|
( ~ df-sb is also suitable, but its mixing of free and bound variables
|
|
is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sb7h $p |- ( [ y / x ] ph <->
|
|
E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
|
|
( nfi sb7f ) ABCDADEFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
sb10f.1 $e |- F/ x ph $.
|
|
$( Hao Wang's identity axiom P6 in Irving Copi, _Symbolic Logic_ (5th ed.,
|
|
1979), p. 328. In traditional predicate calculus, this is a sole axiom
|
|
for identity from which the usual ones can be derived. (Contributed by
|
|
NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) $)
|
|
sb10f $p |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) ) $=
|
|
( weq wsb wa wex nfsb sbequ equsex bicomi ) BCFADBGZHBIADCGZNOBCADCBEJABC
|
|
DKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( An identity law for substitution. Used in proof of Theorem 9.7 of
|
|
[Megill] p. 449 (p. 16 of the preprint). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sbid2v $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
|
|
( nfv sbid2 ) ABCABDE $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x ph $.
|
|
$( Elimination of substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbelx $p |- ( ph <-> E. x ( x = y /\ [ x / y ] ph ) ) $=
|
|
( wsb weq wa wex sbid2v sb5 bitr3i ) AACBDZBCDBCEKFBGABCHKBCIJ $.
|
|
$}
|
|
|
|
${
|
|
$( Note: A more general case could also be proved with
|
|
"$d x z $. $d y w $. $d x ph $. $d y ph $.", but with more
|
|
difficulty. $)
|
|
$d x y z $. $d w y $. $d x y ph $.
|
|
$( Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) $)
|
|
sbel2x $p |- ( ph <-> E. x E. y ( ( x = z /\ y = w ) /\
|
|
[ y / w ] [ x / z ] ph ) ) $=
|
|
( weq wsb wa wex sbelx anbi2i exbii exdistr 3bitr4i anass 2exbii bitr4i )
|
|
ABDFZCEFZADBGZECGZHZHZCIBIZRSHUAHZCIBIRTHZBIRUBCIZHZBIAUDUFUHBTUGRTCEJKLA
|
|
BDJRUBBCMNUEUCBCRSUAOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( A theorem used in elimination of disjoint variable restriction on ` x `
|
|
and ` y ` by replacing it with a distinctor ` -. A. x x = z ` .
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
sbal1 $p |- ( -. A. x x = z ->
|
|
( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
|
|
( weq wal wn wsb wb wi sbequ12 sps dral2 bitr3d a1d wa nfa1 al2imi hbnaes
|
|
syl6 nfsb4 nfrd sp sbimi alimi adantl ax-7 dveeq2 alim syl9 sylan9 impbid
|
|
sb4 sb2 ex pm2.61i ) CDEZCFZBDEBFGZABFZCDHZACDHZBFZIZJURVDUSURUTVAVCUQUTV
|
|
AICUTCDKLAVBCDBUQAVBICACDKLMNOURGZUSVDVEUSPVAVCUSVAVCJVEUSVAVABFVCUSVABUT
|
|
CDBABQUAUBVAVBBUTACDABUCUDUETUFVEVCUQAJZBFZCFZUSVAVEVCVFCFZBFZVHVCVJJCDBV
|
|
EVBVIBACDUMRSVFBCUGTVHVAJBDCUSCFVHUQUTJZCFVAUSVGVKCUSUQUQBFVGUTBDCUHUQABU
|
|
IUJRUTCDUNTSUKULUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x z $.
|
|
$( Move universal quantifier in and out of substitution. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
sbal $p |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) $=
|
|
( weq wal wsb wb a16gb sbimi sbequ5 sbbi 3imtr3i bitr3d sbal1 pm2.61i ) B
|
|
DEBFZABFZCDGZACDGZBFZHQTSUAQCDGARHZCDGQTSHQUBCDABDBIJBDCDKARCDLMTBDBINABC
|
|
DOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x z $.
|
|
$( Move existential quantifier in and out of substitution. (Contributed by
|
|
NM, 27-Sep-2003.) $)
|
|
sbex $p |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph ) $=
|
|
( wn wal wsb wex sbn sbal albii bitri xchbinx df-ex sbbii 3bitr4i ) AEZBF
|
|
ZEZCDGZACDGZEZBFZEABHZCDGUABHTRCDGZUCRCDIUEQCDGZBFUCQBCDJUFUBBACDIKLMUDSC
|
|
DABNOUABNP $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
sbalv.1 $e |- ( [ y / x ] ph <-> ps ) $.
|
|
$( Quantify with new variable inside substitution. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
sbalv $p |- ( [ y / x ] A. z ph <-> A. z ps ) $=
|
|
( wal wsb sbal albii bitri ) AEGCDHACDHZEGBEGAECDILBEFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( An equivalent expression for existence. (Contributed by NM,
|
|
2-Feb-2005.) $)
|
|
exsb $p |- ( E. x ph <-> E. y A. x ( x = y -> ph ) ) $=
|
|
( weq wi wal nfv nfa1 ax11v sp com12 impbid cbvex ) ABCDZAEZBFZBCACGOBHNA
|
|
PABCIPNAOBJKLM $.
|
|
|
|
$( An equivalent expression for existence. Obsolete as of 19-Jun-2017.
|
|
(Contributed by NM, 2-Feb-2005.) (New usage is discouraged.) $)
|
|
exsbOLD $p |- ( E. x ph <-> E. y A. x ( x = y -> ph ) ) $=
|
|
( wex wsb cv wceq wi wal nfv sb8e sb6 exbii bitri ) ABDABCEZCDBFCFGAHBIZC
|
|
DABCACJKOPCABCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y w $. $d z w ph $.
|
|
$( An equivalent expression for double existence. (Contributed by NM,
|
|
2-Feb-2005.) $)
|
|
2exsb $p |- ( E. x E. y ph <->
|
|
E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) ) $=
|
|
( wex weq wi wal wa exsb exbii excom bitri impexp albii 19.21v bitr2i ) A
|
|
CFZBFZCEGZAHZCIZBFZEFZBDGZUAJAHZCIZBIZEFDFZTUCEFZBFUESUKBACEKLUCBEMNUEUID
|
|
FZEFUJUDULEUDUFUCHZBIZDFULUCBDKUNUIDUMUHBUHUFUBHZCIUMUGUOCUFUAAOPUFUBCQRP
|
|
LNLUIEDMNN $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $. $d x z $. $d y z $.
|
|
dvelimALT.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelimALT.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Version of ~ dvelim that doesn't use ~ ax-10 . (See ~ dvelimh for a
|
|
version that doesn't use ~ ax-11 .) (Contributed by NM, 17-May-2008.)
|
|
(New usage is discouraged.) (Proof modification is discouraged.) $)
|
|
dvelimALT $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( weq wal wn wi ax-17 ax16ALT a1d wa hbn1 hban ax12o imp a1i hbimd hbald
|
|
ex pm2.61i equsalh albii 3imtr3g ) CDHZCIJZEDHZAKZEIZULCIBBCIUIUKCEUIELCE
|
|
HZCIZUIUKUKCIKZKUNUOUIUKCEMNUNJZUIUOUPUIOZUJACUPUICUMCPUHCPQUPUIUJUJCIKED
|
|
CRSAACIKUQFTUAUCUDUBABEDBELGUEZULBCURUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d z y $. $d z x $.
|
|
$( Move quantifier in and out of substitution. (Contributed by NM,
|
|
2-Jan-2002.) $)
|
|
sbal2 $p |- ( -. A. x x = y ->
|
|
( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
|
|
( weq wal wn wi wsb alcom nfnae wnf wb dveeq1 nfd 19.21t syl syl5rbbr sb6
|
|
albid albii 3bitr4g ) BCEBFGZCDEZABFZHZCFZUDAHZCFZBFZUECDIACDIZBFUJUHBFZC
|
|
FUCUGUHCBJUCULUFCBCCKUCUDBLULUFMUCUDBBCBKBCDNOUDABPQTRUECDSUKUIBACDSUAUB
|
|
$.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
|
|
The "metalogical completeness theorem", Theorem 9.7 of [Megill] p. 448, uses
|
|
a different but (logically and metalogically) equivalent set of axiom schemes
|
|
for its proof. In order to show that our axiomatization is also
|
|
metalogically complete, we derive the axiom schemes of that paper in this
|
|
section (or mention where they are derived, if they have already been derived
|
|
as therorems above). Additionally, we re-derive our axiomatization from the
|
|
one in the paper, showing that the two systems are equivalent.
|
|
|
|
The 14 predicate calculus axioms used by the paper are ~ ax-5o , ~ ax-4 ,
|
|
~ ax-7 , ~ ax-6o , ~ ax-8 , ~ ax-12o , ~ ax-9o , ~ ax-10o , ~ ax-13 ,
|
|
~ ax-14 , ~ ax-15 , ~ ax-11o , ~ ax-16 , and ~ ax-17 . Like ours, it
|
|
includes the rule of generalization ( ~ ax-gen ).
|
|
|
|
The ones we need to prove from our axioms are ~ ax-5o , ~ ax-4 ,
|
|
~ ax-6o , ~ ax-12o , ~ ax-9o , ~ ax-10o , ~ ax-15 , ~ ax-11o ,
|
|
and ~ ax-16 . The theorems showing the derivations of those axioms,
|
|
which have all been proved earlier, are ~ ax5o , ~ ax4 (also called
|
|
~ sp ), ~ ax6o , ~ ax12o , ~ ax9o , ~ ax10o , ~ ax15 , ~ ax11o ,
|
|
~ ax16 , and ~ ax10 . In addition, ~ ax-10 was an intermediate axiom we
|
|
adopted at one time, and we show its proof in this section as
|
|
~ ax10from10o .
|
|
|
|
This section also includes a few miscellaneous legacy theorems such as
|
|
~ hbequid use the older axioms.
|
|
|
|
Note: The axioms and theorems in this section should not be used outside of
|
|
this section. Inside this section, we may use the external axioms ~ ax-gen ,
|
|
~ ax-17 , ~ ax-8 , ~ ax-9 , ~ ax-13 , and ~ ax-14 since they are common to
|
|
both our current and the older axiomatizations. (These are the ones that
|
|
were never revised.)
|
|
|
|
The following newer axioms may NOT be used in this section until we
|
|
have proved them from the older axioms: ~ ax-5 , ~ ax-6 , ~ ax-9 ,
|
|
~ ax-11 , and ~ ax-12 . However, once we have rederived an axiom
|
|
(e.g. theorem ~ ax5 for axiom ~ ax-5 ), we may make use of theorems
|
|
outside of this section that make use of the rederived axiom (e.g. we
|
|
may use theorem ~ alimi , which uses ~ ax-5 , after proving ~ ax5 ).
|
|
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
These older axiom schemes are obsolete and should not be used outside of this
|
|
section. They are proved above as theorems ax5o , ~ sp , ~ ax6o , ~ ax9o ,
|
|
~ ax10o , ~ ax10 , ~ ax11o , ~ ax12o , ~ ax15 , and ~ ax16 .
|
|
|
|
$)
|
|
|
|
$( Axiom of Specialization. A quantified wff implies the wff without a
|
|
quantifier (i.e. an instance, or special case, of the generalized wff).
|
|
In other words if something is true for all ` x ` , it is true for any
|
|
specific ` x ` (that would typically occur as a free variable in the wff
|
|
substituted for ` ph ` ). (A free variable is one that does not occur in
|
|
the scope of a quantifier: ` x ` and ` y ` are both free in ` x = y ` ,
|
|
but only ` x ` is free in ` A. y x = y ` .) This is one of the axioms of
|
|
what we call "pure" predicate calculus ( ~ ax-4 through ~ ax-7 plus rule
|
|
~ ax-gen ). Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint).
|
|
Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined
|
|
in the last paragraph on p. 77).
|
|
|
|
Note that the converse of this axiom does not hold in general, but a
|
|
weaker inference form of the converse holds and is expressed as rule
|
|
~ ax-gen . Conditional forms of the converse are given by ~ ax-12 ,
|
|
~ ax-15 , ~ ax-16 , and ~ ax-17 .
|
|
|
|
Unlike the more general textbook Axiom of Specialization, we cannot choose
|
|
a variable different from ` x ` for the special case. For use, that
|
|
requires the assistance of equality axioms, and we deal with it later
|
|
after we introduce the definition of proper substitution - see ~ stdpc4 .
|
|
|
|
An interesting alternate axiomatization uses ~ ax467 and ~ ax-5o in place
|
|
of ~ ax-4 , ~ ax-5 , ~ ax-6 , and ~ ax-7 .
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ sp . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-4 $a |- ( A. x ph -> ph ) $.
|
|
|
|
$( Axiom of Quantified Implication. This axiom moves a quantifier from
|
|
outside to inside an implication, quantifying ` ps ` . Notice that ` x `
|
|
must not be a free variable in the antecedent of the quantified
|
|
implication, and we express this by binding ` ph ` to "protect" the axiom
|
|
from a ` ph ` containing a free ` x ` . One of the 4 axioms of "pure"
|
|
predicate calculus. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the
|
|
preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5
|
|
of [Mendelson] p. 69.
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax5o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-5o $a |- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) ) $.
|
|
|
|
$( Axiom of Quantified Negation. This axiom is used to manipulate negated
|
|
quantifiers. One of the 4 axioms of pure predicate calculus. Equivalent
|
|
to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An
|
|
alternate axiomatization could use ~ ax467 in place of ~ ax-4 , ~ ax-6o ,
|
|
and ~ ax-7 .
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax6o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-6o $a |- ( -. A. x -. A. x ph -> ph ) $.
|
|
|
|
$( A variant of ~ ax9 . Axiom scheme C10' in [Megill] p. 448 (p. 16 of the
|
|
preprint).
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax9o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-9o $a |- ( A. x ( x = y -> A. x ph ) -> ph ) $.
|
|
|
|
$( Axiom ~ ax-10o ("o" for "old") was the original version of ~ ax-10 ,
|
|
before it was discovered (in May 2008) that the shorter ~ ax-10 could
|
|
replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of
|
|
the preprint).
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax10o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-10o $a |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $.
|
|
|
|
$( Axiom of Quantifier Substitution. One of the equality and substitution
|
|
axioms of predicate calculus with equality. Appears as Lemma L12 in
|
|
[Megill] p. 445 (p. 12 of the preprint).
|
|
|
|
The original version of this axiom was ~ ax-10o ("o" for "old") and was
|
|
replaced with this shorter ~ ax-10 in May 2008. The old axiom is proved
|
|
from this one as theorem ~ ax10o . Conversely, this axiom is proved from
|
|
~ ax-10o as theorem ~ ax10from10o .
|
|
|
|
This axiom was proved redundant in July 2015. See theorem ~ ax10 .
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax10 . (Contributed by NM, 16-May-2008.)
|
|
(New usage is discouraged.) $)
|
|
ax-10 $a |- ( A. x x = y -> A. y y = x ) $.
|
|
|
|
$( Axiom ~ ax-11o ("o" for "old") was the original version of ~ ax-11 ,
|
|
before it was discovered (in Jan. 2007) that the shorter ~ ax-11 could
|
|
replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of
|
|
the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of
|
|
[Monk2] p. 105, from which it can be proved by cases. To understand this
|
|
theorem more easily, think of " ` -. A. x x = y -> ` ..." as informally
|
|
meaning "if ` x ` and ` y ` are distinct variables then..." The
|
|
antecedent becomes false if the same variable is substituted for ` x ` and
|
|
` y ` , ensuring the theorem is sound whenever this is the case. In some
|
|
later theorems, we call an antecedent of the form ` -. A. x x = y ` a
|
|
"distinctor."
|
|
|
|
Interestingly, if the wff expression substituted for ` ph ` contains no
|
|
wff variables, the resulting statement _can_ be proved without invoking
|
|
this axiom. This means that even though this axiom is _metalogically_
|
|
independent from the others, it is not _logically_ independent.
|
|
Specifically, we can prove any wff-variable-free instance of axiom
|
|
~ ax-11o (from which the ~ ax-11 instance follows by theorem ~ ax11 .)
|
|
The proof is by induction on formula length, using ~ ax11eq and ~ ax11el
|
|
for the basis steps and ~ ax11indn , ~ ax11indi , and ~ ax11inda for the
|
|
induction steps. (This paragraph is true provided we use ~ ax-10o in
|
|
place of ~ ax-10 .)
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax11o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-11o $a |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $.
|
|
|
|
$( Axiom of Quantifier Introduction. One of the equality and substitution
|
|
axioms of predicate calculus with equality. Informally, it says that
|
|
whenever ` z ` is distinct from ` x ` and ` y ` , and ` x = y ` is true,
|
|
then ` x = y ` quantified with ` z ` is also true. In other words, ` z `
|
|
is irrelevant to the truth of ` x = y ` . Axiom scheme C9' in [Megill]
|
|
p. 448 (p. 16 of the preprint). It apparently does not otherwise appear
|
|
in the literature but is easily proved from textbook predicate calculus by
|
|
cases.
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax12o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-12o $a |- ( -. A. z z = x -> ( -. A. z z = y ->
|
|
( x = y -> A. z x = y ) ) ) $.
|
|
|
|
$( Axiom of Quantifier Introduction. One of the equality and substitution
|
|
axioms for a non-logical predicate in our predicate calculus with
|
|
equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint).
|
|
It is redundant if we include ~ ax-17 ; see theorem ~ ax15 . Alternately,
|
|
~ ax-17 becomes unnecessary in principle with this axiom, but we lose the
|
|
more powerful metalogic afforded by ~ ax-17 . We retain ~ ax-15 here to
|
|
provide completeness for systems with the simpler metalogic that results
|
|
from omitting ~ ax-17 , which might be easier to study for some
|
|
theoretical purposes.
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax15 . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-15 $a |- ( -. A. z z = x -> ( -. A. z z = y ->
|
|
( x e. y -> A. z x e. y ) ) ) $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Axiom of Distinct Variables. The only axiom of predicate calculus
|
|
requiring that variables be distinct (if we consider ~ ax-17 to be a
|
|
metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p.
|
|
16 of the preprint). It apparently does not otherwise appear in the
|
|
literature but is easily proved from textbook predicate calculus by
|
|
cases. It is a somewhat bizarre axiom since the antecedent is always
|
|
false in set theory (see dtru in set.mm), but nonetheless it is
|
|
technically necessary as you can see from its uses.
|
|
|
|
This axiom is redundant if we include ~ ax-17 ; see theorem ~ ax16 .
|
|
Alternately, ~ ax-17 becomes logically redundant in the presence of this
|
|
axiom, but without ~ ax-17 we lose the more powerful metalogic that
|
|
results from being able to express the concept of a setvar variable not
|
|
occurring in a wff (as opposed to just two setvar variables being
|
|
distinct). We retain ~ ax-16 here to provide logical completeness for
|
|
systems with the simpler metalogic that results from omitting ~ ax-17 ,
|
|
which might be easier to study for some theoretical purposes.
|
|
|
|
This axiom is obsolete and should no longer be used. It is proved above
|
|
as theorem ~ ax16 . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
ax-16 $a |- ( A. x x = y -> ( ph -> A. x ph ) ) $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Theorems ~ ax11 and ~ ax12from12o require some intermediate theorems that are
|
|
included in this section.
|
|
|
|
$)
|
|
|
|
$( This theorem repeats ~ sp under the name ~ ax4 , so that the metamath
|
|
program's "verify markup" command will check that it matches axiom scheme
|
|
~ ax-4 . It is preferred that references to this theorem use the name
|
|
~ sp . (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.)
|
|
(Proof modification is discouraged.) $)
|
|
ax4 $p |- ( A. x ph -> ph ) $=
|
|
( sp ) ABC $.
|
|
|
|
$( Rederivation of axiom ~ ax-5 from ~ ax-5o and other older axioms. See
|
|
~ ax5o for the derivation of ~ ax-5o from ~ ax-5 . (Contributed by NM,
|
|
23-May-2008.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax5 $p |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $=
|
|
( wi wal ax-5o ax-4 syl5 mpg syl ) ABDZCEZACEZBDZCEZMBCEDLNDLODCKNCFMALBACG
|
|
KCGHIABCFJ $.
|
|
|
|
$( Rederivation of axiom ~ ax-6 from ~ ax-6o and other older axioms. See
|
|
~ ax6o for the derivation of ~ ax-6o from ~ ax-6 . (Contributed by NM,
|
|
23-May-2008.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax6 $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
|
|
( wal wn wi ax-5o ax-4 id mpg nsyl ax-6o nsyl4 ) ABCZBCZDZBCZMDZBCZMPQEPREB
|
|
OQBFPNMOBGMMEMNEBAMBFMHIJIMBKL $.
|
|
|
|
$( Rederivation of axiom ~ ax-9 from ~ ax-9o and other older axioms. See
|
|
~ ax9o for the derivation of ~ ax-9o from ~ ax-9 . Lemma L18 in [Megill]
|
|
p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax9from9o $p |- -. A. x -. x = y $=
|
|
( weq wn wal wi ax-9o ax-6o con4i mpg ) ABCZKDZAEDZAEZFMAMABGNKLAHIJ $.
|
|
|
|
$( ` x ` is not free in ` A. x ph ` . Example in Appendix in [Megill] p. 450
|
|
(p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed
|
|
by NM, 5-Aug-1993.) (New usage is discouraged.) $)
|
|
hba1-o $p |- ( A. x ph -> A. x A. x ph ) $=
|
|
( wal wn ax-4 con2i ax6 con1i alimi 3syl ) ABCZKDZBCZDZNBCKBCMKLBEFLBGNKBKM
|
|
ABGHIJ $.
|
|
|
|
${
|
|
a5i-o.1 $e |- ( A. x ph -> ps ) $.
|
|
$( Inference version of ~ ax-5o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
a5i-o $p |- ( A. x ph -> A. x ps ) $=
|
|
( wal hba1-o alrimih ) ACEBCACFDG $.
|
|
$}
|
|
|
|
$( Commutation law for identical variable specifiers. The antecedent and
|
|
consequent are true when ` x ` and ` y ` are substituted with the same
|
|
variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version
|
|
of ~ aecom using ~ ax-10o . Unlike ~ ax10from10o , this version does not
|
|
require ~ ax-17 . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
aecom-o $p |- ( A. x x = y -> A. y y = x ) $=
|
|
( weq wal ax-10o pm2.43i equcomi alimi syl ) ABCZADZJBDZBACZBDKLJABEFJMBABG
|
|
HI $.
|
|
|
|
${
|
|
alequcoms-o.1 $e |- ( A. x x = y -> ph ) $.
|
|
$( A commutation rule for identical variable specifiers. Version of
|
|
~ aecoms using ax-10o . (Contributed by NM, 5-Aug-1993.)
|
|
(New usage is discouraged.) $)
|
|
aecoms-o $p |- ( A. y y = x -> ph ) $=
|
|
( weq wal aecom-o syl ) CBECFBCEBFACBGDH $.
|
|
$}
|
|
|
|
$( All variables are effectively bound in an identical variable specifier.
|
|
Version of ~ hbae using ~ ax-10o . (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is disccouraged.) (New usage is discouraged.) $)
|
|
hbae-o $p |- ( A. x x = y -> A. z A. x x = y ) $=
|
|
( weq wal wi wn ax-4 ax-12o syl7 ax-10o aecoms-o pm2.43i syl5 pm2.61ii ax-7
|
|
a5i-o syl ) ABDZAEZSCEZAETCESUAACADCEZCBDCEZTUAFZTSUBGUCGUASAHABCIJUDACSACK
|
|
LUDBCTSBEZBCDBEUATUESABKMSBCKNLOQSACPR $.
|
|
|
|
${
|
|
dral1-o.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of
|
|
~ dral1 using ~ ax-10o . (Contributed by NM, 24-Nov-1994.)
|
|
(New usage is discouraged.) $)
|
|
dral1-o $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $=
|
|
( weq wal hbae-o biimpd alimdh ax-10o syld biimprd wi aecoms-o impbid ) C
|
|
DFCGZACGZBDGZQRBCGSQABCCDCHQABEIJBCDKLQSADGZRQBADCDDHQABEMJTRNDCADCKOLP
|
|
$.
|
|
$}
|
|
|
|
$( Rederivation of axiom ~ ax-11 from ~ ax-11o , ~ ax-10o , and other older
|
|
axioms. The proof does not require ~ ax-16 or ~ ax-17 . See theorem
|
|
~ ax11o for the derivation of ~ ax-11o from ~ ax-11 .
|
|
|
|
An open problem is whether we can prove this using ~ ax-10 instead of
|
|
~ ax-10o .
|
|
|
|
This proof uses newer axioms ~ ax-5 and ~ ax-9 , but since these are
|
|
proved from the older axioms above, this is acceptable and lets us avoid
|
|
having to reprove several earlier theorems to use ~ ax-5o and ~ ax-9o .
|
|
(Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax11 $p |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $=
|
|
( weq wal wi biidd dral1-o ax-1 alimi syl6bir a1d ax-4 ax-11o syl7 pm2.61i
|
|
wn ) BCDZBEZRACEZRAFZBEZFZFSUCRSTABEUBAABCSAGHAUABARIJKLTASQRUBACMABCNOP $.
|
|
|
|
$( Derive ~ ax-12 from ~ ax-12o and other older axioms.
|
|
|
|
This proof uses newer axioms ~ ax-5 and ~ ax-9 , but since these are
|
|
proved from the older axioms above, this is acceptable and lets us avoid
|
|
having to reprove several earlier theorems to use ~ ax-5o and ~ ax-9o .
|
|
(Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax12from12o $p |- ( -. x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( weq wn wal wi wa ax-4 con3i adantr equtrr equcoms con3rr3 imp nsyl ax-12o
|
|
sylc ex pm2.43d ) ABDZEZBCDZUCAFZUBUCUCUDGZUBUCHZUAAFZEZACDZAFZEUEUBUHUCUGU
|
|
AUAAIJKUFUIUJUBUCUIEUCUIUAUIUAGCBCBALMNOUIAIPBCAQRST $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Legacy theorems using obsolete axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
These theorems were mostly intended to study properties of the older axiom
|
|
schemes and are not useful outside of this section. They should not be
|
|
used outside of this section. They may be deleted when they are deemed to no
|
|
longer be of interest.
|
|
|
|
$)
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Axiom to quantify a variable over a formula in which it does not occur.
|
|
Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as
|
|
Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of
|
|
[Monk2] p. 113.
|
|
|
|
(This theorem simply repeats ~ ax-17 so that we can include the
|
|
following note, which applies only to the obsolete axiomatization.)
|
|
|
|
This axiom is _logically_ redundant in the (logically complete)
|
|
predicate calculus axiom system consisting of ~ ax-gen , ~ ax-5o ,
|
|
~ ax-4 , ~ ax-7 , ~ ax-6o , ~ ax-8 , ~ ax-12o , ~ ax-9o , ~ ax-10o ,
|
|
~ ax-13 , ~ ax-14 , ~ ax-15 , ~ ax-11o , and ~ ax-16 : in that system,
|
|
we can derive any instance of ~ ax-17 not containing wff variables by
|
|
induction on formula length, using ~ ax17eq and ~ ax17el for the basis
|
|
together ~ hbn , ~ hbal , and ~ hbim . However, if we omit this axiom,
|
|
our development would be quite inconvenient since we could work only
|
|
with specific instances of wffs containing no wff variables - this axiom
|
|
introduces the concept of a setvar variable not occurring in a wff (as
|
|
opposed to just two setvar variables being distinct). (Contributed by
|
|
NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification
|
|
discouraged.) $)
|
|
ax17o $p |- ( ph -> A. x ph ) $=
|
|
( ax-17 ) ABC $.
|
|
$}
|
|
|
|
$( Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This
|
|
is often an axiom of equality in textbook systems, but we don't need it as
|
|
an axiom since it can be proved from our other axioms (although the proof,
|
|
as you can see below, is not as obvious as you might think). This proof
|
|
uses only axioms without distinct variable conditions and thus requires no
|
|
dummy variables. A simpler proof, similar to Tarki's, is possible if we
|
|
make use of ~ ax-17 ; see the proof of ~ equid . See ~ equid1ALT for an
|
|
alternate proof. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
equid1 $p |- x = x $=
|
|
( weq wal wn wi ax-5o ax-4 ax-12o sylc mpg ax-9o syl ax-6o pm2.61i ) AABZAC
|
|
ZDZACZOROPEZACZORSERTEAQSAFRQQSQAGZUAAAAHIJOAAKLOAMN $.
|
|
|
|
${
|
|
sps-o.1 $e |- ( ph -> ps ) $.
|
|
$( Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
sps-o $p |- ( A. x ph -> ps ) $=
|
|
( wal ax-4 syl ) ACEABACFDG $.
|
|
$}
|
|
|
|
$( Bound-variable hypothesis builder for ` x = x ` . This theorem tells us
|
|
that any variable, including ` x ` , is effectively not free in
|
|
` x = x ` , even though ` x ` is technically free according to the
|
|
traditional definition of free variable. (The proof does not use
|
|
~ ax-9o .) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf
|
|
Lammen, 23-Mar-2014.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
hbequid $p |- ( x = x -> A. y x = x ) $=
|
|
( weq wal wi ax-12o ax-8 pm2.43i alimi a1d pm2.61ii ) BACZBDZMAACZNBDZEAABF
|
|
MONLNBLNBAAGHIJZPK $.
|
|
|
|
$( Bound-variable hypothesis builder for ` x = x ` . This theorem tells us
|
|
that any variable, including ` x ` , is effectively not free in
|
|
` x = x ` , even though ` x ` is technically free according to the
|
|
traditional definition of free variable. (The proof uses only ~ ax-5 ,
|
|
~ ax-8 , ~ ax-12o , and ~ ax-gen . This shows that this can be proved
|
|
without ~ ax9 , even though the theorem ~ equid cannot be. A shorter
|
|
proof using ~ ax9 is obtainable from ~ equid and ~ hbth .) Remark added
|
|
2-Dec-2015 NM: This proof does implicitly use ~ ax9v , which is used for
|
|
the derivation of ~ ax12o , unless we consider ~ ax-12o the starting axiom
|
|
rather than ~ ax-12 . (Contributed by NM, 13-Jan-2011.) (Revised by
|
|
Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nfequid-o $p |- F/ y x = x $=
|
|
( weq hbequid nfi ) AACBABDE $.
|
|
|
|
$( Proof of a single axiom that can replace ~ ax-4 and ~ ax-6o . See
|
|
~ ax46to4 and ~ ax46to6 for the re-derivation of those axioms.
|
|
(Contributed by Scott Fenton, 12-Sep-2005.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax46 $p |- ( ( A. x -. A. x ph -> A. x ph ) -> ph ) $=
|
|
( wal wn ax-6o ax-4 ja ) ABCZDBCHAABEABFG $.
|
|
|
|
$( Re-derivation of ~ ax-4 from ~ ax46 . Only propositional calculus is used
|
|
for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax46to4 $p |- ( A. x ph -> ph ) $=
|
|
( wal wn wi ax-1 ax46 syl ) ABCZIDBCZIEAIJFABGH $.
|
|
|
|
$( Re-derivation of ~ ax-6o from ~ ax46 . Only propositional calculus is
|
|
used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax46to6 $p |- ( -. A. x -. A. x ph -> ph ) $=
|
|
( wal wn wi pm2.21 ax46 syl ) ABCZDBCZDJIEAJIFABGH $.
|
|
|
|
$( Proof of a single axiom that can replace both ~ ax-6o and ~ ax-7 . See
|
|
~ ax67to6 and ~ ax67to7 for the re-derivation of those axioms.
|
|
(Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax67 $p |- ( -. A. x -. A. y A. x ph -> A. y ph ) $=
|
|
( wal wn ax-7 con3i alimi ax-6o syl ) ABDCDZEZBDZEACDZBDZEZBDZENQMPLBKOACBF
|
|
GHGNBIJ $.
|
|
|
|
$( ` x ` is not free in ` A. x ph ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
nfa1-o $p |- F/ x A. x ph $=
|
|
( wal hba1-o nfi ) ABCBABDE $.
|
|
|
|
$( Re-derivation of ~ ax-6o from ~ ax67 . Note that ~ ax-6o and ~ ax-7 are
|
|
not used by the re-derivation. (Contributed by NM, 18-Nov-2006.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax67to6 $p |- ( -. A. x -. A. x ph -> ph ) $=
|
|
( wal wn hba1-o con3i alimi ax67 ax-4 3syl ) ABCZDZBCZDKBCZDZBCZDKAPMOLBKNA
|
|
BEFGFABBHABIJ $.
|
|
|
|
$( Re-derivation of ~ ax-7 from ~ ax67 . Note that ~ ax-6o and ~ ax-7 are
|
|
not used by the re-derivation. (Contributed by NM, 18-Nov-2006.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax67to7 $p |- ( A. x A. y ph -> A. y A. x ph ) $=
|
|
( wal wn ax67to6 con4i ax67 alimi syl ) ACDBDZKEZCDEZCDZABDZCDNKLCFGMOCACBH
|
|
IJ $.
|
|
|
|
$( Proof of a single axiom that can replace ~ ax-4 , ~ ax-6o , and ~ ax-7 in
|
|
a subsystem that includes these axioms plus ~ ax-5o and ~ ax-gen (and
|
|
propositional calculus). See ~ ax467to4 , ~ ax467to6 , and ~ ax467to7 for
|
|
the re-derivation of those axioms. This theorem extends the idea in Scott
|
|
Fenton's ~ ax46 . (Contributed by NM, 18-Nov-2006.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax467 $p |- ( ( A. x A. y -. A. x A. y ph -> A. x ph ) -> ph ) $=
|
|
( wal wn ax-4 ax6 ax-6o con1i alimi ax-7 3syl nsyl4 ja ) ACDZBDEZCDBDZABDAO
|
|
AQACFOEZRCDPBDZCDQACGRSCSOOBHIJPCBKLMABFN $.
|
|
|
|
$( Re-derivation of ~ ax-4 from ~ ax467 . Only propositional calculus is
|
|
used by the re-derivation. (Contributed by NM, 19-Nov-2006.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax467to4 $p |- ( A. x ph -> ph ) $=
|
|
( wal wn wi ax-1 ax467 syl ) ABCZIBCDBCBCZIEAIJFABBGH $.
|
|
|
|
$( Re-derivation of ~ ax-6o from ~ ax467 . Note that ~ ax-6o and ~ ax-7 are
|
|
not used by the re-derivation. The use of ~ alimi (which uses ~ ax-4 ) is
|
|
allowed since we have already proved ~ ax467to4 . (Contributed by NM,
|
|
19-Nov-2006.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax467to6 $p |- ( -. A. x -. A. x ph -> ph ) $=
|
|
( wal wn wi hba1-o con3i alimi sps-o pm2.21 ax467 3syl ) ABCZDZBCZDMBCZDZBC
|
|
ZBCZDSMEASOROBQNBMPABFGHIGSMJABBKL $.
|
|
|
|
$( Re-derivation of ~ ax-7 from ~ ax467 . Note that ~ ax-6o and ~ ax-7 are
|
|
not used by the re-derivation. The use of ~ alimi (which uses ~ ax-4 ) is
|
|
allowed since we have already proved ~ ax467to4 . (Contributed by NM,
|
|
19-Nov-2006.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax467to7 $p |- ( A. x A. y ph -> A. y A. x ph ) $=
|
|
( wal wn ax467to6 con4i wi pm2.21 ax467 syl alimi nsyl4 ) ACDBDZNEZCDZEZCDZ
|
|
ABDZCDRNOCFGQSCPBDZEZBDSPUAABUATSHATSIABCJKLPBFMLK $.
|
|
|
|
$( ~ equid with existential quantifier without using ~ ax-4 or ~ ax-17 .
|
|
(Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen,
|
|
27-Feb-2014.) (Proof modification is discouraged.) $)
|
|
equidqe $p |- -. A. y -. x = x $=
|
|
( weq wn wal ax9from9o ax-8 pm2.43i con3i alimi mto ) AACZDZBEBACZDZBEBAFMO
|
|
BNLNLBAAGHIJK $.
|
|
|
|
$( A special case of ~ ax-4 without using ~ ax-4 or ~ ax-17 . (Contributed
|
|
by NM, 13-Jan-2011.) (Proof modification is discouraged.) $)
|
|
ax4sp1 $p |- ( A. y -. x = x -> -. x = x ) $=
|
|
( weq wn wal equidqe pm2.21i ) AACDZBEHABFG $.
|
|
|
|
$( ~ equid with universal quantifier without using ~ ax-4 or ~ ax-17 .
|
|
(Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
equidq $p |- A. y x = x $=
|
|
( weq wal wn equidqe ax6 hbequid con3i alrimih mt3 ) AACZBDZLEZBDABFMENBLBG
|
|
LMABHIJK $.
|
|
|
|
$( Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68.
|
|
Alternate proof of ~ equid1 from older axioms ~ ax-6o and ~ ax-9o .
|
|
(Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
equid1ALT $p |- x = x $=
|
|
( weq wal wn wi ax-12o pm2.43i alimi ax-9o syl ax-6o pm2.61i ) AABZACZDZACZ
|
|
MPMNEZACMOQAOQAAAFGHMAAIJMAKL $.
|
|
|
|
$( Rederivation of ~ ax-10 from original version ~ ax-10o . See theorem
|
|
~ ax10o for the derivation of ~ ax-10o from ~ ax-10 .
|
|
|
|
This theorem should not be referenced in any proof. Instead, use ~ ax-10
|
|
above so that uses of ~ ax-10 can be more easily identified, or use
|
|
~ aecom-o when this form is needed for studies involving ~ ax-10o and
|
|
omitting ~ ax-17 . (Contributed by NM, 16-May-2008.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax10from10o $p |- ( A. x x = y -> A. y y = x ) $=
|
|
( weq wal ax-10o pm2.43i equcomi alimi syl ) ABCZADZJBDZBACZBDKLJABEFJMBABG
|
|
HI $.
|
|
|
|
${
|
|
nalequcoms-o.1 $e |- ( -. A. x x = y -> ph ) $.
|
|
$( A commutation rule for distinct variable specifiers. Version of
|
|
~ naecoms using ~ ax-10o . (Contributed by NM, 2-Jan-2002.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
naecoms-o $p |- ( -. A. y y = x -> ph ) $=
|
|
( weq wal aecom-o nsyl4 con1i ) ACBECFZBCEBFJABCGDHI $.
|
|
$}
|
|
|
|
$( All variables are effectively bound in a distinct variable specifier.
|
|
Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of ~ hbnae
|
|
using ~ ax-10o . (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
hbnae-o $p |- ( -. A. x x = y -> A. z -. A. x x = y ) $=
|
|
( weq wal hbae-o hbn ) ABDAECABCFG $.
|
|
|
|
${
|
|
dvelimf-o.1 $e |- ( ph -> A. x ph ) $.
|
|
dvelimf-o.2 $e |- ( ps -> A. z ps ) $.
|
|
dvelimf-o.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
|
|
$( Proof of ~ dvelimh that uses ~ ax-10o but not ~ ax-11o , ~ ax-10 , or
|
|
~ ax-11 . Version of ~ dvelimh using ~ ax-10o instead of ~ ax10o .
|
|
(Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
dvelimf-o $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
|
|
( weq wal wn wi hba1-o ax-10o aecoms-o syl5 a1d wa hbnae-o hban imp hbimd
|
|
ax-12o a1i hbald ex pm2.61i equsalh albii 3imtr3g ) CDICJKZEDIZALZEJZUNCJ
|
|
ZBBCJCEICJZUKUNUOLZLUPUQUKUNUNEJZUPUOUMEMURUOLECUNECNOPQUPKZUKUQUSUKRZUMC
|
|
EUSUKECEESCDESTUTULACUSUKCCECSCDCSTUSUKULULCJLEDCUCUAAACJLUTFUDUBUEUFUGAB
|
|
EDGHUHZUNBCVAUIUJ $.
|
|
$}
|
|
|
|
${
|
|
dral2-o.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of
|
|
~ dral2 using ~ ax-10o . (Contributed by NM, 27-Feb-2005.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
dral2-o $p |- ( A. x x = y -> ( A. z ph <-> A. z ps ) ) $=
|
|
( weq wal hbae-o albidh ) CDGCHABECDEIFJ $.
|
|
$}
|
|
|
|
${
|
|
$d t u v $. $d t u x y $. $d u w $.
|
|
$( A "distinctor elimination" lemma with no restrictions on variables in
|
|
the consequent, proved without using ~ ax-16 . Version of ~ aev using
|
|
~ ax-10o . (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew
|
|
Salmon, 21-Jun-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
aev-o $p |- ( A. x x = y -> A. z w = v ) $=
|
|
( vt vu weq hbae-o ax-8 spimv alrimih equcomi syl6 aecoms-o a5i-o aecom-o
|
|
wal 3syl ) ABHZARZDEHZCABCIUAFBHZFRZGEHZGRZUBUAUCFABFITUCAFAFBJKLUDFGHZFR
|
|
ZEGHZERUFUCUGFUGBFBFHZUGBGBGHUJGFHUGBGFJGFMNKOPUHUIEFGEIUGUIFEFEGJKLEGQSU
|
|
EUBGDGDEJKSL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( Theorem to add distinct quantifier to atomic formula. (This theorem
|
|
demonstrates the induction basis for ~ ax-17 considered as a
|
|
metatheorem. Do not use it for later proofs - use ~ ax-17 instead, to
|
|
avoid reference to the redundant axiom ~ ax-16 .) (Contributed by NM,
|
|
5-Aug-1993.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax17eq $p |- ( x = y -> A. z x = y ) $=
|
|
( weq wal wi ax-12o ax-16 pm2.61ii ) CADCECBDCEABDZJCEFABCGJCAHJCBHI $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w y $.
|
|
$( Quantifier introduction when one pair of variables is distinct. Version
|
|
of ~ dveeq2 using ~ ax-11o . (Contributed by NM, 2-Jan-2002.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
dveeq2-o $p |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $=
|
|
( vw weq ax-17 equequ2 dvelimf-o ) CDEZCBEZABDIAFJDFDBCGH $.
|
|
|
|
$( Version of ~ dveeq2 using ~ ax-16 instead of ~ ax-17 . TO DO: Recover
|
|
proof from older set.mm to remove use of ~ ax-17 . (Contributed by NM,
|
|
29-Apr-2008.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
dveeq2-o16 $p |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $=
|
|
( vw weq ax17eq equequ2 dvelimALT ) CDECBEABDCDAFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( A generalization of axiom ~ ax-16 . Version of ~ a16g using ~ ax-10o .
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
a16g-o $p |- ( A. x x = y -> ( ph -> A. z ph ) ) $=
|
|
( weq wal aev-o ax-16 biidd dral1-o biimprd sylsyld ) BCEBFDBEDFZAABFZADF
|
|
ZBCDDBGABCHMONAADBMAIJKL $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w y $.
|
|
$( Quantifier introduction when one pair of variables is distinct. Version
|
|
of ~ dveeq1 using ax-10o . (Contributed by NM, 2-Jan-2002.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
dveeq1-o $p |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( vw weq ax-17 equequ1 dvelimf-o ) DCEZBCEZABDIAFJDFDBCGH $.
|
|
|
|
$( Version of ~ dveeq1 using ~ ax-16 instead of ~ ax-17 . (Contributed by
|
|
NM, 29-Apr-2008.) TO DO: Recover proof from older set.mm to remove use
|
|
of ~ ax-17 . (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
dveeq1-o16 $p |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $=
|
|
( vw weq ax17eq equequ1 dvelimh ) DCEBCEABDDCAFBCDFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( Theorem to add distinct quantifier to atomic formula. This theorem
|
|
demonstrates the induction basis for ~ ax-17 considered as a
|
|
metatheorem.) (Contributed by NM, 5-Aug-1993.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax17el $p |- ( x e. y -> A. z x e. y ) $=
|
|
( weq wal wel wi ax-15 ax-16 pm2.61ii ) CADCECBDCEABFZKCEGABCHKCAIKCBIJ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x z w $.
|
|
$( This theorem shows that, given ~ ax-16 , we can derive a version of
|
|
~ ax-10 . However, it is weaker than ~ ax-10 because it has a distinct
|
|
variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax10-16 $p |- ( A. x x = z -> A. z z = x ) $=
|
|
( vw weq wal ax-16 alrimiv a5i-o equequ1 cbvalv a1i imbi12d albidv biimpi
|
|
wi wb wex nfa1-o 19.23 a7s albii pm2.27 ax-mp alimi equequ2 spv sps-o syl
|
|
a9ev sylbi 3syl ) ABDZAEZACDZUNAEZOZCEZAEZBCDZUSBEZOZCEZBEZBADZBEULUQAUMU
|
|
PCUNABFGHURVCUQVBABULUPVACULUNUSUOUTABCIZUOUTPULUNUSABVEJKLMJNVBVDBVAVDCB
|
|
VABEZCEUSBQZUTOZCEZVDVFVHCUSUTBUSBRSUAVIUTCEVDVHUTCVGVHUTOBCUIVGUTUBUCUDU
|
|
SVDBCUSCEVDBUSVDCACABUEUFUGTUHUJTHUK $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w y $.
|
|
$( Version of ~ dveel2 using ~ ax-16 instead of ~ ax-17 . (Contributed by
|
|
NM, 10-May-2008.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
dveel2ALT $p |- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) ) $=
|
|
( vw wel ax17el elequ2 dvelimh ) CDECBEABDCDAFCBDFDBCGH $.
|
|
$}
|
|
|
|
${
|
|
ax11f.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Basis step for constructing a substitution instance of ~ ax-11o without
|
|
using ~ ax-11o . We can start with any formula ` ph ` in which ` x ` is
|
|
not free. (Contributed by NM, 21-Jan-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax11f $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( weq wal wn wi ax-1 alrimih 2a1i ) BCEZBFGLALAHZBFHAMBDALIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x u v $. $d y u v $. $d z u v $. $d w u v $.
|
|
$( Basis step for constructing a substitution instance of ~ ax-11o without
|
|
using ~ ax-11o . Atomic formula for equality predicate. (Contributed
|
|
by NM, 22-Jan-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax11eq $p |- ( -. A. x x = y ->
|
|
( x = y -> ( z = w -> A. x ( x = y -> z = w ) ) ) ) $=
|
|
( vu vv weq wal wn wi wa 19.26 a1i wb equequ1 equequ2 sps-o imbi12d exp32
|
|
imbi2d equid ax-gen sylan9bb nfa1-o adantr sylbir ad2antll impcom adantrr
|
|
albid mpbii ax12o equtrr alimi syl6 sylbid adantll dral2-o ad2antrr mpbid
|
|
imp biimprcd adantlr ad2antlr wex alrimiv adantl dveeq2-o im2anan9 sylibr
|
|
a9ev ax-1 syl exlimdv mpi a1d 4cases ) ACGZAHZADGZAHZABGZAHIZWBCDGZWBWDJZ
|
|
AHZJZJZJZVSWAKVRVTKZAHZWIVRVTALWKWCWBWGWKWCWBKZKAAGZWBWMJZAHZJZWGWOWMWNAW
|
|
MWBAUAMUBMWKWPWGNWLWKWMWDWOWFWJWMWDNAVRWMCAGZVTWDACAOADCPZUCQZWKWNWEAWJAU
|
|
DWKWMWDWBWSTUJRUEUKSUFVSWAIZKZWCWBWGXAWLKVTWBVTJZAHZJZWGWTWLXDVSWTWLKZVTB
|
|
DGZXCWBVTXFNWTWCABDOUGXEXFXFAHZXCWTWCXFXGJZWBWCWTXHBDAULUHUIXFXBABDAUMUNU
|
|
OUPUQVSXDWGNWTWLVSVTWDXCWFVRVTWDNAACDOQZXBWEACAVSVTWDWBXITURRUSUTSVSIZWAK
|
|
ZWCWBWGXKWLKWQWBWQJZAHZJZWGXJWLXNWAXJWLKZWQCBGZXMWBWQXPNXJWCABCPZUGXOXPXP
|
|
AHZXMXJWCXPXRJZWBXJWCXSCBAULVAUIXPXLAWBWQXPXQVBUNUOUPVCWAXNWGNXJWLWAWQWDX
|
|
MWFVTWQWDNAWRQZXLWEADAWAWQWDWBXTTURRVDUTSXJWTKZWHWCYAWGWBYAEDGZEVEWGEDVKY
|
|
AYBWGEYAFCGZFVEYBWGJZFCVKYAYCYDFYAYCYBWGYAYCYBKZKZFEGZWBYGJZAHZJWGYGYHAYG
|
|
WBVLVFYFYGWDYIWFYEYGWDNZYAYCYGCEGYBWDFCEOEDCPUCZVGYFYEAHZYIWFNYFYCAHZYBAH
|
|
ZKZYLYAYEYOXJYCYMWTYBYNACFVHADEVHVIVAYCYBALVJYLYHWEAYEAUDYLYGWDWBYEYJAYKQ
|
|
TUJVMRUKSVNVOVNVOVPVPVQ $.
|
|
$}
|
|
|
|
${
|
|
$d x u v $. $d y u v $. $d z u v $. $d w u v $.
|
|
$( Basis step for constructing a substitution instance of ~ ax-11o without
|
|
using ~ ax-11o . Atomic formula for membership predicate. (Contributed
|
|
by NM, 22-Jan-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax11el $p |- ( -. A. x x = y ->
|
|
( x = y -> ( z e. w -> A. x ( x = y -> z e. w ) ) ) ) $=
|
|
( vv vu weq wal wn wel wi wa wb elequ1 elequ2 adantl sps-o imbi2d imbi12d
|
|
exp32 19.26 bitrd ax-17 biimprcd alimi syl6 adantr sylbid sylan9bb nfa1-o
|
|
albid mpbid sylbir ad2antll ax-15 impcom adantrr adantll dral2-o ad2antrr
|
|
dvelimf-o imp adantlr ad2antlr a9ev ax-1 alrimiv dveeq2-o im2anan9 sylibr
|
|
wex syl mpbii exlimdv mpi a1d 4cases ) ACGZAHZADGZAHZABGZAHIZWBCDJZWBWDKZ
|
|
AHZKZKZKZVSWALVRVTLZAHZWIVRVTAUAWKWCWBWGWKWCWBLZLAAJZWBWMKZAHZKZWGWLWPWKW
|
|
LWMBBJZWOWBWMWQMWCWBWMBAJWQABANABBOUBZPWCWQWOKWBWCWQWQAHWOEEJZWQABEWSAUCW
|
|
QEUCEBGWSBEJWQEBENEBBOUBVAWQWNAWBWMWQWRUDUEUFUGUHPWKWPWGMWLWKWMWDWOWFWJWM
|
|
WDMAVRWMCAJZVTWDACANADCOZUIQZWKWNWEAWJAUJWKWMWDWBXBRUKSUGULTUMVSWAIZLZWCW
|
|
BWGXDWLLADJZWBXEKZAHZKZWGXCWLXHVSXCWLLZXEBDJZXGWBXEXJMXCWCABDNZUNXIXJXJAH
|
|
ZXGXCWCXJXLKZWBWCXCXMBDAUOUPUQXJXFAWBXEXJXKUDUEUFUHURVSXHWGMXCWLVSXEWDXGW
|
|
FVRXEWDMAACDNQZXFWEACAVSXEWDWBXNRUSSUTULTVSIZWALZWCWBWGXPWLLWTWBWTKZAHZKZ
|
|
WGXOWLXSWAXOWLLZWTCBJZXRWBWTYAMXOWCABCOZUNXTYAYAAHZXRXOWCYAYCKZWBXOWCYDCB
|
|
AUOVBUQYAXQAWBWTYAYBUDUEUFUHVCWAXSWGMXOWLWAWTWDXRWFVTWTWDMAXAQZXQWEADAWAW
|
|
TWDWBYERUSSVDULTXOXCLZWHWCYFWGWBYFFDGZFVKWGFDVEYFYGWGFYFECGZEVKYGWGKZECVE
|
|
YFYHYIEYFYHYGWGYFYHYGLZLZEFJZWBYLKZAHZKWGYLYMAYLWBVFVGYKYLWDYNWFYJYLWDMZY
|
|
FYHYLCFJYGWDECFNFDCOUIZPYKYJAHZYNWFMYKYHAHZYGAHZLZYQYFYJYTXOYHYRXCYGYSACE
|
|
VHADFVHVIVBYHYGAUAVJYQYMWEAYJAUJYQYLWDWBYJYOAYPQRUKVLSVMTVNVOVNVOVPVPVQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
ax11indn.1 $e |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $.
|
|
$( Induction step for constructing a substitution instance of ~ ax-11o
|
|
without using ~ ax-11o . Negation case. (Contributed by NM,
|
|
21-Jan-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax11indn $p |- ( -. A. x x = y ->
|
|
( x = y -> ( -. ph -> A. x ( x = y -> -. ph ) ) ) ) $=
|
|
( weq wal wn wi wa 19.8a exanali hbn1 con3 syl6 com23 alrimdh syl5bi syl5
|
|
wex exp3a ) BCEZBFGZUAAGZUAUCHZBFZUAUCIZUFBSZUBUEUFBJUGUAAHZBFZGZUBUEUAAB
|
|
KUBUJUDBUABLUHBLUBUAUJUCUBUAAUIHUJUCHDAUIMNOPQRT $.
|
|
|
|
${
|
|
ax11indi.2 $e |- ( -. A. x x = y ->
|
|
( x = y -> ( ps -> A. x ( x = y -> ps ) ) ) ) $.
|
|
$( Induction step for constructing a substitution instance of ~ ax-11o
|
|
without using ~ ax-11o . Implication case. (Contributed by NM,
|
|
21-Jan-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax11indi $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ( ph -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) ) ) ) $=
|
|
( weq wal wn wi wa ax11indn imp pm2.21 imim2i alimi syl6 ax-1 jad ex )
|
|
CDGZCHIZUAABJZUAUCJZCHZJUBUAKZABUEUFAIZUAUGJZCHZUEUBUAUGUIJACDELMUHUDCU
|
|
GUCUAABNOPQUFBUABJZCHZUEUBUABUKJFMUJUDCBUCUABAROPQST $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ax11indalem.1 $e |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $.
|
|
$( Lemma for ~ ax11inda2 and ~ ax11inda . (Contributed by NM,
|
|
24-Jan-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax11indalem $p |- ( -. A. y y = z -> ( -. A. x x = y ->
|
|
( x = y -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) ) ) ) $=
|
|
( weq wal wn wi ax-1 a5i-o a1i biidd a1d aecom-o con3i imp hbnae-o hban
|
|
wa dral1-o imbi2d dral2-o 3imtr4d adantr simplr ax12o syl2an adantlr ax-4
|
|
aecoms-o hba1-o sylan2 alimdh syl2anc ax-7 wb nfdh 19.21t albidh ad2antrr
|
|
wnf syl syl5ib syld exp31 pm2.61ian ) BDFBGZCDFCGZHZBCFZBGHZVKADGZVKVMIZB
|
|
GZIZIZIZVHVRVJVHVQVLVHVPVKVPDBDBFDGZABGZVKVTIZBGZVMVOVTWBIVSAWABVTVKJKLAA
|
|
DBVSAMUAZVNWADBBVSVMVTVKWCUBUCUDUKNNUEVHHZVJTZVLVKVPWEVLTVKTZVMVKAIZBGZDG
|
|
ZVOWFVLVKDGZVMWIIWEVLVKUFWEVKWJVLWEVKWJWDVSHZDCFDGZHZVKWJIZVJVSVHDBOPWLVI
|
|
DCOPWKWMWNBCDUGQUHZQUIVLWJTAWHDVLWJDBCDRVKDULSWJVLVKAWHIZVKDUJVLVKWPEQUMU
|
|
NUOWEWIVOIVLVKWIWGDGZBGWEVOWGDBUPWEWQVNBWDVJBBDBRCDBRSWEVKDVBWQVNUQWEVKDW
|
|
DVJDBDDRCDDRSWOURVKADUSVCUTVDVAVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d z y $.
|
|
ax11inda2.1 $e |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $.
|
|
$( A proof of ~ ax11inda2 that is slightly more direct. (Contributed by
|
|
NM, 4-May-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax11inda2ALT $p |- ( -. A. x x = y ->
|
|
( x = y -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) ) ) $=
|
|
( weq wal wn wi a5i-o a1i biidd dral1-o imbi2d dral2-o a1d wa imp hbnae-o
|
|
ax-1 3imtr4d aecoms-o simplr dveeq1-o naecoms-o hba1-o hban sylan2 alimdh
|
|
adantlr ax-4 syl2anc ax-7 wnf nfdh 19.21t syl albidh syl5ib ad2antrr syld
|
|
wb exp31 pm2.61i ) BDFBGZBCFZBGHZVFADGZVFVHIZBGZIZIZIVEVLVGVEVKVFVKDBDBFD
|
|
GZABGZVFVNIZBGZVHVJVNVPIVMAVOBVNVFTJKAADBVMALMZVIVODBBVMVHVNVFVQNOUAUBPPV
|
|
EHZVGVFVKVRVGQVFQZVHVFAIZBGZDGZVJVSVGVFDGZVHWBIVRVGVFUCVRVFWCVGVRVFWCVFWC
|
|
IDBDBCUDUEZRUJVGWCQAWADVGWCDBCDSVFDUFUGWCVGVFAWAIZVFDUKVGVFWEERUHUIULVRWB
|
|
VJIVGVFWBVTDGZBGVRVJVTDBUMVRWFVIBBDBSVRVFDUNWFVIVBVRVFDBDDSWDUOVFADUPUQUR
|
|
USUTVAVCVD $.
|
|
|
|
$( Induction step for constructing a substitution instance of ~ ax-11o
|
|
without using ~ ax-11o . Quantification case. When ` z ` and ` y ` are
|
|
distinct, this theorem avoids the dummy variables needed by the more
|
|
general ~ ax11inda . (Contributed by NM, 24-Jan-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax11inda2 $p |- ( -. A. x x = y ->
|
|
( x = y -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) ) ) $=
|
|
( weq wal wn wi ax-1 a16g-o syl5 a1d ax11indalem pm2.61i ) CDFCGZBCFZBGHZ
|
|
QADGZQSIZBGZIZIZIPUCRPUBQSTPUASQJTCDBKLMMABCDENO $.
|
|
$}
|
|
|
|
${
|
|
$d w ph $. $d w x $. $d w y $. $d w z $.
|
|
ax11inda.1 $e |- ( -. A. x x = w ->
|
|
( x = w -> ( ph -> A. x ( x = w -> ph ) ) ) ) $.
|
|
$( Induction step for constructing a substitution instance of ~ ax-11o
|
|
without using ~ ax-11o . Quantification case. (When ` z ` and ` y `
|
|
are distinct, ~ ax11inda2 may be used instead to avoid the dummy
|
|
variable ` w ` in the proof.) (Contributed by NM, 24-Jan-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax11inda $p |- ( -. A. x x = y ->
|
|
( x = y -> ( A. z ph -> A. x ( x = y -> A. z ph ) ) ) ) $=
|
|
( weq wal wn wi wex a9ev wa ax11inda2 wb dveeq2-o imp albidh syl imbi12d
|
|
hba1-o equequ2 sps-o notbid adantl imbi1d imbi2d mpbii ex exlimdv pm2.43i
|
|
mpi ) BCGZBHZIZUMADHZUMUPJZBHZJZJZUOECGZEKUOUTJZECLUOVAVBEUOVAVBUOVAMZBEG
|
|
ZBHZIZVDUPVDUPJZBHZJZJZJVBABEDFNVCVFUOVJUTVCVABHZVFUOOUOVAVKBCEPQZVKVEUNV
|
|
KVDUMBVABUAZVAVDUMOZBECBUBZUCZRUDSVCVDUMVIUSVAVNUOVOUEVCVHURUPVCVKVHUROVL
|
|
VKVGUQBVMVKVDUMUPVPUFRSUGTTUHUIUJULUK $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
ax11v2-o.1 $e |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) $.
|
|
$( Recovery of ~ ax-11o from ~ ax11v without using ~ ax-11o . The
|
|
hypothesis is even weaker than ~ ax11v , with ` z ` both distinct from
|
|
` x ` _and_ not occurring in ` ph ` . Thus, the hypothesis provides an
|
|
alternate axiom that can be used in place of ~ ax-11o . (Contributed by
|
|
NM, 2-Feb-2007.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax11v2-o $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( weq wal wn wex wi wa wb equequ2 adantl dveeq2-o imp nfa1-o imbi1d sps-o
|
|
a9ev albid syl imbi2d imbi12d mpbii ex exlimdv mpi ) BCFZBGHZDCFZDIUIAUIA
|
|
JZBGZJZJZDCTUJUKUODUJUKUOUJUKKZBDFZAUQAJZBGZJZJUOEUPUQUIUTUNUKUQUILUJDCBM
|
|
ZNUPUSUMAUPUKBGZUSUMLUJUKVBBCDOPVBURULBUKBQUKURULLBUKUQUIAVARSUAUBUCUDUEU
|
|
FUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
ax11a2-o.1 $e |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) ) $.
|
|
$( Derive ~ ax-11o from a hypothesis in the form of ~ ax-11 , without using
|
|
~ ax-11 or ~ ax-11o . The hypothesis is even weaker than ~ ax-11 , with
|
|
` z ` both distinct from ` x ` and not occurring in ` ph ` . Thus, the
|
|
hypothesis provides an alternate axiom that can be used in place of
|
|
~ ax-11 , if we also hvae ~ ax-10o which this proof uses . As theorem
|
|
~ ax11 shows, the distinct variable conditions are optional. An open
|
|
problem is whether we can derive this with ~ ax-10 instead of
|
|
~ ax-10o . (Contributed by NM, 2-Feb-2007.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
ax11a2-o $p |- ( -. A. x x = y ->
|
|
( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
|
|
( wal weq wi ax-17 syl5 ax11v2-o ) ABCDAADFBDGZLAHBFADIEJK $.
|
|
$}
|
|
|
|
$( Show that ~ ax-10o can be derived from ~ ax-10 . An open problem is
|
|
whether this theorem can be derived from ~ ax-10 and the others when
|
|
~ ax-11 is replaced with ~ ax-11o . See theorem ~ ax10from10o for the
|
|
rederivation of ~ ax-10 from ~ ax10o .
|
|
|
|
Normally, ~ ax10o should be used rather than ~ ax-10o or ~ ax10o-o ,
|
|
except by theorems specifically studying the latter's properties.
|
|
(Contributed by NM, 16-May-2008.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
ax10o-o $p |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $=
|
|
( weq wal wi ax-10 ax11 equcoms sps-o pm2.27 al2imi sylsyld ) BCDZBECBDZCEA
|
|
BEZOAFZCEZACEBCGNPRFZBSCBACBHIJOQACOAKLM $.
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Existential uniqueness
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$( Declare new symbols needed for uniqueness notation. $)
|
|
$c E! $. $( Backwards E exclamation point. $)
|
|
$c E* $. $( Backwards E superscript *. $)
|
|
|
|
$( Extend wff definition to include existential uniqueness ("there exists a
|
|
unique ` x ` such that ` ph ` "). $)
|
|
weu $a wff E! x ph $.
|
|
|
|
$( Extend wff definition to include uniqueness ("there exists at most one
|
|
` x ` such that ` ph ` "). $)
|
|
wmo $a wff E* x ph $.
|
|
|
|
${
|
|
$d w x y $. $d x z $. $d y ph $. $d w z ph $.
|
|
$( A soundness justification theorem for ~ df-eu , showing that the
|
|
definition is equivalent to itself with its dummy variable renamed.
|
|
Note that ` y ` and ` z ` needn't be distinct variables. See
|
|
~ eujustALT for a proof that provides an example of how it can be
|
|
achieved through the use of ~ dvelim . (Contributed by NM,
|
|
11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
eujust $p |- ( E. y A. x ( ph <-> x = y )
|
|
<-> E. z A. x ( ph <-> x = z ) ) $=
|
|
( vw cv wceq wb wal wex equequ2 bibi2d albidv cbvexv bitri ) ABFZCFZGZHZB
|
|
IZCJAPEFZGZHZBIZEJAPDFZGZHZBIZDJTUDCEQUAGZSUCBUIRUBACEBKLMNUDUHEDUAUEGZUC
|
|
UGBUJUBUFAEDBKLMNO $.
|
|
|
|
$( A soundness justification theorem for ~ df-eu , showing that the
|
|
definition is equivalent to itself with its dummy variable renamed.
|
|
Note that ` y ` and ` z ` needn't be distinct variables. While this
|
|
isn't strictly necessary for soundness, the proof provides an example of
|
|
how it can be achieved through the use of ~ dvelim . (Contributed by
|
|
NM, 11-Mar-2010.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
eujustALT $p |- ( E. y A. x ( ph <-> x = y )
|
|
<-> E. z A. x ( ph <-> x = z ) ) $=
|
|
( vw weq wal wb wex equequ2 bibi2d albidv wn hbnae wi ax-17 notbid dvelim
|
|
sps df-ex drex1 alrimih naecoms a1i cbv2h syl 3bitr4g pm2.61i ) CDFZCGZAB
|
|
CFZHZBGZCIZABDFZHZBGZDIZHUMUQCDUIUMUQHCUIULUPBUIUKUOACDBJKLZSUAUJMZUMMZCG
|
|
ZMUQMZDGZMUNURUTVBVDUTUTDGZCGVBVDHUTVECCDCNCDDNUBUTVAVCCDVAVADGODCABEFZHZ
|
|
BGZMZVADCEVIDPECFZVHUMVJVGULBVJVFUKAECBJKLQRUCVIVCCDEVICPEDFZVHUQVKVGUPBV
|
|
KVFUOAEDBJKLQRUIVAVCHOUTUIUMUQUSQUDUEUFQUMCTUQDTUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Define existential uniqueness, i.e. "there exists exactly one ` x `
|
|
such that ` ph ` ." Definition 10.1 of [BellMachover] p. 97; also
|
|
Definition *14.02 of [WhiteheadRussell] p. 175. Other possible
|
|
definitions are given by ~ eu1 , ~ eu2 , ~ eu3 , and ~ eu5 (which in
|
|
some cases we show with a hypothesis ` ph -> A. y ph ` in place of a
|
|
distinct variable condition on ` y ` and ` ph ` ). Double uniqueness is
|
|
tricky: ` E! x E! y ph ` does not mean "exactly one ` x ` and one
|
|
` y ` " (see ~ 2eu4 ). (Contributed by NM, 12-Aug-1993.) $)
|
|
df-eu $a |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) $.
|
|
$}
|
|
|
|
$( Define "there exists at most one ` x ` such that ` ph ` ." Here we define
|
|
it in terms of existential uniqueness. Notation of [BellMachover] p. 460,
|
|
whose definition we show as ~ mo3 . For other possible definitions see
|
|
~ mo2 and ~ mo4 . (Contributed by NM, 8-Mar-1995.) $)
|
|
df-mo $a |- ( E* x ph <-> ( E. x ph -> E! x ph ) ) $.
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
euf.1 $e |- F/ y ph $.
|
|
$( A version of the existential uniqueness definition with a hypothesis
|
|
instead of a distinct variable condition. (Contributed by NM,
|
|
12-Aug-1993.) $)
|
|
euf $p |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) $=
|
|
( vz weu cv wceq wb wal wex df-eu nfv nfbi nfal bibi2d albidv cbvex bitri
|
|
equequ2 ) ABFABGZEGZHZIZBJZEKAUACGZHZIZBJZCKABELUEUIECUDCBAUCCDUCCMNOUHEB
|
|
AUGEAEMUGEMNOUBUFHZUDUHBUJUCUGAECBTPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $. $d y ch $.
|
|
eubid.1 $e |- F/ x ph $.
|
|
eubid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for uniqueness quantifier (deduction rule).
|
|
(Contributed by NM, 9-Jul-1994.) $)
|
|
eubid $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
|
|
( vy cv wceq wb wal wex weu bibi1d albid exbidv df-eu 3bitr4g ) ABDHGHIZJ
|
|
ZDKZGLCSJZDKZGLBDMCDMAUAUCGATUBDEABCSFNOPBDGQCDGQR $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
eubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for uniqueness quantifier (deduction rule).
|
|
(Contributed by NM, 9-Jul-1994.) $)
|
|
eubidv $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
|
|
( nfv eubid ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
eubii.1 $e |- ( ph <-> ps ) $.
|
|
$( Introduce uniqueness quantifier to both sides of an equivalence.
|
|
(Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro,
|
|
6-Oct-2016.) $)
|
|
eubii $p |- ( E! x ph <-> E! x ps ) $=
|
|
( weu wb wtru a1i eubidv trud ) ACEBCEFGABCABFGDHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Bound-variable hypothesis builder for uniqueness. (Contributed by NM,
|
|
9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfeu1 $p |- F/ x E! x ph $=
|
|
( vy weu cv wceq wb wal wex df-eu nfa1 nfex nfxfr ) ABDABECEFGZBHZCIBABCJ
|
|
OBCNBKLM $.
|
|
$}
|
|
|
|
$( Bound-variable hypothesis builder for "at most one." (Contributed by NM,
|
|
8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfmo1 $p |- F/ x E* x ph $=
|
|
( wmo wex weu wi df-mo nfe1 nfeu1 nfim nfxfr ) ABCABDZABEZFBABGLMBABHABIJK
|
|
$.
|
|
|
|
${
|
|
$d y z $. $d z ph $. $d z ps $.
|
|
nfeud2.1 $e |- F/ y ph $.
|
|
nfeud2.2 $e |- ( ( ph /\ -. A. x x = y ) -> F/ x ps ) $.
|
|
$( Bound-variable hypothesis builder for uniqueness. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
nfeud2 $p |- ( ph -> F/ x E! y ps ) $=
|
|
( vz weu cv wceq wb wal wex df-eu nfv wn wa nfnae nfan wnf adantlr ancoms
|
|
nfeqf adantll nfbid nfald2 nfexd2 nfxfrd ) BDHBDIZGIZJZKZDLZGMACBDGNAUMCG
|
|
AGOACIZUJJCLPZQZULCDAUODECGDRSUPUNUIJCLPZQBUKCAUQBCTUOFUAUOUQUKCTZAUQUOUR
|
|
DGCUCUBUDUEUFUGUH $.
|
|
|
|
$( Bound-variable hypothesis builder for uniqueness. (Contributed by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
nfmod2 $p |- ( ph -> F/ x E* y ps ) $=
|
|
( wmo wex weu wi df-mo nfexd2 nfeud2 nfimd nfxfrd ) BDGBDHZBDIZJACBDKAPQC
|
|
ABCDEFLABCDEFMNO $.
|
|
$}
|
|
|
|
${
|
|
nfeud.1 $e |- F/ y ph $.
|
|
nfeud.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfeu . (Contributed by NM, 15-Feb-2013.)
|
|
(Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfeud $p |- ( ph -> F/ x E! y ps ) $=
|
|
( wnf cv wceq wal wn adantr nfeud2 ) ABCDEABCGCHDHICJKFLM $.
|
|
|
|
$( Bound-variable hypothesis builder for "at most one." (Contributed by
|
|
Mario Carneiro, 14-Nov-2016.) $)
|
|
nfmod $p |- ( ph -> F/ x E* y ps ) $=
|
|
( wnf cv wceq wal wn adantr nfmod2 ) ABCDEABCGCHDHICJKFLM $.
|
|
$}
|
|
|
|
${
|
|
nfeu.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for "at most one." Note that ` x `
|
|
and ` y ` needn't be distinct (this makes the proof more difficult).
|
|
(Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfeu $p |- F/ x E! y ph $=
|
|
( weu wnf wtru nftru a1i nfeud trud ) ACEBFGABCCHABFGDIJK $.
|
|
|
|
$( Bound-variable hypothesis builder for "at most one." (Contributed by
|
|
NM, 9-Mar-1995.) $)
|
|
nfmo $p |- F/ x E* y ph $=
|
|
( wmo wnf wtru nftru a1i nfmod trud ) ACEBFGABCCHABFGDIJK $.
|
|
$}
|
|
|
|
${
|
|
$d w y z $. $d ph z w $. $d w x z $.
|
|
sb8eu.1 $e |- F/ y ph $.
|
|
$( Variable substitution in uniqueness quantifier. (Contributed by NM,
|
|
7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
sb8eu $p |- ( E! x ph <-> E! y [ y / x ] ph ) $=
|
|
( vz vw cv wceq wb wal wex wsb weu nfv sb8 sbbi nfsb equsb3 nfxfr df-eu
|
|
nfbi sbequ cbval sblbis albii 3bitri exbii 3bitr4i ) ABGEGZHZIZBJZEKABCLZ
|
|
CGUIHZIZCJZEKABMUMCMULUPEULUKBFLZFJUKBCLZCJUPUKBFUKFNOUQURFCUQABFLZUJBFLZ
|
|
ICAUJBFPUSUTCABFCDQUTFGUIHZCFBERVACNSUASURFNUKFCBUBUCURUOCUJUNABCCBERUDUE
|
|
UFUGABETUMCETUH $.
|
|
|
|
$( Variable substitution in uniqueness quantifier. (Contributed by
|
|
Alexander van der Vekens, 17-Jun-2017.) $)
|
|
sb8mo $p |- ( E* x ph <-> E* y [ y / x ] ph ) $=
|
|
( wex weu wi wsb wmo sb8e sb8eu imbi12i df-mo 3bitr4i ) ABEZABFZGABCHZCEZ
|
|
QCFZGABIQCIORPSABCDJABCDKLABMQCMN $.
|
|
$}
|
|
|
|
${
|
|
cbveu.1 $e |- F/ y ph $.
|
|
cbveu.2 $e |- F/ x ps $.
|
|
cbveu.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
cbveu $p |- ( E! x ph <-> E! y ps ) $=
|
|
( weu wsb sb8eu sbie eubii bitri ) ACHACDIZDHBDHACDEJNBDABCDFGKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eu1.1 $e |- F/ y ph $.
|
|
$( An alternate way to express uniqueness used by some authors. Exercise
|
|
2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised
|
|
by Mario Carneiro, 7-Oct-2016.) $)
|
|
eu1 $p |- ( E! x ph <->
|
|
E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) ) $=
|
|
( wsb weu cv wb wal wex wi wa nfs1v euf sb8eu equcom imbi2i albii 3bitr4i
|
|
wceq sb6rf anbi12i ancom albiim exbii ) ABCEZCFUFCGZBGZTZHCIZBJABFAUFUHUG
|
|
TZKZCIZLZBJUFCBABCMNABCDOUNUJBUMALUFUIKZCIZUIUFKCIZLUNUJUMUPAUQULUOCUKUIU
|
|
FBCPQRABCDUAUBAUMUCUFUICUDSUES $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
mo.1 $e |- F/ y ph $.
|
|
$( Equivalent definitions of "there exists at most one." (Contributed by
|
|
NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
mo $p |- ( E. y A. x ( ph -> x = y ) <->
|
|
A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
|
|
( vz cv wceq wi wal wex wsb wa nfv nfim nfal equequ2 cbv3 sylbir wn nfn
|
|
imbi2d albidv cbvex nfs1 sbequ2 ax-8 imim12d aaan sylibr prth equtr2 syl6
|
|
ancli 2alimi syl exlimiv nfa2 exp3a com3r alimd com12 eximd alnex equcoms
|
|
sp sbequ1 con3d pm2.21 alimi 19.8a 3syl pm2.61d1 impbii ) ABFZCFZGZHZBIZC
|
|
JZAABCKZLZVPHZCIZBIZVSAVNEFZGZHZBIZEJWDWHVRECWGCBAWFCDWFCMNZOVREMWEVOGZWG
|
|
VQBWJWFVPAECBPUAUBUCWHWDEWHWGVTVOWEGZHZLZCIBIZWDWHWHWLCIZLWNWHWOWGWLBCWIV
|
|
TWKBABCDUDZWKBMNZVPVTAWFWKABCUEBCEUFUGQUMWGWLBCWIWQUHUIWMWBBCWMWAWFWKLVPA
|
|
WFVTWKUJBCEUKULUNUOUPRWDVTCJZVSWDVTVRCWBCBUQVTWDVRVTWCVQBWPWCAVTVPWCAVTVP
|
|
WBCVEURUSUTVAVBWRSVTSZCIZVSVTCVCWTASZBIVRVSWSXACBVTBWPTACDTVOVNGAVTAVTHBC
|
|
ABCVFVDVGQXAVQBAVPVHVIVRCVJVKRVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d ph y $.
|
|
$( Existential uniqueness implies existence. (Contributed by NM,
|
|
15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
euex $p |- ( E! x ph -> E. x ph ) $=
|
|
( vy weu wsb cv wceq wi wal wa wex nfv eu1 exsimpl sylbi ) ABDAABCEBFCFGH
|
|
CIZJBKABKABCACLMAPBNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eumo0.1 $e |- F/ y ph $.
|
|
$( Existential uniqueness implies "at most one." (Contributed by NM,
|
|
8-Jul-1994.) $)
|
|
eumo0 $p |- ( E! x ph -> E. y A. x ( ph -> x = y ) ) $=
|
|
( weu weq wb wal wex wi euf bi1 alimi eximi sylbi ) ABEABCFZGZBHZCIAPJZBH
|
|
ZCIABCDKRTCQSBAPLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eu2.1 $e |- F/ y ph $.
|
|
$( An alternate way of defining existential uniqueness. Definition 6.10 of
|
|
[TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) $)
|
|
eu2 $p |- ( E! x ph <->
|
|
( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
|
|
( weu wex wsb wa weq wi wal euex eumo0 mo sylib 19.29r impexp albii 19.21
|
|
jca bitri anbi2i abai bitr4i exbii eu1 sylibr impbii ) ABEZABFZAABCGZHBCI
|
|
ZJZCKZBKZHZUIUJUOABLUIAULJBKCFUOABCDMABCDNOTUPAUKULJZCKZHZBFZUIUPAUNHZBFU
|
|
TAUNBPVAUSBVAAAURJZHUSUNVBAUNAUQJZCKVBUMVCCAUKULQRAUQCDSUAUBAURUCUDUEOABC
|
|
DUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
eu3.1 $e |- F/ y ph $.
|
|
$( An alternate way to express existential uniqueness. (Contributed by NM,
|
|
8-Jul-1994.) $)
|
|
eu3 $p |- ( E! x ph <->
|
|
( E. x ph /\ E. y A. x ( ph -> x = y ) ) ) $=
|
|
( weu wex wsb wa weq wi wal eu2 mo anbi2i bitr4i ) ABEABFZAABCGHBCIZJCKBK
|
|
ZHPAQJBKCFZHABCDLSRPABCDMNO $.
|
|
$}
|
|
|
|
${
|
|
euor.1 $e |- F/ x ph $.
|
|
$( Introduce a disjunct into a uniqueness quantifier. (Contributed by NM,
|
|
21-Oct-2005.) $)
|
|
euor $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $=
|
|
( wn weu wo nfn biorf eubid biimpa ) AEZBCFABGZCFLBMCACDHABIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Introduce a disjunct into a uniqueness quantifier. (Contributed by NM,
|
|
23-Mar-1995.) $)
|
|
euorv $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $=
|
|
( nfv euor ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
mo2.1 $e |- F/ y ph $.
|
|
$( Alternate definition of "at most one." (Contributed by NM,
|
|
8-Mar-1995.) $)
|
|
mo2 $p |- ( E* x ph <-> E. y A. x ( ph -> x = y ) ) $=
|
|
( wmo wex weu wi weq wal df-mo alnex pm2.21 alimi 19.8a syl sylbir eumo0
|
|
wn ja eu3 simplbi2com impbii bitri ) ABEABFZABGZHZABCIZHZBJZCFZABKUGUKUEU
|
|
FUKUESASZBJZUKABLUMUJUKULUIBAUHMNUJCOPQABCDRTUFUEUKABCDUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d w x z $. $d w y z $. $d w ph $.
|
|
$( Substitution into "at most one". (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
sbmo $p |- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph ) $=
|
|
( vw cv wceq wal wex wsb wmo sbex nfv sblim sbalv exbii bitri mo2 sbbii
|
|
wi 3bitr4i ) ADFEFGZTZDHZEIZBCJZABCJZUBTZDHZEIZADKZBCJUGDKUFUDBCJZEIUJUDE
|
|
BCLULUIEUCUHBCDAUBBCUBBMNOPQUKUEBCADEAEMRSUGDEUGEMRUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
mo3.1 $e |- F/ y ph $.
|
|
$( Alternate definition of "at most one." Definition of [BellMachover]
|
|
p. 460, except that definition has the side condition that ` y ` not
|
|
occur in ` ph ` in place of our hypothesis. (Contributed by NM,
|
|
8-Mar-1995.) $)
|
|
mo3 $p |- ( E* x ph <->
|
|
A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
|
|
( wmo weq wi wal wex wsb wa mo2 mo bitri ) ABEABCFZGBHCIAABCJKOGCHBHABCDL
|
|
ABCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
mo4f.1 $e |- F/ x ps $.
|
|
mo4f.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( "At most one" expressed using implicit substitution. (Contributed by
|
|
NM, 10-Apr-2004.) $)
|
|
mo4f $p |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) ) $=
|
|
( wmo wsb wa weq wi wal nfv mo3 sbie anbi2i imbi1i 2albii bitri ) ACGAACD
|
|
HZIZCDJZKZDLCLABIZUBKZDLCLACDADMNUCUECDUAUDUBTBAABCDEFOPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d x ps $.
|
|
mo4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( "At most one" expressed using implicit substitution. (Contributed by
|
|
NM, 26-Jul-1995.) $)
|
|
mo4 $p |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) ) $=
|
|
( nfv mo4f ) ABCDBCFEG $.
|
|
$}
|
|
|
|
${
|
|
mobid.1 $e |- F/ x ph $.
|
|
mobid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for "at most one" quantifier (deduction rule).
|
|
(Contributed by NM, 8-Mar-1995.) $)
|
|
mobid $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
|
|
( wex weu wi wmo exbid eubid imbi12d df-mo 3bitr4g ) ABDGZBDHZICDGZCDHZIB
|
|
DJCDJAPRQSABCDEFKABCDEFLMBDNCDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
mobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for "at most one" quantifier (deduction rule).
|
|
(Contributed by Mario Carneiro, 7-Oct-2016.) $)
|
|
mobidv $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
|
|
( nfv mobid ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
mobii.1 $e |- ( ps <-> ch ) $.
|
|
$( Formula-building rule for "at most one" quantifier (inference rule).
|
|
(Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro,
|
|
17-Oct-2016.) $)
|
|
mobii $p |- ( E* x ps <-> E* x ch ) $=
|
|
( wmo wb wtru a1i mobidv trud ) ACEBCEFGABCABFGDHIJ $.
|
|
$}
|
|
|
|
${
|
|
cbvmo.1 $e |- F/ y ph $.
|
|
cbvmo.2 $e |- F/ x ps $.
|
|
cbvmo.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
cbvmo $p |- ( E* x ph <-> E* y ps ) $=
|
|
( wex weu wi wmo cbvex cbveu imbi12i df-mo 3bitr4i ) ACHZACIZJBDHZBDIZJAC
|
|
KBDKQSRTABCDEFGLABCDEFGMNACOBDOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Uniqueness in terms of "at most one." (Contributed by NM,
|
|
23-Mar-1995.) $)
|
|
eu5 $p |- ( E! x ph <-> ( E. x ph /\ E* x ph ) ) $=
|
|
( vy weu wex cv wceq wi wal wa wmo nfv eu3 mo2 anbi2i bitr4i ) ABDABEZABF
|
|
CFGHBICEZJQABKZJABCACLZMSRQABCTNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d x ps $.
|
|
eu4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Uniqueness using implicit substitution. (Contributed by NM,
|
|
26-Jul-1995.) $)
|
|
eu4 $p |- ( E! x ph <-> ( E. x ph /\
|
|
A. x A. y ( ( ph /\ ps ) -> x = y ) ) ) $=
|
|
( weu wex wmo wa weq wi wal eu5 mo4 anbi2i bitri ) ACFACGZACHZIQABICDJKDL
|
|
CLZIACMRSQABCDENOP $.
|
|
$}
|
|
|
|
$( Existential uniqueness implies "at most one." (Contributed by NM,
|
|
23-Mar-1995.) $)
|
|
eumo $p |- ( E! x ph -> E* x ph ) $=
|
|
( weu wex wmo eu5 simprbi ) ABCABDABEABFG $.
|
|
|
|
${
|
|
eumoi.1 $e |- E! x ph $.
|
|
$( "At most one" inferred from existential uniqueness. (Contributed by NM,
|
|
5-Apr-1995.) $)
|
|
eumoi $p |- E* x ph $=
|
|
( weu wmo eumo ax-mp ) ABDABECABFG $.
|
|
$}
|
|
|
|
$( Existence in terms of "at most one" and uniqueness. (Contributed by NM,
|
|
5-Apr-2004.) $)
|
|
exmoeu $p |- ( E. x ph <-> ( E* x ph -> E! x ph ) ) $=
|
|
( wex wmo weu wi df-mo biimpi com12 biimpri euex imim12i peirce syl impbii
|
|
) ABCZABDZABEZFZQPRQPRFZABGZHISTPFPTQRPQTUAJABKLPRMNO $.
|
|
|
|
$( Existence implies "at most one" is equivalent to uniqueness. (Contributed
|
|
by NM, 5-Apr-2004.) $)
|
|
exmoeu2 $p |- ( E. x ph -> ( E* x ph <-> E! x ph ) ) $=
|
|
( weu wex wmo eu5 baibr ) ABCABDABEABFG $.
|
|
|
|
$( Absorption of existence condition by "at most one." (Contributed by NM,
|
|
4-Nov-2002.) $)
|
|
moabs $p |- ( E* x ph <-> ( E. x ph -> E* x ph ) ) $=
|
|
( wex weu wi wmo pm5.4 df-mo imbi2i 3bitr4ri ) ABCZKABDZEZEMKABFZENKLGNMKAB
|
|
HZIOJ $.
|
|
|
|
$( Something exists or at most one exists. (Contributed by NM,
|
|
8-Mar-1995.) $)
|
|
exmo $p |- ( E. x ph \/ E* x ph ) $=
|
|
( wex wmo wn weu wi pm2.21 df-mo sylibr orri ) ABCZABDZLELABFZGMLNHABIJK $.
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $.
|
|
$( "At most one" is preserved through implication (notice wff reversal).
|
|
(Contributed by NM, 22-Apr-1995.) $)
|
|
moim $p |- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) ) $=
|
|
( vy wi wal cv wceq wex wmo imim1 al2imi eximdv nfv mo2 3imtr4g ) ABEZCFZ
|
|
BCGDGHZEZCFZDIASEZCFZDIBCJACJRUAUCDQTUBCABSKLMBCDBDNOACDADNOP $.
|
|
$}
|
|
|
|
${
|
|
immoi.1 $e |- ( ph -> ps ) $.
|
|
$( "At most one" is preserved through implication (notice wff reversal).
|
|
(Contributed by NM, 15-Feb-2006.) $)
|
|
moimi $p |- ( E* x ps -> E* x ph ) $=
|
|
( wi wmo moim mpg ) ABEBCFACFECABCGDH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x y ph $. $d y ps $.
|
|
$( Move antecedent outside of "at most one." (Contributed by NM,
|
|
28-Jul-1995.) $)
|
|
morimv $p |- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) ) $=
|
|
( vy wi wmo cv wceq wal wex ax-1 a1i imim1d alimdv eximdv nfv mo2 3imtr4g
|
|
com12 ) AABEZCFZBCFZATCGDGHZEZCIZDJBUCEZCIZDJUAUBAUEUGDAUDUFCABTUCBTEABAK
|
|
LMNOTCDTDPQBCDBDPQRS $.
|
|
$}
|
|
|
|
$( Uniqueness implies "at most one" through implication. (Contributed by NM,
|
|
22-Apr-1995.) $)
|
|
euimmo $p |- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) ) $=
|
|
( weu wmo wi wal eumo moim syl5 ) BCDBCEABFCGACEBCHABCIJ $.
|
|
|
|
$( Add existential uniqueness quantifiers to an implication. Note the
|
|
reversed implication in the antecedent. (Contributed by NM,
|
|
19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
euim $p |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) ) $=
|
|
( wex wi wal wa weu wmo ax-1 euimmo anim12ii eu5 syl6ibr ) ACDZABECFZGBCHZO
|
|
ACIZGACHOQOPROQJABCKLACMN $.
|
|
|
|
$( "At most one" is still the case when a conjunct is added. (Contributed by
|
|
NM, 22-Apr-1995.) $)
|
|
moan $p |- ( E* x ph -> E* x ( ps /\ ph ) ) $=
|
|
( wa simpr moimi ) BADACBAEF $.
|
|
|
|
${
|
|
moani.1 $e |- E* x ph $.
|
|
$( "At most one" is still true when a conjunct is added. (Contributed by
|
|
NM, 9-Mar-1995.) $)
|
|
moani $p |- E* x ( ps /\ ph ) $=
|
|
( wmo wa moan ax-mp ) ACEBAFCEDABCGH $.
|
|
$}
|
|
|
|
$( "At most one" is still the case when a disjunct is removed. (Contributed
|
|
by NM, 5-Apr-2004.) $)
|
|
moor $p |- ( E* x ( ph \/ ps ) -> E* x ph ) $=
|
|
( wo orc moimi ) AABDCABEF $.
|
|
|
|
$( "At most one" imports disjunction to conjunction. (Contributed by NM,
|
|
5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
mooran1 $p |- ( ( E* x ph \/ E* x ps ) -> E* x ( ph /\ ps ) ) $=
|
|
( wmo wa simpl moimi moan jaoi ) ACDABEZCDBCDJACABFGBACHI $.
|
|
|
|
$( "At most one" exports disjunction to conjunction. (Contributed by NM,
|
|
5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
mooran2 $p |- ( E* x ( ph \/ ps ) -> ( E* x ph /\ E* x ps ) ) $=
|
|
( wo wmo moor olc moimi jca ) ABDZCEACEBCEABCFBJCBAGHI $.
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $.
|
|
moanim.1 $e |- F/ x ph $.
|
|
$( Introduction of a conjunct into "at most one" quantifier. (Contributed
|
|
by NM, 3-Dec-2001.) $)
|
|
moanim $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
|
|
( vy wa weq wi wal wex wmo impexp albii 19.21 bitri nfv mo2 imbi2i 19.37v
|
|
exbii bitr4i 3bitr4i ) ABFZCEGZHZCIZEJABUDHZCIZHZEJZUCCKABCKZHZUFUIEUFAUG
|
|
HZCIUIUEUMCABUDLMAUGCDNOTUCCEUCEPQULAUHEJZHUJUKUNABCEBEPQRAUHESUAUB $.
|
|
|
|
$( Introduction of a conjunct into uniqueness quantifier. (Contributed by
|
|
NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
euan $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $=
|
|
( wa weu wex wmo simpl exlimi adantr simpr eximi nfe1 ancrd impbid2 mobid
|
|
a1d biimpa eu5 jca32 anbi2i 3imtr4i ibar eubid impbii ) ABEZCFZABCFZEZUGC
|
|
GZUGCHZEZABCGZBCHZEZEUHUJUMAUNUOUKAULUGACDABIJZKUKUNULUGBCABLZMKUKULUOUKU
|
|
GBCUGCNUKUGBURUKBAUKABUQROPQSUAUGCTUIUPABCTUBUCAUIUHABUGCDABUDUESUF $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Introduction of a conjunct into "at most one" quantifier. (Contributed
|
|
by NM, 23-Mar-1995.) $)
|
|
moanimv $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
|
|
( nfv moanim ) ABCACDE $.
|
|
$}
|
|
|
|
$( Nested "at most one" and uniqueness quantifiers. (Contributed by NM,
|
|
25-Jan-2006.) $)
|
|
moaneu $p |- E* x ( ph /\ E! x ph ) $=
|
|
( weu wa wmo wi eumo nfeu1 moanim mpbir ancom mobii ) AABCZDZBEMADZBEZPMABE
|
|
FABGMABABHIJNOBAMKLJ $.
|
|
|
|
$( Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.) $)
|
|
moanmo $p |- E* x ( ph /\ E* x ph ) $=
|
|
( wmo wa wi id nfmo1 moanim mpbir ancom mobii ) AABCZDZBCLADZBCZOLLELFLABAB
|
|
GHIMNBALJKI $.
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Introduction of a conjunct into uniqueness quantifier. (Contributed by
|
|
NM, 23-Mar-1995.) $)
|
|
euanv $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $=
|
|
( nfv euan ) ABCACDE $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y ps $.
|
|
$( "At most one" picks a variable value, eliminating an existential
|
|
quantifier. (Contributed by NM, 27-Jan-1997.) $)
|
|
mopick $p |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) ) $=
|
|
( vy wa wex wmo wi wsb nfv nfs1v nfan cv wceq sbequ12 anbi12d cbvex sylbi
|
|
wal mo3 sp sps sbequ2 imim2i exp3a com4t imp syl5 exlimiv impcom ) ABEZCF
|
|
ZACGZABHZULACDIZBCDIZEZDFUMUNHZUKUQCDUKDJUOUPCACDKBCDKLCMDMNZAUOBUPACDOBC
|
|
DOPQUQURDUMAUOEZUSHZUQUNUMVADSZCSVAACDADJTVBVACVADUAUBRUOUPVAUNHVAAUOUPBV
|
|
AAUOUPBHZUSVCUTBCDUCUDUEUFUGUHUIRUJ $.
|
|
$}
|
|
|
|
$( Existential uniqueness "picks" a variable value for which another wff is
|
|
true. If there is only one thing ` x ` such that ` ph ` is true, and
|
|
there is also an ` x ` (actually the same one) such that ` ph ` and ` ps `
|
|
are both true, then ` ph ` implies ` ps ` regardless of ` x ` . This
|
|
theorem can be useful for eliminating existential quantifiers in a
|
|
hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192.
|
|
(Contributed by NM, 10-Jul-1994.) $)
|
|
eupick $p |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) ) $=
|
|
( weu wmo wa wex wi eumo mopick sylan ) ACDACEABFCGABHACIABCJK $.
|
|
|
|
$( Version of ~ eupick with closed formulas. (Contributed by NM,
|
|
6-Sep-2008.) $)
|
|
eupicka $p |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) ) $=
|
|
( weu wa wex wi nfeu1 nfe1 nfan eupick alrimi ) ACDZABEZCFZEABGCMOCACHNCIJA
|
|
BCKL $.
|
|
|
|
$( Existential uniqueness "pick" showing wff equivalence. (Contributed by
|
|
NM, 25-Nov-1994.) $)
|
|
eupickb $p |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) ->
|
|
( ph <-> ps ) ) $=
|
|
( weu wa wex w3a wi eupick 3adant2 3simpc pm3.22 eximi anim2i 3syl impbid )
|
|
ACDZBCDZABEZCFZGZABQTABHRABCIJUARTERBAEZCFZEBAHQRTKTUCRSUBCABLMNBACIOP $.
|
|
|
|
$( Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew
|
|
Salmon, 11-Jul-2011.) $)
|
|
eupickbi $p |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) ) $=
|
|
( weu wa wex wi wal eupicka ex nfa1 wb ancl simpl impbid1 eubid euex syl6bi
|
|
sps com12 impbid ) ACDZABEZCFZABGZCHZUBUDUFABCIJUFUBUDUFUBUCCDUDUFAUCCUECKU
|
|
EAUCLCUEAUCABMABNOSPUCCQRTUA $.
|
|
|
|
$( "At most one" can show the existence of a common value. In this case we
|
|
can infer existence of conjunction from a conjunction of existence, and it
|
|
is one way to achieve the converse of ~ 19.40 . (Contributed by NM,
|
|
5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
mopick2 $p |- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) ->
|
|
E. x ( ph /\ ps /\ ch ) ) $=
|
|
( wmo wa wex w3a nfmo1 nfe1 mopick ancld anim1d df-3an syl6ibr eximd 3impia
|
|
nfan ) ADEZABFZDGZACFZDGABCHZDGSUAFZUBUCDSUADADITDJRUDUBTCFUCUDATCUDABABDKL
|
|
MABCNOPQ $.
|
|
|
|
$( Introduce or eliminate a disjunct in a uniqueness quantifier.
|
|
(Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
euor2 $p |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) ) $=
|
|
( wex wn wo nfe1 nfn wb 19.8a con3i orel1 olc impbid1 syl eubid ) ACDZEZABF
|
|
ZBCQCACGHRAEZSBIAQACJKTSBABLBAMNOP $.
|
|
|
|
${
|
|
moexex.1 $e |- F/ y ph $.
|
|
$( "At most one" double quantification. (Contributed by NM,
|
|
3-Dec-2001.) $)
|
|
moexex $p |- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) $=
|
|
( wmo wal wa wex wi nfmo1 nfa1 nfe1 nfmo nfim mopick ex exlimi wn a1d ori
|
|
com3r alrimd moim spsd syl6 nfex exsimpl con3i exmo syl pm2.61i imp ) ACF
|
|
ZBDFZCGZABHZCIZDFZACIZUNUPUSJZJZAVBCUNVACACKUPUSCUOCLURCDUQCMNOOAUNURBJZD
|
|
GZVAAUNVCDEADCENUNURABUNURABJABCPQUBUCVDUOUSCURBDUDUEUFRUTSZVAUNVEUSUPVEU
|
|
RDIZSUSVFUTURUTDADCEUGABCUHRUIVFUSURDUJUAUKTTULUM $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $.
|
|
$( "At most one" double quantification. (Contributed by NM,
|
|
26-Jan-1997.) $)
|
|
moexexv $p |- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) $=
|
|
( nfv moexex ) ABCDADEF $.
|
|
$}
|
|
|
|
$( Double quantification with "at most one." (Contributed by NM,
|
|
3-Dec-2001.) $)
|
|
2moex $p |- ( E* x E. y ph -> A. y E* x ph ) $=
|
|
( wex wmo nfe1 nfmo 19.8a moimi alrimi ) ACDZBEABECKCBACFGAKBACHIJ $.
|
|
|
|
$( Double quantification with existential uniqueness. (Contributed by NM,
|
|
3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
2euex $p |- ( E! x E. y ph -> E. y E! x ph ) $=
|
|
( wex weu wmo wa eu5 excom nfe1 19.8a moimi df-mo sylib eximd syl5bi impcom
|
|
nfmo wi sylbi ) ACDZBEUABDZUABFZGABEZCDZUABHUCUBUEUBABDZCDUCUEABCIUCUFUDCUA
|
|
CBACJRUCABFUFUDSAUABACKLABMNOPQT $.
|
|
|
|
$( Double quantification with existential uniqueness and "at most one."
|
|
(Contributed by NM, 3-Dec-2001.) $)
|
|
2eumo $p |- ( E! x E* y ph -> E* x E! y ph ) $=
|
|
( weu wmo wi euimmo eumo mpg ) ACDZACEZFKBDJBEFBJKBGACHI $.
|
|
|
|
$( Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) $)
|
|
2eu2ex $p |- ( E! x E! y ph -> E. x E. y ph ) $=
|
|
( weu wex euex eximi syl ) ACDZBDIBEACEZBEIBFIJBACFGH $.
|
|
|
|
$( A condition allowing swap of "at most one" and existential quantifiers.
|
|
(Contributed by NM, 10-Apr-2004.) $)
|
|
2moswap $p |- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) ) $=
|
|
( wmo wal wex wa nfe1 moexex expcom 19.8a pm4.71ri exbii mobii syl6ibr ) AC
|
|
DBEZACFZBDZQAGZBFZCDZABFZCDRPUAQABCACHIJUBTCASBAQACKLMNO $.
|
|
|
|
$( A condition allowing swap of uniqueness and existential quantifiers.
|
|
(Contributed by NM, 10-Apr-2004.) $)
|
|
2euswap $p |- ( A. x E* y ph -> ( E! x E. y ph -> E! y E. x ph ) ) $=
|
|
( wmo wal wex wa weu wi excomim a1i 2moswap anim12d eu5 3imtr4g ) ACDBEZACF
|
|
ZBFZQBDZGABFZCFZTCDZGQBHTCHPRUASUBRUAIPABCJKABCLMQBNTCNO $.
|
|
|
|
$( Double existential uniqueness implies double uniqueness quantification.
|
|
(Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro,
|
|
22-Dec-2016.) $)
|
|
2exeu $p |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph ) $=
|
|
( wex weu wa wmo eumo euex moimi syl 2euex anim12ci eu5 sylibr ) ACDZBEZABD
|
|
CEZFACEZBDZSBGZFSBEQUARTQPBGUAPBHSPBACIJKACBLMSBNO $.
|
|
|
|
${
|
|
$d x y z w v u $. $d z w v u ph $.
|
|
$( Two equivalent expressions for double "at most one." (Contributed by
|
|
NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) $)
|
|
2mo $p |- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <->
|
|
A. x A. y A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) ->
|
|
( x = z /\ y = w ) ) ) $=
|
|
( vv vu cv wceq wa wi wal wex nfv albii bitri 2alimi syl sylbir wn imbi2d
|
|
wsb equequ2 bi2anan9 2albidv cbvex2v nfs1v nfim sbequ12 sylan9bbr equequ1
|
|
nfsb imbi12d cbval2 biimpi ancli alcom aaan nfal sylibr prth anim12i an4s
|
|
equtr2 syl6 exlimivv alrot4 pm3.21 alimd com12 alimi exim sylbi alnex nfn
|
|
imim1d notbid pm2.21 19.8a 19.23bi pm2.61d1 impbii ) ABHZDHZIZCHZEHZIZJZK
|
|
ZCLZBLZEMZDMZAACEUBZBDUBZJZWIKZELDLZCLBLZWNAWCFHZIZWFGHZIZJZKZCLZBLZGMFMW
|
|
TXHWLFGDEXAWDIZXCWGIZJZXFWJBCXKXEWIAXIXBWEXJXDWHFDBUCGECUCUDUAUEUFXHWTFGX
|
|
HXFWPWDXAIZWGXCIZJZKZJZELZDLZCLZBLZWTXHXHXOELZDLZJZXTXHYBXHYBXFXOBCDEXFDN
|
|
XFENZWPXNBWOBDUGZXNBNUHZWPXNCWOBDCACEUGULZXNCNUHZWIAWPXEXNWHAWOWEWPACEUIW
|
|
OBDUIUJZWEXBXLWHXDXMBDFUKCEGUKUDUMUNUOUPXTXGYAJZDLZBLYCXSYKBXSXQCLZDLYKXQ
|
|
CDUQYLYJDXFXOCEYDYHUROPOXGYABDXGDNXOBEYFUSURPUTXRWSBCXPWRDEXPWQXEXNJWIAXE
|
|
WPXNVAXBXLXDXMWIXBXLJWEXDXMJWHBDFVDCEGVDVBVCVEQQRVFSWTWPEMZDMZWNWTYMWMKZD
|
|
LZYNWNKWTWRCLZBLZELZDLYPWRBCDEVGYSYODYSWPWLKZELYOYRYTEWPYRWLWPYQWKBYEWPWR
|
|
WJCYGWPAWQWIWPAVHVPVIVIVJVKWPWLEVLRVKVMYMWMDVLRYNTZWPTZELZDLZWNUUDYMTZDLU
|
|
UAUUCUUEDWPEVNOYMDVNPUUDWLWNUUDATZCLBLWLUUFUUBBCDEUUFDNUUFENWPBYEVOWPCYGV
|
|
OWIAWPYIVQUNUUFWJBCAWIVRQSWLWNEWMDVSVTRSWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d z w ph $. $d x y ps $. $d x y z w $.
|
|
2mos.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Double "exists at most one", using implicit substitution. (Contributed
|
|
by NM, 10-Feb-2005.) $)
|
|
2mos $p |- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <->
|
|
A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) ) $=
|
|
( cv wceq wa wi wal wex wsb 2mo nfv sbrim wb sbie 2albii pm5.74d pm5.74ri
|
|
expcom bitr3i anbi2i imbi1i bitri ) ACHEHIZDHFHIZJZKDLCLFMEMAADFNZCENZJZU
|
|
JKZFLELZDLCLABJZUJKZFLELZDLCLACDEFOUOURCDUNUQEFUMUPUJULBAUKBCEBCPUHUKBUHU
|
|
KKUHAKZDFNUHBKZUHADFUHDPQUSUTDFUTDPUIUHABUHUIABRGUCUASUDUBSUEUFTTUG $.
|
|
$}
|
|
|
|
$( Double existential uniqueness. This theorem shows a condition under which
|
|
a "naive" definition matches the correct one. (Contributed by NM,
|
|
3-Dec-2001.) $)
|
|
2eu1 $p |- ( A. x E* y ph ->
|
|
( E! x E! y ph <-> ( E! x E. y ph /\ E! y E. x ph ) ) ) $=
|
|
( wmo wal weu wex wa wi eu5 exbii mobii anbi12i bitri simprbi anim2i ancoms
|
|
sp sylib com12 moimi nfa1 moanim ancrd 2moswap imdistani syl6 syl excom jca
|
|
2eu2ex jctild an4 syl6ibr 2exeu impbid1 ) ACDZBEZACFZBFZACGZBFZABGZCFZHZUTU
|
|
RVEUTURVABGZVCCGZHZVABDZVCCDZHZHZVEUTURVKVHUTVAUQHZBDZURVKIUTVMBGZVNUTUSBGZ
|
|
USBDZHVOVNHUSBJVPVOVQVNUSVMBACJZKUSVMBVRLMNOVNURVIURHVKVNURVIVNURVAHZBDURVI
|
|
IVSVMBVAURVMURUQVAUQBRPQUAURVABUQBUBUCSUDVIURVJURVIVJABCUETUFUGUHUTVFVGABCU
|
|
KZUTVFVGVTABCUISUJULVEVFVIHZVGVJHZHVLVBWAVDWBVABJVCCJMVFVIVGVJUMNUNTABCUOUP
|
|
$.
|
|
|
|
$( Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) $)
|
|
2eu2 $p |- ( E! y E. x ph -> ( E! x E! y ph <-> E! x E. y ph ) ) $=
|
|
( wex weu wmo wal wi eumo 2moex 2eu1 simpl syl6bi 3syl 2exeu expcom impbid
|
|
wa ) ABDZCEZACEBEZACDBEZTSCFACFBGZUAUBHSCIACBJUCUAUBTRUBABCKUBTLMNUBTUAABCO
|
|
PQ $.
|
|
|
|
$( Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) $)
|
|
2eu3 $p |- ( A. x A. y ( E* x ph \/ E* y ph ) ->
|
|
( ( E! x E! y ph /\ E! y E! x ph ) <-> ( E! x E. y ph /\ E! y E. x ph ) ) ) $=
|
|
( wmo wo wal weu wa wex wb nfmo1 19.31 albii nfal 19.32 bitri wi 2eu1 2exeu
|
|
biimpd ancom syl6ib adantld adantrd jaoi ancoms jca impbid1 sylbi ) ABDZACD
|
|
ZECFZBFZUJCFZUKBFZEZACGBGZABGCGZHZACIBGZABICGZHZJUMUNUKEZBFUPULVCBUJUKCACKL
|
|
MUNUKBUJBCABKNOPUPUSVBUNUSVBQUOUNURVBUQUNURVAUTHZVBUNURVDACBRTVAUTUAUBUCUOU
|
|
QVBURUOUQVBABCRTUDUEVBUQURABCSVAUTURACBSUFUGUHUI $.
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $.
|
|
$( This theorem provides us with a definition of double existential
|
|
uniqueness ("exactly one ` x ` and exactly one ` y ` "). Naively one
|
|
might think (incorrectly) that it could be defined by
|
|
` E! x E! y ph ` . See ~ 2eu1 for a condition under which the naive
|
|
definition holds and ~ 2exeu for a one-way implication. See ~ 2eu5 and
|
|
~ 2eu8 for alternate definitions. (Contributed by NM, 3-Dec-2001.) $)
|
|
2eu4 $p |- ( ( E! x E. y ph /\ E! y E. x ph ) <->
|
|
( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) ) $=
|
|
( wex weu wa cv wceq wi wal nfv eu3 anbi12i anbi2i bitri 19.26 nfa1 albii
|
|
an4 excom anidm 19.3 jcab bitr4i bitr2i alcom 19.23v 2albii 3bitri 2exbii
|
|
nfe1 nfim aaan eeanv ) ACFZBGZABFZCGZHUQBFZUQBIDIJZKZBLZDFZHZUSCFZUSCIEIJ
|
|
ZKZCLZEFZHZHVAVGHZVEVKHZHVAAVBVHHKZCLZBLZEFDFZHURVFUTVLUQBDUQDMNUSCEUSEMN
|
|
OVAVEVGVKUAVMVAVNVRVMVAVAHVAVGVAVAACBUBPVAUCQVRVDVJHZEFDFVNVQVSDEVQAVBKZC
|
|
LZAVHKZBLZHZCLZBLZVCVIHZCLBLVSVQWAWBCLZBLZHZBLZWFWKWABLZWIBLZHZVQWAWIBRWN
|
|
WLWIHZVQWMWIWLWIBWHBSUDPVQWAWHHZBLWOVPWPBVPVTWBHZCLWPVOWQCAVBVHUETVTWBCRQ
|
|
TWAWHBRQUFUGWEWJBWEWACLZWCCLZHWJWAWCCRWRWAWSWIWACVTCSUDWBCBUHOQTUFWDWGBCW
|
|
AVCWCVIAVBCUIAVHBUIOUJVCVIBCUQVBCACUMVBCMUNUSVHBABUMVHBMUNUOUKULVDVJDEUPU
|
|
GOUK $.
|
|
|
|
$( An alternate definition of double existential uniqueness (see ~ 2eu4 ).
|
|
A mistake sometimes made in the literature is to use ` E! x E! y ` to
|
|
mean "exactly one ` x ` and exactly one ` y ` ." (For example, see
|
|
Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is
|
|
actually a weaker assertion, as can be seen by expanding out the formal
|
|
definitions. This theorem shows that the erroneous definition can be
|
|
repaired by conjoining ` A. x E* y ph ` as an additional condition. The
|
|
correct definition apparently has never been published. ( ` E* ` means
|
|
"exists at most one.") (Contributed by NM, 26-Oct-2003.) $)
|
|
2eu5 $p |- ( ( E! x E! y ph /\ A. x E* y ph ) <->
|
|
( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) ) $=
|
|
( weu wmo wal wa wex weq 2eu1 pm5.32ri eumo adantl 2moex syl pm4.71i 2eu4
|
|
wi 3bitr2i ) ACFBFZACGBHZIACJZBFZABJZCFZIZUCIUHUDBJABDKCEKITCHBHEJDJIUCUB
|
|
UHABCLMUHUCUHUFCGZUCUGUIUEUFCNOACBPQRABCDESUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u $. $d z w v u ph $.
|
|
$( Two equivalent expressions for double existential uniqueness.
|
|
(Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro,
|
|
17-Oct-2016.) $)
|
|
2eu6 $p |- ( ( E! x E. y ph /\ E! y E. x ph ) <->
|
|
E. z E. w A. x A. y ( ph <-> ( x = z /\ y = w ) ) ) $=
|
|
( vu vv wex wa cv wceq wi wal wsb nfv nfsb sbequ12 equequ2 nfim bitri weu
|
|
wb nfs1v sylan9bbr cbvex2 bi2anan9 imbi2d 2albidv cbvex2v equequ1 imbi12d
|
|
2eu4 cbval2 2exbii 19.29r2 syl2anb 2albiim ancom nfan sbco2 sbcom2 bitr3i
|
|
2mo syl6bb anbi2d equcom anbi12i imbi2i impexp 2albii 19.21-2 anbi2i abai
|
|
sbbii bitr4i 2sb6 anbi1i sylibr bi2 2alimi 2eximi 2exsb bi1 jca impbii )
|
|
ACHZBUAABHCUAIWFBHZABJZDJZKZCJZEJZKZIZLZCMBMZEHDHZIZAWNUBZCMBMZEHDHZABCDE
|
|
ULWRXAWRACENZBDNZXCXCEFNZDGNZIZWIGJZKZWLFJZKZIZLZFMGMZIZEHDHZXAWGXCEHDHXM
|
|
EMDMZXOWQAXCBCDEADOAEOZXBBDUCZXBBDCACEUCPZWMAXBWJXCACEQXBBDQUDZUEWQAWHXGK
|
|
ZWKXIKZIZLZCMBMZFHGHZXPWPYEDEGFXKWOYDBCXKWNYCAXHWJYAXJWMYBDGBREFCRUFUGUHU
|
|
IYFXCXKLZEMDMZFHGHXPYEYHGFYDYGBCDEYDDOYDEOXCXKBXRXKBOZSXCXKCXSXKCOZSWNAXC
|
|
YCXKXTWJYAXHWMYBXJBDGUJCEFUJUFUKUMUNXCDEGFVCTTXCXMDEUOUPXAWNALZCMBMZWPIZE
|
|
HDHXOWTYMDEWTWPYLIYMAWNBCUQWPYLURTUNXNYMDEXNXCWPIZYMXNXCXCWPLZIYNXMYOXCXM
|
|
XCWOLZCMBMZYOXMXCAIZWIWHKZWLWKKZIZLZCMBMYQUUBXLBCGFUUBGOUUBFOXFXKBXCXEBXR
|
|
XDDGBXCEFBXRPPUSYISXFXKCXCXECXSXDDGCXCEFCXSPPUSYJSYCYRXFUUAXKYCAXEXCYCAAC
|
|
FNZBGNZXEYBAUUCYAUUDACFQUUCBGQUDUUDXBEFNZBGNZXEUUEUUCBGACFEXQUTVNUUFUUEBD
|
|
NZDGNXEUUEBGDUUEDOUTUUGXDDGXBEFBDVAVNVBVBVDVEYAYSXHYBYTXJBGDRCFERUFUKUMUU
|
|
BYPBCUUBYRWNLYPUUAWNYRYSWJYTWMDBVFECVFVGVHXCAWNVITVJVBXCWOBCXRXSVKTVLXCWP
|
|
VMVOXCYLWPABCDEVPVQTUNVOVRXAWGWQXAYLEHDHWGWTYLDEWSYKBCAWNVSVTWAABCDEWBVRW
|
|
TWPDEWSWOBCAWNWCVTWAWDWET $.
|
|
$}
|
|
|
|
$( Two equivalent expressions for double existential uniqueness.
|
|
(Contributed by NM, 19-Feb-2005.) $)
|
|
2eu7 $p |- ( ( E! x E. y ph /\ E! y E. x ph ) <->
|
|
E! x E! y ( E. x ph /\ E. y ph ) ) $=
|
|
( wex weu wa nfe1 nfeu euan ancom eubii 3bitri 3bitr4ri ) ABDZCEZACDZFZBEOP
|
|
BEZFNPFZCEZBEROFOPBNBCABGHITQBTPNFZCEPOFQSUACNPJKPNCACGIPOJLKROJM $.
|
|
|
|
$( Two equivalent expressions for double existential uniqueness. Curiously,
|
|
we can put ` E! ` on either of the internal conjuncts but not both. We
|
|
can also commute ` E! x E! y ` using ~ 2eu7 . (Contributed by NM,
|
|
20-Feb-2005.) $)
|
|
2eu8 $p |- ( E! x E! y ( E. x ph /\ E. y ph ) <->
|
|
E! x E! y ( E! x ph /\ E. y ph ) ) $=
|
|
( wex wa 2eu2 pm5.32i nfeu1 nfeu euan ancom eubii nfe1 3bitri 3bitr4ri 2eu7
|
|
weu 3bitr3ri ) ACDZBQZABQZCQZEZTABDZCQZEUASEZCQZBQZUDSECQBQTUBUEACBFGUBSEZB
|
|
QUBTEUHUCUBSBUABCABHIJUGUIBUGSUAEZCQSUBEUIUFUJCUASKLSUACACMJSUBKNLTUBKOABCP
|
|
R $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Equality has existential uniqueness. Special case of ~ eueq1 proved
|
|
using only predicate calculus. (Contributed by Stefan Allan,
|
|
4-Dec-2008.) $)
|
|
euequ1 $p |- E! x x = y $=
|
|
( vz weq weu wex wa wi wal a9ev equtr2 gen2 equequ1 eu4 mpbir2an ) ABDZAE
|
|
PAFPCBDZGACDHZCIAIABJRACACBKLPQACACBMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Two ways to express "only one thing exists." The left-hand side
|
|
requires only one variable to express this. Both sides are false in set
|
|
theory; see theorem dtru in set.mm. (Contributed by NM, 5-Apr-2004.) $)
|
|
exists1 $p |- ( E! x x = x <-> A. x x = y ) $=
|
|
( cv wceq weu wb wal wex df-eu equid bicom bitri albii exbii nfae 3bitr2i
|
|
tbt 19.9 ) ACZSDZAETSBCDZFZAGZBHUAAGZBHUDTABIUDUCBUAUBAUAUATFUBTUAAJQUATK
|
|
LMNUDBABBORP $.
|
|
|
|
$( A condition implying that at least two things exist. (Contributed by
|
|
NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
exists2 $p |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x ) $=
|
|
( vy wex wn cv wceq weu wal nfeu1 nfa1 wi exists1 ax16 sylbi exlimd com12
|
|
alex syl6ib con2d imp ) ABDZAEBDZBFZUDGZBHZEUBUFUCUBUFABIZUCEUFUBUGUFAUGB
|
|
UEBJABKUFUDCFGBIAUGLBCMABCNOPQABRSTUA $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Other axiomatizations related to classical predicate calculus
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Predicate calculus with all distinct variables
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Distinct variable version of ~ ax-7 . (Contributed by Mario Carneiro,
|
|
14-Aug-2015.) $)
|
|
ax-7d $a |- ( A. x A. y ph -> A. y A. x ph ) $.
|
|
|
|
$( Distinct variable version of ~ ax-8 . (Contributed by Mario Carneiro,
|
|
14-Aug-2015.) $)
|
|
ax-8d $a |- ( x = y -> ( x = z -> y = z ) ) $.
|
|
|
|
$( Distinct variable version of ~ ax9 , equal variables case. (Contributed
|
|
by Mario Carneiro, 14-Aug-2015.) $)
|
|
ax-9d1 $a |- -. A. x -. x = x $.
|
|
|
|
$( Distinct variable version of ~ ax9 , distinct variables case.
|
|
(Contributed by Mario Carneiro, 14-Aug-2015.) $)
|
|
ax-9d2 $a |- -. A. x -. x = y $.
|
|
|
|
$( Distinct variable version of ~ ax10 . (Contributed by Mario Carneiro,
|
|
14-Aug-2015.) $)
|
|
ax-10d $a |- ( A. x x = y -> A. y y = x ) $.
|
|
|
|
$( Distinct variable version of ~ ax-11 . (Contributed by Mario Carneiro,
|
|
14-Aug-2015.) $)
|
|
ax-11d $a |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Aristotelian logic: Assertic syllogisms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Model the Aristotelian assertic syllogisms using modern notation.
|
|
This section shows that the Aristotelian assertic syllogisms can be proven
|
|
with our axioms of logic, and also provides generally useful theorems.
|
|
|
|
In antiquity Aristotelian logic and Stoic logic
|
|
(see ~ mpto1 ) were the leading logical systems.
|
|
Aristotelian logic became the leading system in medieval Europe;
|
|
this section models this system (including later refinements to it).
|
|
Aristotle defined syllogisms very generally
|
|
("a discourse in which certain (specific) things having been supposed,
|
|
something different from the things supposed results of necessity
|
|
because these things are so")
|
|
Aristotle, _Prior Analytics_ 24b18-20.
|
|
However, in _Prior Analytics_ he limits himself to
|
|
categorical syllogisms that consist of three categorical propositions
|
|
with specific structures. The syllogisms are the valid subset of
|
|
the possible combinations of these structures.
|
|
The medieval schools used vowels to identify the types of terms
|
|
(a=all, e=none, i=some, and o=some are not), and named the different
|
|
syllogisms with Latin words that had the vowels in the intended order.
|
|
|
|
"There is a surprising amount of scholarly debate
|
|
about how best to formalize Aristotle's syllogisms..." according to
|
|
_Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic_,
|
|
Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28.
|
|
For example, Lukasiewicz believes it is important to note that
|
|
"Aristotle does not introduce singular terms or premisses into his system".
|
|
Lukasiewicz also believes that Aristotelian syllogisms are
|
|
predicates (having a true/false value), not inference rules:
|
|
"The characteristic sign of an inference is the word 'therefore'...
|
|
no syllogism is formulated by Aristotle primarily as an inference,
|
|
but they are all implications."
|
|
Jan Lukasiewicz, _Aristotle's Syllogistic from the Standpoint of
|
|
Modern Formal Logic_, Second edition, Oxford, 1957, page 1-2.
|
|
Lukasiewicz devised a specialized prefix notation for representing
|
|
Aristotelian syllogisms instead of using standard predicate logic notation.
|
|
|
|
We instead translate each Aristotelian syllogism into an inference rule,
|
|
and each rule is defined using standard predicate logic notation and
|
|
predicates. The predicates are represented by wff variables
|
|
that may depend on the quantified variable ` x ` .
|
|
Our translation is essentially identical to the one
|
|
use in Rini page 18, Table 2 "Non-Modal Syllogisms in
|
|
Lower Predicate Calculus (LPC)", which uses
|
|
standard predicate logic with predicates. Rini states,
|
|
"the crucial point is that we capture the meaning Aristotle intends,
|
|
and the method by which we represent that meaning is less important."
|
|
There are two differences: we make the existence criteria explicit, and
|
|
we use ` ph ` , ` ps ` , and ` ch ` in the order they appear
|
|
(a common Metamath convention).
|
|
Patzig also uses standard predicate logic notation and predicates
|
|
(though he interprets them as conditional propositions, not as
|
|
inference rules); see
|
|
Gunther Patzig, _Aristotle's Theory of the Syllogism_ second edition, 1963,
|
|
English translation by Jonathan Barnes, 1968, page 38.
|
|
Terms such as "all" and "some" are translated into predicate logic
|
|
using the aproach devised by Frege and Russell.
|
|
"Frege (and Russell) devised an ingenious procedure for regimenting
|
|
binary quantifiers like "every" and "some" in terms of unary quantifiers
|
|
like "everything" and "something": they formalized sentences of the form
|
|
"Some A is B" and "Every A is B" as
|
|
exists x (Ax and Bx) and all x (Ax implies Bx), respectively."
|
|
"Quantifiers and Quantification", _Stanford Encyclopedia of Philosophy_,
|
|
~ http://plato.stanford.edu/entries/quantification/ .
|
|
See _Principia Mathematica_ page 22 and *10 for more information
|
|
(especially *10.3 and *10.26).
|
|
|
|
Expressions of the form "no ` ph ` is ` ps ` " are consistently translated as
|
|
` A. x ( ph -> -. ps ) ` . These can also be expressed as
|
|
` -. E. x ( ph /\ ps ) ` , per ~ alinexa .
|
|
We translate "all ` ph ` is ` ps ` " to ` A. x ( ph -> ps ) ` ,
|
|
"some ` ph ` is ` ps ` " to ` E. x ( ph /\ ps ) ` , and
|
|
"some ` ph ` is not ` ps ` " to ` E. x ( ph /\ -. ps ) ` .
|
|
It is traditional to use the singular verb "is", not the plural
|
|
verb "are", in the generic expressions.
|
|
By convention the major premise is listed first.
|
|
|
|
In traditional Aristotelian syllogisms the predicates
|
|
have a restricted form ("x is a ..."); those predicates
|
|
could be modeled in modern notation by constructs such as
|
|
` x = A ` , ` x e. A ` , or ` x C_ A ` .
|
|
Here we use wff variables instead of specialized restricted forms.
|
|
This generalization makes the syllogisms more useful
|
|
in more circumstances. In addition, these expressions make
|
|
it clearer that the syllogisms of Aristolean logic are the
|
|
forerunners of predicate calculus. If we used restricted forms
|
|
like ` x e. A ` instead, we would not only unnecessarily limit
|
|
their use, but we would also need to use set and class axioms,
|
|
making their relationship to predicate calculus less clear.
|
|
|
|
There are some widespread misconceptions about the existential
|
|
assumptions made by Aristotle (aka "existential import").
|
|
Aristotle was not trying to develop something exactly corresponding
|
|
to modern logic. Aristotle devised "a companion-logic for science.
|
|
He relegates fictions like fairy godmothers and mermaids and unicorns to
|
|
the realms of poetry and literature. In his mind, they exist outside the
|
|
ambit of science. This is why he leaves no room for such non-existent
|
|
entities in his logic. This is a thoughtful choice, not an inadvertent
|
|
omission. Technically, Aristotelian science is a search for definitions,
|
|
where a definition is "a phrase signifying a thing's essence."
|
|
(Topics, I.5.102a37, Pickard-Cambridge.)...
|
|
Because non-existent entities cannot be anything, they do not, in
|
|
Aristotle's mind, possess an essence... This is why he leaves
|
|
no place for fictional entities like goat-stags (or unicorns)."
|
|
Source: Louis F. Groarke, "Aristotle: Logic",
|
|
section 7. (Existential Assumptions),
|
|
_Internet Encyclopedia of Philosophy_ (A Peer-Reviewed Academic Resource),
|
|
~ http://www.iep.utm.edu/aris-log/ .
|
|
Thus, some syllogisms have "extra" existence
|
|
hypotheses that do not directly appear in Aristotle's original materials
|
|
(since they were always assumed); they are added where they are needed.
|
|
This affects ~ barbari , ~ celaront , ~ cesaro , ~ camestros , ~ felapton ,
|
|
~ darapti , ~ calemos , ~ fesapo , and ~ bamalip .
|
|
|
|
These are only the _assertic_ syllogisms.
|
|
Aristotle also defined modal syllogisms that deal with modal
|
|
qualifiers such as "necessarily" and "possibly".
|
|
Historically Aristotelian modal syllogisms were not as widely used.
|
|
For more about modal syllogisms in a modern context, see Rini as well as
|
|
_Aristotle's Modal Syllogistic_ by Marko Malink, Harvard
|
|
University Press, November 2013. We do not treat them further here.
|
|
|
|
Aristotelean logic is essentially the forerunner of predicate calculus
|
|
(as well as set theory since it discusses membership in groups),
|
|
while Stoic logic is essentially the forerunner of propositional calculus.
|
|
$)
|
|
|
|
$( Figure 1. Aristotelian syllogisms are grouped by "figures",
|
|
which doesn't matter for our purposes but is a reasonable way
|
|
to order them. $)
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Barbara", e.g.,
|
|
"All men are mortal". By convention, the major premise is first. $)
|
|
barbara.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Barbara, e.g., "Socrates is a man". $)
|
|
barbara.min $e |- A. x ( ch -> ph ) $.
|
|
$( "Barbara", one of the fundamental syllogisms of Aristotelian logic. All
|
|
` ph ` is ` ps ` , and all ` ch ` is ` ph ` , therefore all ` ch ` is
|
|
` ps ` . (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.)
|
|
For example, given "All men are mortal" and "Socrates is a man", we can
|
|
prove "Socrates is mortal". If H is the set of men, M is the set of
|
|
mortal beings, and S is Socrates, these word phrases can be represented
|
|
as ` A. x ( x e. H -> x e. M ) ` (all men are mortal) and
|
|
` A. x ( x = S -> x e. H ) ` (Socrates is a man) therefore
|
|
` A. x ( x = S -> x e. M ) ` (Socrates is mortal). Russell and
|
|
Whitehead note that the "syllogism in Barbara is derived..." from
|
|
~ syl . (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most
|
|
of the proof is in ~ alsyl . There are a legion of sources for Barbara,
|
|
including ~ http://www.friesian.com/aristotl.htm ,
|
|
~ http://plato.stanford.edu/entries/aristotle-logic/ , and
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 24-Aug-2016.) $)
|
|
barbara $p |- A. x ( ch -> ps ) $=
|
|
( wi wal alsyl mp2an ) CAGDHABGDHCBGDHFECABDIJ $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Celarent", e.g.,
|
|
"No reptiles have fur". $)
|
|
celarent.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Celarent, e.g., "All snakes are reptiles". $)
|
|
celarent.min $e |- A. x ( ch -> ph ) $.
|
|
$( "Celarent", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , and all ` ch ` is ` ph ` , therefore no ` ch ` is ` ps ` . (In
|
|
Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example,
|
|
given the "No reptiles have fur" and "All snakes are reptiles",
|
|
therefore "No snakes have fur". Example from
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
celarent $p |- A. x ( ch -> -. ps ) $=
|
|
( wn barbara ) ABGCDEFH $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Darii", e.g.,
|
|
"All rabbits have fur". $)
|
|
darii.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Darii, e.g., "Some pets are rabbits." $)
|
|
darii.min $e |- E. x ( ch /\ ph ) $.
|
|
$( "Darii", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and some ` ch ` is ` ph ` , therefore some ` ch ` is ` ps ` .
|
|
(In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For
|
|
example, given "All rabbits have fur" and "Some pets are rabbits",
|
|
therefore "Some pets have fur". Example from
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 24-Aug-2016.) $)
|
|
darii $p |- E. x ( ch /\ ps ) $=
|
|
( wa wex wi spi anim2i eximi ax-mp ) CAGZDHCBGZDHFNODABCABIDEJKLM $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Ferio" ("Ferioque"),
|
|
e.g., "No homework is fun". $)
|
|
ferio.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Ferio, e.g., "Some reading is homework." $)
|
|
ferio.min $e |- E. x ( ch /\ ph ) $.
|
|
$( "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No
|
|
` ph ` is ` ps ` , and some ` ch ` is ` ph ` , therefore some ` ch ` is
|
|
not ` ps ` . (In Aristotelian notation, EIO-1: MeP and SiM therefore
|
|
SoP.) For example, given "No homework is fun" and "Some reading is
|
|
homework", therefore "Some reading is not fun". This is essentially a
|
|
logical axiom in Aristotelian logic. Example from
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
ferio $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn darii ) ABGCDEFH $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Barbari", e.g.,
|
|
e.g., "All men are mortal". $)
|
|
barbari.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Barbari, e.g., "All Greeks are men." $)
|
|
barbari.min $e |- A. x ( ch -> ph ) $.
|
|
$( Existence premise for Barbari, e.g., "Greeks exist." $)
|
|
barbari.e $e |- E. x ch $.
|
|
$( "Barbari", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , all ` ch ` is ` ph ` , and some ` ch ` exist, therefore some
|
|
` ch ` is ` ps ` . (In Aristotelian notation, AAI-1: MaP and SaM
|
|
therefore SiP.) For example, given "All men are mortal", "All Greeks are
|
|
men", and "Greeks exist", therefore "Some Greeks are mortal". Note the
|
|
existence hypothesis (to prove the "some" in the conclusion). Example
|
|
from ~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David
|
|
A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler,
|
|
30-Aug-2016.) $)
|
|
barbari $p |- E. x ( ch /\ ps ) $=
|
|
( wex wa wi barbara spi ancli eximi ax-mp ) CDHCBIZDHGCPDCBCBJDABCDEFKLMN
|
|
O $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Celaront", e.g.,
|
|
e.g., "No reptiles have fur". $)
|
|
celaront.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Celaront, e.g., "All Snakes are reptiles." $)
|
|
celaront.min $e |- A. x ( ch -> ph ) $.
|
|
$( Existence premise for Celaront, e.g., "Snakes exist." $)
|
|
celaront.e $e |- E. x ch $.
|
|
$( "Celaront", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , all ` ch ` is ` ph ` , and some ` ch ` exist, therefore some
|
|
` ch ` is not ` ps ` . (In Aristotelian notation, EAO-1: MeP and SaM
|
|
therefore SoP.) For example, given "No reptiles have fur", "All snakes
|
|
are reptiles.", and "Snakes exist.", prove "Some snakes have no fur".
|
|
Note the existence hypothesis. Example from
|
|
~ https://en.wikipedia.org/wiki/Syllogism . (Contributed by David A.
|
|
Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
celaront $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn barbari ) ABHCDEFGI $.
|
|
$}
|
|
|
|
$( Figure 2 $)
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Cesare" $)
|
|
cesare.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Cesare $)
|
|
cesare.min $e |- A. x ( ch -> ps ) $.
|
|
$( "Cesare", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , and all ` ch ` is ` ps ` , therefore no ` ch ` is ` ph ` . (In
|
|
Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to
|
|
~ celarent . (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised
|
|
by David A. Wheeler, 13-Nov-2016.) $)
|
|
cesare $p |- A. x ( ch -> -. ph ) $=
|
|
( wn wi spi nsyl3 ax-gen ) CAGHDABCABGHDEICBHDFIJK $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Camestres" $)
|
|
camestres.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Camestres $)
|
|
camestres.min $e |- A. x ( ch -> -. ps ) $.
|
|
$( "Camestres", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and no ` ch ` is ` ps ` , therefore no ` ch ` is ` ph ` . (In
|
|
Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed
|
|
by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler,
|
|
2-Sep-2016.) $)
|
|
camestres $p |- A. x ( ch -> -. ph ) $=
|
|
( wn wi spi nsyl ax-gen ) CAGHDCBACBGHDFIABHDEIJK $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Festino" $)
|
|
festino.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Festino $)
|
|
festino.min $e |- E. x ( ch /\ ps ) $.
|
|
$( "Festino", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , and some ` ch ` is ` ps ` , therefore some ` ch ` is not
|
|
` ph ` . (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.)
|
|
(Contributed by David A. Wheeler, 25-Nov-2016.) $)
|
|
festino $p |- E. x ( ch /\ -. ph ) $=
|
|
( wa wex wn wi spi con2i anim2i eximi ax-mp ) CBGZDHCAIZGZDHFPRDBQCABABIJ
|
|
DEKLMNO $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Baroco" $)
|
|
baroco.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for Baroco $)
|
|
baroco.min $e |- E. x ( ch /\ -. ps ) $.
|
|
$( "Baroco", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and some ` ch ` is not ` ps ` , therefore some ` ch ` is not
|
|
` ph ` . (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.)
|
|
For example, "All informative things are useful", "Some websites are not
|
|
useful", therefore "Some websites are not informative." (Contributed by
|
|
David A. Wheeler, 28-Aug-2016.) $)
|
|
baroco $p |- E. x ( ch /\ -. ph ) $=
|
|
( wn wa wex wi spi con3i anim2i eximi ax-mp ) CBGZHZDICAGZHZDIFQSDPRCABAB
|
|
JDEKLMNO $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Cesaro" $)
|
|
cesaro.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for Cesaro $)
|
|
cesaro.min $e |- A. x ( ch -> ps ) $.
|
|
$( Existence premise for Cesaro $)
|
|
cesaro.e $e |- E. x ch $.
|
|
$( "Cesaro", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , all ` ch ` is ` ps ` , and ` ch ` exist, therefore some ` ch `
|
|
is not ` ph ` . (In Aristotelian notation, EAO-2: PeM and SaM
|
|
therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
cesaro $p |- E. x ( ch /\ -. ph ) $=
|
|
( wex wn wa wi spi nsyl3 ancli eximi ax-mp ) CDHCAIZJZDHGCRDCQABCABIKDELC
|
|
BKDFLMNOP $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Camestros" $)
|
|
camestros.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
camestros.min $e |- A. x ( ch -> -. ps ) $.
|
|
$( Existence premise for Camestros $)
|
|
camestros.e $e |- E. x ch $.
|
|
$( "Camestros", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , no ` ch ` is ` ps ` , and ` ch ` exist, therefore some ` ch `
|
|
is not ` ph ` . (In Aristotelian notation, AEO-2: PaM and SeM
|
|
therefore SoP.) For example, "All horses have hooves", "No humans have
|
|
hooves", and humans exist, therefore "Some humans are not horses".
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A.
|
|
Wheeler, 2-Sep-2016.) $)
|
|
camestros $p |- E. x ( ch /\ -. ph ) $=
|
|
( wex wn wa wi spi nsyl ancli eximi ax-mp ) CDHCAIZJZDHGCRDCQCBACBIKDFLAB
|
|
KDELMNOP $.
|
|
$}
|
|
|
|
$( Figure 3 $)
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Datisi" $)
|
|
datisi.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
datisi.min $e |- E. x ( ph /\ ch ) $.
|
|
$( "Datisi", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and some ` ph ` is ` ch ` , therefore some ` ch ` is ` ps ` .
|
|
(In Aristotelian notation, AII-3: MaP and MiS therefore SiP.)
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) $)
|
|
datisi $p |- E. x ( ch /\ ps ) $=
|
|
( wa wex simpr wi spi adantr jca eximi ax-mp ) ACGZDHCBGZDHFPQDPCBACIABCA
|
|
BJDEKLMNO $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Disamis" $)
|
|
disamis.maj $e |- E. x ( ph /\ ps ) $.
|
|
$( Minor premise for $)
|
|
disamis.min $e |- A. x ( ph -> ch ) $.
|
|
$( "Disamis", one of the syllogisms of Aristotelian logic. Some ` ph ` is
|
|
` ps ` , and all ` ph ` is ` ch ` , therefore some ` ch ` is ` ps ` .
|
|
(In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.)
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) $)
|
|
disamis $p |- E. x ( ch /\ ps ) $=
|
|
( wa wex wi spi anim1i eximi ax-mp ) ABGZDHCBGZDHENODACBACIDFJKLM $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Ferison" $)
|
|
ferison.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for $)
|
|
ferison.min $e |- E. x ( ph /\ ch ) $.
|
|
$( "Ferison", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , and some ` ph ` is ` ch ` , therefore some ` ch ` is not
|
|
` ps ` . (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.)
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A.
|
|
Wheeler, 2-Sep-2016.) $)
|
|
ferison $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn datisi ) ABGCDEFH $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Bocardo" $)
|
|
bocardo.maj $e |- E. x ( ph /\ -. ps ) $.
|
|
$( Minor premise for $)
|
|
bocardo.min $e |- A. x ( ph -> ch ) $.
|
|
$( "Bocardo", one of the syllogisms of Aristotelian logic. Some ` ph ` is
|
|
not ` ps ` , and all ` ph ` is ` ch ` , therefore some ` ch ` is not
|
|
` ps ` . (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.)
|
|
For example, "Some cats have no tails", "All cats are mammals",
|
|
therefore "Some mammals have no tails". A reorder of ~ disamis ; prefer
|
|
using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(New usage is discouraged.) $)
|
|
bocardo $p |- E. x ( ch /\ -. ps ) $=
|
|
( wn disamis ) ABGCDEFH $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Felapton" $)
|
|
felapton.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for $)
|
|
felapton.min $e |- A. x ( ph -> ch ) $.
|
|
$( Existence premise for Felapton $)
|
|
felapton.e $e |- E. x ph $.
|
|
$( "Felapton", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , all ` ph ` is ` ch ` , and some ` ph ` exist, therefore some
|
|
` ch ` is not ` ps ` . (In Aristotelian notation, EAO-3: MeP and MaS
|
|
therefore SoP.) For example, "No flowers are animals" and "All flowers
|
|
are plants", therefore "Some plants are not animals". (Contributed by
|
|
David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler,
|
|
2-Sep-2016.) $)
|
|
felapton $p |- E. x ( ch /\ -. ps ) $=
|
|
( wex wn wa wi spi jca eximi ax-mp ) ADHCBIZJZDHGAQDACPACKDFLAPKDELMNO $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Darapti" $)
|
|
darapti.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
darapti.min $e |- A. x ( ph -> ch ) $.
|
|
$( Existence premise for Darapti $)
|
|
darapti.e $e |- E. x ph $.
|
|
$( "Darapti", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , all ` ph ` is ` ch ` , and some ` ph ` exist, therefore some
|
|
` ch ` is ` ps ` . (In Aristotelian notation, AAI-3: MaP and MaS
|
|
therefore SiP.) For example, "All squares are rectangles" and "All
|
|
squares are rhombuses", therefore "Some rhombuses are rectangles".
|
|
(Contributed by David A. Wheeler, 28-Aug-2016.) $)
|
|
darapti $p |- E. x ( ch /\ ps ) $=
|
|
( wex wa wi spi jca eximi ax-mp ) ADHCBIZDHGAODACBACJDFKABJDEKLMN $.
|
|
$}
|
|
|
|
$( Figure 4 $)
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Calemes" $)
|
|
calemes.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
calemes.min $e |- A. x ( ps -> -. ch ) $.
|
|
$( "Calemes", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , and no ` ps ` is ` ch ` , therefore no ` ch ` is ` ph ` . (In
|
|
Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed
|
|
by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler,
|
|
2-Sep-2016.) $)
|
|
calemes $p |- A. x ( ch -> -. ph ) $=
|
|
( wn wi spi con2i nsyl ax-gen ) CAGHDCBABCBCGHDFIJABHDEIKL $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Dimatis" $)
|
|
dimatis.maj $e |- E. x ( ph /\ ps ) $.
|
|
$( Minor premise for $)
|
|
dimatis.min $e |- A. x ( ps -> ch ) $.
|
|
$( "Dimatis", one of the syllogisms of Aristotelian logic. Some ` ph ` is
|
|
` ps ` , and all ` ps ` is ` ch ` , therefore some ` ch ` is ` ph ` .
|
|
(In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For
|
|
example, "Some pets are rabbits.", "All rabbits have fur", therefore
|
|
"Some fur bearing animals are pets". Like ~ darii with positions
|
|
interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) $)
|
|
dimatis $p |- E. x ( ch /\ ph ) $=
|
|
( wa wex wi spi adantl simpl jca eximi ax-mp ) ABGZDHCAGZDHEPQDPCABCABCID
|
|
FJKABLMNO $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Fresison" $)
|
|
fresison.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for $)
|
|
fresison.min $e |- E. x ( ps /\ ch ) $.
|
|
$( "Fresison", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` (PeM), and some ` ps ` is ` ch ` (MiS), therefore some ` ch ` is
|
|
not ` ph ` (SoP). (In Aristotelian notation, EIO-4: PeM and MiS
|
|
therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
fresison $p |- E. x ( ch /\ -. ph ) $=
|
|
( wa wex wn simpr wi spi con2i adantr jca eximi ax-mp ) BCGZDHCAIZGZDHFRT
|
|
DRCSBCJBSCABABIKDELMNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Calemos" $)
|
|
calemos.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
calemos.min $e |- A. x ( ps -> -. ch ) $.
|
|
$( Existence premise for Calemos $)
|
|
calemos.e $e |- E. x ch $.
|
|
$( "Calemos", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` (PaM), no ` ps ` is ` ch ` (MeS), and ` ch ` exist, therefore
|
|
some ` ch ` is not ` ph ` (SoP). (In Aristotelian notation, AEO-4: PaM
|
|
and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
calemos $p |- E. x ( ch /\ -. ph ) $=
|
|
( wex wn wa wi spi con2i nsyl ancli eximi ax-mp ) CDHCAIZJZDHGCSDCRCBABCB
|
|
CIKDFLMABKDELNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Fesapo" $)
|
|
fesapo.maj $e |- A. x ( ph -> -. ps ) $.
|
|
$( Minor premise for $)
|
|
fesapo.min $e |- A. x ( ps -> ch ) $.
|
|
$( Existence premise for Fesapo $)
|
|
fesapo.e $e |- E. x ps $.
|
|
$( "Fesapo", one of the syllogisms of Aristotelian logic. No ` ph ` is
|
|
` ps ` , all ` ps ` is ` ch ` , and ` ps ` exist, therefore some ` ch `
|
|
is not ` ph ` . (In Aristotelian notation, EAO-4: PeM and MaS
|
|
therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|
|
(Revised by David A. Wheeler, 2-Sep-2016.) $)
|
|
fesapo $p |- E. x ( ch /\ -. ph ) $=
|
|
( wex wn wa wi spi con2i jca eximi ax-mp ) BDHCAIZJZDHGBRDBCQBCKDFLABABIK
|
|
DELMNOP $.
|
|
$}
|
|
|
|
${
|
|
$( Major premise for the Aristotelian syllogism "Bamalip" $)
|
|
bamalip.maj $e |- A. x ( ph -> ps ) $.
|
|
$( Minor premise for $)
|
|
bamalip.min $e |- A. x ( ps -> ch ) $.
|
|
$( Existence premise for Bamalip $)
|
|
bamalip.e $e |- E. x ph $.
|
|
$( "Bamalip", one of the syllogisms of Aristotelian logic. All ` ph ` is
|
|
` ps ` , all ` ps ` is ` ch ` , and ` ph ` exist, therefore some ` ch `
|
|
is ` ph ` . (In Aristotelian notation, AAI-4: PaM and MaS therefore
|
|
SiP.) Like ~ barbari . (Contributed by David A. Wheeler,
|
|
28-Aug-2016.) $)
|
|
bamalip $p |- E. x ( ch /\ ph ) $=
|
|
( wex wa wi spi syl ancri eximi ax-mp ) ADHCAIZDHGAPDACABCABJDEKBCJDFKLMN
|
|
O $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Intuitionistic logic
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
Intuitionistic (constructive) logic is similar to classical logic
|
|
with the notable omission of ~ ax-3 and theorems such as ~ exmid or
|
|
~ peirce . We mostly treat intuitionistic logic in a separate file, iset.mm,
|
|
which is known as the Intuitionistic Logic Explorer on the web site. However,
|
|
iset.mm has a number of additional axioms (mainly to replace definitions like
|
|
~ df-or and ~ df-ex which are not valid in intitionistic logic) and we want
|
|
to prove those axioms here to demonstrate that adding those axioms in iset.mm
|
|
does not make iset.mm any less consistent than set.mm.
|
|
|
|
$)
|
|
|
|
$( Specialization (intuitionistic logic axiom ax-4). This is just ~ sp by
|
|
another name. (Contributed by Jim Kingdon, 31-Dec-2017.) $)
|
|
axi4 $p |- ( A. x ph -> ph ) $=
|
|
( sp ) ABC $.
|
|
|
|
$( Converse of ax-5o (intuitionistic logic axiom ax-i5r). (Contributed by
|
|
Jim Kingdon, 31-Dec-2017.) $)
|
|
axi5r $p |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) ) $=
|
|
( wal wi hba1 hbim sp imim2i alimi syl ) ACDZBCDZEZNCDLBEZCDLMCACFBCFGNOCMB
|
|
LBCHIJK $.
|
|
|
|
$( ` x ` is not free in ` A. x ph ` (intuitionistic logic axiom ax-ial).
|
|
(Contributed by Jim Kingdon, 31-Dec-2017.) $)
|
|
axial $p |- ( A. x ph -> A. x A. x ph ) $=
|
|
( hba1 ) ABC $.
|
|
|
|
$( ` x ` is bound in ` E. x ph ` (intuitionistic logic axiom ax-ie1).
|
|
(Contributed by Jim Kingdon, 31-Dec-2017.) $)
|
|
axie1 $p |- ( E. x ph -> A. x E. x ph ) $=
|
|
( hbe1 ) ABC $.
|
|
|
|
$( A key property of existential quantification (intuitionistic logic axiom
|
|
ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.) $)
|
|
axie2 $p |- ( A. x ( ps -> A. x ps ) ->
|
|
( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
|
|
( wal wi wnf wex wb df-nf 19.23t sylbir ) BBCDECDBCFABECDACGBEHBCIABCJK $.
|
|
|
|
$( Axiom of existence (intuitionistic logic axiom ax-i9). In classical
|
|
logic, this is equivalent to ~ ax-9 but in intuitionistic logic it needs
|
|
to be stated using the existential quantifier. (Contributed by Jim
|
|
Kingdon, 31-Dec-2017.) $)
|
|
axi9 $p |- E. x x = y $=
|
|
( a9e ) ABC $.
|
|
|
|
$( Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This
|
|
is just ~ ax10 by another name. (Contributed by Jim Kingdon,
|
|
31-Dec-2017.) $)
|
|
axi10 $p |- ( A. x x = y -> A. y y = x ) $=
|
|
( ax10 ) ABC $.
|
|
|
|
$( Axiom of Variable Substitution for Existence (intuitionistic logic axiom
|
|
ax-i11e). This can be derived from ~ ax-11 in a classical context but a
|
|
separate axiom is needed for intuitionistic predicate calculus.
|
|
(Contributed by Jim Kingdon, 31-Dec-2017.) $)
|
|
axi11e $p |- ( x = y -> ( E. x ( x = y /\ ph ) -> E. y ph ) ) $=
|
|
( weq wex wa wn wal wi ax-11 alnex alinexa 3imtr3g con4d ) BCDZACEZOAFBEZOA
|
|
GZCHORIBHPGQGRBCJACKOABLMN $.
|
|
|
|
$( Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12).
|
|
|
|
In classical logic, this is mostly a restatement of ~ ax12o (with one
|
|
additional quantifier). But in intuitionistic logic, changing the
|
|
negations and implications to disjunctions makes it stronger.
|
|
|
|
(Contributed by Jim Kingdon, 31-Dec-2017.) $)
|
|
axi12 $p |- ( A. z z = x \/ ( A. z z = y \/
|
|
A. z ( x = y -> A. z x = y ) ) ) $=
|
|
( cv wceq wal wo wi wn ax12o df-or imbi2i mpbir orass ax-gen nfa1 nfor mpbi
|
|
19.32 ) CDZADZEZCFZTBDZEZCFZGZUAUDEZUHCFHZCFZGZUCUFUJGGUGUIGZCFUKULCULUCUFU
|
|
IGZGZUNUCIZUMHZUPUOUFIUIHZHABCJUMUQUOUFUIKLMUCUMKMUCUFUINMOUGUICUCUFCUBCPUE
|
|
CPQSRUCUFUJNR $.
|
|
|
|
$( End $[ set-pred.mm $] $)
|
|
|
|
|
|
$(
|
|
###############################################################################
|
|
NEW FOUNDATIONS (NF) SET THEORY
|
|
###############################################################################
|
|
|
|
Here we introduce New Foundations set theory.
|
|
We first introduce the axiom of extensionality in ~ ax-ext .
|
|
We later add set construction axioms from
|
|
Hailperin, such as ~ ax-nin ,
|
|
that are designed to implement the
|
|
Stratification Axiom from Quine.
|
|
|
|
We then introduce ordered pairs, relationships, and functions.
|
|
Note that the definition of an ordered pair (in ~ df-op ) is different
|
|
than the Kuratowski ordered pair definition (in ~ df-opk )
|
|
typically used in ZFC, because the Kuratowski definition is not type-level.
|
|
|
|
We conclude with orderings.
|
|
$)
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
NF Set Theory - start with the Axiom of Extensionality
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce the Axiom of Extensionality
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It
|
|
states that two sets are identical if they contain the same elements.
|
|
Axiom Ext of [BellMachover] p. 461.
|
|
|
|
Set theory can also be formulated with a _single_ primitive predicate
|
|
` e. ` on top of traditional predicate calculus _without_ equality. In
|
|
that case the Axiom of Extensionality becomes
|
|
` ( A. w ( w e. x <-> w e. y ) -> ( x e. z -> y e. z ) ) ` , and
|
|
equality ` x = y ` is _defined_ as ` A. w ( w e. x <-> w e. y ) ` . All
|
|
of the usual axioms of equality then become theorems of set theory.
|
|
See, for example, Axiom 1 of [TakeutiZaring] p. 8.
|
|
|
|
To use the above "equality-free" version of Extensionality with
|
|
Metamath's logical axioms, we would rewrite ~ ax-8 through ~ ax-16 with
|
|
equality expanded according to the above definition. Some of those
|
|
axioms could be proved from set theory and would be redundant. Not all
|
|
of them are redundant, since our axioms of predicate calculus make
|
|
essential use of equality for the proper substitution that is a
|
|
primitive notion in traditional predicate calculus. A study of such an
|
|
axiomatization would be an interesting project for someone exploring the
|
|
foundations of logic.
|
|
|
|
_General remarks_: Our set theory axioms are presented using defined
|
|
connectives ( ` <-> ` , ` E. ` , etc.) for convenience. However, it is
|
|
implicitly understood that the actual axioms use only the primitive
|
|
connectives ` -> ` , ` -. ` , ` A. ` , ` = ` , and ` e. ` . It is
|
|
straightforward to establish the equivalence between the actual axioms
|
|
and the ones we display, and we will not do so.
|
|
|
|
It is important to understand that strictly speaking, all of our set
|
|
theory axioms are really schemes that represent an infinite number of
|
|
actual axioms. This is inherent in the design of Metamath
|
|
("metavariable math"), which manipulates only metavariables. For
|
|
example, the metavariable ` x ` in ~ ax-ext can represent any actual
|
|
variable _v1_, _v2_, _v3_,... . Distinct variable restrictions ($d)
|
|
prevent us from substituting say _v1_ for both ` x ` and ` z ` . This
|
|
is in contrast to typical textbook presentations that present actual
|
|
axioms (except for ZFC Replacement ax-rep in set.mm, which involves a
|
|
wff metavariable). In practice, though, the theorems and proofs are
|
|
essentially the same. The $d restrictions make each of the infinite
|
|
axioms generated by the ~ ax-ext scheme exactly logically equivalent to
|
|
each other and in particular to the actual axiom of the textbook
|
|
version. (Contributed by NM, 5-Aug-1993.) $)
|
|
ax-ext $a |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $.
|
|
|
|
$( The Axiom of Extensionality ( ~ ax-ext ) restated so that it postulates
|
|
the existence of a set ` z ` given two arbitrary sets ` x ` and ` y ` .
|
|
This way to express it follows the general idea of the other ZFC axioms,
|
|
which is to postulate the existence of sets given other sets.
|
|
(Contributed by NM, 28-Sep-2003.) $)
|
|
axext2 $p |- E. z ( ( z e. x <-> z e. y ) -> x = y ) $=
|
|
( cv wcel wb wceq wi wex wal ax-ext 19.36v mpbir ) CDZADZENBDZEFZOPGZHCIQ
|
|
CJRHABCKQRCLM $.
|
|
$}
|
|
|
|
${
|
|
$d z x w $. $d z y w $.
|
|
$( A generalization of the Axiom of Extensionality in which ` x ` and ` y `
|
|
need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof
|
|
shortened by Andrew Salmon, 12-Aug-2011.) $)
|
|
axext3 $p |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $=
|
|
( vw cv wcel wb wal wi elequ2 bibi1d albidv equequ1 imbi12d ax-ext chvarv
|
|
wceq ) CEZDEZFZRBEZFZGZCHZSUAQZIRAEZFZUBGZCHZUFUAQZIDASUFQZUDUIUEUJUKUCUH
|
|
CUKTUGUBDACJKLDABMNDBCOP $.
|
|
|
|
$( A bidirectional version of Extensionality. Although this theorem
|
|
"looks" like it is just a definition of equality, it requires the Axiom
|
|
of Extensionality for its proof under our axiomatization. See the
|
|
comments for ~ ax-ext and ~ df-cleq . (Contributed by NM,
|
|
14-Nov-2008.) $)
|
|
axext4 $p |- ( x = y <-> A. z ( z e. x <-> z e. y ) ) $=
|
|
( cv wceq wcel wb wal elequ2 alrimiv axext3 impbii ) ADZBDZEZCDZMFPNFGZCH
|
|
OQCABCIJABCKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d ph z $.
|
|
bm1.1.1 $e |- F/ x ph $.
|
|
$( Any set defined by a property is the only set defined by that property.
|
|
Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM,
|
|
30-Jun-1994.) $)
|
|
bm1.1 $p |- ( E. x A. y ( y e. x <-> ph ) ->
|
|
E! x A. y ( y e. x <-> ph ) ) $=
|
|
( vz cv wcel wb wal wex wsb wa wceq wi weu nfv nfbi nfal elequ2 bibi1d
|
|
albidv sbie 19.26 biantr alimi ax-ext syl sylbir sylan2b gen2 jctr sylibr
|
|
eu2 ) CFZBFZGZAHZCIZBJZUSURURBEKZLUOEFZMZNZEIBIZLURBOUSVDVCBEUTURUNVAGZAH
|
|
ZCIZVBURVGBEVFBCVEABVEBPDQRVBUQVFCVBUPVEABECSTUAUBURVGLUQVFLZCIZVBUQVFCUC
|
|
VIUPVEHZCIVBVHVJCUPAVEUDUEBECUFUGUHUIUJUKURBEUREPUMUL $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Class abstractions (a.k.a. class builders)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new constants use in class definition. $)
|
|
$c { $. $( Left brace $)
|
|
$c | $. $( Vertical bar $)
|
|
$c } $. $( Right brace $)
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@c class @. @( Class variable type @)
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( Declare symbols as variables $)
|
|
$v ./\ $.
|
|
$v .\/ $.
|
|
$v .<_ $.
|
|
$v .< $.
|
|
$v .+ $.
|
|
$v .- $.
|
|
$v .X. $.
|
|
$v ./ $.
|
|
$v .^ $.
|
|
$v .0. $.
|
|
$v .1. $.
|
|
$v .|| $.
|
|
$v .~ $.
|
|
$v ._|_ $.
|
|
$v .+^ $.
|
|
$v .+b $.
|
|
$v .(+) $.
|
|
$v .* $.
|
|
$v .x. $.
|
|
$v .xb $.
|
|
$v ., $.
|
|
$v .(x) $.
|
|
$v .0b $.
|
|
|
|
$( Declare variable symbols that will be used to represent classes. Note
|
|
that later on ` R ` , ` S ` , ` F ` and ` G ` denote relations and
|
|
functions, but these letters serve as mnemonics only and in fact behave
|
|
no differently from the variables ` A ` through ` D ` . $)
|
|
$v A $.
|
|
$v B $.
|
|
$v C $.
|
|
$v D $.
|
|
$v P $.
|
|
$v Q $.
|
|
$v R $.
|
|
$v S $.
|
|
$v T $.
|
|
$v U $.
|
|
|
|
$( Introduce the class builder or class abstraction notation ("the class of
|
|
sets ` x ` such that ` ph ` is true"). Our class variables ` A ` ,
|
|
` B ` , etc. range over class builders (implicitly in the case of defined
|
|
class terms such as ~ df-nul ). Note that a setvar variable can be
|
|
expressed as a class builder per theorem ~ cvjust , justifying the
|
|
assignment of setvar variables to class variables via the use of ~ cv . $)
|
|
cab $a class { x | ph } $.
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( A setvar variable is a class expression. The syntax " ` class x ` " can
|
|
be viewed as an abbreviation for " ` class { y | y e. x } ` " (a special
|
|
case of ~ cab ), where ` y ` is distinct from ` x ` . See the discussion
|
|
under the definition of class in [Jech] p. 4. Note that
|
|
` { y | y e. x } = x ` by ~ cvjust . @)
|
|
cv @a class x @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
$( $j primitive 'cv' 'wceq' 'wcel' 'cab'; $)
|
|
|
|
$( Let ` A ` be a class variable. $)
|
|
cA $f class A $.
|
|
$( Let ` B ` be a class variable. $)
|
|
cB $f class B $.
|
|
$( Let ` C ` be a class variable. $)
|
|
cC $f class C $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pa $f class .|| $.
|
|
|
|
$( Let ` D ` be a class variable. $)
|
|
cD $f class D $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.dv $f class ./ $.
|
|
|
|
$( Let ` P ` be a class variable. $)
|
|
cP $f class P $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pl $f class .+ $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pd $f class .+^ $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pb $f class .+b $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.po $f class .(+) $.
|
|
|
|
$( Let ` Q ` be a class variable. $)
|
|
cQ $f class Q $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.sm $f class .~ $.
|
|
|
|
$( Let ` R ` be a class variable. $)
|
|
cR $f class R $.
|
|
$( Let ` S ` be a class variable. $)
|
|
cS $f class S $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.lt $f class .< $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.xb $f class .xb $.
|
|
|
|
$( Let ` T ` be a class variable. $)
|
|
cT $f class T $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.x $f class .x. $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.xp $f class .X. $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.xo $f class .(x) $.
|
|
|
|
$( Let ` U ` be a class variable. $)
|
|
cU $f class U $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.1 $f class .1. $.
|
|
|
|
$v e $.
|
|
$v f $.
|
|
$v g $.
|
|
$v h $.
|
|
$v i $.
|
|
$v j $.
|
|
$v k $.
|
|
$v m $.
|
|
$v n $.
|
|
$v o $.
|
|
$v E $.
|
|
$v F $.
|
|
$v G $.
|
|
$v H $.
|
|
$v I $.
|
|
$v J $.
|
|
$v K $.
|
|
$v L $.
|
|
$v M $.
|
|
$v N $.
|
|
$v V $.
|
|
$v W $.
|
|
$v X $.
|
|
$v Y $.
|
|
$v Z $.
|
|
$v O $.
|
|
$v s $.
|
|
$v r $.
|
|
$v q $.
|
|
$v p $.
|
|
$v a $.
|
|
$v b $.
|
|
$v c $.
|
|
$v d $.
|
|
$v l $.
|
|
|
|
$( Let ` e ` be an individual variable. $)
|
|
ve $f setvar e $.
|
|
$( Let ` f ` be an individual variable. $)
|
|
vf $f setvar f $.
|
|
$( Let ` g ` be an individual variable. $)
|
|
vg $f setvar g $.
|
|
$( Let ` h ` be an individual variable. $)
|
|
vh $f setvar h $.
|
|
$( Let ` i ` be an individual variable. $)
|
|
vi $f setvar i $.
|
|
$( Let ` j ` be an individual variable. $)
|
|
vj $f setvar j $.
|
|
$( Let ` k ` be an individual variable. $)
|
|
vk $f setvar k $.
|
|
$( Let ` m ` be an individual variable. $)
|
|
vm $f setvar m $.
|
|
$( Let ` n ` be an individual variable. $)
|
|
vn $f setvar n $.
|
|
$( Let ` o ` be an individual variable. $)
|
|
vo $f setvar o $.
|
|
$( Let ` E ` be a class variable. $)
|
|
cE $f class E $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.ex $f class .^ $.
|
|
|
|
$( Let ` F ` be a class variable. $)
|
|
cF $f class F $.
|
|
$( Let ` G ` be a class variable. $)
|
|
cG $f class G $.
|
|
$( Let ` H ` be a class variable. $)
|
|
cH $f class H $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.xi $f class ., $.
|
|
|
|
$( Let ` I ` be a class variable. $)
|
|
cI $f class I $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.as $f class .* $.
|
|
|
|
$( Let ` J ` be a class variable. $)
|
|
cJ $f class J $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.or $f class .\/ $.
|
|
|
|
$( Let ` K ` be a class variable. $)
|
|
cK $f class K $.
|
|
$( Let ` L ` be a class variable. $)
|
|
cL $f class L $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.le $f class .<_ $.
|
|
|
|
$( Let ` M ` be a class variable. $)
|
|
cM $f class M $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.an $f class ./\ $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.mi $f class .- $.
|
|
|
|
$( Let ` N ` be a class variable. $)
|
|
cN $f class N $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.pe $f class ._|_ $.
|
|
|
|
$( Let ` O ` be a class variable. $)
|
|
cO $f class O $.
|
|
$( Let ` V ` be a class variable. $)
|
|
cV $f class V $.
|
|
$( Let ` W ` be a class variable. $)
|
|
cW $f class W $.
|
|
$( Let ` X ` be a class variable. $)
|
|
cX $f class X $.
|
|
$( Let ` Y ` be a class variable. $)
|
|
cY $f class Y $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.0 $f class .0. $.
|
|
|
|
$( Define a connective symbol for use as a class variable. $)
|
|
c.0b $f class .0b $.
|
|
|
|
$( Let ` Z ` be a class variable. $)
|
|
cZ $f class Z $.
|
|
$( Let ` s ` be an individual variable. $)
|
|
vs $f setvar s $.
|
|
$( Let ` r ` be an individual variable. $)
|
|
vr $f setvar r $.
|
|
$( Let ` q ` be an individual variable. $)
|
|
vq $f setvar q $.
|
|
$( Let ` p ` be an individual variable. $)
|
|
vp $f setvar p $.
|
|
$( Let ` a ` be an individual variable. $)
|
|
va $f setvar a $.
|
|
$( Let ` b ` be an individual variable. $)
|
|
vb $f setvar b $.
|
|
$( Let ` c ` be an individual variable. $)
|
|
vc $f setvar c $.
|
|
$( Let ` d ` be an individual variable. $)
|
|
vd $f setvar d $.
|
|
$( Let ` l ` be an individual variable. $)
|
|
vl $f setvar l $.
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( Extend wff definition to include class equality. @)
|
|
wceq @a wff A = B @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( Extend wff definition to include the membership connective between
|
|
classes. @)
|
|
wcel @a wff A e. B @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( Define class abstraction notation (so-called by Quine), also called a
|
|
"class builder" in the literature. ` x ` and ` y ` need not be distinct.
|
|
Definition 2.1 of [Quine] p. 16. Typically, ` ph ` will have ` y ` as a
|
|
free variable, and " ` { y | ph } ` " is read "the class of all sets ` y `
|
|
such that ` ph ( y ) ` is true." We do not define ` { y | ph } ` in
|
|
isolation but only as part of an expression that extends or "overloads"
|
|
the ` e. ` relationship.
|
|
|
|
This is our first use of the ` e. ` symbol to connect classes instead of
|
|
sets. The syntax definition ~ wcel , which extends or "overloads" the
|
|
~ wel definition connecting setvar variables, requires that both sides of
|
|
` e. ` be a class. In ~ df-cleq and ~ df-clel , we introduce a new kind
|
|
of variable (class variable) that can substituted with expressions such as
|
|
` { y | ph } ` . In the present definition, the ` x ` on the left-hand
|
|
side is a setvar variable. Syntax definition ~ cv allows us to substitute
|
|
a setvar variable ` x ` for a class variable: all sets are classes by
|
|
~ cvjust (but not necessarily vice-versa). For a full description of how
|
|
classes are introduced and how to recover the primitive language, see the
|
|
discussion in Quine (and under ~ abeq2 for a quick overview).
|
|
|
|
Because class variables can be substituted with compound expressions and
|
|
setvar variables cannot, it is often useful to convert a theorem
|
|
containing a free setvar variable to a more general version with a class
|
|
variable. This is done with theorems such as ~ vtoclg which is used, for
|
|
example, to convert elirrv in set.mm to elirr in set.mm.
|
|
|
|
This is called the "axiom of class comprehension" by [Levy] p. 338, who
|
|
treats the theory of classes as an extralogical extension to our logic and
|
|
set theory axioms. He calls the construction ` { y | ph } ` a "class
|
|
term".
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
df-clab $a |- ( x e. { y | ph } <-> [ x / y ] ph ) $.
|
|
|
|
$( Simplification of class abstraction notation when the free and bound
|
|
variables are identical. (Contributed by NM, 5-Aug-1993.) $)
|
|
abid $p |- ( x e. { x | ph } <-> ph ) $=
|
|
( cv cab wcel wsb df-clab sbid bitri ) BCABDEABBFAABBGABHI $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
hbab1 $p |- ( y e. { x | ph } -> A. x y e. { x | ph } ) $=
|
|
( cv cab wcel wsb df-clab hbs1 hbxfrbi ) CDABEFABCGBACBHABCIJ $.
|
|
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfsab1 $p |- F/ x y e. { x | ph } $=
|
|
( cv cab wcel hbab1 nfi ) CDABEFBABCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $.
|
|
hbab.1 $e |- ( ph -> A. x ph ) $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by NM, 1-Mar-1995.) $)
|
|
hbab $p |- ( z e. { y | ph } -> A. x z e. { y | ph } ) $=
|
|
( cv cab wcel wsb df-clab hbsb hbxfrbi ) DFACGHACDIBADCJACDBEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $.
|
|
nfsab.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfsab $p |- F/ x z e. { y | ph } $=
|
|
( cv cab wcel nfri hbab nfi ) DFACGHBABCDABEIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x y z $.
|
|
df-cleq.1 $e |- ( A. x ( x e. y <-> x e. z ) -> y = z ) $.
|
|
$( Define the equality connective between classes. Definition 2.7 of
|
|
[Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4
|
|
provides its justification and methods for eliminating it. Note that
|
|
its elimination will not necessarily result in a single wff in the
|
|
original language but possibly a "scheme" of wffs.
|
|
|
|
This is an example of a somewhat "risky" definition, meaning that it has
|
|
a more complex than usual soundness justification (outside of Metamath),
|
|
because it "overloads" or reuses the existing equality symbol rather
|
|
than introducing a new symbol. This allows us to make statements that
|
|
may not hold for the original symbol. For example, it permits us to
|
|
deduce ` y = z <-> A. x ( x e. y <-> x e. z ) ` , which is not a theorem
|
|
of logic but rather presupposes the Axiom of Extensionality (see theorem
|
|
~ axext4 ). We therefore include this axiom as a hypothesis, so that
|
|
the use of Extensionality is properly indicated.
|
|
|
|
We could avoid this complication by introducing a new symbol, say =_2,
|
|
in place of ` = ` . This would also have the advantage of making
|
|
elimination of the definition straightforward, so that we could
|
|
eliminate Extensionality as a hypothesis. We would then also have the
|
|
advantage of being able to identify in various proofs exactly where
|
|
Extensionality truly comes into play rather than just being an artifact
|
|
of a definition. One of our theorems would then be ` x ` =_2
|
|
` y <-> x = y ` by invoking Extensionality.
|
|
|
|
However, to conform to literature usage, we retain this overloaded
|
|
definition. This also makes some proofs shorter and probably easier to
|
|
read, without the constant switching between two kinds of equality.
|
|
|
|
See also comments under ~ df-clab , ~ df-clel , and ~ abeq2 .
|
|
|
|
In the form of ~ dfcleq , this is called the "axiom of extensionality"
|
|
by [Levy] p. 338, who treats the theory of classes as an extralogical
|
|
extension to our logic and set theory axioms.
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class . (Contributed by NM,
|
|
15-Sep-1993.) $)
|
|
df-cleq $a |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x y z $.
|
|
$( The same as ~ df-cleq with the hypothesis removed using the Axiom of
|
|
Extensionality ~ ax-ext . (Contributed by NM, 15-Sep-1993.) $)
|
|
dfcleq $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
|
|
( vy vz ax-ext df-cleq ) ADEBCDEAFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Every setvar is a class. Proposition 4.9 of [TakeutiZaring] p. 13.
|
|
This theorem shows that a setvar variable can be expressed as a class
|
|
abstraction. This provides a motivation for the class syntax
|
|
construction ~ cv , which allows us to substitute a setvar variable for
|
|
a class variable. See also ~ cab and ~ df-clab . Note that this is not
|
|
a rigorous justification, because ~ cv is used as part of the proof of
|
|
this theorem, but a careful argument can be made outside of the
|
|
formalism of Metamath, for example as is done in Chapter 4 of Takeuti
|
|
and Zaring. See also the discussion under the definition of class in
|
|
[Jech] p. 4 showing that "Every set can be considered to be a class."
|
|
(Contributed by NM, 7-Nov-2006.) $)
|
|
cvjust $p |- x = { y | y e. x } $=
|
|
( vz cv wcel cab wceq wb dfcleq wsb df-clab elsb3 bitr2i mpgbir ) ADZBDOE
|
|
ZBFZGCDZOEZRQEZHCCOQITPBCJSPCBKCBALMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Define the membership connective between classes. Theorem 6.3 of
|
|
[Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we
|
|
adopt as a definition. See these references for its metalogical
|
|
justification. Note that like ~ df-cleq it extends or "overloads" the
|
|
use of the existing membership symbol, but unlike ~ df-cleq it does not
|
|
strengthen the set of valid wffs of logic when the class variables are
|
|
replaced with setvar variables (see ~ cleljust ), so we don't include
|
|
any set theory axiom as a hypothesis. See also comments about the
|
|
syntax under ~ df-clab . Alternate definitions of ` A e. B ` (but that
|
|
require either ` A ` or ` B ` to be a set) are shown by ~ clel2 ,
|
|
~ clel3 , and ~ clel4 .
|
|
|
|
This is called the "axiom of membership" by [Levy] p. 338, who treats
|
|
the theory of classes as an extralogical extension to our logic and set
|
|
theory axioms.
|
|
|
|
For a general discussion of the theory of classes, see
|
|
~ http://us.metamath.org/mpeuni/mmset.html#class . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
df-clel $a |- ( A e. B <-> E. x ( x = A /\ x e. B ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
eqriv.1 $e |- ( x e. A <-> x e. B ) $.
|
|
$( Infer equality of classes from equivalence of membership. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
eqriv $p |- A = B $=
|
|
( wceq cv wcel wb dfcleq mpgbir ) BCEAFZBGKCGHAABCIDJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
eqrdv.1 $e |- ( ph -> ( x e. A <-> x e. B ) ) $.
|
|
$( Deduce equality of classes from equivalence of membership. (Contributed
|
|
by NM, 17-Mar-1996.) $)
|
|
eqrdv $p |- ( ph -> A = B ) $=
|
|
( cv wcel wb wal wceq alrimiv dfcleq sylibr ) ABFZCGNDGHZBICDJAOBEKBCDLM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
eqrdav.1 $e |- ( ( ph /\ x e. A ) -> x e. C ) $.
|
|
eqrdav.2 $e |- ( ( ph /\ x e. B ) -> x e. C ) $.
|
|
eqrdav.3 $e |- ( ( ph /\ x e. C ) -> ( x e. A <-> x e. B ) ) $.
|
|
$( Deduce equality of classes from an equivalence of membership that
|
|
depends on the membership variable. (Contributed by NM, 7-Nov-2008.) $)
|
|
eqrdav $p |- ( ph -> A = B ) $=
|
|
( cv wcel wa biimpd impancom mpd wi exbiri com23 imp impbida eqrdv ) ABCD
|
|
ABIZCJZUADJZAUBKUAEJZUCFAUDUBUCAUDKUBUCHLMNAUCKUDUBGAUCUDUBOAUDUCUBAUDUBU
|
|
CHPQRNST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine]
|
|
p. 41.
|
|
|
|
This law is thought to have originated with Aristotle (_Metaphysics_,
|
|
Zeta, 17, 1041 a, 10-20: "Therefore, inquiring why a thing is itself,
|
|
it's inquiring nothing; ... saying that the thing is itself constitutes
|
|
the sole reasoning and the sole cause, in every case, to the question of
|
|
why the man is man or the musician musician."). (Thanks to Stefan Allan
|
|
and Benoît Jubin for this information.) (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Benoît Jubin, 14-Oct-2017.) $)
|
|
eqid $p |- A = A $=
|
|
( vx cv wcel biid eqriv ) BAABCADEF $.
|
|
$}
|
|
|
|
$( Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) $)
|
|
eqidd $p |- ( ph -> A = A ) $=
|
|
( wceq eqid a1i ) BBCABDE $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Commutative law for class equality. Theorem 6.5 of [Quine] p. 41.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
eqcom $p |- ( A = B <-> B = A ) $=
|
|
( vx cv wcel wb wal wceq bicom albii dfcleq 3bitr4i ) CDZAEZMBEZFZCGONFZC
|
|
GABHBAHPQCNOIJCABKCBAKL $.
|
|
$}
|
|
|
|
${
|
|
eqcoms.1 $e |- ( A = B -> ph ) $.
|
|
$( Inference applying commutative law for class equality to an antecedent.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
eqcoms $p |- ( B = A -> ph ) $=
|
|
( wceq eqcom sylbi ) CBEBCEACBFDG $.
|
|
$}
|
|
|
|
${
|
|
eqcomi.1 $e |- A = B $.
|
|
$( Inference from commutative law for class equality. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eqcomi $p |- B = A $=
|
|
( wceq eqcom mpbi ) ABDBADCABEF $.
|
|
$}
|
|
|
|
${
|
|
eqcomd.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction from commutative law for class equality. (Contributed by NM,
|
|
15-Aug-1994.) $)
|
|
eqcomd $p |- ( ph -> B = A ) $=
|
|
( wceq eqcom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality implies equivalence of equalities. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eqeq1 $p |- ( A = B -> ( A = C <-> B = C ) ) $=
|
|
( vx wceq cv wcel wb wal dfcleq biimpi 19.21bi bibi1d albidv 3bitr4g ) AB
|
|
EZDFZAGZQCGZHZDIQBGZSHZDIACEBCEPTUBDPRUASPRUAHZDPUCDIDABJKLMNDACJDBCJO $.
|
|
$}
|
|
|
|
${
|
|
eqeq1i.1 $e |- A = B $.
|
|
$( Inference from equality to equivalence of equalities. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqeq1i $p |- ( A = C <-> B = C ) $=
|
|
( wceq wb eqeq1 ax-mp ) ABEACEBCEFDABCGH $.
|
|
$}
|
|
|
|
${
|
|
eqeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction from equality to equivalence of equalities. (Contributed by
|
|
NM, 27-Dec-1993.) $)
|
|
eqeq1d $p |- ( ph -> ( A = C <-> B = C ) ) $=
|
|
( wceq wb eqeq1 syl ) ABCFBDFCDFGEBCDHI $.
|
|
$}
|
|
|
|
$( Equality implies equivalence of equalities. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eqeq2 $p |- ( A = B -> ( C = A <-> C = B ) ) $=
|
|
( wceq eqeq1 eqcom 3bitr4g ) ABDACDBCDCADCBDABCECAFCBFG $.
|
|
|
|
${
|
|
eqeq2i.1 $e |- A = B $.
|
|
$( Inference from equality to equivalence of equalities. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqeq2i $p |- ( C = A <-> C = B ) $=
|
|
( wceq wb eqeq2 ax-mp ) ABECAECBEFDABCGH $.
|
|
$}
|
|
|
|
${
|
|
eqeq2d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction from equality to equivalence of equalities. (Contributed by
|
|
NM, 27-Dec-1993.) $)
|
|
eqeq2d $p |- ( ph -> ( C = A <-> C = B ) ) $=
|
|
( wceq wb eqeq2 syl ) ABCFDBFDCFGEBCDHI $.
|
|
$}
|
|
|
|
$( Equality relationship among 4 classes. (Contributed by NM,
|
|
3-Aug-1994.) $)
|
|
eqeq12 $p |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) ) $=
|
|
( wceq eqeq1 eqeq2 sylan9bb ) ABEACEBCECDEBDEABCFCDBGH $.
|
|
|
|
${
|
|
eqeq12i.1 $e |- A = B $.
|
|
eqeq12i.2 $e |- C = D $.
|
|
$( A useful inference for substituting definitions into an equality.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
eqeq12i $p |- ( A = C <-> B = D ) $=
|
|
( wceq wb eqeq12 mp2an ) ABGCDGACGBDGHEFABCDIJ $.
|
|
|
|
$( Theorem eqeq12i is the congruence law for equality. $)
|
|
$( $j congruence 'eqeq12i'; $)
|
|
$}
|
|
|
|
${
|
|
eqeq12d.1 $e |- ( ph -> A = B ) $.
|
|
eqeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( A useful inference for substituting definitions into an equality.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
eqeq12d $p |- ( ph -> ( A = C <-> B = D ) ) $=
|
|
( wceq wb eqeq12 syl2anc ) ABCHDEHBDHCEHIFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
eqeqan12d.1 $e |- ( ph -> A = B ) $.
|
|
eqeqan12d.2 $e |- ( ps -> C = D ) $.
|
|
$( A useful inference for substituting definitions into an equality.
|
|
(Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
eqeqan12d $p |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) ) $=
|
|
( wceq wb eqeq12 syl2an ) ACDIEFICEIDFIJBGHCDEFKL $.
|
|
$}
|
|
|
|
${
|
|
eqeqan12rd.1 $e |- ( ph -> A = B ) $.
|
|
eqeqan12rd.2 $e |- ( ps -> C = D ) $.
|
|
$( A useful inference for substituting definitions into an equality.
|
|
(Contributed by NM, 9-Aug-1994.) $)
|
|
eqeqan12rd $p |- ( ( ps /\ ph ) -> ( A = C <-> B = D ) ) $=
|
|
( wceq wb eqeqan12d ancoms ) ABCEIDFIJABCDEFGHKL $.
|
|
$}
|
|
|
|
$( Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring]
|
|
p. 13. (Contributed by NM, 25-Jan-2004.) $)
|
|
eqtr $p |- ( ( A = B /\ B = C ) -> A = C ) $=
|
|
( wceq eqeq1 biimpar ) ABDACDBCDABCEF $.
|
|
|
|
$( A transitive law for class equality. (Contributed by NM, 20-May-2005.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
eqtr2 $p |- ( ( A = B /\ A = C ) -> B = C ) $=
|
|
( wceq eqcom eqtr sylanb ) ABDBADACDBCDABEBACFG $.
|
|
|
|
$( A transitive law for class equality. (Contributed by NM, 20-May-2005.) $)
|
|
eqtr3 $p |- ( ( A = C /\ B = C ) -> A = B ) $=
|
|
( wceq eqcom eqtr sylan2b ) BCDACDCBDABDBCEACBFG $.
|
|
|
|
${
|
|
eqtri.1 $e |- A = B $.
|
|
eqtri.2 $e |- B = C $.
|
|
$( An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) $)
|
|
eqtri $p |- A = C $=
|
|
( wceq eqeq2i mpbi ) ABFACFDBCAEGH $.
|
|
$}
|
|
|
|
${
|
|
eqtr2i.1 $e |- A = B $.
|
|
eqtr2i.2 $e |- B = C $.
|
|
$( An equality transitivity inference. (Contributed by NM,
|
|
21-Feb-1995.) $)
|
|
eqtr2i $p |- C = A $=
|
|
( eqtri eqcomi ) ACABCDEFG $.
|
|
$}
|
|
|
|
${
|
|
eqtr3i.1 $e |- A = B $.
|
|
eqtr3i.2 $e |- A = C $.
|
|
$( An equality transitivity inference. (Contributed by NM, 6-May-1994.) $)
|
|
eqtr3i $p |- B = C $=
|
|
( eqcomi eqtri ) BACABDFEG $.
|
|
$}
|
|
|
|
${
|
|
eqtr4i.1 $e |- A = B $.
|
|
eqtr4i.2 $e |- C = B $.
|
|
$( An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) $)
|
|
eqtr4i $p |- A = C $=
|
|
( eqcomi eqtri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
$( Register '=' as an equality for its type (class). $)
|
|
$( $j equality 'wceq' from 'eqid' 'eqcomi' 'eqtri'; $)
|
|
|
|
${
|
|
3eqtri.1 $e |- A = B $.
|
|
3eqtri.2 $e |- B = C $.
|
|
3eqtri.3 $e |- C = D $.
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
29-Aug-1993.) $)
|
|
3eqtri $p |- A = D $=
|
|
( eqtri ) ABDEBCDFGHH $.
|
|
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtrri $p |- D = A $=
|
|
( eqtri eqtr2i ) ACDABCEFHGI $.
|
|
$}
|
|
|
|
${
|
|
3eqtr2i.1 $e |- A = B $.
|
|
3eqtr2i.2 $e |- C = B $.
|
|
3eqtr2i.3 $e |- C = D $.
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
3-Aug-2006.) $)
|
|
3eqtr2i $p |- A = D $=
|
|
( eqtr4i eqtri ) ACDABCEFHGI $.
|
|
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr2ri $p |- D = A $=
|
|
( eqtr4i eqtr2i ) ACDABCEFHGI $.
|
|
$}
|
|
|
|
${
|
|
3eqtr3i.1 $e |- A = B $.
|
|
3eqtr3i.2 $e |- A = C $.
|
|
3eqtr3i.3 $e |- B = D $.
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr3i $p |- C = D $=
|
|
( eqtr3i ) BCDABCEFHGH $.
|
|
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
15-Aug-2004.) $)
|
|
3eqtr3ri $p |- D = C $=
|
|
( eqtr3i ) BDCGABCEFHH $.
|
|
$}
|
|
|
|
${
|
|
3eqtr4i.1 $e |- A = B $.
|
|
3eqtr4i.2 $e |- C = A $.
|
|
3eqtr4i.3 $e |- D = B $.
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr4i $p |- C = D $=
|
|
( eqtr4i ) CADFDBAGEHH $.
|
|
|
|
$( An inference from three chained equalities. (Contributed by NM,
|
|
2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr4ri $p |- D = C $=
|
|
( eqtr4i ) DACDBAGEHFH $.
|
|
$}
|
|
|
|
${
|
|
eqtrd.1 $e |- ( ph -> A = B ) $.
|
|
eqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
eqtrd $p |- ( ph -> A = C ) $=
|
|
( wceq eqeq2d mpbid ) ABCGBDGEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
eqtr2d.1 $e |- ( ph -> A = B ) $.
|
|
eqtr2d.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
18-Oct-1999.) $)
|
|
eqtr2d $p |- ( ph -> C = A ) $=
|
|
( eqtrd eqcomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
eqtr3d.1 $e |- ( ph -> A = B ) $.
|
|
eqtr3d.2 $e |- ( ph -> A = C ) $.
|
|
$( An equality transitivity equality deduction. (Contributed by NM,
|
|
18-Jul-1995.) $)
|
|
eqtr3d $p |- ( ph -> B = C ) $=
|
|
( eqcomd eqtrd ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
eqtr4d.1 $e |- ( ph -> A = B ) $.
|
|
eqtr4d.2 $e |- ( ph -> C = B ) $.
|
|
$( An equality transitivity equality deduction. (Contributed by NM,
|
|
18-Jul-1995.) $)
|
|
eqtr4d $p |- ( ph -> A = C ) $=
|
|
( eqcomd eqtrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
3eqtrd.1 $e |- ( ph -> A = B ) $.
|
|
3eqtrd.2 $e |- ( ph -> B = C ) $.
|
|
3eqtrd.3 $e |- ( ph -> C = D ) $.
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
29-Oct-1995.) $)
|
|
3eqtrd $p |- ( ph -> A = D ) $=
|
|
( eqtrd ) ABCEFACDEGHII $.
|
|
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtrrd $p |- ( ph -> D = A ) $=
|
|
( eqtrd eqtr2d ) ABDEABCDFGIHJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr2d.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr2d.2 $e |- ( ph -> C = B ) $.
|
|
3eqtr2d.3 $e |- ( ph -> C = D ) $.
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3eqtr2d $p |- ( ph -> A = D ) $=
|
|
( eqtr4d eqtrd ) ABDEABCDFGIHJ $.
|
|
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-2006.) $)
|
|
3eqtr2rd $p |- ( ph -> D = A ) $=
|
|
( eqtr4d eqtr2d ) ABDEABCDFGIHJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr3d.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr3d.2 $e |- ( ph -> A = C ) $.
|
|
3eqtr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr3d $p |- ( ph -> C = D ) $=
|
|
( eqtr3d ) ACDEABCDFGIHI $.
|
|
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
14-Jan-2006.) $)
|
|
3eqtr3rd $p |- ( ph -> D = C ) $=
|
|
( eqtr3d ) ACEDHABCDFGII $.
|
|
$}
|
|
|
|
${
|
|
3eqtr4d.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr4d.2 $e |- ( ph -> C = A ) $.
|
|
3eqtr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
3eqtr4d $p |- ( ph -> C = D ) $=
|
|
( eqtr4d ) ADBEGAECBHFII $.
|
|
|
|
$( A deduction from three chained equalities. (Contributed by NM,
|
|
21-Sep-1995.) $)
|
|
3eqtr4rd $p |- ( ph -> D = C ) $=
|
|
( eqtr4d ) AEBDAECBHFIGI $.
|
|
$}
|
|
|
|
${
|
|
syl5eq.1 $e |- A = B $.
|
|
syl5eq.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5eq $p |- ( ph -> A = C ) $=
|
|
( wceq a1i eqtrd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl5req.1 $e |- A = B $.
|
|
syl5req.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
syl5req $p |- ( ph -> C = A ) $=
|
|
( syl5eq eqcomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5eqr.1 $e |- B = A $.
|
|
syl5eqr.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl5eqr $p |- ( ph -> A = C ) $=
|
|
( eqcomi syl5eq ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5reqr.1 $e |- B = A $.
|
|
syl5reqr.2 $e |- ( ph -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
syl5reqr $p |- ( ph -> C = A ) $=
|
|
( eqcomi syl5req ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6eq.1 $e |- ( ph -> A = B ) $.
|
|
syl6eq.2 $e |- B = C $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl6eq $p |- ( ph -> A = C ) $=
|
|
( wceq a1i eqtrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6req.1 $e |- ( ph -> A = B ) $.
|
|
syl6req.2 $e |- B = C $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
syl6req $p |- ( ph -> C = A ) $=
|
|
( syl6eq eqcomd ) ABDABCDEFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6eqr.1 $e |- ( ph -> A = B ) $.
|
|
syl6eqr.2 $e |- C = B $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) $)
|
|
syl6eqr $p |- ( ph -> A = C ) $=
|
|
( eqcomi syl6eq ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6reqr.1 $e |- ( ph -> A = B ) $.
|
|
syl6reqr.2 $e |- C = B $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
syl6reqr $p |- ( ph -> C = A ) $=
|
|
( eqcomi syl6req ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
sylan9eq.1 $e |- ( ph -> A = B ) $.
|
|
sylan9eq.2 $e |- ( ps -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
|
|
(Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
sylan9eq $p |- ( ( ph /\ ps ) -> A = C ) $=
|
|
( wceq eqtr syl2an ) ACDHDEHCEHBFGCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan9req.1 $e |- ( ph -> B = A ) $.
|
|
sylan9req.2 $e |- ( ps -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM,
|
|
23-Jun-2007.) $)
|
|
sylan9req $p |- ( ( ph /\ ps ) -> A = C ) $=
|
|
( eqcomd sylan9eq ) ABCDEADCFHGI $.
|
|
$}
|
|
|
|
${
|
|
sylan9eqr.1 $e |- ( ph -> A = B ) $.
|
|
sylan9eqr.2 $e |- ( ps -> B = C ) $.
|
|
$( An equality transitivity deduction. (Contributed by NM, 8-May-1994.) $)
|
|
sylan9eqr $p |- ( ( ps /\ ph ) -> A = C ) $=
|
|
( wceq sylan9eq ancoms ) ABCEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr3g.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr3g.2 $e |- A = C $.
|
|
3eqtr3g.3 $e |- B = D $.
|
|
$( A chained equality inference, useful for converting from definitions.
|
|
(Contributed by NM, 15-Nov-1994.) $)
|
|
3eqtr3g $p |- ( ph -> C = D ) $=
|
|
( syl5eqr syl6eq ) ADCEADBCGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr3a.1 $e |- A = B $.
|
|
3eqtr3a.2 $e |- ( ph -> A = C ) $.
|
|
3eqtr3a.3 $e |- ( ph -> B = D ) $.
|
|
$( A chained equality inference, useful for converting from definitions.
|
|
(Contributed by Mario Carneiro, 6-Nov-2015.) $)
|
|
3eqtr3a $p |- ( ph -> C = D ) $=
|
|
( syl5eq eqtr3d ) ABDEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr4g.1 $e |- ( ph -> A = B ) $.
|
|
3eqtr4g.2 $e |- C = A $.
|
|
3eqtr4g.3 $e |- D = B $.
|
|
$( A chained equality inference, useful for converting to definitions.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
3eqtr4g $p |- ( ph -> C = D ) $=
|
|
( syl5eq syl6eqr ) ADCEADBCGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3eqtr4a.1 $e |- A = B $.
|
|
3eqtr4a.2 $e |- ( ph -> C = A ) $.
|
|
3eqtr4a.3 $e |- ( ph -> D = B ) $.
|
|
$( A chained equality inference, useful for converting to definitions.
|
|
(Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) $)
|
|
3eqtr4a $p |- ( ph -> C = D ) $=
|
|
( syl6eq eqtr4d ) ADCEADBCGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
eq2tr.1 $e |- ( A = C -> D = F ) $.
|
|
eq2tr.2 $e |- ( B = D -> C = G ) $.
|
|
$( A compound transitive inference for class equality. (Contributed by NM,
|
|
22-Jan-2004.) $)
|
|
eq2tri $p |- ( ( A = C /\ B = F ) <-> ( B = D /\ A = G ) ) $=
|
|
( wceq wa ancom eqeq2d pm5.32i 3bitr3i ) ACIZBDIZJPOJOBEIZJPAFIZJOPKOPQOD
|
|
EBGLMPORPCFAHLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality implies equivalence of membership. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eleq1 $p |- ( A = B -> ( A e. C <-> B e. C ) ) $=
|
|
( vx wceq cv wcel wa wex eqeq2 anbi1d exbidv df-clel 3bitr4g ) ABEZDFZAEZ
|
|
PCGZHZDIPBEZRHZDIACGBCGOSUADOQTRABPJKLDACMDBCMN $.
|
|
|
|
$( Equality implies equivalence of membership. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
eleq2 $p |- ( A = B -> ( C e. A <-> C e. B ) ) $=
|
|
( vx wceq cv wcel wa wex wal dfcleq biimpi 19.21bi anbi2d df-clel 3bitr4g
|
|
wb exbidv ) ABEZDFZCEZTAGZHZDIUATBGZHZDICAGCBGSUCUEDSUBUDUASUBUDQZDSUFDJD
|
|
ABKLMNRDCAODCBOP $.
|
|
$}
|
|
|
|
$( Equality implies equivalence of membership. (Contributed by NM,
|
|
31-May-1999.) $)
|
|
eleq12 $p |- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) ) $=
|
|
( wceq wcel eleq1 eleq2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.
|
|
|
|
${
|
|
eleq1i.1 $e |- A = B $.
|
|
$( Inference from equality to equivalence of membership. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleq1i $p |- ( A e. C <-> B e. C ) $=
|
|
( wceq wcel wb eleq1 ax-mp ) ABEACFBCFGDABCHI $.
|
|
|
|
$( Inference from equality to equivalence of membership. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleq2i $p |- ( C e. A <-> C e. B ) $=
|
|
( wceq wcel wb eleq2 ax-mp ) ABECAFCBFGDABCHI $.
|
|
|
|
${
|
|
eleq12i.2 $e |- C = D $.
|
|
$( Inference from equality to equivalence of membership. (Contributed by
|
|
NM, 31-May-1994.) $)
|
|
eleq12i $p |- ( A e. C <-> B e. D ) $=
|
|
( wcel eleq2i eleq1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
|
|
$( Theorem eleq12i is the congruence law for elementhood. $)
|
|
$( $j congruence 'eleq12i'; $)
|
|
$}
|
|
$}
|
|
|
|
${
|
|
eleq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction from equality to equivalence of membership. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleq1d $p |- ( ph -> ( A e. C <-> B e. C ) ) $=
|
|
( wceq wcel wb eleq1 syl ) ABCFBDGCDGHEBCDIJ $.
|
|
|
|
$( Deduction from equality to equivalence of membership. (Contributed by
|
|
NM, 27-Dec-1993.) $)
|
|
eleq2d $p |- ( ph -> ( C e. A <-> C e. B ) ) $=
|
|
( wceq wcel wb eleq2 syl ) ABCFDBGDCGHEBCDIJ $.
|
|
|
|
${
|
|
eleq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Deduction from equality to equivalence of membership. (Contributed by
|
|
NM, 31-May-1994.) $)
|
|
eleq12d $p |- ( ph -> ( A e. C <-> B e. D ) ) $=
|
|
( wcel eleq2d eleq1d bitrd ) ABDHBEHCEHADEBGIABCEFJK $.
|
|
$}
|
|
$}
|
|
|
|
$( A transitive-type law relating membership and equality. (Contributed by
|
|
NM, 9-Apr-1994.) $)
|
|
eleq1a $p |- ( A e. B -> ( C = A -> C e. B ) ) $=
|
|
( wceq wcel eleq1 biimprcd ) CADCBEABECABFG $.
|
|
|
|
${
|
|
eqeltr.1 $e |- A = B $.
|
|
eqeltr.2 $e |- B e. C $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqeltri $p |- A e. C $=
|
|
( wcel eleq1i mpbir ) ACFBCFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
eqeltrr.1 $e |- A = B $.
|
|
eqeltrr.2 $e |- A e. C $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqeltrri $p |- B e. C $=
|
|
( eqcomi eqeltri ) BACABDFEG $.
|
|
$}
|
|
|
|
${
|
|
eleqtr.1 $e |- A e. B $.
|
|
eleqtr.2 $e |- B = C $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleqtri $p |- A e. C $=
|
|
( wcel eleq2i mpbi ) ABFACFDBCAEGH $.
|
|
$}
|
|
|
|
${
|
|
eleqtrr.1 $e |- A e. B $.
|
|
eleqtrr.2 $e |- C = B $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eleqtrri $p |- A e. C $=
|
|
( eqcomi eleqtri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
eqeltrd.1 $e |- ( ph -> A = B ) $.
|
|
eqeltrd.2 $e |- ( ph -> B e. C ) $.
|
|
$( Substitution of equal classes into membership relation, deduction form.
|
|
(Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
eqeltrd $p |- ( ph -> A e. C ) $=
|
|
( wcel eleq1d mpbird ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqeltrrd.1 $e |- ( ph -> A = B ) $.
|
|
eqeltrrd.2 $e |- ( ph -> A e. C ) $.
|
|
$( Deduction that substitutes equal classes into membership. (Contributed
|
|
by NM, 14-Dec-2004.) $)
|
|
eqeltrrd $p |- ( ph -> B e. C ) $=
|
|
( eqcomd eqeltrd ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
eleqtrd.1 $e |- ( ph -> A e. B ) $.
|
|
eleqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( Deduction that substitutes equal classes into membership. (Contributed
|
|
by NM, 14-Dec-2004.) $)
|
|
eleqtrd $p |- ( ph -> A e. C ) $=
|
|
( wcel eleq2d mpbid ) ABCGBDGEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
eleqtrrd.1 $e |- ( ph -> A e. B ) $.
|
|
eleqtrrd.2 $e |- ( ph -> C = B ) $.
|
|
$( Deduction that substitutes equal classes into membership. (Contributed
|
|
by NM, 14-Dec-2004.) $)
|
|
eleqtrrd $p |- ( ph -> A e. C ) $=
|
|
( eqcomd eleqtrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
3eltr3.1 $e |- A e. B $.
|
|
3eltr3.2 $e |- A = C $.
|
|
3eltr3.3 $e |- B = D $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr3i $p |- C e. D $=
|
|
( eleqtri eqeltrri ) ACDFABDEGHI $.
|
|
$}
|
|
|
|
${
|
|
3eltr4.1 $e |- A e. B $.
|
|
3eltr4.2 $e |- C = A $.
|
|
3eltr4.3 $e |- D = B $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr4i $p |- C e. D $=
|
|
( eleqtrri eqeltri ) CADFABDEGHI $.
|
|
$}
|
|
|
|
${
|
|
3eltr3d.1 $e |- ( ph -> A e. B ) $.
|
|
3eltr3d.2 $e |- ( ph -> A = C ) $.
|
|
3eltr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr3d $p |- ( ph -> C e. D ) $=
|
|
( eleqtrd eqeltrrd ) ABDEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3eltr4d.1 $e |- ( ph -> A e. B ) $.
|
|
3eltr4d.2 $e |- ( ph -> C = A ) $.
|
|
3eltr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr4d $p |- ( ph -> C e. D ) $=
|
|
( eleqtrrd eqeltrd ) ADBEGABCEFHIJ $.
|
|
$}
|
|
|
|
${
|
|
3eltr3g.1 $e |- ( ph -> A e. B ) $.
|
|
3eltr3g.2 $e |- A = C $.
|
|
3eltr3g.3 $e |- B = D $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr3g $p |- ( ph -> C e. D ) $=
|
|
( wcel eleq12i sylib ) ABCIDEIFBDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3eltr4g.1 $e |- ( ph -> A e. B ) $.
|
|
3eltr4g.2 $e |- C = A $.
|
|
3eltr4g.3 $e |- D = B $.
|
|
$( Substitution of equal classes into membership relation. (Contributed by
|
|
Mario Carneiro, 6-Jan-2017.) $)
|
|
3eltr4g $p |- ( ph -> C e. D ) $=
|
|
( wcel eleq12i sylibr ) ABCIDEIFDBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
syl5eqel.1 $e |- A = B $.
|
|
syl5eqel.2 $e |- ( ph -> B e. C ) $.
|
|
$( B membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl5eqel $p |- ( ph -> A e. C ) $=
|
|
( wceq a1i eqeltrd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl5eqelr.1 $e |- B = A $.
|
|
syl5eqelr.2 $e |- ( ph -> B e. C ) $.
|
|
$( B membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl5eqelr $p |- ( ph -> A e. C ) $=
|
|
( eqcomi syl5eqel ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl5eleq.1 $e |- A e. B $.
|
|
syl5eleq.2 $e |- ( ph -> B = C ) $.
|
|
$( B membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl5eleq $p |- ( ph -> A e. C ) $=
|
|
( wcel a1i eleqtrd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl5eleqr.1 $e |- A e. B $.
|
|
syl5eleqr.2 $e |- ( ph -> C = B ) $.
|
|
$( B membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl5eleqr $p |- ( ph -> A e. C ) $=
|
|
( eqcomd syl5eleq ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl6eqel.1 $e |- ( ph -> A = B ) $.
|
|
syl6eqel.2 $e |- B e. C $.
|
|
$( A membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl6eqel $p |- ( ph -> A e. C ) $=
|
|
( wcel a1i eqeltrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6eqelr.1 $e |- ( ph -> B = A ) $.
|
|
syl6eqelr.2 $e |- B e. C $.
|
|
$( A membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl6eqelr $p |- ( ph -> A e. C ) $=
|
|
( eqcomd syl6eqel ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6eleq.1 $e |- ( ph -> A e. B ) $.
|
|
syl6eleq.2 $e |- B = C $.
|
|
$( A membership and equality inference. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl6eleq $p |- ( ph -> A e. C ) $=
|
|
( wceq a1i eleqtrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6eleqr.1 $e |- ( ph -> A e. B ) $.
|
|
syl6eleqr.2 $e |- C = B $.
|
|
$( A membership and equality inference. (Contributed by NM,
|
|
24-Apr-2005.) $)
|
|
syl6eleqr $p |- ( ph -> A e. C ) $=
|
|
( eqcomi syl6eleq ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
eleq2s.1 $e |- ( A e. B -> ph ) $.
|
|
eleq2s.2 $e |- C = B $.
|
|
$( Substitution of equal classes into a membership antecedent.
|
|
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
eleq2s $p |- ( A e. C -> ph ) $=
|
|
( wcel eleq2i sylbi ) BDGBCGADCBFHEI $.
|
|
$}
|
|
|
|
${
|
|
eqneltrd.1 $e |- ( ph -> A = B ) $.
|
|
eqneltrd.2 $e |- ( ph -> -. B e. C ) $.
|
|
$( If a class is not an element of another class, an equal class is also
|
|
not an element. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
eqneltrd $p |- ( ph -> -. A e. C ) $=
|
|
( wcel eleq1d mtbird ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqneltrrd.1 $e |- ( ph -> A = B ) $.
|
|
eqneltrrd.2 $e |- ( ph -> -. A e. C ) $.
|
|
$( If a class is not an element of another class, an equal class is also
|
|
not an element. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
eqneltrrd $p |- ( ph -> -. B e. C ) $=
|
|
( wcel eleq1d mtbid ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
neleqtrd.1 $e |- ( ph -> -. C e. A ) $.
|
|
neleqtrd.2 $e |- ( ph -> A = B ) $.
|
|
$( If a class is not an element of another class, it is also not an element
|
|
of an equal class. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
neleqtrd $p |- ( ph -> -. C e. B ) $=
|
|
( wcel eleq2d mtbid ) ADBGDCGEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
neleqtrrd.1 $e |- ( ph -> -. C e. B ) $.
|
|
neleqtrrd.2 $e |- ( ph -> A = B ) $.
|
|
$( If a class is not an element of another class, it is also not an element
|
|
of an equal class. Deduction form. (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
neleqtrrd $p |- ( ph -> -. C e. A ) $=
|
|
( wcel eleq2d mtbird ) ADBGDCGEABCDFHI $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
cleqh.1 $e |- ( y e. A -> A. x y e. A ) $.
|
|
cleqh.2 $e |- ( y e. B -> A. x y e. B ) $.
|
|
$( Establish equality between classes, using bound-variable hypotheses
|
|
instead of distinct variable conditions. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
cleqh $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
|
|
( wceq cv wcel wb wal dfcleq ax-17 wi wa dfbi2 hbim weq eleq1 cbv3h hban
|
|
hbxfrbi bibi12d biimpd equcoms biimprd impbii bitr4i ) CDGBHZCIZUIDIZJZBK
|
|
ZAHZCIZUNDIZJZAKZBCDLURUMUQULABUQBMZULUJUKNZUKUJNZOAUJUKPUTVAAUJUKAEFQUKU
|
|
JAFEQUAUBZABRZUQULVCUOUJUPUKUNUICSUNUIDSUCZUDTULUQBAVBUSBARUQULUQULJABVDU
|
|
EUFTUGUH $.
|
|
$}
|
|
|
|
$( A way of showing two classes are not equal. (Contributed by NM,
|
|
1-Apr-1997.) $)
|
|
nelneq $p |- ( ( A e. C /\ -. B e. C ) -> -. A = B ) $=
|
|
( wcel wceq eleq1 biimpcd con3and ) ACDZABEZBCDZJIKABCFGH $.
|
|
|
|
$( A way of showing two classes are not equal. (Contributed by NM,
|
|
12-Jan-2002.) $)
|
|
nelneq2 $p |- ( ( A e. B /\ -. A e. C ) -> -. B = C ) $=
|
|
( wcel wceq eleq2 biimpcd con3and ) ABDZBCEZACDZJIKBCAFGH $.
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Lemma for ~ eqsb3 . (Contributed by Rodolfo Medina, 28-Apr-2010.)
|
|
(Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
eqsb3lem $p |- ( [ x / y ] y = A <-> x = A ) $=
|
|
( cv wceq nfv eqeq1 sbie ) BDZCEADZCEZBAKBFIJCGH $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d w y $. $d w A $. $d x w $.
|
|
$( Substitution applied to an atomic wff (class version of ~ equsb3 ).
|
|
(Contributed by Rodolfo Medina, 28-Apr-2010.) $)
|
|
eqsb3 $p |- ( [ x / y ] y = A <-> x = A ) $=
|
|
( vw cv wceq wsb eqsb3lem sbbii nfv sbco2 3bitr3i ) BECFZBDGZDAGDECFZDAGM
|
|
BAGAECFNODADBCHIMBADMDJKADCHL $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d w y $. $d w A $. $d w x $.
|
|
$( Substitution applied to an atomic wff (class version of ~ elsb3 ).
|
|
(Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by
|
|
Andrew Salmon, 14-Jun-2011.) $)
|
|
clelsb3 $p |- ( [ x / y ] y e. A <-> x e. A ) $=
|
|
( vw cv wcel wsb nfv sbco2 eleq1 sbie sbbii 3bitr3i ) DEZCFZDBGZBAGODAGBE
|
|
ZCFZBAGAEZCFZODABOBHIPRBAORDBRDHNQCJKLOTDATDHNSCJKM $.
|
|
$}
|
|
|
|
${
|
|
hbxfr.1 $e |- A = B $.
|
|
hbxfr.2 $e |- ( y e. B -> A. x y e. B ) $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. See ~ hbxfrbi for equivalence version. (Contributed by NM,
|
|
21-Aug-2007.) $)
|
|
hbxfreq $p |- ( y e. A -> A. x y e. A ) $=
|
|
( cv wcel eleq2i hbxfrbi ) BGZCHKDHACDKEIFJ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x z $.
|
|
hblem.1 $e |- ( y e. A -> A. x y e. A ) $.
|
|
$( Change the free variable of a hypothesis builder. Lemma for ~ nfcrii .
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
hblem $p |- ( z e. A -> A. x z e. A ) $=
|
|
( cv wcel wsb wal hbsb clelsb3 albii 3imtr3i ) BFDGZBCHZOAICFDGZPAINBCAEJ
|
|
CBDKZOPAQLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
$( Equality of a class variable and a class abstraction (also called a
|
|
class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the
|
|
relationship between expressions with class abstractions and expressions
|
|
with class variables. Note that ~ abbi and its relatives are among
|
|
those useful for converting theorems with class variables to equivalent
|
|
theorems with wff variables, by first substituting a class abstraction
|
|
for each class variable.
|
|
|
|
Class variables can always be eliminated from a theorem to result in an
|
|
equivalent theorem with wff variables, and vice-versa. The idea is
|
|
roughly as follows. To convert a theorem with a wff variable ` ph `
|
|
(that has a free variable ` x ` ) to a theorem with a class variable
|
|
` A ` , we substitute ` x e. A ` for ` ph ` throughout and simplify,
|
|
where ` A ` is a new class variable not already in the wff. An example
|
|
is the conversion of zfauscl in set.mm to inex1 in set.mm (look at the
|
|
instance of zfauscl that occurs in the proof of inex1 ). Conversely, to
|
|
convert a theorem with a class variable ` A ` to one with ` ph ` , we
|
|
substitute ` { x | ph } ` for ` A ` throughout and simplify, where ` x `
|
|
and ` ph ` are new setvar and wff variables not already in the wff. An
|
|
example is cp in set.mm , which derives a formula containing wff
|
|
variables from substitution instances of the class variables in its
|
|
equivalent formulation cplem2 in set.mm. For more information on class
|
|
variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
abeq2 $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
|
|
( vy cab wceq cv wcel wb wal ax-17 hbab1 cleqh abid bibi2i albii bitri )
|
|
CABEZFBGZCHZSRHZIZBJTAIZBJBDCRDGCHBKABDLMUBUCBUAATABNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Equality of a class variable and a class abstraction. (Contributed by
|
|
NM, 20-Aug-1993.) $)
|
|
abeq1 $p |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) ) $=
|
|
( cab wceq cv wcel wb wal abeq2 eqcom bicom albii 3bitr4i ) CABDZEBFCGZAH
|
|
ZBIOCEAPHZBIABCJOCKRQBAPLMN $.
|
|
$}
|
|
|
|
${
|
|
abeqi.1 $e |- A = { x | ph } $.
|
|
$( Equality of a class variable and a class abstraction (inference rule).
|
|
(Contributed by NM, 3-Apr-1996.) $)
|
|
abeq2i $p |- ( x e. A <-> ph ) $=
|
|
( cv wcel cab eleq2i abid bitri ) BEZCFKABGZFACLKDHABIJ $.
|
|
$}
|
|
|
|
${
|
|
abeqri.1 $e |- { x | ph } = A $.
|
|
$( Equality of a class variable and a class abstraction (inference rule).
|
|
(Contributed by NM, 31-Jul-1994.) $)
|
|
abeq1i $p |- ( ph <-> x e. A ) $=
|
|
( cv cab wcel abid eleq2i bitr3i ) ABEZABFZGKCGABHLCKDIJ $.
|
|
$}
|
|
|
|
${
|
|
abeqd.1 $e |- ( ph -> A = { x | ps } ) $.
|
|
$( Equality of a class variable and a class abstraction (deduction).
|
|
(Contributed by NM, 16-Nov-1995.) $)
|
|
abeq2d $p |- ( ph -> ( x e. A <-> ps ) ) $=
|
|
( cv wcel cab eleq2d abid syl6bb ) ACFZDGLBCHZGBADMLEIBCJK $.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d ps y $. $d x y $.
|
|
$( Equivalent wff's correspond to equal class abstractions. (Contributed
|
|
by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) $)
|
|
abbi $p |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } ) $=
|
|
( vy cab wceq wcel wal dfcleq nfsab1 nfbi nfv wsb df-clab sbequ12r syl5bb
|
|
cv wb bibi12d cbval bitr2i ) ACEZBCEZFDQZUBGZUDUCGZRZDHABRZCHDUBUCIUGUHDC
|
|
UEUFCACDJBCDJKUHDLUDCQFZUEAUFBUEACDMUIAADCNADCOPUFBCDMUIBBDCNBDCOPSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
abbiri.1 $e |- ( x e. A <-> ph ) $.
|
|
$( Equality of a class variable and a class abstraction (inference rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
abbi2i $p |- A = { x | ph } $=
|
|
( cab wceq cv wcel wb abeq2 mpgbir ) CABEFBGCHAIBABCJDK $.
|
|
$}
|
|
|
|
${
|
|
abbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Equivalent wff's yield equal class abstractions (inference rule).
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
abbii $p |- { x | ph } = { x | ps } $=
|
|
( wb cab wceq abbi mpgbi ) ABEACFBCFGCABCHDI $.
|
|
|
|
$( Theorem abbii is the congruence law for class abstraction. $)
|
|
$( $j congruence 'abbii'; $)
|
|
$}
|
|
|
|
${
|
|
abbid.1 $e |- F/ x ph $.
|
|
abbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal class abstractions (deduction rule).
|
|
(Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
abbid $p |- ( ph -> { x | ps } = { x | ch } ) $=
|
|
( wb wal cab wceq alrimi abbi sylib ) ABCGZDHBDICDIJANDEFKBCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
abbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal class abstractions (deduction rule).
|
|
(Contributed by NM, 10-Aug-1993.) $)
|
|
abbidv $p |- ( ph -> { x | ps } = { x | ch } ) $=
|
|
( nfv abbid ) ABCDADFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d ph x $.
|
|
abbirdv.1 $e |- ( ph -> ( x e. A <-> ps ) ) $.
|
|
$( Deduction from a wff to a class abstraction. (Contributed by NM,
|
|
9-Jul-1994.) $)
|
|
abbi2dv $p |- ( ph -> A = { x | ps } ) $=
|
|
( cv wcel wb wal cab wceq alrimiv abeq2 sylibr ) ACFDGBHZCIDBCJKAOCELBCDM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d ph x $.
|
|
abbildv.1 $e |- ( ph -> ( ps <-> x e. A ) ) $.
|
|
$( Deduction from a wff to a class abstraction. (Contributed by NM,
|
|
9-Jul-1994.) $)
|
|
abbi1dv $p |- ( ph -> { x | ps } = A ) $=
|
|
( cv wcel wb wal cab wceq alrimiv abeq1 sylibr ) ABCFDGHZCIBCJDKAOCELBCDM
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.
|
|
(Contributed by NM, 26-Dec-1993.) $)
|
|
abid2 $p |- { x | x e. A } = A $=
|
|
( cv wcel cab biid abbi2i eqcomi ) BACBDZAEIABIFGH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d ph z $. $d ps z $.
|
|
cbvab.1 $e |- F/ y ph $.
|
|
cbvab.2 $e |- F/ x ps $.
|
|
cbvab.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by Andrew Salmon, 11-Jul-2011.) $)
|
|
cbvab $p |- { x | ph } = { y | ps } $=
|
|
( vz cab wsb cv wcel nfsb wceq wb equcoms bicomd sbie sbequ df-clab eqriv
|
|
syl5bbr 3bitr4i ) HACIZBDIZACHJBDHJZHKZUDLUGUELAUFCHBDHCFMABDCJCKZUGNUFBA
|
|
DCEDKUHNABABOCDGPQRBCHDSUBRAHCTBHDTUCUA $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $.
|
|
cbvabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 26-May-1999.) $)
|
|
cbvabv $p |- { x | ph } = { y | ps } $=
|
|
( nfv cbvab ) ABCDADFBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
$( Membership of a class variable in a class abstraction. (Contributed by
|
|
NM, 23-Dec-1993.) $)
|
|
clelab $p |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) ) $=
|
|
( vy cv wceq cab wcel wex wsb df-clab anbi2i exbii df-clel nfv nfs1v nfan
|
|
wa eqeq1 sbequ12 anbi12d cbvex 3bitr4i ) DEZCFZUDABGZHZRZDIUEABDJZRZDICUF
|
|
HBEZCFZARZBIUHUJDUGUIUEADBKLMDCUFNUMUJBDUMDOUEUIBUEBOABDPQUKUDFULUEAUIUKU
|
|
DCSABDTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y ph $. $d x y $.
|
|
$( Membership of a class abstraction in another class. (Contributed by NM,
|
|
17-Jan-2006.) $)
|
|
clabel $p |- ( { x | ph } e. A <->
|
|
E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) ) $=
|
|
( cab wcel cv wceq wa wex wb wal df-clel abeq2 anbi2ci exbii bitri ) ABEZ
|
|
DFCGZRHZSDFZIZCJUABGSFAKBLZIZCJCRDMUBUDCTUCUAABSNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z x $. $d z y $.
|
|
$( The right-hand side of the second equality is a way of representing
|
|
proper substitution of ` y ` for ` x ` into a class variable.
|
|
(Contributed by NM, 14-Sep-2003.) $)
|
|
sbab $p |- ( x = y -> A = { z | [ y / x ] z e. A } ) $=
|
|
( cv wceq wcel wsb sbequ12 abbi2dv ) AEBEFCEDGZABHCDKABIJ $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Class form not-free predicate
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c F/_ $. $( Underlined not-free symbol. $)
|
|
|
|
$( Extend wff definition to include the not-free predicate for classes. $)
|
|
wnfc $a wff F/_ x A $.
|
|
|
|
${
|
|
$d x y z $. $d y z A $.
|
|
$( Justification theorem for ~ df-nfc . (Contributed by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
nfcjust $p |- ( A. y F/ x y e. A <-> A. z F/ x z e. A ) $=
|
|
( cv wcel wnf wceq nfv eleq1 nfbidf cbvalv ) BEZDFZAGCEZDFZAGBCMOHZNPAQAI
|
|
MODJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Define the not-free predicate for classes. This is read " ` x ` is not
|
|
free in ` A ` ". Not-free means that the value of ` x ` cannot affect
|
|
the value of ` A ` , e.g., any occurrence of ` x ` in ` A ` is
|
|
effectively bound by a "for all" or something that expands to one (such
|
|
as "there exists"). It is defined in terms of the not-free predicate
|
|
~ df-nf for wffs; see that definition for more information.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
df-nfc $a |- ( F/_ x A <-> A. y F/ x y e. A ) $.
|
|
|
|
${
|
|
nfci.1 $e |- F/ x y e. A $.
|
|
$( Deduce that a class ` A ` does not have ` x ` free in it.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfci $p |- F/_ x A $=
|
|
( wnfc cv wcel wnf df-nfc mpgbir ) ACEBFCGAHBABCIDJ $.
|
|
$}
|
|
|
|
${
|
|
nfcii.1 $e |- ( y e. A -> A. x y e. A ) $.
|
|
$( Deduce that a class ` A ` does not have ` x ` free in it.
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcii $p |- F/_ x A $=
|
|
( cv wcel nfi nfci ) ABCBECFADGH $.
|
|
$}
|
|
|
|
$( Consequence of the not-free predicate. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfcr $p |- ( F/_ x A -> F/ x y e. A ) $=
|
|
( wnfc cv wcel wnf wal df-nfc sp sylbi ) ACDBECFAGZBHLABCILBJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z A $.
|
|
nfcri.1 $e |- F/_ x A $.
|
|
$( Consequence of the not-free predicate. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfcrii $p |- ( y e. A -> A. x y e. A ) $=
|
|
( vz cv wcel wnfc wnf nfcr ax-mp nfri hblem ) AEBCEFCGZAACHNAIDAECJKLM $.
|
|
|
|
$( Consequence of the not-free predicate. (Note that unlike ~ nfcr , this
|
|
does not require ` y ` and ` A ` to be disjoint.) (Contributed by Mario
|
|
Carneiro, 11-Aug-2016.) $)
|
|
nfcri $p |- F/ x y e. A $=
|
|
( cv wcel nfcrii nfi ) BECFAABCDGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
nfcd.1 $e |- F/ y ph $.
|
|
nfcd.2 $e |- ( ph -> F/ x y e. A ) $.
|
|
$( Deduce that a class ` A ` does not have ` x ` free in it. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcd $p |- ( ph -> F/_ x A ) $=
|
|
( cv wcel wnf wal wnfc alrimi df-nfc sylibr ) ACGDHBIZCJBDKAOCEFLBCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
nfceqi.1 $e |- A = B $.
|
|
$( Equality theorem for class not-free. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfceqi $p |- ( F/_ x A <-> F/_ x B ) $=
|
|
( vy cv wcel wnf wal wnfc eleq2i nfbii albii df-nfc 3bitr4i ) EFZBGZAHZEI
|
|
PCGZAHZEIABJACJRTEQSABCPDKLMAEBNAECNO $.
|
|
|
|
${
|
|
nfcxfr.2 $e |- F/_ x B $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcxfr $p |- F/_ x A $=
|
|
( wnfc nfceqi mpbir ) ABFACFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
nfcxfrd.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( A utility lemma to transfer a bound-variable hypothesis builder into a
|
|
definition. (Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcxfrd $p |- ( ph -> F/_ x A ) $=
|
|
( wnfc nfceqi sylibr ) ABDGBCGFBCDEHI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $. $d B y $. $d ph y $.
|
|
nfceqdf.1 $e |- F/ x ph $.
|
|
nfceqdf.2 $e |- ( ph -> A = B ) $.
|
|
$( An equality theorem for effectively not free. (Contributed by Mario
|
|
Carneiro, 14-Oct-2016.) $)
|
|
nfceqdf $p |- ( ph -> ( F/_ x A <-> F/_ x B ) ) $=
|
|
( vy cv wcel wnf wal wnfc eleq2d nfbidf albidv df-nfc 3bitr4g ) AGHZCIZBJ
|
|
ZGKRDIZBJZGKBCLBDLATUBGASUABEACDRFMNOBGCPBGDPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( If ` x ` is disjoint from ` A ` , then ` x ` is not free in ` A ` .
|
|
(Contributed by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfcv $p |- F/_ x A $=
|
|
( vy cv wcel nfv nfci ) ACBCDBEAFG $.
|
|
|
|
$( If ` x ` is disjoint from ` A ` , then ` x ` is not free in ` A ` .
|
|
(Contributed by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfcvd $p |- ( ph -> F/_ x A ) $=
|
|
( wnfc nfcv a1i ) BCDABCEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y ph $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfab1 $p |- F/_ x { x | ph } $=
|
|
( vy cab nfsab1 nfci ) BCABDABCEF $.
|
|
|
|
$( ` x ` is bound in ` F/_ x A ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfnfc1 $p |- F/ x F/_ x A $=
|
|
( vy wnfc cv wcel wnf wal df-nfc nfnf1 nfal nfxfr ) ABDCEBFZAGZCHAACBINAC
|
|
MAJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
nfab.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 11-Aug-2016.) $)
|
|
nfab $p |- F/_ x { y | ph } $=
|
|
( vz cab nfsab nfci ) BEACFABCEDGH $.
|
|
$}
|
|
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 14-Oct-2016.) $)
|
|
nfaba1 $p |- F/_ x { y | A. x ph } $=
|
|
( wal nfa1 nfab ) ABDBCABEF $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z B $.
|
|
nfnfc.1 $e |- F/_ x A $.
|
|
$( Hypothesis builder for ` F/_ y A ` . (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfnfc $p |- F/ x F/_ y A $=
|
|
( vz wnfc cv wcel wnf wal df-nfc nfcri nfnf nfal nfxfr ) BCFEGCHZBIZEJABE
|
|
CKQAEPABAECDLMNO $.
|
|
|
|
nfeq.2 $e |- F/_ x B $.
|
|
$( Hypothesis builder for equality. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfeq $p |- F/ x A = B $=
|
|
( vz wceq cv wcel wb wal dfcleq nfcri nfbi nfal nfxfr ) BCGFHZBIZQCIZJZFK
|
|
AFBCLTAFRSAAFBDMAFCEMNOP $.
|
|
|
|
$( Hypothesis builder for elementhood. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfel $p |- F/ x A e. B $=
|
|
( vz wcel cv wceq wa wex df-clel nfcv nfeq nfcri nfan nfex nfxfr ) BCGFHZ
|
|
BIZSCGZJZFKAFBCLUBAFTUAAASBASMDNAFCEOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
nfeq1.1 $e |- F/_ x A $.
|
|
$( Hypothesis builder for equality, special case. (Contributed by Mario
|
|
Carneiro, 10-Oct-2016.) $)
|
|
nfeq1 $p |- F/ x A = B $=
|
|
( nfcv nfeq ) ABCDACEF $.
|
|
|
|
$( Hypothesis builder for elementhood, special case. (Contributed by Mario
|
|
Carneiro, 10-Oct-2016.) $)
|
|
nfel1 $p |- F/ x A e. B $=
|
|
( nfcv nfel ) ABCDACEF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
nfeq2.1 $e |- F/_ x B $.
|
|
$( Hypothesis builder for equality, special case. (Contributed by Mario
|
|
Carneiro, 10-Oct-2016.) $)
|
|
nfeq2 $p |- F/ x A = B $=
|
|
( nfcv nfeq ) ABCABEDF $.
|
|
|
|
$( Hypothesis builder for elementhood, special case. (Contributed by Mario
|
|
Carneiro, 10-Oct-2016.) $)
|
|
nfel2 $p |- F/ x A e. B $=
|
|
( nfcv nfel ) ABCABEDF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
nfeqd.1 $e |- ( ph -> F/_ x A ) $.
|
|
$( Consequence of the not-free predicate. (Contributed by Mario Carneiro,
|
|
11-Aug-2016.) $)
|
|
nfcrd $p |- ( ph -> F/ x y e. A ) $=
|
|
( wnfc cv wcel wnf nfcr syl ) ABDFCGDHBIEBCDJK $.
|
|
|
|
$d y ph $.
|
|
nfeqd.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( Hypothesis builder for equality. (Contributed by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfeqd $p |- ( ph -> F/ x A = B ) $=
|
|
( vy wceq cv wcel wb wal dfcleq nfv nfcrd nfbid nfald nfxfrd ) CDHGIZCJZS
|
|
DJZKZGLABGCDMAUBBGAGNATUABABGCEOABGDFOPQR $.
|
|
|
|
$( Hypothesis builder for elementhood. (Contributed by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfeld $p |- ( ph -> F/ x A e. B ) $=
|
|
( vy wcel cv wceq wa wex df-clel nfv nfcvd nfeqd nfcrd nfand nfexd nfxfrd
|
|
) CDHGIZCJZUADHZKZGLABGCDMAUDBGAGNAUBUCBABUACABUAOEPABGDFQRST $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $. $d w A $. $d w B $.
|
|
drnfc1.1 $e |- ( A. x x = y -> A = B ) $.
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
(Contributed by Mario Carneiro, 8-Oct-2016.) $)
|
|
drnfc1 $p |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) ) $=
|
|
( vw cv wceq wal wcel wnf wnfc eleq2d drnf1 dral2 df-nfc 3bitr4g ) AGBGHA
|
|
IZFGZCJZAKZFISDJZBKZFIACLBDLUAUCABFTUBABRCDSEMNOAFCPBFDPQ $.
|
|
|
|
$( Formula-building lemma for use with the Distinctor Reduction Theorem.
|
|
(Contributed by Mario Carneiro, 8-Oct-2016.) $)
|
|
drnfc2 $p |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) ) $=
|
|
( vw cv wceq wal wcel wnf wnfc eleq2d drnf2 dral2 df-nfc 3bitr4g ) AHBHIA
|
|
JZGHZDKZCLZGJTEKZCLZGJCDMCEMUBUDABGUAUCABCSDETFNOPCGDQCGEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $. $d z ps $.
|
|
nfabd2.1 $e |- F/ y ph $.
|
|
nfabd2.2 $e |- ( ( ph /\ -. A. x x = y ) -> F/ x ps ) $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 8-Oct-2016.) $)
|
|
nfabd2 $p |- ( ph -> F/_ x { y | ps } ) $=
|
|
( vz cv wceq wal cab wnfc wn wa nfv wcel wsb df-clab nfnae nfan nfxfrd ex
|
|
nfsbd nfcd nfab1 eqidd drnfc1 mpbiri pm2.61d2 ) ACHDHICJZCBDKZLZAUJMZULAU
|
|
MNZCGUKUNGOGHUKPBDGQUNCBGDRUNBDGCAUMDECDDSTFUCUAUDUBUJULDUKLBDUECDUKUKUJU
|
|
KUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
nfabd.1 $e |- F/ y ph $.
|
|
nfabd.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( Bound-variable hypothesis builder for a class abstraction. (Contributed
|
|
by Mario Carneiro, 8-Oct-2016.) $)
|
|
nfabd $p |- ( ph -> F/_ x { y | ps } ) $=
|
|
( wnf cv wceq wal wn adantr nfabd2 ) ABCDEABCGCHDHICJKFLM $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $. $d w A $. $d w B $. $d w ph $.
|
|
dvelimdc.1 $e |- F/ x ph $.
|
|
dvelimdc.2 $e |- F/ z ph $.
|
|
dvelimdc.3 $e |- ( ph -> F/_ x A ) $.
|
|
dvelimdc.4 $e |- ( ph -> F/_ z B ) $.
|
|
dvelimdc.5 $e |- ( ph -> ( z = y -> A = B ) ) $.
|
|
$( Deduction form of ~ dvelimc . (Contributed by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
dvelimdc $p |- ( ph -> ( -. A. x x = y -> F/_ x B ) ) $=
|
|
( vw cv wceq wal wn wnfc wa wcel nfcrd nfv wnf wb eleq2 syl6 dvelimdf imp
|
|
nfcd ex ) ABMCMZNBOPZBFQAUKRZBLFULLUAAUKLMZFSZBUBAUMESZUNBCDGHABLEITADLFJ
|
|
TADMUJNEFNUOUNUCKEFUMUDUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
dvelimc.1 $e |- F/_ x A $.
|
|
dvelimc.2 $e |- F/_ z B $.
|
|
dvelimc.3 $e |- ( z = y -> A = B ) $.
|
|
$( Version of ~ dvelim for classes. (Contributed by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
dvelimc $p |- ( -. A. x x = y -> F/_ x B ) $=
|
|
( cv wceq wal wn wnfc wi wtru nftru a1i dvelimdc trud ) AIBIZJAKLAEMNOABC
|
|
DEAPCPADMOFQCEMOGQCITJDEJNOHQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $.
|
|
$( If ` x ` and ` y ` are distinct, then ` x ` is not free in ` y ` .
|
|
(Contributed by Mario Carneiro, 8-Oct-2016.) $)
|
|
nfcvf $p |- ( -. A. x x = y -> F/_ x y ) $=
|
|
( vz cv nfcv wceq id dvelimc ) ABCCDZBDZAIECJEIJFGH $.
|
|
|
|
$( If ` x ` and ` y ` are distinct, then ` y ` is not free in ` x ` .
|
|
(Contributed by Mario Carneiro, 5-Dec-2016.) $)
|
|
nfcvf2 $p |- ( -. A. x x = y -> F/_ y x ) $=
|
|
( cv wnfc nfcvf naecoms ) BACDBABAEF $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
cleqf.1 $e |- F/_ x A $.
|
|
cleqf.2 $e |- F/_ x B $.
|
|
$( Establish equality between classes, using bound-variable hypotheses
|
|
instead of distinct variable conditions. (Contributed by NM,
|
|
5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
cleqf $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
|
|
( vy wceq cv wcel wb wal dfcleq nfv nfcri nfbi eleq1 bibi12d cbval bitr4i
|
|
) BCGFHZBIZTCIZJZFKAHZBIZUDCIZJZAKFBCLUGUCAFUGFMUAUBAAFBDNAFCENOUDTGUEUAU
|
|
FUBUDTBPUDTCPQRS $.
|
|
$}
|
|
|
|
${
|
|
abid2f.1 $e |- F/_ x A $.
|
|
$( A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.
|
|
(Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
abid2f $p |- { x | x e. A } = A $=
|
|
( cv wcel cab wceq wb wal nfab1 cleqf abid bibi2i albii bitri biid mpgbir
|
|
eqcomi ) BADZBEZAFZBUAGZTTHZAUBTSUAEZHZAIUCAIABUACTAJKUEUCAUDTTTALMNOTPQR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d v A $. $d x v z $. $d y v z $. $d v ph $.
|
|
sbabel.1 $e |- F/_ x A $.
|
|
$( Theorem to move a substitution in and out of a class abstraction.
|
|
(Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
sbabel $p |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A ) $=
|
|
( vv cv cab wceq wcel wa wex wsb wb wal sbf abeq2 sbbii 3bitr4i sbex sban
|
|
nfv sbrbis sbalv nfcri anbi12i bitri exbii df-clel ) GHZADIZJZUKEKZLZGMZB
|
|
CNZUKABCNZDIZJZUNLZGMZULEKZBCNUSEKUQUOBCNZGMVBUOGBCUAVDVAGVDUMBCNZUNBCNZL
|
|
VAUMUNBCUBVEUTVFUNDHUKKZAOZDPZBCNVGUROZDPVEUTVHVJBCDVGVGABCVGBCVGBUCQUDUE
|
|
UMVIBCADUKRSURDUKRTUNBCBGEFUFQUGUHUIUHVCUPBCGULEUJSGUSEUJT $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Negated equality and membership
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new connectives. $)
|
|
$c =/= $. $( Not equal to (equal sign with slash through it). $)
|
|
$c e/ $. $( Not an element of (epsilon with slash through it). $)
|
|
|
|
$( Extend wff notation to include inequality. $)
|
|
wne $a wff A =/= B $.
|
|
|
|
$( Extend wff notation to include negated membership. $)
|
|
wnel $a wff A e/ B $.
|
|
|
|
$( Define inequality. (Contributed by NM, 5-Aug-1993.) $)
|
|
df-ne $a |- ( A =/= B <-> -. A = B ) $.
|
|
|
|
$( Define negated membership. (Contributed by NM, 7-Aug-1994.) $)
|
|
df-nel $a |- ( A e/ B <-> -. A e. B ) $.
|
|
|
|
$( Negation of inequality. (Contributed by NM, 9-Jun-2006.) $)
|
|
nne $p |- ( -. A =/= B <-> A = B ) $=
|
|
( wceq wne wn df-ne con2bii bicomi ) ABCZABDZEJIABFGH $.
|
|
|
|
$( No class is unequal to itself. (Contributed by Stefan O'Rear,
|
|
1-Jan-2015.) $)
|
|
neirr $p |- -. A =/= A $=
|
|
( wne wn wceq eqid nne mpbir ) AABCAADAEAAFG $.
|
|
|
|
$( Excluded middle with equality and inequality. (Contributed by NM,
|
|
3-Feb-2012.) $)
|
|
exmidne $p |- ( A = B \/ A =/= B ) $=
|
|
( wceq wne wo wn exmid df-ne orbi2i mpbir ) ABCZABDZEKKFZEKGLMKABHIJ $.
|
|
|
|
$( Law of noncontradiction with equality and inequality. (Contributed by NM,
|
|
3-Feb-2012.) $)
|
|
nonconne $p |- -. ( A = B /\ A =/= B ) $=
|
|
( wceq wne wa wn pm3.24 df-ne anbi2i mtbir ) ABCZABDZEKKFZEKGLMKABHIJ $.
|
|
|
|
$( Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) $)
|
|
neeq1 $p |- ( A = B -> ( A =/= C <-> B =/= C ) ) $=
|
|
( wceq wn wne eqeq1 notbid df-ne 3bitr4g ) ABDZACDZEBCDZEACFBCFKLMABCGHACIB
|
|
CIJ $.
|
|
|
|
$( Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) $)
|
|
neeq2 $p |- ( A = B -> ( C =/= A <-> C =/= B ) ) $=
|
|
( wceq wn wne eqeq2 notbid df-ne 3bitr4g ) ABDZCADZECBDZECAFCBFKLMABCGHCAIC
|
|
BIJ $.
|
|
|
|
${
|
|
neeq1i.1 $e |- A = B $.
|
|
$( Inference for inequality. (Contributed by NM, 29-Apr-2005.) $)
|
|
neeq1i $p |- ( A =/= C <-> B =/= C ) $=
|
|
( wceq wne wb neeq1 ax-mp ) ABEACFBCFGDABCHI $.
|
|
|
|
$( Inference for inequality. (Contributed by NM, 29-Apr-2005.) $)
|
|
neeq2i $p |- ( C =/= A <-> C =/= B ) $=
|
|
( wceq wne wb neeq2 ax-mp ) ABECAFCBFGDABCHI $.
|
|
|
|
neeq12i.2 $e |- C = D $.
|
|
$( Inference for inequality. (Contributed by NM, 24-Jul-2012.) $)
|
|
neeq12i $p |- ( A =/= C <-> B =/= D ) $=
|
|
( wne neeq2i neeq1i bitri ) ACGADGBDGCDAFHABDEIJ $.
|
|
$}
|
|
|
|
${
|
|
neeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction for inequality. (Contributed by NM, 25-Oct-1999.) $)
|
|
neeq1d $p |- ( ph -> ( A =/= C <-> B =/= C ) ) $=
|
|
( wceq wne wb neeq1 syl ) ABCFBDGCDGHEBCDIJ $.
|
|
|
|
$( Deduction for inequality. (Contributed by NM, 25-Oct-1999.) $)
|
|
neeq2d $p |- ( ph -> ( C =/= A <-> C =/= B ) ) $=
|
|
( wceq wne wb neeq2 syl ) ABCFDBGDCGHEBCDIJ $.
|
|
|
|
neeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Deduction for inequality. (Contributed by NM, 24-Jul-2012.) $)
|
|
neeq12d $p |- ( ph -> ( A =/= C <-> B =/= D ) ) $=
|
|
( wne neeq1d neeq2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
neneqd.1 $e |- ( ph -> A =/= B ) $.
|
|
$( Deduction eliminating inequality definition. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
neneqd $p |- ( ph -> -. A = B ) $=
|
|
( wne wceq wn df-ne sylib ) ABCEBCFGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
eqnetr.1 $e |- A = B $.
|
|
eqnetr.2 $e |- B =/= C $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
eqnetri $p |- A =/= C $=
|
|
( wne neeq1i mpbir ) ACFBCFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
eqnetrd.1 $e |- ( ph -> A = B ) $.
|
|
eqnetrd.2 $e |- ( ph -> B =/= C ) $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
eqnetrd $p |- ( ph -> A =/= C ) $=
|
|
( wne neeq1d mpbird ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqnetrr.1 $e |- A = B $.
|
|
eqnetrr.2 $e |- A =/= C $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
eqnetrri $p |- B =/= C $=
|
|
( eqcomi eqnetri ) BACABDFEG $.
|
|
$}
|
|
|
|
${
|
|
eqnetrrd.1 $e |- ( ph -> A = B ) $.
|
|
eqnetrrd.2 $e |- ( ph -> A =/= C ) $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
eqnetrrd $p |- ( ph -> B =/= C ) $=
|
|
( eqcomd eqnetrd ) ACBDABCEGFH $.
|
|
$}
|
|
|
|
${
|
|
neeqtr.1 $e |- A =/= B $.
|
|
neeqtr.2 $e |- B = C $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
neeqtri $p |- A =/= C $=
|
|
( wne neeq2i mpbi ) ABFACFDBCAEGH $.
|
|
$}
|
|
|
|
${
|
|
neeqtrd.1 $e |- ( ph -> A =/= B ) $.
|
|
neeqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
neeqtrd $p |- ( ph -> A =/= C ) $=
|
|
( wne neeq2d mpbid ) ABCGBDGEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
neeqtrr.1 $e |- A =/= B $.
|
|
neeqtrr.2 $e |- C = B $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
neeqtrri $p |- A =/= C $=
|
|
( eqcomi neeqtri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
neeqtrrd.1 $e |- ( ph -> A =/= B ) $.
|
|
neeqtrrd.2 $e |- ( ph -> C = B ) $.
|
|
$( Substitution of equal classes into an inequality. (Contributed by NM,
|
|
4-Jul-2012.) $)
|
|
neeqtrrd $p |- ( ph -> A =/= C ) $=
|
|
( eqcomd neeqtrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5eqner.1 $e |- B = A $.
|
|
syl5eqner.2 $e |- ( ph -> B =/= C ) $.
|
|
$( B chained equality inference for inequality. (Contributed by NM,
|
|
6-Jun-2012.) $)
|
|
syl5eqner $p |- ( ph -> A =/= C ) $=
|
|
( wne neeq1i sylib ) ACDGBDGFCBDEHI $.
|
|
$}
|
|
|
|
${
|
|
3netr3d.1 $e |- ( ph -> A =/= B ) $.
|
|
3netr3d.2 $e |- ( ph -> A = C ) $.
|
|
3netr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( Substitution of equality into both sides of an inequality. (Contributed
|
|
by NM, 24-Jul-2012.) $)
|
|
3netr3d $p |- ( ph -> C =/= D ) $=
|
|
( wne neeq12d mpbid ) ABCIDEIFABDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3netr4d.1 $e |- ( ph -> A =/= B ) $.
|
|
3netr4d.2 $e |- ( ph -> C = A ) $.
|
|
3netr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( Substitution of equality into both sides of an inequality. (Contributed
|
|
by NM, 24-Jul-2012.) $)
|
|
3netr4d $p |- ( ph -> C =/= D ) $=
|
|
( wne neeq12d mpbird ) ADEIBCIFADBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
3netr3g.1 $e |- ( ph -> A =/= B ) $.
|
|
3netr3g.2 $e |- A = C $.
|
|
3netr3g.3 $e |- B = D $.
|
|
$( Substitution of equality into both sides of an inequality. (Contributed
|
|
by NM, 24-Jul-2012.) $)
|
|
3netr3g $p |- ( ph -> C =/= D ) $=
|
|
( wne neeq12i sylib ) ABCIDEIFBDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3netr4g.1 $e |- ( ph -> A =/= B ) $.
|
|
3netr4g.2 $e |- C = A $.
|
|
3netr4g.3 $e |- D = B $.
|
|
$( Substitution of equality into both sides of an inequality. (Contributed
|
|
by NM, 14-Jun-2012.) $)
|
|
3netr4g $p |- ( ph -> C =/= D ) $=
|
|
( wne neeq12i sylibr ) ABCIDEIFDBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
necon3abii.1 $e |- ( A = B <-> ph ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
9-Nov-2007.) $)
|
|
necon3abii $p |- ( A =/= B <-> -. ph ) $=
|
|
( wne wceq df-ne xchbinx ) BCEBCFABCGDH $.
|
|
$}
|
|
|
|
${
|
|
necon3bbii.1 $e |- ( ph <-> A = B ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
13-Apr-2007.) $)
|
|
necon3bbii $p |- ( -. ph <-> A =/= B ) $=
|
|
( wne wn wceq bicomi necon3abii ) BCEAFABCABCGDHIH $.
|
|
$}
|
|
|
|
${
|
|
necon3bii.1 $e |- ( A = B <-> C = D ) $.
|
|
$( Inference from equality to inequality. (Contributed by NM,
|
|
23-Feb-2005.) $)
|
|
necon3bii $p |- ( A =/= B <-> C =/= D ) $=
|
|
( wne wceq wn necon3abii df-ne bitr4i ) ABFCDGZHCDFLABEICDJK $.
|
|
$}
|
|
|
|
${
|
|
necon3abid.1 $e |- ( ph -> ( A = B <-> ps ) ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
21-Mar-2007.) $)
|
|
necon3abid $p |- ( ph -> ( A =/= B <-> -. ps ) ) $=
|
|
( wne wceq wn df-ne notbid syl5bb ) CDFCDGZHABHCDIALBEJK $.
|
|
$}
|
|
|
|
${
|
|
necon3bbid.1 $e |- ( ph -> ( ps <-> A = B ) ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
2-Jun-2007.) $)
|
|
necon3bbid $p |- ( ph -> ( -. ps <-> A =/= B ) ) $=
|
|
( wne wn wceq bicomd necon3abid ) ACDFBGABCDABCDHEIJI $.
|
|
$}
|
|
|
|
${
|
|
necon3bid.1 $e |- ( ph -> ( A = B <-> C = D ) ) $.
|
|
$( Deduction from equality to inequality. (Contributed by NM,
|
|
23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon3bid $p |- ( ph -> ( A =/= B <-> C =/= D ) ) $=
|
|
( wne wceq wn df-ne necon3bbid syl5bb ) BCGBCHZIADEGBCJAMDEFKL $.
|
|
$}
|
|
|
|
${
|
|
necon3ad.1 $e |- ( ph -> ( ps -> A = B ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon3ad $p |- ( ph -> ( A =/= B -> -. ps ) ) $=
|
|
( wne wceq wn nne syl6ibr con2d ) ABCDFZABCDGLHECDIJK $.
|
|
$}
|
|
|
|
${
|
|
necon3bd.1 $e |- ( ph -> ( A = B -> ps ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon3bd $p |- ( ph -> ( -. ps -> A =/= B ) ) $=
|
|
( wne wn wceq nne syl5bi con1d ) ACDFZBLGCDHABCDIEJK $.
|
|
$}
|
|
|
|
${
|
|
necon3d.1 $e |- ( ph -> ( A = B -> C = D ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
10-Jun-2006.) $)
|
|
necon3d $p |- ( ph -> ( C =/= D -> A =/= B ) ) $=
|
|
( wne wceq wn necon3ad df-ne syl6ibr ) ADEGBCHZIBCGAMDEFJBCKL $.
|
|
$}
|
|
|
|
${
|
|
necon3i.1 $e |- ( A = B -> C = D ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
9-Aug-2006.) $)
|
|
necon3i $p |- ( C =/= D -> A =/= B ) $=
|
|
( wceq wi wne id necon3d ax-mp ) ABFCDFGZCDHABHGELABCDLIJK $.
|
|
$}
|
|
|
|
${
|
|
necon3ai.1 $e |- ( ph -> A = B ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon3ai $p |- ( A =/= B -> -. ph ) $=
|
|
( wne wceq wn nne sylibr con2i ) ABCEZABCFKGDBCHIJ $.
|
|
$}
|
|
|
|
${
|
|
necon3bi.1 $e |- ( A = B -> ph ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon3bi $p |- ( -. ph -> A =/= B ) $=
|
|
( wne wn wceq nne sylbi con1i ) BCEZAKFBCGABCHDIJ $.
|
|
$}
|
|
|
|
${
|
|
necon1ai.1 $e |- ( -. ph -> A = B ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
12-Feb-2007.) $)
|
|
necon1ai $p |- ( A =/= B -> ph ) $=
|
|
( wne wceq wn df-ne con1i sylbi ) BCEBCFZGABCHAKDIJ $.
|
|
$}
|
|
|
|
${
|
|
necon1bi.1 $e |- ( A =/= B -> ph ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon1bi $p |- ( -. ph -> A = B ) $=
|
|
( wn wne wceq con3i nne sylib ) AEBCFZEBCGKADHBCIJ $.
|
|
$}
|
|
|
|
${
|
|
necon1i.1 $e |- ( A =/= B -> C = D ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
18-Mar-2007.) $)
|
|
necon1i $p |- ( C =/= D -> A = B ) $=
|
|
( wceq wn wne df-ne sylbir necon1ai ) ABFZCDLGABHCDFABIEJK $.
|
|
$}
|
|
|
|
${
|
|
necon2ai.1 $e |- ( A = B -> -. ph ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon2ai $p |- ( ph -> A =/= B ) $=
|
|
( wne wn wceq nne sylbi con4i ) BCEZAKFBCGAFBCHDIJ $.
|
|
$}
|
|
|
|
${
|
|
necon2bi.1 $e |- ( ph -> A =/= B ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
1-Apr-2007.) $)
|
|
necon2bi $p |- ( A = B -> -. ph ) $=
|
|
( wceq neneqd con2i ) ABCEABCDFG $.
|
|
$}
|
|
|
|
${
|
|
necon2i.1 $e |- ( A = B -> C =/= D ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
18-Mar-2007.) $)
|
|
necon2i $p |- ( C = D -> A =/= B ) $=
|
|
( wceq neneqd necon2ai ) CDFABABFCDEGH $.
|
|
$}
|
|
|
|
${
|
|
necon2ad.1 $e |- ( ph -> ( A = B -> -. ps ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon2ad $p |- ( ph -> ( ps -> A =/= B ) ) $=
|
|
( wne wn wceq nne syl5bi con4d ) ACDFZBLGCDHABGCDIEJK $.
|
|
$}
|
|
|
|
${
|
|
necon2bd.1 $e |- ( ph -> ( ps -> A =/= B ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
13-Apr-2007.) $)
|
|
necon2bd $p |- ( ph -> ( A = B -> -. ps ) ) $=
|
|
( wceq wne wn df-ne syl6ib con2d ) ABCDFZABCDGLHECDIJK $.
|
|
$}
|
|
|
|
${
|
|
necon2d.1 $e |- ( ph -> ( A = B -> C =/= D ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
28-Dec-2008.) $)
|
|
necon2d $p |- ( ph -> ( C = D -> A =/= B ) ) $=
|
|
( wceq wne wn df-ne syl6ib necon2ad ) ADEGZBCABCGDEHMIFDEJKL $.
|
|
$}
|
|
|
|
${
|
|
necon1abii.1 $e |- ( -. ph <-> A = B ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
17-Mar-2007.) $)
|
|
necon1abii $p |- ( A =/= B <-> ph ) $=
|
|
( wne wceq wn df-ne con1bii bitri ) BCEBCFZGABCHAKDIJ $.
|
|
$}
|
|
|
|
${
|
|
necon1bbii.1 $e |- ( A =/= B <-> ph ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
17-Mar-2007.) $)
|
|
necon1bbii $p |- ( -. ph <-> A = B ) $=
|
|
( wceq wn wne df-ne bitr3i con1bii ) BCEZAKFBCGABCHDIJ $.
|
|
$}
|
|
|
|
${
|
|
necon1abid.1 $e |- ( ph -> ( -. ps <-> A = B ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by NM,
|
|
21-Aug-2007.) $)
|
|
necon1abid $p |- ( ph -> ( A =/= B <-> ps ) ) $=
|
|
( wne wceq wn df-ne con1bid syl5bb ) CDFCDGZHABCDIABLEJK $.
|
|
$}
|
|
|
|
${
|
|
necon1bbid.1 $e |- ( ph -> ( A =/= B <-> ps ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
31-Jan-2008.) $)
|
|
necon1bbid $p |- ( ph -> ( -. ps <-> A = B ) ) $=
|
|
( wceq wn wne df-ne syl5bbr con1bid ) ACDFZBLGCDHABCDIEJK $.
|
|
$}
|
|
|
|
${
|
|
necon2abii.1 $e |- ( A = B <-> -. ph ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
2-Mar-2007.) $)
|
|
necon2abii $p |- ( ph <-> A =/= B ) $=
|
|
( wne wceq wn bicomi necon1abii ) BCEAABCBCFAGDHIH $.
|
|
$}
|
|
|
|
${
|
|
necon2bbii.1 $e |- ( ph <-> A =/= B ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
13-Apr-2007.) $)
|
|
necon2bbii $p |- ( A = B <-> -. ph ) $=
|
|
( wn wceq wne bicomi necon1bbii ) AEBCFABCABCGDHIH $.
|
|
$}
|
|
|
|
${
|
|
necon2abid.1 $e |- ( ph -> ( A = B <-> -. ps ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by NM,
|
|
18-Jul-2007.) $)
|
|
necon2abid $p |- ( ph -> ( ps <-> A =/= B ) ) $=
|
|
( wceq wn wne con2bid df-ne syl6bbr ) ABCDFZGCDHALBEICDJK $.
|
|
$}
|
|
|
|
${
|
|
necon2bbid.1 $e |- ( ph -> ( ps <-> A =/= B ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by NM,
|
|
13-Apr-2007.) $)
|
|
necon2bbid $p |- ( ph -> ( A = B <-> -. ps ) ) $=
|
|
( wceq wne wn df-ne syl6bb con2bid ) ABCDFZABCDGLHECDIJK $.
|
|
$}
|
|
|
|
${
|
|
necon4ai.1 $e |- ( A =/= B -> -. ph ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon4ai $p |- ( ph -> A = B ) $=
|
|
( wne wn wceq con2i nne sylib ) ABCEZFBCGKADHBCIJ $.
|
|
$}
|
|
|
|
${
|
|
necon4i.1 $e |- ( A =/= B -> C =/= D ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon4i $p |- ( C = D -> A = B ) $=
|
|
( wceq wne wn necon2bi nne sylib ) CDFABGZHABFLCDEIABJK $.
|
|
$}
|
|
|
|
${
|
|
necon4ad.1 $e |- ( ph -> ( A =/= B -> -. ps ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon4ad $p |- ( ph -> ( ps -> A = B ) ) $=
|
|
( wne wn wceq con2d nne syl6ib ) ABCDFZGCDHALBEICDJK $.
|
|
$}
|
|
|
|
${
|
|
necon4bd.1 $e |- ( ph -> ( -. ps -> A =/= B ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon4bd $p |- ( ph -> ( A = B -> ps ) ) $=
|
|
( wceq wne wn nne con1d syl5bir ) CDFCDGZHABCDIABLEJK $.
|
|
$}
|
|
|
|
${
|
|
necon4d.1 $e |- ( ph -> ( A =/= B -> C =/= D ) ) $.
|
|
$( Contrapositive inference for inequality. (Contributed by NM,
|
|
2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon4d $p |- ( ph -> ( C = D -> A = B ) ) $=
|
|
( wceq wne wn necon2bd nne syl6ib ) ADEGBCHZIBCGAMDEFJBCKL $.
|
|
$}
|
|
|
|
${
|
|
necon4abid.1 $e |- ( ph -> ( A =/= B <-> -. ps ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
11-Jan-2008.) $)
|
|
necon4abid $p |- ( ph -> ( A = B <-> ps ) ) $=
|
|
( wceq wn wne df-ne syl5bbr con4bid ) ACDFZBLGCDHABGCDIEJK $.
|
|
$}
|
|
|
|
${
|
|
necon4bbid.1 $e |- ( ph -> ( -. ps <-> A =/= B ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
9-May-2012.) $)
|
|
necon4bbid $p |- ( ph -> ( ps <-> A = B ) ) $=
|
|
( wceq wn wne bicomd necon4abid ) ACDFBABCDABGCDHEIJI $.
|
|
$}
|
|
|
|
${
|
|
necon4bid.1 $e |- ( ph -> ( A =/= B <-> C =/= D ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
29-Jun-2007.) $)
|
|
necon4bid $p |- ( ph -> ( A = B <-> C = D ) ) $=
|
|
( wceq wne wn necon2bbid nne syl6rbb ) ADEGBCHZIBCGAMDEFJBCKL $.
|
|
$}
|
|
|
|
${
|
|
necon1ad.1 $e |- ( ph -> ( -. ps -> A = B ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by NM,
|
|
2-Apr-2007.) $)
|
|
necon1ad $p |- ( ph -> ( A =/= B -> ps ) ) $=
|
|
( wne wceq wn df-ne con1d syl5bi ) CDFCDGZHABCDIABLEJK $.
|
|
$}
|
|
|
|
${
|
|
necon1bd.1 $e |- ( ph -> ( A =/= B -> ps ) ) $.
|
|
$( Contrapositive deduction for inequality. (Contributed by NM,
|
|
21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon1bd $p |- ( ph -> ( -. ps -> A = B ) ) $=
|
|
( wn wne wceq con3d nne syl6ib ) ABFCDGZFCDHALBEICDJK $.
|
|
$}
|
|
|
|
${
|
|
necon1d.1 $e |- ( ph -> ( A =/= B -> C = D ) ) $.
|
|
$( Contrapositive law deduction for inequality. (Contributed by NM,
|
|
28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
necon1d $p |- ( ph -> ( C =/= D -> A = B ) ) $=
|
|
( wne wceq wn nne syl6ibr necon4ad ) ADEGZBCABCGDEHMIFDEJKL $.
|
|
$}
|
|
|
|
${
|
|
neneqad.1 $e |- ( ph -> -. A = B ) $.
|
|
$( If it is not the case that two classes are equal, they are unequal.
|
|
Converse of ~ neneqd . One-way deduction form of ~ df-ne .
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
neneqad $p |- ( ph -> A =/= B ) $=
|
|
( wceq con2i necon2ai ) ABCABCEDFG $.
|
|
$}
|
|
|
|
$( Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.) $)
|
|
nebi $p |- ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) ) $=
|
|
( wceq wb wne id necon3bid necon4bid impbii ) ABECDEFZABGCDGFZLABCDLHIMABCD
|
|
MHJK $.
|
|
|
|
$( Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew
|
|
Salmon, 3-Jun-2011.) $)
|
|
pm13.18 $p |- ( ( A = B /\ A =/= C ) -> B =/= C ) $=
|
|
( wceq wne eqeq1 biimprd necon3d imp ) ABDZACEBCEJBCACJACDBCDABCFGHI $.
|
|
|
|
$( Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew
|
|
Salmon, 3-Jun-2011.) $)
|
|
pm13.181 $p |- ( ( A = B /\ B =/= C ) -> A =/= C ) $=
|
|
( wceq wne eqcom pm13.18 sylanb ) ABDBADBCEACEABFBACGH $.
|
|
|
|
${
|
|
pm2.21ddne.1 $e |- ( ph -> A = B ) $.
|
|
pm2.21ddne.2 $e |- ( ph -> A =/= B ) $.
|
|
$( A contradiction implies anything. Equality/inequality deduction form.
|
|
(Contributed by David Moews, 28-Feb-2017.) $)
|
|
pm2.21ddne $p |- ( ph -> ps ) $=
|
|
( wceq neneqd pm2.21dd ) ACDGBEACDFHI $.
|
|
$}
|
|
|
|
${
|
|
pm2.61ne.1 $e |- ( A = B -> ( ps <-> ch ) ) $.
|
|
pm2.61ne.2 $e |- ( ( ph /\ A =/= B ) -> ps ) $.
|
|
pm2.61ne.3 $e |- ( ph -> ch ) $.
|
|
$( Deduction eliminating an inequality in an antecedent. (Contributed by
|
|
NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
pm2.61ne $p |- ( ph -> ps ) $=
|
|
( wne wi expcom wn wceq nne syl5ibr sylbi pm2.61i ) DEIZABJZARBGKRLDEMZSD
|
|
ENABTCHFOPQ $.
|
|
$}
|
|
|
|
${
|
|
pm2.61ine.1 $e |- ( A = B -> ph ) $.
|
|
pm2.61ine.2 $e |- ( A =/= B -> ph ) $.
|
|
$( Inference eliminating an inequality in an antecedent. (Contributed by
|
|
NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
pm2.61ine $p |- ph $=
|
|
( wne wn wceq nne sylbi pm2.61i ) BCFZAELGBCHABCIDJK $.
|
|
$}
|
|
|
|
${
|
|
pm2.61dne.1 $e |- ( ph -> ( A = B -> ps ) ) $.
|
|
pm2.61dne.2 $e |- ( ph -> ( A =/= B -> ps ) ) $.
|
|
$( Deduction eliminating an inequality in an antecedent. (Contributed by
|
|
NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) $)
|
|
pm2.61dne $p |- ( ph -> ps ) $=
|
|
( wne wn wceq nne syl5bi pm2.61d ) ACDGZBFMHCDIABCDJEKL $.
|
|
$}
|
|
|
|
${
|
|
pm2.61dane.1 $e |- ( ( ph /\ A = B ) -> ps ) $.
|
|
pm2.61dane.2 $e |- ( ( ph /\ A =/= B ) -> ps ) $.
|
|
$( Deduction eliminating an inequality in an antecedent. (Contributed by
|
|
NM, 30-Nov-2011.) $)
|
|
pm2.61dane $p |- ( ph -> ps ) $=
|
|
( wceq ex wne pm2.61dne ) ABCDACDGBEHACDIBFHJ $.
|
|
$}
|
|
|
|
${
|
|
pm2.61da2ne.1 $e |- ( ( ph /\ A = B ) -> ps ) $.
|
|
pm2.61da2ne.2 $e |- ( ( ph /\ C = D ) -> ps ) $.
|
|
pm2.61da2ne.3 $e |- ( ( ph /\ ( A =/= B /\ C =/= D ) ) -> ps ) $.
|
|
$( Deduction eliminating two inequalities in an antecedent. (Contributed
|
|
by NM, 29-May-2013.) $)
|
|
pm2.61da2ne $p |- ( ph -> ps ) $=
|
|
( wne wa wceq adantlr anassrs pm2.61dane ) ABCDGACDJZKBEFAEFLBPHMAPEFJBIN
|
|
OO $.
|
|
$}
|
|
|
|
${
|
|
pm2.61da3ne.1 $e |- ( ( ph /\ A = B ) -> ps ) $.
|
|
pm2.61da3ne.2 $e |- ( ( ph /\ C = D ) -> ps ) $.
|
|
pm2.61da3ne.3 $e |- ( ( ph /\ E = F ) -> ps ) $.
|
|
pm2.61da3ne.4 $e |- ( ( ph /\ ( A =/= B /\ C =/= D /\ E =/= F ) )
|
|
-> ps ) $.
|
|
$( Deduction eliminating three inequalities in an antecedent. (Contributed
|
|
by NM, 15-Jun-2013.) $)
|
|
pm2.61da3ne $p |- ( ph -> ps ) $=
|
|
( wne wa wceq adantlr simpll simplrl simplrr simpr pm2.61dane pm2.61da2ne
|
|
syl13anc ) ABCDEFIJACDMZEFMZNZNZBGHAGHOBUFKPUGGHMZNAUDUEUHBAUFUHQAUDUEUHR
|
|
AUDUEUHSUGUHTLUCUAUB $.
|
|
$}
|
|
|
|
$( Commutation of inequality. (Contributed by NM, 14-May-1999.) $)
|
|
necom $p |- ( A =/= B <-> B =/= A ) $=
|
|
( eqcom necon3bii ) ABBAABCD $.
|
|
|
|
${
|
|
necomi.1 $e |- A =/= B $.
|
|
$( Inference from commutative law for inequality. (Contributed by NM,
|
|
17-Oct-2012.) $)
|
|
necomi $p |- B =/= A $=
|
|
( wne necom mpbi ) ABDBADCABEF $.
|
|
$}
|
|
|
|
${
|
|
necomd.1 $e |- ( ph -> A =/= B ) $.
|
|
$( Deduction from commutative law for inequality. (Contributed by NM,
|
|
12-Feb-2008.) $)
|
|
necomd $p |- ( ph -> B =/= A ) $=
|
|
( wne necom sylib ) ABCECBEDBCFG $.
|
|
$}
|
|
|
|
$( Logical OR with an equality. (Contributed by NM, 29-Apr-2007.) $)
|
|
neor $p |- ( ( A = B \/ ps ) <-> ( A =/= B -> ps ) ) $=
|
|
( wceq wo wn wi wne df-or df-ne imbi1i bitr4i ) BCDZAEMFZAGBCHZAGMAIONABCJK
|
|
L $.
|
|
|
|
$( A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.) $)
|
|
neanior $p |- ( ( A =/= B /\ C =/= D ) <-> -. ( A = B \/ C = D ) ) $=
|
|
( wne wa wceq wn wo df-ne anbi12i pm4.56 bitri ) ABEZCDEZFABGZHZCDGZHZFPRIH
|
|
NQOSABJCDJKPRLM $.
|
|
|
|
$( A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) $)
|
|
ne3anior $p |- ( ( A =/= B /\ C =/= D /\ E =/= F )
|
|
<-> -. ( A = B \/ C = D \/ E = F ) ) $=
|
|
( wne w3a wn w3o wceq 3anor nne 3orbi123i xchbinx ) ABGZCDGZEFGZHPIZQIZRIZJ
|
|
ABKZCDKZEFKZJPQRLSUBTUCUAUDABMCDMEFMNO $.
|
|
|
|
$( A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.) $)
|
|
neorian $p |- ( ( A =/= B \/ C =/= D ) <-> -. ( A = B /\ C = D ) ) $=
|
|
( wne wo wceq wn wa df-ne orbi12i ianor bitr4i ) ABEZCDEZFABGZHZCDGZHZFPRIH
|
|
NQOSABJCDJKPRLM $.
|
|
|
|
${
|
|
nemtbir.1 $e |- A =/= B $.
|
|
nemtbir.2 $e |- ( ph <-> A = B ) $.
|
|
$( An inference from an inequality, related to modus tollens. (Contributed
|
|
by NM, 13-Apr-2007.) $)
|
|
nemtbir $p |- -. ph $=
|
|
( wceq wne wn df-ne mpbi mtbir ) ABCFZBCGLHDBCIJEK $.
|
|
$}
|
|
|
|
$( Two classes are different if they don't contain the same element.
|
|
(Contributed by NM, 3-Feb-2012.) $)
|
|
nelne1 $p |- ( ( A e. B /\ -. A e. C ) -> B =/= C ) $=
|
|
( wcel wn wne wceq eleq2 biimpcd necon3bd imp ) ABDZACDZEBCFLMBCBCGLMBCAHIJ
|
|
K $.
|
|
|
|
$( Two classes are different if they don't belong to the same class.
|
|
(Contributed by NM, 25-Jun-2012.) $)
|
|
nelne2 $p |- ( ( A e. C /\ -. B e. C ) -> A =/= B ) $=
|
|
( wcel wn wne wceq eleq1 biimpcd necon3bd imp ) ACDZBCDZEABFLMABABGLMABCHIJ
|
|
K $.
|
|
|
|
$( Equality theorem for negated membership. (Contributed by NM,
|
|
20-Nov-1994.) $)
|
|
neleq1 $p |- ( A = B -> ( A e/ C <-> B e/ C ) ) $=
|
|
( wceq wcel wn wnel eleq1 notbid df-nel 3bitr4g ) ABDZACEZFBCEZFACGBCGLMNAB
|
|
CHIACJBCJK $.
|
|
|
|
$( Equality theorem for negated membership. (Contributed by NM,
|
|
20-Nov-1994.) $)
|
|
neleq2 $p |- ( A = B -> ( C e/ A <-> C e/ B ) ) $=
|
|
( wceq wcel wn wnel eleq2 notbid df-nel 3bitr4g ) ABDZCAEZFCBEZFCAGCBGLMNAB
|
|
CHICAJCBJK $.
|
|
|
|
${
|
|
neleq12d.1 $e |- ( ph -> A = B ) $.
|
|
neleq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality theorem for negated membership. (Contributed by FL,
|
|
10-Aug-2016.) $)
|
|
neleq12d $p |- ( ph -> ( A e/ C <-> B e/ D ) ) $=
|
|
( wnel wceq wb neleq1 syl neleq2 bitrd ) ABDHZCDHZCEHZABCIOPJFBCDKLADEIPQ
|
|
JGDECMLN $.
|
|
$}
|
|
|
|
${
|
|
nfne.1 $e |- F/_ x A $.
|
|
nfne.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for inequality. (Contributed by NM,
|
|
10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfne $p |- F/ x A =/= B $=
|
|
( wne wceq wn df-ne nfeq nfn nfxfr ) BCFBCGZHABCIMAABCDEJKL $.
|
|
$}
|
|
|
|
${
|
|
nfnel.1 $e |- F/_ x A $.
|
|
nfnel.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for inequality. (Contributed by David
|
|
Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfnel $p |- F/ x A e/ B $=
|
|
( wnel wcel wn df-nel nfel nfn nfxfr ) BCFBCGZHABCIMAABCDEJKL $.
|
|
$}
|
|
|
|
${
|
|
nfned.1 $e |- ( ph -> F/_ x A ) $.
|
|
nfned.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( Bound-variable hypothesis builder for inequality. (Contributed by NM,
|
|
10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfned $p |- ( ph -> F/ x A =/= B ) $=
|
|
( wne wceq wn df-ne nfeqd nfnd nfxfrd ) CDGCDHZIABCDJANBABCDEFKLM $.
|
|
$}
|
|
|
|
${
|
|
nfneld.1 $e |- ( ph -> F/_ x A ) $.
|
|
nfneld.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( Bound-variable hypothesis builder for inequality. (Contributed by David
|
|
Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfneld $p |- ( ph -> F/ x A e/ B ) $=
|
|
( wnel wcel wn df-nel nfeld nfnd nfxfrd ) CDGCDHZIABCDJANBABCDEFKLM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Restricted quantification
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend wff notation to include restricted universal quantification. $)
|
|
wral $a wff A. x e. A ph $.
|
|
|
|
$( Extend wff notation to include restricted existential quantification. $)
|
|
wrex $a wff E. x e. A ph $.
|
|
|
|
$( Extend wff notation to include restricted existential uniqueness. $)
|
|
wreu $a wff E! x e. A ph $.
|
|
|
|
$( Extend wff notation to include restricted "at most one." $)
|
|
wrmo $a wff E* x e. A ph $.
|
|
|
|
$( Extend class notation to include the restricted class abstraction (class
|
|
builder). $)
|
|
crab $a class { x e. A | ph } $.
|
|
|
|
$( Define restricted universal quantification. Special case of Definition
|
|
4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.) $)
|
|
df-ral $a |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) ) $.
|
|
|
|
$( Define restricted existential quantification. Special case of Definition
|
|
4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.) $)
|
|
df-rex $a |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) ) $.
|
|
|
|
$( Define restricted existential uniqueness. (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
df-reu $a |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) ) $.
|
|
|
|
$( Define restricted "at most one". (Contributed by NM, 16-Jun-2017.) $)
|
|
df-rmo $a |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) ) $.
|
|
|
|
$( Define a restricted class abstraction (class builder), which is the class
|
|
of all ` x ` in ` A ` such that ` ph ` is true. Definition of
|
|
[TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.) $)
|
|
df-rab $a |- { x e. A | ph } = { x | ( x e. A /\ ph ) } $.
|
|
|
|
$( Relationship between restricted universal and existential quantifiers.
|
|
(Contributed by NM, 21-Jan-1997.) $)
|
|
ralnex $p |- ( A. x e. A -. ph <-> -. E. x e. A ph ) $=
|
|
( wn wral cv wcel wi wal wrex df-ral wa wex alinexa df-rex xchbinxr bitri )
|
|
ADZBCEBFCGZRHBIZABCJZDRBCKTSALBMUASABNABCOPQ $.
|
|
|
|
$( Relationship between restricted universal and existential quantifiers.
|
|
(Contributed by NM, 21-Jan-1997.) $)
|
|
rexnal $p |- ( E. x e. A -. ph <-> -. A. x e. A ph ) $=
|
|
( wn wrex cv wcel wa wex wral df-rex wi wal exanali df-ral xchbinxr bitri )
|
|
ADZBCEBFCGZRHBIZABCJZDRBCKTSALBMUASABNABCOPQ $.
|
|
|
|
$( Relationship between restricted universal and existential quantifiers.
|
|
(Contributed by NM, 21-Jan-1997.) $)
|
|
dfral2 $p |- ( A. x e. A ph <-> -. E. x e. A -. ph ) $=
|
|
( wn wrex wral rexnal con2bii ) ADBCEABCFABCGH $.
|
|
|
|
$( Relationship between restricted universal and existential quantifiers.
|
|
(Contributed by NM, 21-Jan-1997.) $)
|
|
dfrex2 $p |- ( E. x e. A ph <-> -. A. x e. A -. ph ) $=
|
|
( wn wral wrex ralnex con2bii ) ADBCEABCFABCGH $.
|
|
|
|
${
|
|
ralbida.1 $e |- F/ x ph $.
|
|
ralbida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 6-Oct-2003.) $)
|
|
ralbida $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
|
|
( cv wcel wi wal wral pm5.74da albid df-ral 3bitr4g ) ADHEIZBJZDKQCJZDKBD
|
|
ELCDELARSDFAQBCGMNBDEOCDEOP $.
|
|
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 6-Oct-2003.) $)
|
|
rexbida $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
|
|
( cv wcel wa wex wrex pm5.32da exbid df-rex 3bitr4g ) ADHEIZBJZDKQCJZDKBD
|
|
ELCDELARSDFAQBCGMNBDEOCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralbidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 4-Mar-1997.) $)
|
|
ralbidva $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
|
|
( nfv ralbida ) ABCDEADGFH $.
|
|
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 9-Mar-1997.) $)
|
|
rexbidva $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
|
|
( nfv rexbida ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
ralbid.1 $e |- F/ x ph $.
|
|
ralbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 27-Jun-1998.) $)
|
|
ralbid $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
|
|
( wb cv wcel adantr ralbida ) ABCDEFABCHDIEJGKL $.
|
|
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 27-Jun-1998.) $)
|
|
rexbid $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
|
|
( wb cv wcel adantr rexbida ) ABCDEFABCHDIEJGKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 20-Nov-1994.) $)
|
|
ralbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
|
|
( nfv ralbid ) ABCDEADGFH $.
|
|
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 20-Nov-1994.) $)
|
|
rexbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
|
|
( nfv rexbid ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralbidv2.1 $e |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 6-Apr-1997.) $)
|
|
ralbidv2 $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
|
|
( cv wcel wi wal wral albidv df-ral 3bitr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
|
|
AQRDGMBDENCDFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rexbidv2.1 $e |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 22-May-1999.) $)
|
|
rexbidv2 $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
|
|
( cv wcel wa wex wrex exbidv df-rex 3bitr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
|
|
AQRDGMBDENCDFNO $.
|
|
$}
|
|
|
|
${
|
|
ralbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Inference adding restricted universal quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario
|
|
Carneiro, 17-Oct-2016.) $)
|
|
ralbii $p |- ( A. x e. A ph <-> A. x e. A ps ) $=
|
|
( wral wb wtru a1i ralbidv trud ) ACDFBCDFGHABCDABGHEIJK $.
|
|
|
|
$( Inference adding restricted existential quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario
|
|
Carneiro, 17-Oct-2016.) $)
|
|
rexbii $p |- ( E. x e. A ph <-> E. x e. A ps ) $=
|
|
( wrex wb wtru a1i rexbidv trud ) ACDFBCDFGHABCDABGHEIJK $.
|
|
|
|
$( Inference adding two restricted universal quantifiers to both sides of
|
|
an equivalence. (Contributed by NM, 1-Aug-2004.) $)
|
|
2ralbii $p |- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps ) $=
|
|
( wral ralbii ) ADFHBDFHCEABDFGII $.
|
|
|
|
$( Inference adding two restricted existential quantifiers to both sides of
|
|
an equivalence. (Contributed by NM, 11-Nov-1995.) $)
|
|
2rexbii $p |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps ) $=
|
|
( wrex rexbii ) ADFHBDFHCEABDFGII $.
|
|
$}
|
|
|
|
${
|
|
ralbii2.1 $e |- ( ( x e. A -> ph ) <-> ( x e. B -> ps ) ) $.
|
|
$( Inference adding different restricted universal quantifiers to each side
|
|
of an equivalence. (Contributed by NM, 15-Aug-2005.) $)
|
|
ralbii2 $p |- ( A. x e. A ph <-> A. x e. B ps ) $=
|
|
( cv wcel wi wal wral albii df-ral 3bitr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
|
|
CFLACDMBCEMN $.
|
|
$}
|
|
|
|
${
|
|
rexbii2.1 $e |- ( ( x e. A /\ ph ) <-> ( x e. B /\ ps ) ) $.
|
|
$( Inference adding different restricted existential quantifiers to each
|
|
side of an equivalence. (Contributed by NM, 4-Feb-2004.) $)
|
|
rexbii2 $p |- ( E. x e. A ph <-> E. x e. B ps ) $=
|
|
( cv wcel wa wex wrex exbii df-rex 3bitr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
|
|
CFLACDMBCEMN $.
|
|
$}
|
|
|
|
${
|
|
raleqbii.1 $e |- A = B $.
|
|
raleqbii.2 $e |- ( ps <-> ch ) $.
|
|
$( Equality deduction for restricted universal quantifier, changing both
|
|
formula and quantifier domain. Inference form. (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
raleqbii $p |- ( A. x e. A ps <-> A. x e. B ch ) $=
|
|
( cv wcel eleq2i imbi12i ralbii2 ) ABCDECHZDIMEIABDEMFJGKL $.
|
|
|
|
$( Equality deduction for restricted existential quantifier, changing both
|
|
formula and quantifier domain. Inference form. (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
rexeqbii $p |- ( E. x e. A ps <-> E. x e. B ch ) $=
|
|
( cv wcel eleq2i anbi12i rexbii2 ) ABCDECHZDIMEIABDEMFJGKL $.
|
|
$}
|
|
|
|
${
|
|
ralbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Inference adding restricted universal quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 26-Nov-2000.) $)
|
|
ralbiia $p |- ( A. x e. A ph <-> A. x e. A ps ) $=
|
|
( cv wcel pm5.74i ralbii2 ) ABCDDCFDGABEHI $.
|
|
|
|
$( Inference adding restricted existential quantifier to both sides of an
|
|
equivalence. (Contributed by NM, 26-Oct-1999.) $)
|
|
rexbiia $p |- ( E. x e. A ph <-> E. x e. A ps ) $=
|
|
( cv wcel pm5.32i rexbii2 ) ABCDDCFDGABEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
2rexbiia.1 $e |- ( ( x e. A /\ y e. B ) -> ( ph <-> ps ) ) $.
|
|
$( Inference adding two restricted existential quantifiers to both sides of
|
|
an equivalence. (Contributed by NM, 1-Aug-2004.) $)
|
|
2rexbiia $p |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps ) $=
|
|
( wrex cv wcel rexbidva rexbiia ) ADFHBDFHCECIEJABDFGKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
r2alf.1 $e |- F/_ y A $.
|
|
$( Double restricted universal quantification. (Contributed by Mario
|
|
Carneiro, 14-Oct-2016.) $)
|
|
r2alf $p |- ( A. x e. A A. y e. B ph <->
|
|
A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
|
|
( wral cv wi wal wa df-ral nfcri 19.21 impexp albii imbi2i 3bitr4i bitr4i
|
|
wcel ) ACEGZBDGBHDTZUAIZBJUBCHETZKAIZCJZBJUABDLUFUCBUBUDAIZIZCJUBUGCJZIUF
|
|
UCUBUGCCBDFMNUEUHCUBUDAOPUAUIUBACELQRPS $.
|
|
|
|
$( Double restricted existential quantification. (Contributed by Mario
|
|
Carneiro, 14-Oct-2016.) $)
|
|
r2exf $p |- ( E. x e. A E. y e. B ph <->
|
|
E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
|
|
( wrex cv wcel wex df-rex nfcri 19.42 anass exbii anbi2i 3bitr4i bitr4i
|
|
wa ) ACEGZBDGBHDIZTSZBJUACHEIZSASZCJZBJTBDKUEUBBUAUCASZSZCJUAUFCJZSUEUBUA
|
|
UFCCBDFLMUDUGCUAUCANOTUHUAACEKPQOR $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Double restricted universal quantification. (Contributed by NM,
|
|
19-Nov-1995.) $)
|
|
r2al $p |- ( A. x e. A A. y e. B ph <->
|
|
A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
|
|
( nfcv r2alf ) ABCDECDFG $.
|
|
|
|
$( Double restricted existential quantification. (Contributed by NM,
|
|
11-Nov-1995.) $)
|
|
r2ex $p |- ( E. x e. A E. y e. B ph <->
|
|
E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
|
|
( nfcv r2exf ) ABCDECDFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
2ralbida.1 $e |- F/ x ph $.
|
|
2ralbida.2 $e |- F/ y ph $.
|
|
2ralbida.3 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifier (deduction
|
|
rule). (Contributed by NM, 24-Feb-2004.) $)
|
|
2ralbida $p |- ( ph ->
|
|
( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
|
|
( wral cv wcel wa nfv nfan wb anassrs ralbida ) ABEGKCEGKDFHADLFMZNBCEGAT
|
|
EITEOPATELGMBCQJRSS $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d y A $.
|
|
2ralbidva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifiers (deduction
|
|
rule). (Contributed by NM, 4-Mar-1997.) $)
|
|
2ralbidva $p |- ( ph ->
|
|
( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
|
|
( nfv 2ralbida ) ABCDEFGADIAEIHJ $.
|
|
|
|
$( Formula-building rule for restricted existential quantifiers (deduction
|
|
rule). (Contributed by NM, 15-Dec-2004.) $)
|
|
2rexbidva $p |- ( ph ->
|
|
( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) ) $=
|
|
( wrex cv wcel wa wb anassrs rexbidva ) ABEGICEGIDFADJFKZLBCEGAPEJGKBCMHN
|
|
OO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
2ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted universal quantifiers (deduction
|
|
rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon
|
|
Jaroszewicz, 16-Mar-2007.) $)
|
|
2ralbidv $p |- ( ph ->
|
|
( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
|
|
( wral ralbidv ) ABEGICEGIDFABCEGHJJ $.
|
|
|
|
$( Formula-building rule for restricted existential quantifiers (deduction
|
|
rule). (Contributed by NM, 28-Jan-2006.) $)
|
|
2rexbidv $p |- ( ph ->
|
|
( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) ) $=
|
|
( wrex rexbidv ) ABEGICEGIDFABCEGHJJ $.
|
|
|
|
$( Formula-building rule for restricted quantifiers (deduction rule).
|
|
(Contributed by NM, 28-Jan-2006.) $)
|
|
rexralbidv $p |- ( ph ->
|
|
( E. x e. A A. y e. B ps <-> E. x e. A A. y e. B ch ) ) $=
|
|
( wral ralbidv rexbidv ) ABEGICEGIDFABCEGHJK $.
|
|
$}
|
|
|
|
$( A transformation of restricted quantifiers and logical connectives.
|
|
(Contributed by NM, 4-Sep-2005.) $)
|
|
ralinexa $p |- ( A. x e. A ( ph -> -. ps ) <-> -. E. x e. A ( ph /\ ps ) ) $=
|
|
( wn wi wral wa wrex imnan ralbii ralnex bitri ) ABEFZCDGABHZEZCDGOCDIENPCD
|
|
ABJKOCDLM $.
|
|
|
|
$( A transformation of restricted quantifiers and logical connectives.
|
|
(Contributed by NM, 4-Sep-2005.) $)
|
|
rexanali $p |- ( E. x e. A ( ph /\ -. ps ) <-> -. A. x e. A ( ph -> ps ) ) $=
|
|
( wn wa wrex wi wral annim rexbii rexnal bitri ) ABEFZCDGABHZEZCDGOCDIENPCD
|
|
ABJKOCDLM $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Two ways to say " ` A ` belongs to ` B ` ." (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
risset $p |- ( A e. B <-> E. x e. B x = A ) $=
|
|
( cv wcel wceq wa wex wrex exancom df-rex df-clel 3bitr4ri ) ADZCEZNBFZGA
|
|
HPOGAHPACIBCEOPAJPACKABCLM $.
|
|
$}
|
|
|
|
${
|
|
hbral.1 $e |- ( y e. A -> A. x y e. A ) $.
|
|
hbral.2 $e |- ( ph -> A. x ph ) $.
|
|
$( Bound-variable hypothesis builder for restricted quantification.
|
|
(Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy,
|
|
13-Dec-2009.) $)
|
|
hbral $p |- ( A. y e. A ph -> A. x A. y e. A ph ) $=
|
|
( wral cv wcel wi wal df-ral hbim hbal hbxfrbi ) ACDGCHDIZAJZCKBACDLQBCPA
|
|
BEFMNO $.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` A. x e. A ph ` . (Contributed by NM,
|
|
18-Oct-1996.) $)
|
|
hbra1 $p |- ( A. x e. A ph -> A. x A. x e. A ph ) $=
|
|
( wral cv wcel wi wal df-ral hba1 hbxfrbi ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
$( ` x ` is not free in ` A. x e. A ph ` . (Contributed by NM,
|
|
18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfra1 $p |- F/ x A. x e. A ph $=
|
|
( wral cv wcel wi wal df-ral nfa1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
${
|
|
nfrald.2 $e |- F/ y ph $.
|
|
nfrald.3 $e |- ( ph -> F/_ x A ) $.
|
|
nfrald.4 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfral . (Contributed by NM, 15-Feb-2013.)
|
|
(Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfrald $p |- ( ph -> F/ x A. y e. A ps ) $=
|
|
( wral cv wcel wi wal df-ral wceq wn wa wnfc nfcvf adantr nfeld wnf nfimd
|
|
adantl nfald2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACJUGOCMPZQZUHBCUKCUGEU
|
|
JCUGRACDSUDACERUJGTUAABCUBUJHTUCUEUF $.
|
|
|
|
$( Deduction version of ~ nfrex . (Contributed by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
nfrexd $p |- ( ph -> F/ x E. y e. A ps ) $=
|
|
( wrex wn wral dfrex2 nfnd nfrald nfxfrd ) BDEIBJZDEKZJACBDELAQCAPCDEFGAB
|
|
CHMNMO $.
|
|
$}
|
|
|
|
${
|
|
nfral.1 $e |- F/_ x A $.
|
|
nfral.2 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for restricted quantification.
|
|
(Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfral $p |- F/ x A. y e. A ph $=
|
|
( wral wnf wtru nftru wnfc a1i nfrald trud ) ACDGBHIABCDCJBDKIELABHIFLMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A y $.
|
|
$( Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the
|
|
antecedent is restricted. Derived automatically from hbra2VD in
|
|
set.mm. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM,
|
|
31-Dec-2011.) $)
|
|
nfra2 $p |- F/ y A. x e. A A. y e. B ph $=
|
|
( wral nfcv nfra1 nfral ) ACEFCBDCDGACEHI $.
|
|
$}
|
|
|
|
${
|
|
nfrex.1 $e |- F/_ x A $.
|
|
nfrex.2 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for restricted quantification.
|
|
(Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro,
|
|
7-Oct-2016.) $)
|
|
nfrex $p |- F/ x E. y e. A ph $=
|
|
( wrex wn wral dfrex2 nfn nfral nfxfr ) ACDGAHZCDIZHBACDJOBNBCDEABFKLKM
|
|
$.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` E. x e. A ph ` . (Contributed by NM,
|
|
19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.) $)
|
|
nfre1 $p |- F/ x E. x e. A ph $=
|
|
( wrex cv wcel wa wex df-rex nfe1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
${
|
|
$d x y z $. $d y z A $. $d z B $.
|
|
$( Triple restricted universal quantification. (Contributed by NM,
|
|
19-Nov-1995.) $)
|
|
r3al $p |- ( A. x e. A A. y e. B A. z e. C ph <->
|
|
A. x A. y A. z ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) ) $=
|
|
( cv wcel wa wi wal wral w3a df-ral r2al ralbii bitri albii 19.21v 3anass
|
|
imbi1i impexp 3bitr4i ) CHFIZDHGIZJZAKZDLZCLZBEMBHEIZUJKZBLADGMCFMZBEMUKU
|
|
EUFNZAKZDLZCLZBLUJBEOUMUJBEACDFGPQUQULBUQUKUIKZCLULUPURCUPUKUHKZDLURUOUSD
|
|
UOUKUGJZAKUSUNUTAUKUEUFUAUBUKUGAUCRSUKUHDTRSUKUICTRSUD $.
|
|
$}
|
|
|
|
$( Universal quantification implies restricted quantification. (Contributed
|
|
by NM, 20-Oct-2006.) $)
|
|
alral $p |- ( A. x ph -> A. x e. A ph ) $=
|
|
( wal cv wcel wi wral ax-1 alimi df-ral sylibr ) ABDBECFZAGZBDABCHANBAMIJAB
|
|
CKL $.
|
|
|
|
$( Restricted existence implies existence. (Contributed by NM,
|
|
11-Nov-1995.) $)
|
|
rexex $p |- ( E. x e. A ph -> E. x ph ) $=
|
|
( wrex cv wcel wa wex df-rex simpr eximi sylbi ) ABCDBECFZAGZBHABHABCINABMA
|
|
JKL $.
|
|
|
|
$( Restricted specialization. (Contributed by NM, 17-Oct-1996.) $)
|
|
rsp $p |- ( A. x e. A ph -> ( x e. A -> ph ) ) $=
|
|
( wral cv wcel wi wal df-ral sp sylbi ) ABCDBECFAGZBHLABCILBJK $.
|
|
|
|
$( Restricted specialization. (Contributed by NM, 12-Oct-1999.) $)
|
|
rspe $p |- ( ( x e. A /\ ph ) -> E. x e. A ph ) $=
|
|
( cv wcel wa wex wrex 19.8a df-rex sylibr ) BDCEAFZLBGABCHLBIABCJK $.
|
|
|
|
$( Restricted specialization. (Contributed by NM, 11-Feb-1997.) $)
|
|
rsp2 $p |- ( A. x e. A A. y e. B ph -> ( ( x e. A /\ y e. B ) -> ph ) ) $=
|
|
( wral cv wcel wi rsp syl6 imp3a ) ACEFZBDFZBGDHZCGEHZANOMPAIMBDJACEJKL $.
|
|
|
|
$( Restricted specialization. (Contributed by FL, 4-Jun-2012.) $)
|
|
rsp2e $p |- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph ) $=
|
|
( cv wcel w3a wrex wa wex simp1 rspe 3adant1 19.8a syl2anc df-rex sylibr )
|
|
BFDGZCFEGZAHZSACEIZJZBKZUBBDIUASUBUDSTALTAUBSACEMNUCBOPUBBDQR $.
|
|
|
|
${
|
|
rspec.1 $e |- A. x e. A ph $.
|
|
$( Specialization rule for restricted quantification. (Contributed by NM,
|
|
19-Nov-1994.) $)
|
|
rspec $p |- ( x e. A -> ph ) $=
|
|
( wral cv wcel wi rsp ax-mp ) ABCEBFCGAHDABCIJ $.
|
|
$}
|
|
|
|
${
|
|
rgen.1 $e |- ( x e. A -> ph ) $.
|
|
$( Generalization rule for restricted quantification. (Contributed by NM,
|
|
19-Nov-1994.) $)
|
|
rgen $p |- A. x e. A ph $=
|
|
( wral cv wcel wi df-ral mpgbir ) ABCEBFCGAHBABCIDJ $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z $.
|
|
rgen2a.1 $e |- ( ( x e. A /\ y e. A ) -> ph ) $.
|
|
$( Generalization rule for restricted quantification. Note that ` x ` and
|
|
` y ` needn't be distinct (and illustrates the use of ~ dvelim ).
|
|
(Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon,
|
|
25-May-2011.) (Proof modification is discouraged. $)
|
|
rgen2a $p |- A. x e. A A. y e. A ph $=
|
|
( vz wral cv wcel wi wal wceq eleq1 ex syl6bi pm2.43d alimi a1d dvelimv
|
|
wn syl6 pm2.61i df-ral sylibr rgen ) ACDGZBDBHZDIZCHZDIZAJZCKZUFUIUGLZCKZ
|
|
UHULJUNULUHUMUKCUMUJAUMUJUHUKUIUGDMUHUJAENZOPQRUNTUHUHCKULFHZDIUHCBFUPUGD
|
|
MSUHUKCUOQUAUBACDUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
rgenw.1 $e |- ph $.
|
|
$( Generalization rule for restricted quantification. (Contributed by NM,
|
|
18-Jun-2014.) $)
|
|
rgenw $p |- A. x e. A ph $=
|
|
( cv wcel a1i rgen ) ABCABECFDGH $.
|
|
|
|
$( Generalization rule for restricted quantification. Note that ` x ` and
|
|
` y ` needn't be distinct. (Contributed by NM, 18-Jun-2014.) $)
|
|
rgen2w $p |- A. x e. A A. y e. B ph $=
|
|
( wral rgenw ) ACEGBDACEFHH $.
|
|
$}
|
|
|
|
${
|
|
mprg.1 $e |- ( A. x e. A ph -> ps ) $.
|
|
mprg.2 $e |- ( x e. A -> ph ) $.
|
|
$( Modus ponens combined with restricted generalization. (Contributed by
|
|
NM, 10-Aug-2004.) $)
|
|
mprg $p |- ps $=
|
|
( wral rgen ax-mp ) ACDGBACDFHEI $.
|
|
$}
|
|
|
|
${
|
|
mprgbir.1 $e |- ( ph <-> A. x e. A ps ) $.
|
|
mprgbir.2 $e |- ( x e. A -> ps ) $.
|
|
$( Modus ponens on biconditional combined with restricted generalization.
|
|
(Contributed by NM, 21-Mar-2004.) $)
|
|
mprgbir $p |- ph $=
|
|
( wral rgen mpbir ) ABCDGBCDFHEI $.
|
|
$}
|
|
|
|
$( Distribution of restricted quantification over implication. (Contributed
|
|
by NM, 9-Feb-1997.) $)
|
|
ralim $p |- ( A. x e. A ( ph -> ps ) ->
|
|
( A. x e. A ph -> A. x e. A ps ) ) $=
|
|
( wi wral cv wcel wal df-ral ax-2 al2imi sylbi 3imtr4g ) ABEZCDFZCGDHZAEZCI
|
|
ZQBEZCIZACDFBCDFPQOEZCISUAEOCDJUBRTCQABKLMACDJBCDJN $.
|
|
|
|
${
|
|
ralimi2.1 $e |- ( ( x e. A -> ph ) -> ( x e. B -> ps ) ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 22-Feb-2004.) $)
|
|
ralimi2 $p |- ( A. x e. A ph -> A. x e. B ps ) $=
|
|
( cv wcel wi wal wral alimi df-ral 3imtr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
|
|
CFLACDMBCEMN $.
|
|
$}
|
|
|
|
${
|
|
ralimia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 19-Jul-1996.) $)
|
|
ralimia $p |- ( A. x e. A ph -> A. x e. A ps ) $=
|
|
( cv wcel a2i ralimi2 ) ABCDDCFDGABEHI $.
|
|
$}
|
|
|
|
${
|
|
ralimiaa.1 $e |- ( ( x e. A /\ ph ) -> ps ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 4-Aug-2007.) $)
|
|
ralimiaa $p |- ( A. x e. A ph -> A. x e. A ps ) $=
|
|
( cv wcel ex ralimia ) ABCDCFDGABEHI $.
|
|
$}
|
|
|
|
${
|
|
ralimi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference quantifying both antecedent and consequent, with strong
|
|
hypothesis. (Contributed by NM, 4-Mar-1997.) $)
|
|
ralimi $p |- ( A. x e. A ph -> A. x e. A ps ) $=
|
|
( wi cv wcel a1i ralimia ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
ral2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference quantifying antecedent, nested antecedent, and consequent,
|
|
with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) $)
|
|
ral2imi $p |- ( A. x e. A ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
|
|
( wral wi ralimi ralim syl ) ADEGBCHZDEGBDEGCDEGHALDEFIBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
ralimdaa.1 $e |- F/ x ph $.
|
|
ralimdaa.2 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) $)
|
|
ralimdaa $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
|
|
( cv wcel wi wal wral ex a2d alimd df-ral 3imtr4g ) ADHEIZBJZDKRCJZDKBDEL
|
|
CDELASTDFARBCARBCJGMNOBDEPCDEPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.) $)
|
|
ralimdva $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
|
|
( nfv ralimdaa ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.) $)
|
|
ralimdv $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
|
|
( wi cv wcel adantr ralimdva ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralimdv2.1 $e |- ( ph -> ( ( x e. A -> ps ) -> ( x e. B -> ch ) ) ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 1-Feb-2005.) $)
|
|
ralimdv2 $p |- ( ph -> ( A. x e. A ps -> A. x e. B ch ) ) $=
|
|
( cv wcel wi wal wral alimdv df-ral 3imtr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
|
|
AQRDGMBDENCDFNO $.
|
|
$}
|
|
|
|
${
|
|
ralrimi.1 $e |- F/ x ph $.
|
|
ralrimi.2 $e |- ( ph -> ( x e. A -> ps ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 10-Oct-1999.) $)
|
|
ralrimi $p |- ( ph -> A. x e. A ps ) $=
|
|
( cv wcel wi wal wral alrimi df-ral sylibr ) ACGDHBIZCJBCDKAOCEFLBCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralrimiv.1 $e |- ( ph -> ( x e. A -> ps ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 22-Nov-1994.) $)
|
|
ralrimiv $p |- ( ph -> A. x e. A ps ) $=
|
|
( nfv ralrimi ) ABCDACFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralrimiva.1 $e |- ( ( ph /\ x e. A ) -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 2-Jan-2006.) $)
|
|
ralrimiva $p |- ( ph -> A. x e. A ps ) $=
|
|
( cv wcel ex ralrimiv ) ABCDACFDGBEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ralrimivw.1 $e |- ( ph -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 18-Jun-2014.) $)
|
|
ralrimivw $p |- ( ph -> A. x e. A ps ) $=
|
|
( cv wcel a1d ralrimiv ) ABCDABCFDGEHI $.
|
|
$}
|
|
|
|
$( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed
|
|
theorem version). (Contributed by NM, 1-Mar-2008.) $)
|
|
r19.21t $p |- ( F/ x ph ->
|
|
( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) ) $=
|
|
( wnf cv wcel wi wal wral bi2.04 albii 19.21t syl5bb df-ral imbi2i 3bitr4g
|
|
) ACEZCFDGZABHZHZCIZASBHZCIZHZTCDJABCDJZHUBAUCHZCIRUEUAUGCSABKLAUCCMNTCDOUF
|
|
UDABCDOPQ $.
|
|
|
|
${
|
|
r19.21.1 $e |- F/ x ph $.
|
|
$( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by Scott Fenton, 30-Mar-2011.) $)
|
|
r19.21 $p |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) $=
|
|
( wnf wi wral wb r19.21t ax-mp ) ACFABGCDHABCDHGIEABCDJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.21v $p |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) $=
|
|
( nfv r19.21 ) ABCDACEF $.
|
|
$}
|
|
|
|
${
|
|
ralrimd.1 $e |- F/ x ph $.
|
|
ralrimd.2 $e |- F/ x ps $.
|
|
ralrimd.3 $e |- ( ph -> ( ps -> ( x e. A -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 16-Feb-2004.) $)
|
|
ralrimd $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
|
|
( cv wcel wi wal wral alrimd df-ral syl6ibr ) ABDIEJCKZDLCDEMABQDFGHNCDEO
|
|
P $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $.
|
|
ralrimdv.1 $e |- ( ph -> ( ps -> ( x e. A -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 27-May-1998.) $)
|
|
ralrimdv $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
|
|
( nfv ralrimd ) ABCDEADGBDGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $.
|
|
ralrimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 2-Feb-2008.) $)
|
|
ralrimdva $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
|
|
( cv wcel wi ex com23 ralrimdv ) ABCDEADGEHZBCAMBCIFJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d y A $.
|
|
ralrimivv.1 $e |- ( ph -> ( ( x e. A /\ y e. B ) -> ps ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with double quantification.) (Contributed by NM,
|
|
24-Jul-2004.) $)
|
|
ralrimivv $p |- ( ph -> A. x e. A A. y e. B ps ) $=
|
|
( wral cv wcel exp3a ralrimdv ralrimiv ) ABDFHCEACIEJZBDFANDIFJBGKLM $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y $. $d A y $.
|
|
ralrimivva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with double quantification.) (Contributed by Jeff
|
|
Madsen, 19-Jun-2011.) $)
|
|
ralrimivva $p |- ( ph -> A. x e. A A. y e. B ps ) $=
|
|
( cv wcel wa ex ralrimivv ) ABCDEFACHEIDHFIJBGKL $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z $. $d A y z $. $d B z $.
|
|
ralrimivvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B /\ z e. C ) ) -> ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with triple quantification.) (Contributed by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
ralrimivvva $p |- ( ph -> A. x e. A A. y e. B A. z e. C ps ) $=
|
|
( wral cv wcel wa 3exp2 imp41 ralrimiva ) ABEHJZDGJCFACKFLZMZQDGSDKGLZMBE
|
|
HARTEKHLZBARTUABINOPPP $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ps $. $d y A $.
|
|
ralrimdvv.1 $e |- ( ph -> ( ps -> ( ( x e. A /\ y e. B ) -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with double quantification.) (Contributed by NM,
|
|
1-Jun-2005.) $)
|
|
ralrimdvv $p |- ( ph -> ( ps -> A. x e. A A. y e. B ch ) ) $=
|
|
( wral wa cv wcel wi imp ralrimivv ex ) ABCEGIDFIABJCDEFGABDKFLEKGLJCMHNO
|
|
P $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ps $. $d y A $.
|
|
ralrimdvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version with double quantification.) (Contributed by NM,
|
|
2-Feb-2008.) $)
|
|
ralrimdvva $p |- ( ph -> ( ps -> A. x e. A A. y e. B ch ) ) $=
|
|
( cv wcel wa wi ex com23 ralrimdvv ) ABCDEFGADIFJEIGJKZBCAPBCLHMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
rgen2.1 $e |- ( ( x e. A /\ y e. B ) -> ph ) $.
|
|
$( Generalization rule for restricted quantification. (Contributed by NM,
|
|
30-May-1999.) $)
|
|
rgen2 $p |- A. x e. A A. y e. B ph $=
|
|
( wral cv wcel ralrimiva rgen ) ACEGBDBHDIACEFJK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d z B $. $d x y z $.
|
|
rgen3.1 $e |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) $.
|
|
$( Generalization rule for restricted quantification. (Contributed by NM,
|
|
12-Jan-2008.) $)
|
|
rgen3 $p |- A. x e. A A. y e. B A. z e. C ph $=
|
|
( wral cv wcel wa 3expa ralrimiva rgen2 ) ADGIBCEFBJEKZCJFKZLADGPQDJGKAHM
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
r19.21bi.1 $e |- ( ph -> A. x e. A ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 20-Nov-1994.) $)
|
|
r19.21bi $p |- ( ( ph /\ x e. A ) -> ps ) $=
|
|
( cv wcel wi wral wal df-ral sylib 19.21bi imp ) ACFDGZBAOBHZCABCDIPCJEBC
|
|
DKLMN $.
|
|
$}
|
|
|
|
${
|
|
rspec2.1 $e |- A. x e. A A. y e. B ph $.
|
|
$( Specialization rule for restricted quantification. (Contributed by NM,
|
|
20-Nov-1994.) $)
|
|
rspec2 $p |- ( ( x e. A /\ y e. B ) -> ph ) $=
|
|
( cv wcel wral rspec r19.21bi ) BGDHACEACEIBDFJK $.
|
|
$}
|
|
|
|
${
|
|
rspec3.1 $e |- A. x e. A A. y e. B A. z e. C ph $.
|
|
$( Specialization rule for restricted quantification. (Contributed by NM,
|
|
20-Nov-1994.) $)
|
|
rspec3 $p |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph ) $=
|
|
( cv wcel wa wral rspec2 r19.21bi 3impa ) BIEJZCIFJZDIGJAPQKADGADGLBCEFHM
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
r19.21be.1 $e |- ( ph -> A. x e. A ps ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 21-Nov-1994.) $)
|
|
r19.21be $p |- A. x e. A ( ph -> ps ) $=
|
|
( wi cv wcel r19.21bi expcom rgen ) ABFCDACGDHBABCDEIJK $.
|
|
$}
|
|
|
|
${
|
|
nrex.1 $e |- ( x e. A -> -. ps ) $.
|
|
$( Inference adding restricted existential quantifier to negated wff.
|
|
(Contributed by NM, 16-Oct-2003.) $)
|
|
nrex $p |- -. E. x e. A ps $=
|
|
( wn wral wrex rgen ralnex mpbi ) AEZBCFABCGEKBCDHABCIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
nrexdv.1 $e |- ( ( ph /\ x e. A ) -> -. ps ) $.
|
|
$( Deduction adding restricted existential quantifier to negated wff.
|
|
(Contributed by NM, 16-Oct-2003.) $)
|
|
nrexdv $p |- ( ph -> -. E. x e. A ps ) $=
|
|
( wn wral wrex ralrimiva ralnex sylib ) ABFZCDGBCDHFALCDEIBCDJK $.
|
|
$}
|
|
|
|
$( Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.)
|
|
(Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
rexim $p |- ( A. x e. A ( ph -> ps ) ->
|
|
( E. x e. A ph -> E. x e. A ps ) ) $=
|
|
( wi wral wn wrex con3 ral2imi con3d dfrex2 3imtr4g ) ABEZCDFZAGZCDFZGBGZCD
|
|
FZGACDHBCDHOSQNRPCDABIJKACDLBCDLM $.
|
|
|
|
${
|
|
reximia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 10-Feb-1997.) $)
|
|
reximia $p |- ( E. x e. A ph -> E. x e. A ps ) $=
|
|
( wi wrex rexim mprg ) ABFACDGBCDGFCDABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
reximi2.1 $e |- ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) ) $.
|
|
$( Inference quantifying both antecedent and consequent, based on Theorem
|
|
19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.) $)
|
|
reximi2 $p |- ( E. x e. A ph -> E. x e. B ps ) $=
|
|
( cv wcel wa wex wrex eximi df-rex 3imtr4i ) CGZDHAIZCJOEHBIZCJACDKBCEKPQ
|
|
CFLACDMBCEMN $.
|
|
$}
|
|
|
|
${
|
|
reximi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference quantifying both antecedent and consequent. (Contributed by
|
|
NM, 18-Oct-1996.) $)
|
|
reximi $p |- ( E. x e. A ph -> E. x e. A ps ) $=
|
|
( wi cv wcel a1i reximia ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
reximdai.1 $e |- F/ x ph $.
|
|
reximdai.2 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 31-Aug-1999.) $)
|
|
reximdai $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
|
|
( wi wral wrex ralrimi rexim syl ) ABCHZDEIBDEJCDEJHANDEFGKBCDELM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximdv2.1 $e |- ( ph -> ( ( x e. A /\ ps ) -> ( x e. B /\ ch ) ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.) $)
|
|
reximdv2 $p |- ( ph -> ( E. x e. A ps -> E. x e. B ch ) ) $=
|
|
( cv wcel wa wex wrex eximdv df-rex 3imtr4g ) ADHZEIBJZDKPFICJZDKBDELCDFL
|
|
AQRDGMBDENCDFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximdvai.1 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.) $)
|
|
reximdvai $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
|
|
( nfv reximdai ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted
|
|
quantifier version with strong hypothesis.) (Contributed by NM,
|
|
24-Jun-1998.) $)
|
|
reximdv $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
|
|
( wi cv wcel a1d reximdvai ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reximdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Deduction quantifying both antecedent and consequent, based on Theorem
|
|
19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.) $)
|
|
reximdva $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
|
|
( cv wcel wi ex reximdvai ) ABCDEADGEHBCIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d x B $.
|
|
$( Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers.
|
|
(Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.12 $p |- ( E. x e. A A. y e. B ph -> A. y e. B E. x e. A ph ) $=
|
|
( wral wrex nfcv nfra1 nfrex cv wcel ax-1 ralrimi rsp reximdv ralimia syl
|
|
com12 ) ACEFZBDGZUACEFABDGZCEFUAUACETCBDCDHACEIJUACKELZMNUAUBCEUCTABDTUCA
|
|
ACEOSPQR $.
|
|
$}
|
|
|
|
$( Closed theorem form of ~ r19.23 . (Contributed by NM, 4-Mar-2013.)
|
|
(Revised by Mario Carneiro, 8-Oct-2016.) $)
|
|
r19.23t $p |- ( F/ x ps ->
|
|
( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) ) $=
|
|
( wnf cv wcel wa wal wex wral wrex 19.23t df-ral impexp albii bitr4i df-rex
|
|
wi imbi1i 3bitr4g ) BCECFDGZAHZBSZCIZUCCJZBSABSZCDKZACDLZBSUCBCMUHUBUGSZCIU
|
|
EUGCDNUDUJCUBABOPQUIUFBACDRTUA $.
|
|
|
|
${
|
|
r19.23.1 $e |- F/ x ps $.
|
|
$( Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
r19.23 $p |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) $=
|
|
( wnf wi wral wrex wb r19.23t ax-mp ) BCFABGCDHACDIBGJEABCDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 31-Aug-1999.) $)
|
|
r19.23v $p |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) $=
|
|
( nfv r19.23 ) ABCDBCEF $.
|
|
$}
|
|
|
|
${
|
|
rexlimi.1 $e |- F/ x ps $.
|
|
rexlimi.2 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof
|
|
shortened by Andrew Salmon, 30-May-2011.) $)
|
|
rexlimi $p |- ( E. x e. A ph -> ps ) $=
|
|
( wi wral wrex rgen r19.23 mpbi ) ABGZCDHACDIBGMCDFJABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
rexlimiv.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 20-Nov-1994.) $)
|
|
rexlimiv $p |- ( E. x e. A ph -> ps ) $=
|
|
( nfv rexlimi ) ABCDBCFEG $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
rexlimiva.1 $e |- ( ( x e. A /\ ph ) -> ps ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 18-Dec-2006.) $)
|
|
rexlimiva $p |- ( E. x e. A ph -> ps ) $=
|
|
( cv wcel ex rexlimiv ) ABCDCFDGABEHI $.
|
|
$}
|
|
|
|
${
|
|
$d ps x $.
|
|
rexlimivw.1 $e |- ( ph -> ps ) $.
|
|
$( Weaker version of ~ rexlimiv . (Contributed by FL, 19-Sep-2011.) $)
|
|
rexlimivw $p |- ( E. x e. A ph -> ps ) $=
|
|
( wi cv wcel a1i rexlimiv ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
rexlimd.1 $e |- F/ x ph $.
|
|
rexlimd.2 $e |- F/ x ch $.
|
|
rexlimd.3 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew
|
|
Salmon, 30-May-2011.) $)
|
|
rexlimd $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( wi wral wrex ralrimi r19.23 sylib ) ABCIZDEJBDEKCIAODEFHLBCDEGMN $.
|
|
$}
|
|
|
|
${
|
|
rexlimd2.1 $e |- F/ x ph $.
|
|
rexlimd2.2 $e |- ( ph -> F/ x ch ) $.
|
|
rexlimd2.3 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Version of ~ rexlimd with deduction version of second hypothesis.
|
|
(Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
rexlimd2 $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( wi wral wrex ralrimi wnf wb r19.23t syl mpbid ) ABCIZDEJZBDEKCIZARDEFHL
|
|
ACDMSTNGBCDEOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdv.1 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric
|
|
Schmidt, 22-Dec-2006.) $)
|
|
rexlimdv $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( nfv rexlimd ) ABCDEADGCDGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdva.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 20-Jan-2007.) $)
|
|
rexlimdva $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( cv wcel wi ex rexlimdv ) ABCDEADGEHBCIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdvaa.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by Mario Carneiro, 15-Jun-2016.) $)
|
|
rexlimdvaa $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( cv wcel expr rexlimdva ) ABCDEADGEHBCFIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdv3a.1 $e |- ( ( ph /\ x e. A /\ ps ) -> ch ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). Frequently-used variant of ~ rexlimdv . (Contributed by NM,
|
|
7-Jun-2015.) $)
|
|
rexlimdv3a $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( cv wcel 3exp rexlimdv ) ABCDEADGEHBCFIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimdvw.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 18-Jun-2014.) $)
|
|
rexlimdvw $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
|
|
( wi cv wcel a1d rexlimdv ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ch $.
|
|
rexlimddv.1 $e |- ( ph -> E. x e. A ps ) $.
|
|
rexlimddv.2 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch ) $.
|
|
$( Restricted existential elimination rule of natural deduction.
|
|
(Contributed by Mario Carneiro, 15-Jun-2016.) $)
|
|
rexlimddv $p |- ( ph -> ch ) $=
|
|
( wrex rexlimdvaa mpd ) ABDEHCFABCDEGIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y ps $. $d y A $.
|
|
rexlimivv.1 $e |- ( ( x e. A /\ y e. B ) -> ( ph -> ps ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier
|
|
version). (Contributed by NM, 17-Feb-2004.) $)
|
|
rexlimivv $p |- ( E. x e. A E. y e. B ph -> ps ) $=
|
|
( wrex cv wcel rexlimdva rexlimiv ) ADFHBCECIEJABDFGKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ch $. $d y A $.
|
|
rexlimdvv.1 $e |- ( ph -> ( ( x e. A /\ y e. B ) -> ( ps -> ch ) ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 22-Jul-2004.) $)
|
|
rexlimdvv $p |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) ) $=
|
|
( wrex cv wcel wa wi expdimp rexlimdv rexlimdva ) ABEGICDFADJFKZLBCEGAQEJ
|
|
GKBCMHNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ch $. $d y A $.
|
|
rexlimdvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) ) $.
|
|
$( Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted
|
|
quantifier version.) (Contributed by NM, 18-Jun-2014.) $)
|
|
rexlimdvva $p |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) ) $=
|
|
( cv wcel wa wi ex rexlimdvv ) ABCDEFGADIFJEIGJKBCLHMN $.
|
|
$}
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.26 $p |- ( A. x e. A ( ph /\ ps ) <->
|
|
( A. x e. A ph /\ A. x e. A ps ) ) $=
|
|
( wa wral simpl ralimi simpr jca pm3.2 ral2imi imp impbii ) ABEZCDFZACDFZBC
|
|
DFZEPQROACDABGHOBCDABIHJQRPABOCDABKLMN $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers.
|
|
(Contributed by NM, 10-Aug-2004.) $)
|
|
r19.26-2 $p |- ( A. x e. A A. y e. B ( ph /\ ps ) <->
|
|
( A. x e. A A. y e. B ph /\ A. x e. A A. y e. B ps ) ) $=
|
|
( wa wral r19.26 ralbii bitri ) ABGDFHZCEHADFHZBDFHZGZCEHMCEHNCEHGLOCEABDFI
|
|
JMNCEIK $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers.
|
|
(Contributed by FL, 22-Nov-2010.) $)
|
|
r19.26-3 $p |- ( A. x e. A ( ph /\ ps /\ ch ) <->
|
|
( A. x e. A ph /\ A. x e. A ps /\ A. x e. A ch ) ) $=
|
|
( w3a wral wa df-3an ralbii r19.26 anbi1i bitr4i 3bitri ) ABCFZDEGABHZCHZDE
|
|
GPDEGZCDEGZHZADEGZBDEGZSFZOQDEABCIJPCDEKTUAUBHZSHUCRUDSABDEKLUAUBSIMN $.
|
|
|
|
$( Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by
|
|
NM, 22-Feb-2004.) $)
|
|
r19.26m $p |- ( A. x ( ( x e. A -> ph ) /\ ( x e. B -> ps ) ) <->
|
|
( A. x e. A ph /\ A. x e. B ps ) ) $=
|
|
( cv wcel wi wa wal wral 19.26 df-ral anbi12i bitr4i ) CFZDGAHZPEGBHZICJQCJ
|
|
ZRCJZIACDKZBCEKZIQRCLUASUBTACDMBCEMNO $.
|
|
|
|
$( Distribute a restricted universal quantifier over a biconditional.
|
|
Theorem 19.15 of [Margaris] p. 90 with restricted quantification.
|
|
(Contributed by NM, 6-Oct-2003.) $)
|
|
ralbi $p |- ( A. x e. A ( ph <-> ps ) ->
|
|
( A. x e. A ph <-> A. x e. A ps ) ) $=
|
|
( wb wral nfra1 cv wcel rsp imp ralbida ) ABEZCDFZABCDMCDGNCHDIMMCDJKL $.
|
|
|
|
$( Split a biconditional and distribute quantifier. (Contributed by NM,
|
|
3-Jun-2012.) $)
|
|
ralbiim $p |- ( A. x e. A ( ph <-> ps ) <->
|
|
( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) ) $=
|
|
( wb wral wi wa dfbi2 ralbii r19.26 bitri ) ABEZCDFABGZBAGZHZCDFNCDFOCDFHMP
|
|
CDABIJNOCDKL $.
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Restricted version of one direction of Theorem 19.27 of [Margaris]
|
|
p. 90. (The other direction doesn't hold when ` A ` is empty.)
|
|
(Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.27av $p |- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) ) $=
|
|
( wral wa cv wcel ax-1 ralrimiv anim2i r19.26 sylibr ) ACDEZBFNBCDEZFABFC
|
|
DEBONBBCDBCGDHIJKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Restricted version of one direction of Theorem 19.28 of [Margaris]
|
|
p. 90. (The other direction doesn't hold when ` A ` is empty.)
|
|
(Contributed by NM, 2-Apr-2004.) $)
|
|
r19.28av $p |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) ) $=
|
|
( wral wa r19.27av ancom ralbii 3imtr4i ) BCDEZAFBAFZCDEAKFABFZCDEBACDGAK
|
|
HMLCDABHIJ $.
|
|
$}
|
|
|
|
$( Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon,
|
|
30-May-2011.) $)
|
|
r19.29 $p |- ( ( A. x e. A ph /\ E. x e. A ps ) ->
|
|
E. x e. A ( ph /\ ps ) ) $=
|
|
( wral wrex wa wi pm3.2 ralimi rexim syl imp ) ACDEZBCDFZABGZCDFZNBPHZCDEOQ
|
|
HARCDABIJBPCDKLM $.
|
|
|
|
$( Variation of Theorem 19.29 of [Margaris] p. 90 with restricted
|
|
quantifiers. (Contributed by NM, 31-Aug-1999.) $)
|
|
r19.29r $p |- ( ( E. x e. A ph /\ A. x e. A ps ) ->
|
|
E. x e. A ( ph /\ ps ) ) $=
|
|
( wral wrex wa r19.29 ancom rexbii 3imtr4i ) BCDEZACDFZGBAGZCDFMLGABGZCDFBA
|
|
CDHMLIONCDABIJK $.
|
|
|
|
$( Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by Scott Fenton, 25-Feb-2011.) $)
|
|
r19.30 $p |- ( A. x e. A ( ph \/ ps ) ->
|
|
( A. x e. A ph \/ E. x e. A ps ) ) $=
|
|
( wn wi wral wrex ralim orcom df-or bitri ralbii dfrex2 orbi2i imor 3bitr4i
|
|
wo 3imtr4i ) BEZAFZCDGTCDGZACDGZFZABRZCDGUCBCDHZRZTACDIUEUACDUEBARUAABJBAKL
|
|
MUCUBEZRUHUCRUGUDUCUHJUFUHUCBCDNOUBUCPQS $.
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers.
|
|
(Contributed by NM, 25-Nov-2003.) $)
|
|
r19.32v $p |- ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) $=
|
|
( wn wi wral wo r19.21v df-or ralbii 3bitr4i ) AEZBFZCDGMBCDGZFABHZCDGAOH
|
|
MBCDIPNCDABJKAOJL $.
|
|
$}
|
|
|
|
$( Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90.
|
|
(Contributed by NM, 20-Sep-2003.) $)
|
|
r19.35 $p |- ( E. x e. A ( ph -> ps ) <->
|
|
( A. x e. A ph -> E. x e. A ps ) ) $=
|
|
( wral wn wi wrex r19.26 annim ralbii df-an 3bitr3i con2bii dfrex2 3bitr4ri
|
|
wa imbi2i ) ACDEZBFZCDEZFZGZABGZFZCDEZFSBCDHZGUDCDHUFUCATQZCDESUAQUFUCFATCD
|
|
IUHUECDABJKSUALMNUGUBSBCDORUDCDOP $.
|
|
|
|
${
|
|
$d x ps $.
|
|
$( One direction of a restricted quantifier version of Theorem 19.36 of
|
|
[Margaris] p. 90. The other direction doesn't hold when ` A ` is
|
|
empty. (Contributed by NM, 22-Oct-2003.) $)
|
|
r19.36av $p |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) ) $=
|
|
( wi wrex wral r19.35 cv wcel idd rexlimiv imim2i sylbi ) ABECDFACDGZBCDF
|
|
ZEOBEABCDHPBOBBCDCIDJBKLMN $.
|
|
$}
|
|
|
|
${
|
|
r19.37.1 $e |- F/ x ph $.
|
|
$( Restricted version of one direction of Theorem 19.37 of [Margaris]
|
|
p. 90. (The other direction doesn't hold when ` A ` is empty.)
|
|
(Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro,
|
|
11-Dec-2016.) $)
|
|
r19.37 $p |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) ) $=
|
|
( wi wrex wral r19.35 cv wcel ax-1 ralrimi imim1i sylbi ) ABFCDGACDHZBCDG
|
|
ZFAQFABCDIAPQAACDEACJDKLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Restricted version of one direction of Theorem 19.37 of [Margaris]
|
|
p. 90. (The other direction doesn't hold when ` A ` is empty.)
|
|
(Contributed by NM, 2-Apr-2004.) $)
|
|
r19.37av $p |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) ) $=
|
|
( nfv r19.37 ) ABCDACEF $.
|
|
$}
|
|
|
|
$( Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90.
|
|
(Contributed by NM, 2-Apr-2004.) $)
|
|
r19.40 $p |- ( E. x e. A ( ph /\ ps ) ->
|
|
( E. x e. A ph /\ E. x e. A ps ) ) $=
|
|
( wa wrex simpl reximi simpr jca ) ABEZCDFACDFBCDFKACDABGHKBCDABIHJ $.
|
|
|
|
${
|
|
r19.41.1 $e |- F/ x ps $.
|
|
$( Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
|
|
(Contributed by NM, 1-Nov-2010.) $)
|
|
r19.41 $p |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) ) $=
|
|
( cv wcel wa wex wrex anass exbii 19.41 bitr3i df-rex anbi1i 3bitr4i ) CF
|
|
DGZABHZHZCIZRAHZCIZBHZSCDJACDJZBHUAUBBHZCIUDUFTCRABKLUBBCEMNSCDOUEUCBACDO
|
|
PQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
$( Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
|
|
(Contributed by NM, 17-Dec-2003.) $)
|
|
r19.41v $p |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) ) $=
|
|
( nfv r19.41 ) ABCDBCEF $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed
|
|
by NM, 27-May-1998.) $)
|
|
r19.42v $p |- ( E. x e. A ( ph /\ ps ) <-> ( ph /\ E. x e. A ps ) ) $=
|
|
( wa wrex r19.41v ancom rexbii 3bitr4i ) BAEZCDFBCDFZAEABEZCDFALEBACDGMKC
|
|
DABHIALHJ $.
|
|
$}
|
|
|
|
$( Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by
|
|
NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.) $)
|
|
r19.43 $p |- ( E. x e. A ( ph \/ ps ) <->
|
|
( E. x e. A ph \/ E. x e. A ps ) ) $=
|
|
( wn wi wrex wral wo r19.35 df-or rexbii ralnex imbi1i bitr4i 3bitr4i ) AEZ
|
|
BFZCDGQCDHZBCDGZFZABIZCDGACDGZTIZQBCDJUBRCDABKLUDUCEZTFUAUCTKSUETACDMNOP $.
|
|
|
|
${
|
|
$d x ps $.
|
|
$( One direction of a restricted quantifier version of Theorem 19.44 of
|
|
[Margaris] p. 90. The other direction doesn't hold when ` A ` is
|
|
empty. (Contributed by NM, 2-Apr-2004.) $)
|
|
r19.44av $p |- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ ps ) ) $=
|
|
( wo wrex r19.43 cv wcel idd rexlimiv orim2i sylbi ) ABECDFACDFZBCDFZENBE
|
|
ABCDGOBNBBCDCHDIBJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Restricted version of one direction of Theorem 19.45 of [Margaris]
|
|
p. 90. (The other direction doesn't hold when ` A ` is empty.)
|
|
(Contributed by NM, 2-Apr-2004.) $)
|
|
r19.45av $p |- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) ) $=
|
|
( wo wrex r19.43 cv wcel idd rexlimiv orim1i sylbi ) ABECDFACDFZBCDFZEAOE
|
|
ABCDGNAOAACDCHDIAJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
ralcomf.1 $e |- F/_ y A $.
|
|
ralcomf.2 $e |- F/_ x B $.
|
|
$( Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
ralcomf $p |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph ) $=
|
|
( cv wcel wa wi wal wral ancomsimp 2albii alcom bitri r2alf 3bitr4i ) BHD
|
|
IZCHEIZJAKZCLBLZUATJAKZBLCLZACEMBDMABDMCEMUCUDCLBLUEUBUDBCTUAANOUDBCPQABC
|
|
DEFRACBEDGRS $.
|
|
|
|
$( Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
rexcomf $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $=
|
|
( cv wcel wa wex wrex ancom anbi1i 2exbii excom bitri r2exf 3bitr4i ) BHD
|
|
IZCHEIZJZAJZCKBKZUATJZAJZBKCKZACELBDLABDLCELUDUFCKBKUGUCUFBCUBUEATUAMNOUF
|
|
BCPQABCDEFRACBEDGRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x B $. $d y A $.
|
|
$( Commutation of restricted quantifiers. (Contributed by NM,
|
|
13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
ralcom $p |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph ) $=
|
|
( nfcv ralcomf ) ABCDECDFBEFG $.
|
|
|
|
$( Commutation of restricted quantifiers. (Contributed by NM,
|
|
19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
rexcom $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $=
|
|
( nfcv rexcomf ) ABCDECDFBEFG $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z B $. $d x y C $.
|
|
$( Swap 1st and 3rd restricted existential quantifiers. (Contributed by
|
|
NM, 8-Apr-2015.) $)
|
|
rexcom13 $p |- ( E. x e. A E. y e. B E. z e. C ph
|
|
<-> E. z e. C E. y e. B E. x e. A ph ) $=
|
|
( wrex rexcom rexbii 3bitri ) ADGHZCFHBEHLBEHZCFHABEHZDGHZCFHNCFHDGHLBCEF
|
|
IMOCFABDEGIJNCDFGIK $.
|
|
$}
|
|
|
|
${
|
|
$d w z A $. $d w z B $. $d w x y C $. $d x y z D $.
|
|
$( Rotate existential restricted quantifiers twice. (Contributed by NM,
|
|
8-Apr-2015.) $)
|
|
rexrot4 $p |- ( E. x e. A E. y e. B E. z e. C E. w e. D ph
|
|
<-> E. z e. C E. w e. D E. x e. A E. y e. B ph ) $=
|
|
( wrex rexcom13 rexbii bitri ) AEIJDHJCGJZBFJACGJZDHJEIJZBFJOBFJEIJDHJNPB
|
|
FACDEGHIKLOBEDFIHKM $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x A $.
|
|
$( Commutation of restricted quantifiers. Note that ` x ` and ` y `
|
|
needn't be distinct (this makes the proof longer). (Contributed by NM,
|
|
24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) $)
|
|
ralcom2 $p |- ( A. x e. A A. y e. A ph -> A. y e. A A. x e. A ph ) $=
|
|
( cv wceq wal wral wi wcel wb eleq1 dral1 df-ral 3bitr4g wa nfnae ralrimi
|
|
nfan ex sps imbi1d bicomd imbi12d biimpd wn nfra2 nfra1 wnfc nfcvf adantr
|
|
nfcvd nfeld nfan1 rsp2 ancomsd expdimp adantll pm2.61i ) BEZCEZFZBGZACDHZ
|
|
BDHZABDHZCDHZIVCVEVGVCUTDJZVDIZBGVADJZVFIZCGVEVGVIVKBCVCVHVJVDVFVBVHVJKBU
|
|
TVADLUAZVCVJAIZCGZVHAIZBGZVDVFVCVPVNVOVMBCVCVHVJAVLUBMUCACDNABDNOUDMVDBDN
|
|
VFCDNOUEVCUFZVEVGVQVEPZVFCDVQVECBCCQABCDDUGSVRVJVFVRVJPABDVRVJBVQVEBBCBQV
|
|
DBDUHSVRBVADVQBVAUIVEBCUJUKVRBDULUMUNVEVJVOVQVEVJVHAVEVHVJAABCDDUOUPUQURR
|
|
TRTUS $.
|
|
$}
|
|
|
|
$( A commutative law for restricted quantifiers that swaps the domain of the
|
|
restriction. (Contributed by NM, 22-Feb-2004.) $)
|
|
ralcom3 $p |- ( A. x e. A ( x e. B -> ph ) <->
|
|
A. x e. B ( x e. A -> ph ) ) $=
|
|
( cv wcel wi wral pm2.04 ralimi2 impbii ) BEZDFZAGZBCHLCFZAGZBDHNPBCDOMAIJP
|
|
NBDCMOAIJK $.
|
|
|
|
${
|
|
$d y A $. $d x B $. $d x y $.
|
|
reean.1 $e |- F/ y ph $.
|
|
reean.2 $e |- F/ x ps $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
|
|
(Proof shortened by Andrew Salmon, 30-May-2011.) $)
|
|
reean $p |- ( E. x e. A E. y e. B ( ph /\ ps ) <->
|
|
( E. x e. A ph /\ E. y e. B ps ) ) $=
|
|
( cv wcel wa wex wrex an4 2exbii nfv nfan eean bitri df-rex r2ex anbi12i
|
|
3bitr4i ) CIEJZDIFJZKABKZKZDLCLZUDAKZCLZUEBKZDLZKZUFDFMCEMACEMZBDFMZKUHUI
|
|
UKKZDLCLUMUGUPCDUDUEABNOUIUKCDUDADUDDPGQUEBCUECPHQRSUFCDEFUAUNUJUOULACETB
|
|
DFTUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $. $d x y $. $d y A $. $d x B $.
|
|
$( Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.) $)
|
|
reeanv $p |- ( E. x e. A E. y e. B ( ph /\ ps ) <->
|
|
( E. x e. A ph /\ E. y e. B ps ) ) $=
|
|
( nfv reean ) ABCDEFADGBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d ph y z $. $d ps x z $. $d ch x y $. $d A y $. $d B x z $.
|
|
$d C x y $.
|
|
$( Rearrange three existential quantifiers. (Contributed by Jeff Madsen,
|
|
11-Jun-2010.) $)
|
|
3reeanv $p |- ( E. x e. A E. y e. B E. z e. C ( ph /\ ps /\ ch )
|
|
<-> ( E. x e. A ph /\ E. y e. B ps /\ E. z e. C ch ) ) $=
|
|
( wa wrex w3a r19.41v reeanv anbi1i bitri df-3an 2rexbii rexbii 3bitr4i )
|
|
ABJZEHKZCFIKZJZDGKZADGKZBEHKZJZUCJZABCLZFIKEHKZDGKUFUGUCLUEUBDGKZUCJUIUBU
|
|
CDGMULUHUCABDEGHNOPUKUDDGUKUACJZFIKEHKUDUJUMEFHIABCQRUACEFHINPSUFUGUCQT
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d ps x $. $d A y $. $d B x $. $d x y $.
|
|
$( Distribute quantification over "or". (Contributed by Jeff Madsen,
|
|
19-Jun-2010.) $)
|
|
2ralor $p |- ( A. x e. A A. y e. B ( ph \/ ps ) <->
|
|
( A. x e. A ph \/ A. y e. B ps ) ) $=
|
|
( wo wral wn wrex wa rexnal anbi12i rexbii bitr3i reeanv 3bitr3ri 3bitr4i
|
|
ioran con4bii ) ABGZDFHZCEHZACEHZBDFHZGZAIZCEJZBIZDFJZKZUDIZUEIZKUCIZUFIU
|
|
HULUJUMACELBDFLMUGUIKZDFJZCEJUBIZCEJUKUNUPUQCEUPUAIZDFJUQURUODFABSNUADFLO
|
|
NUGUICDEFPUBCELQUDUESRT $.
|
|
$}
|
|
|
|
$( ` x ` is not free in ` E! x e. A ph ` . (Contributed by NM,
|
|
19-Mar-1997.) $)
|
|
nfreu1 $p |- F/ x E! x e. A ph $=
|
|
( wreu cv wcel wa weu df-reu nfeu1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
$( ` x ` is not free in ` E* x e. A ph ` . (Contributed by NM,
|
|
16-Jun-2017.) $)
|
|
nfrmo1 $p |- F/ x E* x e. A ph $=
|
|
( wrmo cv wcel wa wmo df-rmo nfmo1 nfxfr ) ABCDBECFAGZBHBABCILBJK $.
|
|
|
|
${
|
|
nfreud.1 $e |- F/ y ph $.
|
|
nfreud.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfreud.3 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfreu . (Contributed by NM, 15-Feb-2013.)
|
|
(Revised by Mario Carneiro, 8-Oct-2016.) $)
|
|
nfreud $p |- ( ph -> F/ x E! y e. A ps ) $=
|
|
( wreu cv wcel wa weu df-reu wceq wal wn wnfc nfcvf adantr adantl nfeud2
|
|
nfeld wnf nfand nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACJUGOCPQZLZUHBCUKCUG
|
|
EUJCUGRACDSUAACERUJGTUCABCUDUJHTUEUBUF $.
|
|
|
|
$( Deduction version of ~ nfrmo . (Contributed by NM, 17-Jun-2017.) $)
|
|
nfrmod $p |- ( ph -> F/ x E* y e. A ps ) $=
|
|
( wrmo cv wcel wa wmo df-rmo weq wal wn wnfc nfcvf adantr nfeld wnf nfand
|
|
adantl nfmod2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCPQZLZUHBCUKCUGEUJC
|
|
UGRACDSUDACERUJGTUAABCUBUJHTUCUEUF $.
|
|
$}
|
|
|
|
${
|
|
nfreu.1 $e |- F/_ x A $.
|
|
nfreu.2 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for restricted uniqueness.
|
|
(Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro,
|
|
8-Oct-2016.) $)
|
|
nfreu $p |- F/ x E! y e. A ph $=
|
|
( wreu wnf wtru nftru wnfc a1i nfreud trud ) ACDGBHIABCDCJBDKIELABHIFLMN
|
|
$.
|
|
|
|
$( Bound-variable hypothesis builder for restricted uniqueness.
|
|
(Contributed by NM, 16-Jun-2017.) $)
|
|
nfrmo $p |- F/ x E* y e. A ph $=
|
|
( wrmo cv wcel wa wmo df-rmo wnf wtru nftru weq wal wn nfcvf a1i adantl
|
|
wnfc nfeld nfand nfmod2 trud nfxfr ) ACDGCHZDIZAJZCKZBACDLUKBMNUJBCCOBCPB
|
|
QRZUJBMNULUIABULBUHDBCSBDUBULETUCABMULFTUDUAUEUFUG $.
|
|
$}
|
|
|
|
$( An "identity" law of concretion for restricted abstraction. Special case
|
|
of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) $)
|
|
rabid $p |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) ) $=
|
|
( cv wcel wa crab df-rab abeq2i ) BDCEAFBABCGABCHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( An "identity" law for restricted class abstraction. (Contributed by NM,
|
|
9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) $)
|
|
rabid2 $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
|
|
( cv wcel wa cab wceq wi wal crab abeq2 pm4.71 albii bitr4i df-rab eqeq2i
|
|
wral wb df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCRUEUBUCSZBJUGUCB
|
|
CLUFUIBUBAMNOUHUDCABCPQABCTUA $.
|
|
$}
|
|
|
|
$( Equivalent wff's correspond to equal restricted class abstractions.
|
|
Closed theorem form of ~ rabbidva . (Contributed by NM, 25-Nov-2013.) $)
|
|
rabbi $p |- ( A. x e. A ( ps <-> ch )
|
|
<-> { x e. A | ps } = { x e. A | ch } ) $=
|
|
( cv wcel wa wb wal wceq wral crab abbi wi df-ral pm5.32 albii bitri df-rab
|
|
cab eqeq12i 3bitr4i ) CEDFZAGZUCBGZHZCIZUDCTZUECTZJABHZCDKZACDLZBCDLZJUDUEC
|
|
MUKUCUJNZCIUGUJCDOUNUFCUCABPQRULUHUMUIACDSBCDSUAUB $.
|
|
|
|
$( Swap with a membership relation in a restricted class abstraction.
|
|
(Contributed by NM, 4-Jul-2005.) $)
|
|
rabswap $p |- { x e. A | x e. B } = { x e. B | x e. A } $=
|
|
( cv wcel wa cab crab ancom abbii df-rab 3eqtr4i ) ADZBEZMCEZFZAGONFZAGOABH
|
|
NACHPQANOIJOABKNACKL $.
|
|
|
|
$( The abstraction variable in a restricted class abstraction isn't free.
|
|
(Contributed by NM, 19-Mar-1997.) $)
|
|
nfrab1 $p |- F/_ x { x e. A | ph } $=
|
|
( crab cv wcel wa cab df-rab nfab1 nfcxfr ) BABCDBECFAGZBHABCILBJK $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $.
|
|
nfrab.1 $e |- F/ x ph $.
|
|
nfrab.2 $e |- F/_ x A $.
|
|
$( A variable not free in a wff remains so in a restricted class
|
|
abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario
|
|
Carneiro, 9-Oct-2016.) $)
|
|
nfrab $p |- F/_ x { y e. A | ph } $=
|
|
( vz crab cv wcel wa cab df-rab wnfc wtru nftru wceq wal wn wnf nfcri a1i
|
|
eleq1 dvelimnf nfand adantl nfabd2 trud nfcxfr ) BACDHCIZDJZAKZCLZACDMBUM
|
|
NOULBCCPBIUJQBRSZULBTOUNUKABGIZDJUKBCGBGDFUAUOUJDUCUDABTUNEUBUEUFUGUHUI
|
|
$.
|
|
$}
|
|
|
|
${
|
|
reubida.1 $e |- F/ x ph $.
|
|
reubida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by Mario Carneiro, 19-Nov-2016.) $)
|
|
reubida $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
|
|
( cv wcel wa weu wreu pm5.32da eubid df-reu 3bitr4g ) ADHEIZBJZDKQCJZDKBD
|
|
ELCDELARSDFAQBCGMNBDEOCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reubidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 13-Nov-2004.) $)
|
|
reubidva $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
|
|
( nfv reubida ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
reubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 17-Oct-1996.) $)
|
|
reubidv $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
|
|
( wb cv wcel adantr reubidva ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
reubiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (inference
|
|
rule). (Contributed by NM, 14-Nov-2004.) $)
|
|
reubiia $p |- ( E! x e. A ph <-> E! x e. A ps ) $=
|
|
( cv wcel wa weu wreu pm5.32i eubii df-reu 3bitr4i ) CFDGZAHZCIOBHZCIACDJ
|
|
BCDJPQCOABEKLACDMBCDMN $.
|
|
$}
|
|
|
|
${
|
|
reubii.1 $e |- ( ph <-> ps ) $.
|
|
$( Formula-building rule for restricted existential quantifier (inference
|
|
rule). (Contributed by NM, 22-Oct-1999.) $)
|
|
reubii $p |- ( E! x e. A ph <-> E! x e. A ps ) $=
|
|
( wb cv wcel a1i reubiia ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
rmobida.1 $e |- F/ x ph $.
|
|
rmobida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobida $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
|
|
( cv wcel wa wmo wrmo pm5.32da mobid df-rmo 3bitr4g ) ADHEIZBJZDKQCJZDKBD
|
|
ELCDELARSDFAQBCGMNBDEOCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rmobidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobidva $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
|
|
( nfv rmobida ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rmobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (deduction
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobidv $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
|
|
( wb cv wcel adantr rmobidva ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
rmobiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Formula-building rule for restricted existential quantifier (inference
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobiia $p |- ( E* x e. A ph <-> E* x e. A ps ) $=
|
|
( cv wcel wa wmo wrmo pm5.32i mobii df-rmo 3bitr4i ) CFDGZAHZCIOBHZCIACDJ
|
|
BCDJPQCOABEKLACDMBCDMN $.
|
|
$}
|
|
|
|
${
|
|
rmobii.1 $e |- ( ph <-> ps ) $.
|
|
$( Formula-building rule for restricted existential quantifier (inference
|
|
rule). (Contributed by NM, 16-Jun-2017.) $)
|
|
rmobii $p |- ( E* x e. A ph <-> E* x e. A ps ) $=
|
|
( wb cv wcel a1i rmobiia ) ABCDABFCGDHEIJ $.
|
|
$}
|
|
|
|
${
|
|
raleq1f.1 $e |- F/_ x A $.
|
|
raleq1f.2 $e |- F/_ x B $.
|
|
$( Equality theorem for restricted universal quantifier, with
|
|
bound-variable hypotheses instead of distinct variable restrictions.
|
|
(Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
raleqf $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $=
|
|
( wceq cv wcel wi wal wral nfeq eleq2 imbi1d albid df-ral 3bitr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
|
|
$( Equality theorem for restricted existential quantifier, with
|
|
bound-variable hypotheses instead of distinct variable restrictions.
|
|
(Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
rexeqf $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
|
|
( wceq cv wcel wa wex wrex nfeq eleq2 anbi1d exbid df-rex 3bitr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
|
|
$( Equality theorem for restricted uniqueness quantifier, with
|
|
bound-variable hypotheses instead of distinct variable restrictions.
|
|
(Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
reueq1f $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $=
|
|
( wceq cv wcel wa weu wreu nfeq eleq2 anbi1d eubid df-reu 3bitr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
|
|
$( Equality theorem for restricted uniqueness quantifier, with
|
|
bound-variable hypotheses instead of distinct variable restrictions.
|
|
(Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoeq1f $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $=
|
|
( wceq cv wcel wa wmo wrmo nfeq eleq2 anbi1d mobid df-rmo 3bitr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for restricted universal quantifier. (Contributed by
|
|
NM, 16-Nov-1995.) $)
|
|
raleq $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $=
|
|
( nfcv raleqf ) ABCDBCEBDEF $.
|
|
|
|
$( Equality theorem for restricted existential quantifier. (Contributed by
|
|
NM, 29-Oct-1995.) $)
|
|
rexeq $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
|
|
( nfcv rexeqf ) ABCDBCEBDEF $.
|
|
|
|
$( Equality theorem for restricted uniqueness quantifier. (Contributed by
|
|
NM, 5-Apr-2004.) $)
|
|
reueq1 $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $=
|
|
( nfcv reueq1f ) ABCDBCEBDEF $.
|
|
|
|
$( Equality theorem for restricted uniqueness quantifier. (Contributed by
|
|
Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoeq1 $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $=
|
|
( nfcv rmoeq1f ) ABCDBCEBDEF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
raleq1i.1 $e |- A = B $.
|
|
$( Equality inference for restricted universal qualifier. (Contributed by
|
|
Paul Chapman, 22-Jun-2011.) $)
|
|
raleqi $p |- ( A. x e. A ph <-> A. x e. B ph ) $=
|
|
( wceq wral wb raleq ax-mp ) CDFABCGABDGHEABCDIJ $.
|
|
|
|
$( Equality inference for restricted existential qualifier. (Contributed
|
|
by Mario Carneiro, 23-Apr-2015.) $)
|
|
rexeqi $p |- ( E. x e. A ph <-> E. x e. B ph ) $=
|
|
( wceq wrex wb rexeq ax-mp ) CDFABCGABDGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
raleq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 13-Nov-2005.) $)
|
|
raleqdv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) ) $=
|
|
( wceq wral wb raleq syl ) ADEGBCDHBCEHIFBCDEJK $.
|
|
|
|
$( Equality deduction for restricted existential quantifier. (Contributed
|
|
by NM, 14-Jan-2007.) $)
|
|
rexeqdv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) ) $=
|
|
( wceq wrex wb rexeq syl ) ADEGBCDHBCEHIFBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
raleqd.1 $e |- ( A = B -> ( ph <-> ps ) ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 16-Nov-1995.) $)
|
|
raleqbi1dv $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ps ) ) $=
|
|
( wceq wral raleq ralbidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
|
|
|
|
$( Equality deduction for restricted existential quantifier. (Contributed
|
|
by NM, 18-Mar-1997.) $)
|
|
rexeqbi1dv $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) ) $=
|
|
( wceq wrex rexeq rexbidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
|
|
|
|
$( Equality deduction for restricted uniqueness quantifier. (Contributed
|
|
by NM, 5-Apr-2004.) $)
|
|
reueqd $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) ) $=
|
|
( wceq wreu reueq1 reubidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
|
|
|
|
$( Equality deduction for restricted uniqueness quantifier. (Contributed
|
|
by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoeqd $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) ) $=
|
|
( wceq wrmo rmoeq1 rmobidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
raleqbidv.1 $e |- ( ph -> A = B ) $.
|
|
raleqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 6-Nov-2007.) $)
|
|
raleqbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
|
|
( wral raleqdv ralbidv bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.
|
|
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
NM, 6-Nov-2007.) $)
|
|
rexeqbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
|
|
( wrex rexeqdv rexbidv bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
raleqbidva.1 $e |- ( ph -> A = B ) $.
|
|
raleqbidva.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
Mario Carneiro, 5-Jan-2017.) $)
|
|
raleqbidva $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
|
|
( wral ralbidva raleqdv bitrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.
|
|
|
|
$( Equality deduction for restricted universal quantifier. (Contributed by
|
|
Mario Carneiro, 5-Jan-2017.) $)
|
|
rexeqbidva $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
|
|
( wrex rexbidva rexeqdv bitrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.
|
|
$}
|
|
|
|
$( Unrestricted "at most one" implies restricted "at most one". (Contributed
|
|
by NM, 16-Jun-2017.) $)
|
|
mormo $p |- ( E* x ph -> E* x e. A ph ) $=
|
|
( wmo cv wcel wa wrmo moan df-rmo sylibr ) ABDBECFZAGBDABCHALBIABCJK $.
|
|
|
|
$( Restricted uniqueness in terms of "at most one." (Contributed by NM,
|
|
23-May-1999.) (Revised by NM, 16-Jun-2017.) $)
|
|
reu5 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) ) $=
|
|
( cv wcel wa weu wex wmo wreu wrex eu5 df-reu df-rex df-rmo anbi12i 3bitr4i
|
|
wrmo ) BDCEAFZBGSBHZSBIZFABCJABCKZABCRZFSBLABCMUBTUCUAABCNABCOPQ $.
|
|
|
|
$( Restricted unique existence implies restricted existence. (Contributed by
|
|
NM, 19-Aug-1999.) $)
|
|
reurex $p |- ( E! x e. A ph -> E. x e. A ph ) $=
|
|
( wreu wrex wrmo reu5 simplbi ) ABCDABCEABCFABCGH $.
|
|
|
|
$( Restricted existential uniqueness implies restricted "at most one."
|
|
(Contributed by NM, 16-Jun-2017.) $)
|
|
reurmo $p |- ( E! x e. A ph -> E* x e. A ph ) $=
|
|
( wreu wrex wrmo reu5 simprbi ) ABCDABCEABCFABCGH $.
|
|
|
|
$( Restricted "at most one" in term of uniqueness. (Contributed by NM,
|
|
16-Jun-2017.) $)
|
|
rmo5 $p |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) ) $=
|
|
( cv wcel wa wmo wex weu wrmo wrex wreu df-mo df-rmo df-rex imbi12i 3bitr4i
|
|
wi df-reu ) BDCEAFZBGTBHZTBIZRABCJABCKZABCLZRTBMABCNUCUAUDUBABCOABCSPQ $.
|
|
|
|
$( Nonexistence implies restricted "at most one". (Contributed by NM,
|
|
17-Jun-2017.) $)
|
|
nrexrmo $p |- ( -. E. x e. A ph -> E* x e. A ph ) $=
|
|
( wrex wn wreu wi wrmo pm2.21 rmo5 sylibr ) ABCDZELABCFZGABCHLMIABCJK $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z ps $. $d z ph $.
|
|
cbvralf.1 $e |- F/_ x A $.
|
|
cbvralf.2 $e |- F/_ y A $.
|
|
cbvralf.3 $e |- F/ y ph $.
|
|
cbvralf.4 $e |- F/ x ps $.
|
|
cbvralf.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro,
|
|
9-Oct-2016.) $)
|
|
cbvralf $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
|
|
( vz cv wcel wi wal wral wsb nfv nfcri nfim nfs1v wceq eleq1 sbequ12 nfsb
|
|
imbi12d cbval sbequ sbie syl6bb bitri df-ral 3bitr4i ) CLZEMZANZCOZDLZEMZ
|
|
BNZDOZACEPBDEPUQKLZEMZACKQZNZKOVAUPVECKUPKRVCVDCCKEFSACKUATUNVBUBUOVCAVDU
|
|
NVBEUCACKUDUFUGVEUTKDVCVDDDKEGSACKDHUETUTKRVBURUBZVCUSVDBVBUREUCVFVDACDQB
|
|
AKDCUHABCDIJUIUJUFUGUKACEULBDEULUM $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro,
|
|
9-Oct-2016.) $)
|
|
cbvrexf $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
|
|
( wn wral wrex nfn cv wceq notbid cbvralf notbii dfrex2 3bitr4i ) AKZCELZ
|
|
KBKZDELZKACEMBDEMUCUEUBUDCDEFGADHNBCINCODOPABJQRSACETBDETUA $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y z A $. $d z ph $. $d z ps $.
|
|
cbvral.1 $e |- F/ y ph $.
|
|
cbvral.2 $e |- F/ x ps $.
|
|
cbvral.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 31-Jul-2003.) $)
|
|
cbvral $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
|
|
( nfcv cbvralf ) ABCDECEIDEIFGHJ $.
|
|
|
|
$( Rule used to change bound variables, using implicit substitution.
|
|
(Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
cbvrex $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
|
|
( nfcv cbvrexf ) ABCDECEIDEIFGHJ $.
|
|
|
|
$( Change the bound variable of a restricted uniqueness quantifier using
|
|
implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.) $)
|
|
cbvreu $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
|
|
( vz cv wcel wa weu wreu wsb nfv sb8eu sban eubii df-reu anbi1i nfsb nfan
|
|
clelsb3 wceq eleq1 sbequ sbie syl6bb anbi12d cbveu bitri 3bitri 3bitr4i )
|
|
CJEKZALZCMZDJZEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZVAUPCIUPIPQVBVEI
|
|
UOACIRSVFIJZEKZVDLZIMVAVEVIIVCVHVDICEUDUASVIUTIDVHVDDVHDPACIDFUBUCUTIPVGU
|
|
RUEZVHUSVDBVGUREUFVJVDACDOBAIDCUGABCDGHUHUIUJUKULUMACETBDETUN $.
|
|
|
|
$( Change the bound variable of restricted "at most one" using implicit
|
|
substitution. (Contributed by NM, 16-Jun-2017.) $)
|
|
cbvrmo $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
|
|
( wrex wreu wi wrmo cbvrex cbvreu imbi12i rmo5 3bitr4i ) ACEIZACEJZKBDEIZ
|
|
BDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d y ph $. $d x ps $.
|
|
cbvralv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change the bound variable of a restricted universal quantifier using
|
|
implicit substitution. (Contributed by NM, 28-Jan-1997.) $)
|
|
cbvralv $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
|
|
( nfv cbvral ) ABCDEADGBCGFH $.
|
|
|
|
$( Change the bound variable of a restricted existential quantifier using
|
|
implicit substitution. (Contributed by NM, 2-Jun-1998.) $)
|
|
cbvrexv $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
|
|
( nfv cbvrex ) ABCDEADGBCGFH $.
|
|
|
|
$( Change the bound variable of a restricted uniqueness quantifier using
|
|
implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by
|
|
Mario Carneiro, 15-Oct-2016.) $)
|
|
cbvreuv $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
|
|
( nfv cbvreu ) ABCDEADGBCGFH $.
|
|
|
|
$( Change the bound variable of a restricted uniqueness quantifier using
|
|
implicit substitution. (Contributed by Alexander van der Vekens,
|
|
17-Jun-2017.) $)
|
|
cbvrmov $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
|
|
( nfv cbvrmo ) ABCDEADGBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d ps y $. $d B x $. $d ch x $. $d x ph y $.
|
|
cbvraldva2.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
|
|
cbvraldva2.2 $e |- ( ( ph /\ x = y ) -> A = B ) $.
|
|
$( Rule used to change the bound variable in a restricted universal
|
|
quantifier with implicit substitution which also changes the quantifier
|
|
domain. Deduction form. (Contributed by David Moews, 1-May-2017.) $)
|
|
cbvraldva2 $p |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) ) $=
|
|
( cv wcel wi wal wral weq wa simpr eleq12d imbi12d df-ral cbvaldva
|
|
3bitr4g ) ADJZFKZBLZDMEJZGKZCLZEMBDFNCEGNAUEUHDEADEOZPZUDUGBCUJUCUFFGAUIQ
|
|
IRHSUABDFTCEGTUB $.
|
|
|
|
$( Rule used to change the bound variable in a restricted existential
|
|
quantifier with implicit substitution which also changes the quantifier
|
|
domain. Deduction form. (Contributed by David Moews, 1-May-2017.) $)
|
|
cbvrexdva2 $p |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) ) $=
|
|
( cv wcel wa wex wrex weq simpr eleq12d anbi12d cbvexdva df-rex 3bitr4g )
|
|
ADJZFKZBLZDMEJZGKZCLZEMBDFNCEGNAUDUGDEADEOZLZUCUFBCUIUBUEFGAUHPIQHRSBDFTC
|
|
EGTUA $.
|
|
$}
|
|
|
|
${
|
|
$d ps y $. $d ch x $. $d A x y $. $d x ph y $.
|
|
cbvraldva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
|
|
$( Rule used to change the bound variable in a restricted universal
|
|
quantifier with implicit substitution. Deduction form. (Contributed by
|
|
David Moews, 1-May-2017.) $)
|
|
cbvraldva $p |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) ) $=
|
|
( weq wa eqidd cbvraldva2 ) ABCDEFFGADEHIFJK $.
|
|
|
|
$( Rule used to change the bound variable in a restricted existential
|
|
quantifier with implicit substitution. Deduction form. (Contributed by
|
|
David Moews, 1-May-2017.) $)
|
|
cbvrexdva $p |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) ) $=
|
|
( weq wa eqidd cbvrexdva2 ) ABCDEFFGADEHIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d z A $. $d x y B $. $d z y B $. $d w B $. $d z ph $.
|
|
$d y ps $. $d x ch $. $d w ch $.
|
|
cbvral2v.1 $e |- ( x = z -> ( ph <-> ch ) ) $.
|
|
cbvral2v.2 $e |- ( y = w -> ( ch <-> ps ) ) $.
|
|
$( Change bound variables of double restricted universal quantification,
|
|
using implicit substitution. (Contributed by NM, 10-Aug-2004.) $)
|
|
cbvral2v $p |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps ) $=
|
|
( wral cv wceq ralbidv cbvralv ralbii bitri ) AEILZDHLCEILZFHLBGILZFHLSTD
|
|
FHDMFMNACEIJOPTUAFHCBEGIKPQR $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A z $. $d B w $. $d B x y $. $d B z y $. $d ch w $.
|
|
$d ch x $. $d ph z $. $d ps y $.
|
|
cbvrex2v.1 $e |- ( x = z -> ( ph <-> ch ) ) $.
|
|
cbvrex2v.2 $e |- ( y = w -> ( ch <-> ps ) ) $.
|
|
$( Change bound variables of double restricted universal quantification,
|
|
using implicit substitution. (Contributed by FL, 2-Jul-2012.) $)
|
|
cbvrex2v $p |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps ) $=
|
|
( wrex weq rexbidv cbvrexv rexbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHDF
|
|
MACEIJNOSTFHCBEGIKOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d w ph $. $d z ps $. $d x ch $. $d v ch $. $d y u th $. $d x A $.
|
|
$d w A $. $d x y B $. $d w y B $. $d v B $. $d x y z C $.
|
|
$d w y z C $. $d v z C $. $d z y C $. $d z C $. $d u C $.
|
|
cbvral3v.1 $e |- ( x = w -> ( ph <-> ch ) ) $.
|
|
cbvral3v.2 $e |- ( y = v -> ( ch <-> th ) ) $.
|
|
cbvral3v.3 $e |- ( z = u -> ( th <-> ps ) ) $.
|
|
$( Change bound variables of triple restricted universal quantification,
|
|
using implicit substitution. (Contributed by NM, 10-May-2005.) $)
|
|
cbvral3v $p |- ( A. x e. A A. y e. B A. z e. C ph <->
|
|
A. w e. A A. v e. B A. u e. C ps ) $=
|
|
( wral cv wceq 2ralbidv cbvralv cbvral2v ralbii bitri ) AGMQFLQZEKQCGMQFL
|
|
QZHKQBJMQILQZHKQUEUFEHKERHRSACFGLMNTUAUFUGHKCBDFGIJLMOPUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d z x A $. $d y A $. $d z y ph $.
|
|
$( Change bound variable by using a substitution. (Contributed by NM,
|
|
20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) $)
|
|
cbvralsv $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $=
|
|
( vz wral wsb nfv nfs1v sbequ12 cbvral nfsb sbequ bitri ) ABDFABEGZEDFABC
|
|
GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $.
|
|
$}
|
|
|
|
${
|
|
$d z x A $. $d y z ph $. $d y A $.
|
|
$( Change bound variable by using a substitution. (Contributed by NM,
|
|
2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) $)
|
|
cbvrexsv $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $=
|
|
( vz wrex wsb nfv nfs1v sbequ12 cbvrex nfsb sbequ bitri ) ABDFABEGZEDFABC
|
|
GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z ph $. $d x z ps $.
|
|
sbralie.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Implicit to explicit substitution that swaps variables in a quantified
|
|
expression. (Contributed by NM, 5-Sep-2004.) $)
|
|
sbralie $p |- ( [ x / y ] A. x e. y ph <-> A. y e. x ps ) $=
|
|
( vz cv wral wsb cbvralsv sbbii nfv raleq sbie bitri sbco2 ralbii ) ACDGZ
|
|
HZDCIZACFIZFCGZHZBDUBHZTUAFRHZDCIUCSUEDCACFRJKUEUCDCUCDLUAFRUBMNOUCUAFDIZ
|
|
DUBHUDUAFDUBJUFBDUBUFACDIBACDFAFLPABCDBCLENOQOO $.
|
|
$}
|
|
|
|
${
|
|
rabbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
|
|
$( Equivalent wff's yield equal restricted class abstractions (inference
|
|
rule). (Contributed by NM, 22-May-1999.) $)
|
|
rabbiia $p |- { x e. A | ph } = { x e. A | ps } $=
|
|
( cv wcel wa cab crab pm5.32i abbii df-rab 3eqtr4i ) CFDGZAHZCIOBHZCIACDJ
|
|
BCDJPQCOABEKLACDMBCDMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rabbidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal restricted class abstractions (deduction
|
|
rule). (Contributed by NM, 28-Nov-2003.) $)
|
|
rabbidva $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
|
|
( wb wral crab wceq ralrimiva rabbi sylib ) ABCGZDEHBDEICDEIJANDEFKBCDELM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
rabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal restricted class abstractions (deduction
|
|
rule). (Contributed by NM, 10-Feb-1995.) $)
|
|
rabbidv $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
|
|
( wb cv wcel adantr rabbidva ) ABCDEABCGDHEIFJK $.
|
|
$}
|
|
|
|
${
|
|
rabeqf.1 $e |- F/_ x A $.
|
|
rabeqf.2 $e |- F/_ x B $.
|
|
$( Equality theorem for restricted class abstractions, with bound-variable
|
|
hypotheses instead of distinct variable restrictions. (Contributed by
|
|
NM, 7-Mar-2004.) $)
|
|
rabeqf $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $=
|
|
( wceq cv wcel wa cab crab nfeq eleq2 anbi1d abbid df-rab 3eqtr4g ) CDGZB
|
|
HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for restricted class abstractions. (Contributed by NM,
|
|
15-Oct-2003.) $)
|
|
rabeq $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $=
|
|
( nfcv rabeqf ) ABCDBCEBDEF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d ph x $.
|
|
rabeqbidv.1 $e |- ( ph -> A = B ) $.
|
|
rabeqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equality of restricted class abstractions. (Contributed by Jeff Madsen,
|
|
1-Dec-2009.) $)
|
|
rabeqbidv $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
|
|
( crab wceq rabeq syl rabbidv eqtrd ) ABDEIZBDFIZCDFIAEFJOPJGBDEFKLABCDFH
|
|
MN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d ph x $.
|
|
rabeqbidva.1 $e |- ( ph -> A = B ) $.
|
|
rabeqbidva.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
|
|
$( Equality of restricted class abstractions. (Contributed by Mario
|
|
Carneiro, 26-Jan-2017.) $)
|
|
rabeqbidva $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
|
|
( crab rabbidva wceq rabeq syl eqtrd ) ABDEICDEIZCDFIZABCDEHJAEFKOPKGCDEF
|
|
LMN $.
|
|
$}
|
|
|
|
${
|
|
rabeqi.1 $e |- A = { x e. B | ph } $.
|
|
$( Inference rule from equality of a class variable and a restricted class
|
|
abstraction. (Contributed by NM, 16-Feb-2004.) $)
|
|
rabeq2i $p |- ( x e. A <-> ( x e. B /\ ph ) ) $=
|
|
( cv wcel crab wa eleq2i rabid bitri ) BFZCGMABDHZGMDGAICNMEJABDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d A z $. $d ph z $. $d ps z $.
|
|
cbvrab.1 $e |- F/_ x A $.
|
|
cbvrab.2 $e |- F/_ y A $.
|
|
cbvrab.3 $e |- F/ y ph $.
|
|
cbvrab.4 $e |- F/ x ps $.
|
|
cbvrab.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule to change the bound variable in a restricted class abstraction,
|
|
using implicit substitution. This version has bound-variable hypotheses
|
|
in place of distinct variable conditions. (Contributed by Andrew
|
|
Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) $)
|
|
cbvrab $p |- { x e. A | ph } = { y e. A | ps } $=
|
|
( vz cv wcel wa cab crab wsb nfv nfcri nfan nfs1v wceq eleq1 sbequ12 nfsb
|
|
anbi12d cbvab sbequ sbie syl6bb eqtri df-rab 3eqtr4i ) CLZEMZANZCOZDLZEMZ
|
|
BNZDOZACEPBDEPUQKLZEMZACKQZNZKOVAUPVECKUPKRVCVDCCKEFSACKUATUNVBUBUOVCAVDU
|
|
NVBEUCACKUDUFUGVEUTKDVCVDDDKEGSACKDHUETUTKRVBURUBZVCUSVDBVBUREUCVFVDACDQB
|
|
AKDCUHABCDIJUIUJUFUGUKACEULBDEULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $. $d x ps $.
|
|
cbvrabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Rule to change the bound variable in a restricted class abstraction,
|
|
using implicit substitution. (Contributed by NM, 26-May-1999.) $)
|
|
cbvrabv $p |- { x e. A | ph } = { y e. A | ps } $=
|
|
( nfcv nfv cbvrab ) ABCDECEGDEGADHBCHFI $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The universal class
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the symbol for the universal class. $)
|
|
$c _V $. $( Letter V (for the universal class) $)
|
|
|
|
$( Extend class notation to include the universal class symbol. $)
|
|
cvv $a class _V $.
|
|
|
|
${
|
|
$d z x $. $d z y $.
|
|
$( Soundness justification theorem for ~ df-v . (Contributed by Rodolfo
|
|
Medina, 27-Apr-2010.) $)
|
|
vjust $p |- { x | x = x } = { y | y = y } $=
|
|
( vz cv wceq cab wsb wcel equid sbt 2th df-clab 3bitr4i eqriv ) CADZOEZAF
|
|
ZBDZREZBFZPACGZSBCGZCDZQHUCTHUAUBPACAIJSBCBIJKPCALSCBLMN $.
|
|
$}
|
|
|
|
$( Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21.
|
|
Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.) $)
|
|
df-v $a |- _V = { x | x = x } $.
|
|
|
|
$( All setvar variables are sets (see ~ isset ). Theorem 6.8 of [Quine]
|
|
p. 43. (Contributed by NM, 5-Aug-1993.) $)
|
|
vex $p |- x e. _V $=
|
|
( cv cvv wcel wceq eqid df-v abeq2i mpbir ) ABZCDJJEZJFKACAGHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Two ways to say " ` A ` is a set": A class ` A ` is a member of the
|
|
universal class ` _V ` (see ~ df-v ) if and only if the class ` A `
|
|
exists (i.e. there exists some set ` x ` equal to class ` A ` ).
|
|
Theorem 6.9 of [Quine] p. 43. _Notational convention_: We will use the
|
|
notational device " ` A e. _V ` " to mean " ` A ` is a set" very
|
|
frequently, for example in ~ uniex . Note the when ` A ` is not a set,
|
|
it is called a proper class. In some theorems, such as ~ uniexg , in
|
|
order to shorten certain proofs we use the more general antecedent
|
|
` A e. V ` instead of ` A e. _V ` to mean " ` A ` is a set."
|
|
|
|
Note that a constant is implicitly considered distinct from all
|
|
variables. This is why ` _V ` is not included in the distinct variable
|
|
list, even though ~ df-clel requires that the expression substituted for
|
|
` B ` not contain ` x ` . (Also, the Metamath spec does not allow
|
|
constants in the distinct variable list.) (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
isset $p |- ( A e. _V <-> E. x x = A ) $=
|
|
( cvv wcel cv wceq wa wex df-clel vex biantru exbii bitr4i ) BCDAEZBFZNCD
|
|
ZGZAHOAHABCIOQAPOAJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d x y $.
|
|
issetf.1 $e |- F/_ x A $.
|
|
$( A version of isset that does not require x and A to be distinct.
|
|
(Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro,
|
|
10-Oct-2016.) $)
|
|
issetf $p |- ( A e. _V <-> E. x x = A ) $=
|
|
( vy cvv wcel cv wceq wex isset nfeq2 nfv eqeq1 cbvex bitri ) BEFDGZBHZDI
|
|
AGZBHZAIDBJQSDAAPBCKSDLPRBMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
isseti.1 $e |- A e. _V $.
|
|
$( A way to say " ` A ` is a set" (inference rule). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
isseti $p |- E. x x = A $=
|
|
( cvv wcel cv wceq wex isset mpbi ) BDEAFBGAHCABIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
issetri.1 $e |- E. x x = A $.
|
|
$( A way to say " ` A ` is a set" (inference rule). (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
issetri $p |- A e. _V $=
|
|
( cvv wcel cv wceq wex isset mpbir ) BDEAFBGAHCABIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( If a class is a member of another class, it is a set. Theorem 6.12 of
|
|
[Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by
|
|
Andrew Salmon, 8-Jun-2011.) $)
|
|
elex $p |- ( A e. B -> A e. _V ) $=
|
|
( vx cv wceq wcel wa wex cvv exsimpl df-clel isset 3imtr4i ) CDZAEZNBFZGC
|
|
HOCHABFAIFOPCJCABKCALM $.
|
|
$}
|
|
|
|
${
|
|
elisseti.1 $e |- A e. B $.
|
|
$( If a class is a member of another class, it is a set. (Contributed by
|
|
NM, 11-Jun-1994.) $)
|
|
elexi $p |- A e. _V $=
|
|
( wcel cvv elex ax-mp ) ABDAEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( An element of a class exists. (Contributed by NM, 1-May-1995.) $)
|
|
elisset $p |- ( A e. V -> E. x x = A ) $=
|
|
( wcel cvv cv wceq wex elex isset sylib ) BCDBEDAFBGAHBCIABJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( If two classes each contain another class, then both contain some set.
|
|
(Contributed by Alan Sare, 24-Oct-2011.) $)
|
|
elex22 $p |- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) ) $=
|
|
( wcel wa cv wceq wi wal eleq1a anim12ii alrimiv elisset adantr exim sylc
|
|
wex ) BCEZBDEZFZAGZBHZUBCEZUBDEZFZIZAJUCARZUFARUAUGASUCUDTUEBCUBKBDUBKLMS
|
|
UHTABCNOUCUFAPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( If a class contains another class, then it contains some set.
|
|
(Contributed by Alan Sare, 25-Sep-2011.) $)
|
|
elex2 $p |- ( A e. B -> E. x x e. B ) $=
|
|
( wcel cv wceq wi wal wex eleq1a alrimiv elisset exim sylc ) BCDZAEZBFZPC
|
|
DZGZAHQAIRAIOSABCPJKABCLQRAMN $.
|
|
$}
|
|
|
|
$( A universal quantifier restricted to the universe is unrestricted.
|
|
(Contributed by NM, 26-Mar-2004.) $)
|
|
ralv $p |- ( A. x e. _V ph <-> A. x ph ) $=
|
|
( cvv wral cv wcel wi wal df-ral vex a1bi albii bitr4i ) ABCDBECFZAGZBHABHA
|
|
BCIAOBNABJKLM $.
|
|
|
|
$( An existential quantifier restricted to the universe is unrestricted.
|
|
(Contributed by NM, 26-Mar-2004.) $)
|
|
rexv $p |- ( E. x e. _V ph <-> E. x ph ) $=
|
|
( cvv wrex cv wcel wa wex df-rex vex biantrur exbii bitr4i ) ABCDBECFZAGZBH
|
|
ABHABCIAOBNABJKLM $.
|
|
|
|
$( A uniqueness quantifier restricted to the universe is unrestricted.
|
|
(Contributed by NM, 1-Nov-2010.) $)
|
|
reuv $p |- ( E! x e. _V ph <-> E! x ph ) $=
|
|
( cvv wreu cv wcel wa weu df-reu vex biantrur eubii bitr4i ) ABCDBECFZAGZBH
|
|
ABHABCIAOBNABJKLM $.
|
|
|
|
$( A uniqueness quantifier restricted to the universe is unrestricted.
|
|
(Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmov $p |- ( E* x e. _V ph <-> E* x ph ) $=
|
|
( cvv wrmo cv wcel wa wmo df-rmo vex biantrur mobii bitr4i ) ABCDBECFZAGZBH
|
|
ABHABCIAOBNABJKLM $.
|
|
|
|
$( A class abstraction restricted to the universe is unrestricted.
|
|
(Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
rabab $p |- { x e. _V | ph } = { x | ph } $=
|
|
( cvv crab cv wcel wa cab df-rab vex biantrur abbii eqtr4i ) ABCDBECFZAGZBH
|
|
ABHABCIAOBNABJKLM $.
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Commutation of restricted and unrestricted universal quantifiers.
|
|
(Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
ralcom4 $p |- ( A. x e. A A. y ph <-> A. y A. x e. A ph ) $=
|
|
( cvv wral wal ralcom ralv ralbii 3bitr3i ) ACEFZBDFABDFZCEFACGZBDFMCGABC
|
|
DEHLNBDACIJMCIK $.
|
|
|
|
$( Commutation of restricted and unrestricted existential quantifiers.
|
|
(Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
rexcom4 $p |- ( E. x e. A E. y ph <-> E. y E. x e. A ph ) $=
|
|
( cvv wrex wex rexcom rexv rexbii 3bitr3i ) ACEFZBDFABDFZCEFACGZBDFMCGABC
|
|
DEHLNBDACIJMCIK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d x y $. $d ph x $.
|
|
$( Specialized existential commutation lemma. (Contributed by Jeff Madsen,
|
|
1-Jun-2011.) $)
|
|
rexcom4a $p |- ( E. x E. y e. A ( ph /\ ps )
|
|
<-> E. y e. A ( ph /\ E. x ps ) ) $=
|
|
( wa wrex wex rexcom4 19.42v rexbii bitr3i ) ABFZDEGCHMCHZDEGABCHFZDEGMDC
|
|
EINODEABCJKL $.
|
|
|
|
$d B x $.
|
|
rexcom4b.1 $e |- B e. _V $.
|
|
$( Specialized existential commutation lemma. (Contributed by Jeff Madsen,
|
|
1-Jun-2011.) $)
|
|
rexcom4b $p |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph ) $=
|
|
( cv wceq wa wrex wex rexcom4a isseti biantru rexbii bitr4i ) ABGEHZICDJB
|
|
KAQBKZIZCDJACDJAQBCDLASCDRABEFMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Closed theorem version of ~ ceqsalg . (Contributed by NM,
|
|
28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) $)
|
|
ceqsalt $p |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V )
|
|
-> ( A. x ( x = A -> ph ) <-> ps ) ) $=
|
|
( wnf cv wceq wb wi wal wcel w3a wex elisset 3ad2ant3 bi1 imim3i 3ad2ant2
|
|
3ad2ant1 al2imi 19.23t sylibd mpid imim2i com23 alimi 19.21t mpbid impbid
|
|
bi2 ) BCFZCGDHZABIZJZCKZDELZMZUMAJZCKZBURUTUMCNZBUQULVAUPCDEOPURUTUMBJZCK
|
|
ZVABJZUPULUTVCJUQUOUSVBCUNABUMABQRUASULUPVCVDIUQUMBCUBTUCUDURBUSJZCKZBUTJ
|
|
ZUPULVFUQUOVECUOUMBAUNBAJUMABUKUEUFUGSULUPVFVGIUQBUSCUHTUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Restricted quantifier version of ~ ceqsalt . (Contributed by NM,
|
|
28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) $)
|
|
ceqsralt $p |- ( ( F/ x ps
|
|
/\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B )
|
|
-> ( A. x e. B ( x = A -> ph ) <-> ps ) ) $=
|
|
( wnf cv wceq wb wi wal wcel w3a wral df-ral eleq1 pm5.32ri imbi1i impexp
|
|
wa 3bitr3i albii a1i syl5bb 19.21v syl6bb biimt 3ad2ant3 ceqsalt 3bitr2d
|
|
) BCFZCGZDHZABIJCKZDELZMZUMAJZCENZUOUQCKZJZUSBUPURUOUQJZCKZUTURULELZUQJZC
|
|
KZUPVBUQCEOVEVBIUPVDVACVCUMTZAJUOUMTZAJVDVAVFVGAUMVCUOULDEPQRVCUMASUOUMAS
|
|
UAUBUCUDUOUQCUEUFUOUKUSUTIUNUOUSUGUHABCDEUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ceqsalg.1 $e |- F/ x ps $.
|
|
ceqsalg.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A representation of explicit substitution of a class for a variable,
|
|
inferred from an implicit substitution hypothesis. (Contributed by NM,
|
|
29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
ceqsalg $p |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) ) $=
|
|
( wcel cv wceq wi wal wex elisset nfa1 biimpd a2i sps exlimd syl5com
|
|
biimprcd alrimi impbid1 ) DEHZCIDJZAKZCLZBUDUECMUGBCDENUGUEBCUFCOFUFUEBKC
|
|
UEABUEABGPQRSTBUFCFUEABGUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ceqsal.1 $e |- F/ x ps $.
|
|
ceqsal.2 $e |- A e. _V $.
|
|
ceqsal.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A representation of explicit substitution of a class for a variable,
|
|
inferred from an implicit substitution hypothesis. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
ceqsal $p |- ( A. x ( x = A -> ph ) <-> ps ) $=
|
|
( cvv wcel cv wceq wi wal wb ceqsalg ax-mp ) DHICJDKALCMBNFABCDHEGOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
ceqsalv.1 $e |- A e. _V $.
|
|
ceqsalv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A representation of explicit substitution of a class for a variable,
|
|
inferred from an implicit substitution hypothesis. (Contributed by NM,
|
|
18-Aug-1993.) $)
|
|
ceqsalv $p |- ( A. x ( x = A -> ph ) <-> ps ) $=
|
|
( nfv ceqsal ) ABCDBCGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
ceqsralv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Restricted quantifier version of ~ ceqsalv . (Contributed by NM,
|
|
21-Jun-2013.) $)
|
|
ceqsralv $p |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) ) $=
|
|
( wnf cv wceq wb wi wal wcel wral nfv ax-gen ceqsralt mp3an12 ) BCGCHDIZA
|
|
BJKZCLDEMSAKCENBJBCOTCFPABCDEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $.
|
|
gencl.1 $e |- ( th <-> E. x ( ch /\ A = B ) ) $.
|
|
gencl.2 $e |- ( A = B -> ( ph <-> ps ) ) $.
|
|
gencl.3 $e |- ( ch -> ph ) $.
|
|
$( Implicit substitution for class with embedded variable. (Contributed by
|
|
NM, 17-May-1996.) $)
|
|
gencl $p |- ( th -> ps ) $=
|
|
( wceq wa wex syl5ib impcom exlimiv sylbi ) DCFGKZLZEMBHSBERCBCARBJINOPQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x R $. $d x ps $. $d y C $. $d y S $. $d y ch $.
|
|
2gencl.1 $e |- ( C e. S <-> E. x e. R A = C ) $.
|
|
2gencl.2 $e |- ( D e. S <-> E. y e. R B = D ) $.
|
|
2gencl.3 $e |- ( A = C -> ( ph <-> ps ) ) $.
|
|
2gencl.4 $e |- ( B = D -> ( ps <-> ch ) ) $.
|
|
2gencl.5 $e |- ( ( x e. R /\ y e. R ) -> ph ) $.
|
|
$( Implicit substitution for class with embedded variable. (Contributed by
|
|
NM, 17-May-1996.) $)
|
|
2gencl $p |- ( ( C e. S /\ D e. S ) -> ch ) $=
|
|
( wcel wi cv wceq wrex wa wex df-rex bitri imbi2d ex gencl com12 impcom )
|
|
IKQZHKQZCULBRULCRESJQZUKEGIUKGITZEJUAUMUNUBEUCMUNEJUDUEUNBCULOUFULUMBUMAR
|
|
UMBRDSJQZULDFHULFHTZDJUAUOUPUBDUCLUPDJUDUEUPABUMNUFUOUMAPUGUHUIUHUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z D $. $d z F $. $d x y R $. $d y z S $. $d x ps $.
|
|
$d y ch $. $d z th $.
|
|
3gencl.1 $e |- ( D e. S <-> E. x e. R A = D ) $.
|
|
3gencl.2 $e |- ( F e. S <-> E. y e. R B = F ) $.
|
|
3gencl.3 $e |- ( G e. S <-> E. z e. R C = G ) $.
|
|
3gencl.4 $e |- ( A = D -> ( ph <-> ps ) ) $.
|
|
3gencl.5 $e |- ( B = F -> ( ps <-> ch ) ) $.
|
|
3gencl.6 $e |- ( C = G -> ( ch <-> th ) ) $.
|
|
3gencl.7 $e |- ( ( x e. R /\ y e. R /\ z e. R ) -> ph ) $.
|
|
$( Implicit substitution for class with embedded variable. (Contributed by
|
|
NM, 17-May-1996.) $)
|
|
3gencl $p |- ( ( D e. S /\ F e. S /\ G e. S ) -> th ) $=
|
|
( wcel wa wi wceq wrex wex df-rex bitri imbi2d 3expia 2gencl com12 3impia
|
|
cv gencl ) KMUCZNMUCZOMUCZDUTURUSUDZDVACUEVADUEGUPLUCZUTGJOUTJOUFZGLUGVBV
|
|
CUDGUHRVCGLUIUJVCCDVAUAUKVAVBCVBAUEVBBUEVBCUEEFHIKNLMPQHKUFABVBSUKINUFBCV
|
|
BTUKEUPLUCFUPLUCVBAUBULUMUNUQUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
cgsexg.1 $e |- ( x = A -> ch ) $.
|
|
cgsexg.2 $e |- ( ch -> ( ph <-> ps ) ) $.
|
|
$( Implicit substitution inference for general classes. (Contributed by
|
|
NM, 26-Aug-2007.) $)
|
|
cgsexg $p |- ( A e. V ->
|
|
( E. x ( ch /\ ph ) <-> ps ) ) $=
|
|
( wcel wa wex biimpa exlimiv cv wceq elisset eximi syl biimprcd ancld
|
|
eximdv syl5com impbid2 ) EFIZCAJZDKZBUEBDCABHLMUDCDKZBUFUDDNEOZDKUGDEFPUH
|
|
CDGQRBCUEDBCACABHSTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y ps $. $d x y A $. $d x y B $.
|
|
cgsex2g.1 $e |- ( ( x = A /\ y = B ) -> ch ) $.
|
|
cgsex2g.2 $e |- ( ch -> ( ph <-> ps ) ) $.
|
|
$( Implicit substitution inference for general classes. (Contributed by
|
|
NM, 26-Jul-1995.) $)
|
|
cgsex2g $p |- ( ( A e. V /\ B e. W ) ->
|
|
( E. x E. y ( ch /\ ph ) <-> ps ) ) $=
|
|
( wcel wa wex biimpa exlimivv cv wceq elisset anim12i eeanv sylibr 2eximi
|
|
syl biimprcd ancld 2eximdv syl5com impbid2 ) FHLZGILZMZCAMZENDNZBUMBDECAB
|
|
KOPULCENDNZBUNULDQFRZEQGRZMZENDNZUOULUPDNZUQENZMUSUJUTUKVADFHSEGISTUPUQDE
|
|
UAUBURCDEJUCUDBCUMDEBCACABKUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d x y z w D $.
|
|
$d x y z w ps $.
|
|
cgsex4g.1 $e |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch ) $.
|
|
cgsex4g.2 $e |- ( ch -> ( ph <-> ps ) ) $.
|
|
$( An implicit substitution inference for 4 general classes. (Contributed
|
|
by NM, 5-Aug-1995.) $)
|
|
cgsex4g $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) ->
|
|
( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) ) $=
|
|
( wcel wa wex cv wceq biimpa exlimivv elisset anim12i eeanv sylibr ee4anv
|
|
2eximi syl biimprcd ancld 2eximdv syl5com impbid2 ) HLPZIMPZQZJLPZKMPZQZQ
|
|
ZCAQZGRFRZERDRZBVCBDEVBBFGCABOUAUBUBVACGRFRZERDRZBVDVADSHTZESITZQZFSJTZGS
|
|
KTZQZQZGRFRZERDRZVFVAVIERDRZVLGRFRZQVOUQVPUTVQUQVGDRZVHERZQVPUOVRUPVSDHLU
|
|
CEIMUCUDVGVHDEUEUFUTVJFRZVKGRZQVQURVTUSWAFJLUCGKMUCUDVJVKFGUEUFUDVIVLDEFG
|
|
UGUFVNVEDEVMCFGNUHUHUIBVEVCDEBCVBFGBCACABOUJUKULULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ceqsex.1 $e |- F/ x ps $.
|
|
ceqsex.2 $e |- A e. _V $.
|
|
ceqsex.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Elimination of an existential quantifier, using implicit substitution.
|
|
(Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro,
|
|
10-Oct-2016.) $)
|
|
ceqsex $p |- ( E. x ( x = A /\ ph ) <-> ps ) $=
|
|
( cv wceq wa wex biimpa exlimi wi wal biimprcd alrimi isseti exintr ee10
|
|
impbii ) CHDIZAJZCKZBUCBCEUBABGLMBUBANZCOUBCKUDBUECEUBABGPQCDFRUBACSTUA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
ceqsexv.1 $e |- A e. _V $.
|
|
ceqsexv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Elimination of an existential quantifier, using implicit substitution.
|
|
(Contributed by NM, 2-Mar-1995.) $)
|
|
ceqsexv $p |- ( E. x ( x = A /\ ph ) <-> ps ) $=
|
|
( nfv ceqsex ) ABCDBCGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
ceqsex2.1 $e |- F/ x ps $.
|
|
ceqsex2.2 $e |- F/ y ch $.
|
|
ceqsex2.3 $e |- A e. _V $.
|
|
ceqsex2.4 $e |- B e. _V $.
|
|
ceqsex2.5 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex2.6 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( Elimination of two existential quantifiers, using implicit
|
|
substitution. (Contributed by Scott Fenton, 7-Jun-2006.) $)
|
|
ceqsex2 $p |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch ) $=
|
|
( cv wceq w3a wex wa exbii ceqsex 3anass 19.42v nfan anbi2d exbidv 3bitri
|
|
bitri nfv nfex ) DNFOZENGOZAPZEQZDQUJUKARZEQZRZDQUKBRZEQZCUMUPDUMUJUNRZEQ
|
|
UPULUSEUJUKAUASUJUNEUBUGSUOURDFUQDEUKBDUKDUHHUCUIJUJUNUQEUJABUKLUDUETBCEG
|
|
IKMTUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x ps $. $d y ch $.
|
|
ceqsex2v.1 $e |- A e. _V $.
|
|
ceqsex2v.2 $e |- B e. _V $.
|
|
ceqsex2v.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex2v.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( Elimination of two existential quantifiers, using implicit
|
|
substitution. (Contributed by Scott Fenton, 7-Jun-2006.) $)
|
|
ceqsex2v $p |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch ) $=
|
|
( nfv ceqsex2 ) ABCDEFGBDLCELHIJKM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x ps $. $d y ch $.
|
|
$d z th $.
|
|
ceqsex3v.1 $e |- A e. _V $.
|
|
ceqsex3v.2 $e |- B e. _V $.
|
|
ceqsex3v.3 $e |- C e. _V $.
|
|
ceqsex3v.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex3v.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ceqsex3v.6 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
$( Elimination of three existential quantifiers, using implicit
|
|
substitution. (Contributed by NM, 16-Aug-2011.) $)
|
|
ceqsex3v $p |- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph )
|
|
<-> th ) $=
|
|
( cv wceq wa wex anass 3anass anbi1i df-3an anbi2i 3bitr4i 2exbii 19.42vv
|
|
w3a bitri exbii 3anbi3d 2exbidv ceqsexv ceqsex2v ) EQHRZFQIRZGQJRZUIZASZG
|
|
TFTZETUPUQURAUIZGTFTZSZETZDVAVDEVAUPVBSZGTFTVDUTVFFGUPUQURSZSZASUPVGASZSU
|
|
TVFUPVGAUAUSVHAUPUQURUBUCVBVIUPUQURAUDUEUFUGUPVBFGUHUJUKVEUQURBUIZGTFTZDV
|
|
CVKEHKUPVBVJFGUPABUQURNULUMUNBCDFGIJLMOPUOUJUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d x y z w D $.
|
|
$d x ps $. $d y ch $. $d z th $. $d w ta $.
|
|
ceqsex4v.1 $e |- A e. _V $.
|
|
ceqsex4v.2 $e |- B e. _V $.
|
|
ceqsex4v.3 $e |- C e. _V $.
|
|
ceqsex4v.4 $e |- D e. _V $.
|
|
ceqsex4v.7 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex4v.8 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ceqsex4v.9 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ceqsex4v.10 $e |- ( w = D -> ( th <-> ta ) ) $.
|
|
$( Elimination of four existential quantifiers, using implicit
|
|
substitution. (Contributed by NM, 23-Sep-2011.) $)
|
|
ceqsex4v $p |- ( E. x E. y E. z E. w
|
|
( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ta ) $=
|
|
( wceq w3a wex 19.42vv 3anass df-3an anbi2i bitr4i 2exbii 3bitr4i 3anbi3d
|
|
cv wa 2exbidv ceqsex2v 3bitri ) FUMJUBZGUMKUBZUNZHUMLUBZIUMMUBZUNZAUCZIUD
|
|
HUDZGUDFUDURUSVAVBAUCZIUDHUDZUCZGUDFUDVAVBCUCZIUDHUDZEVEVHFGUTVFUNZIUDHUD
|
|
UTVGUNVEVHUTVFHIUEVDVKHIVDUTVCAUNZUNVKUTVCAUFVFVLUTVAVBAUGUHUIUJURUSVGUGU
|
|
KUJVGVAVBBUCZIUDHUDVJFGJKNOURVFVMHIURABVAVBRULUOUSVMVIHIUSBCVAVBSULUOUPCD
|
|
EHILMPQTUAUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u D $. $d x y z w v u E $. $d x y z w v u F $. $d x ps $.
|
|
$d y ch $. $d z th $. $d w ta $. $d v et $. $d u ze $.
|
|
ceqsex6v.1 $e |- A e. _V $.
|
|
ceqsex6v.2 $e |- B e. _V $.
|
|
ceqsex6v.3 $e |- C e. _V $.
|
|
ceqsex6v.4 $e |- D e. _V $.
|
|
ceqsex6v.5 $e |- E e. _V $.
|
|
ceqsex6v.6 $e |- F e. _V $.
|
|
ceqsex6v.7 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex6v.8 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ceqsex6v.9 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ceqsex6v.10 $e |- ( w = D -> ( th <-> ta ) ) $.
|
|
ceqsex6v.11 $e |- ( v = E -> ( ta <-> et ) ) $.
|
|
ceqsex6v.12 $e |- ( u = F -> ( et <-> ze ) ) $.
|
|
$( Elimination of six existential quantifiers, using implicit
|
|
substitution. (Contributed by NM, 21-Sep-2011.) $)
|
|
ceqsex6v $p |- ( E. x E. y E. z E. w E. v E. u
|
|
( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph )
|
|
<-> ze ) $=
|
|
( cv wceq w3a wex wa 3anass 3exbii 19.42vvv bitri anbi2d 3exbidv ceqsex3v
|
|
) HULNUMZIULOUMZJULPUMZUNZKULQUMLULRUMMULSUMUNZAUNZMUOLUOKUOZJUOIUOHUOVGV
|
|
HAUPZMUOLUOKUOZUPZJUOIUOHUOZGVJVMHIJVJVGVKUPZMUOLUOKUOVMVIVOKLMVGVHAUQURV
|
|
GVKKLMUSUTURVNVHDUPZMUOLUOKUOZGVLVHBUPZMUOLUOKUOVHCUPZMUOLUOKUOVQHIJNOPTU
|
|
AUBVDVKVRKLMVDABVHUFVAVBVEVRVSKLMVEBCVHUGVAVBVFVSVPKLMVFCDVHUHVAVBVCDEFGK
|
|
LMQRSUCUDUEUIUJUKVCUTUT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u t s A $. $d x y z w v u t s B $. $d x y z w v u t s C $.
|
|
$d x y z w v u t s D $. $d x y z w v u t s E $. $d x y z w v u t s F $.
|
|
$d x y z w v u t s G $. $d x y z w v u t s H $. $d x ps $. $d y ch $.
|
|
$d z th $. $d w ta $. $d v et $. $d u ze $. $d t si $. $d s rh $.
|
|
ceqsex8v.1 $e |- A e. _V $.
|
|
ceqsex8v.2 $e |- B e. _V $.
|
|
ceqsex8v.3 $e |- C e. _V $.
|
|
ceqsex8v.4 $e |- D e. _V $.
|
|
ceqsex8v.5 $e |- E e. _V $.
|
|
ceqsex8v.6 $e |- F e. _V $.
|
|
ceqsex8v.7 $e |- G e. _V $.
|
|
ceqsex8v.8 $e |- H e. _V $.
|
|
ceqsex8v.9 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsex8v.10 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ceqsex8v.11 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ceqsex8v.12 $e |- ( w = D -> ( th <-> ta ) ) $.
|
|
ceqsex8v.13 $e |- ( v = E -> ( ta <-> et ) ) $.
|
|
ceqsex8v.14 $e |- ( u = F -> ( et <-> ze ) ) $.
|
|
ceqsex8v.15 $e |- ( t = G -> ( ze <-> si ) ) $.
|
|
ceqsex8v.16 $e |- ( s = H -> ( si <-> rh ) ) $.
|
|
$( Elimination of eight existential quantifiers, using implicit
|
|
substitution. (Contributed by NM, 23-Sep-2011.) $)
|
|
ceqsex8v $p |- ( E. x E. y E. z E. w E. v E. u E. t E. s
|
|
( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) )
|
|
/\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> rh ) $=
|
|
( cv wceq wa w3a 19.42vv 2exbii bitri 3anass df-3an anbi2i bitr4i 3bitr4i
|
|
wex 3anbi3d 4exbidv ceqsex4v ) JVBQVCZKVBRVCZVDZLVBSVCZMVBTVCZVDZVDZNVBUA
|
|
VCOVBUBVCVDZPVBUCVCUEVBUDVCVDZVDZAVEZUEVNPVNZOVNNVNZMVNLVNZKVNJVNVTWCWEWF
|
|
AVEZUEVNPVNZOVNNVNZVEZMVNLVNZKVNJVNZIWKWPJKWJWOLMWDWLVDZUEVNPVNZOVNNVNZWD
|
|
WNVDZWJWOWTWDWMVDZOVNNVNXAWSXBNOWDWLPUEVFVGWDWMNOVFVHWIWSNOWHWRPUEWHWDWGA
|
|
VDZVDWRWDWGAVIWLXCWDWEWFAVJVKVLVGVGVTWCWNVJVMVGVGWQWEWFEVEZUEVNPVNOVNNVNZ
|
|
IWNWEWFBVEZUEVNPVNOVNNVNWEWFCVEZUEVNPVNOVNNVNWEWFDVEZUEVNPVNOVNNVNXEJKLMQ
|
|
RSTUFUGUHUIVRWLXFNOPUEVRABWEWFUNVOVPVSXFXGNOPUEVSBCWEWFUOVOVPWAXGXHNOPUEW
|
|
ACDWEWFUPVOVPWBXHXDNOPUEWBDEWEWFUQVOVPVQEFGHINOPUEUAUBUCUDUJUKULUMURUSUTV
|
|
AVQVHVH $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x th $. $d y ch $. $d y A $.
|
|
gencbvex.1 $e |- A e. _V $.
|
|
gencbvex.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
|
|
gencbvex.3 $e |- ( A = y -> ( ch <-> th ) ) $.
|
|
gencbvex.4 $e |- ( th <-> E. x ( ch /\ A = y ) ) $.
|
|
$( Change of bound variable using implicit substitution. (Contributed by
|
|
NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
gencbvex $p |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) ) $=
|
|
( cv wceq wa wex excom wb anbi12d bicomd exbii eqcoms ceqsexv simpr eqcom
|
|
19.41v biimpi adantl eximi sylbi adantr ancri impbii bitri 3bitr3i ) FLZG
|
|
MZDBNZNZFOZEOUREOZFOCANZEOUQFOUREFPUSVAEUQVAFGHUQVAQGUOGUOMZVAUQVBCDABJIR
|
|
SUAUBTUTUQFUTUPEOZUQNZUQUPUQEUEVDUQVCUQUCUQVCDVCBDCVBNZEOVCKVEUPEVBUPCVBU
|
|
PGUOUDUFUGUHUIUJUKULUMTUN $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x th $. $d y ch $. $d y A $.
|
|
gencbvex2.1 $e |- A e. _V $.
|
|
gencbvex2.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
|
|
gencbvex2.3 $e |- ( A = y -> ( ch <-> th ) ) $.
|
|
gencbvex2.4 $e |- ( th -> E. x ( ch /\ A = y ) ) $.
|
|
$( Restatement of ~ gencbvex with weaker hypotheses. (Contributed by
|
|
Jeffrey Hankins, 6-Dec-2006.) $)
|
|
gencbvex2 $p |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) ) $=
|
|
( cv wceq wa wex biimpac exlimiv impbii gencbvex ) ABCDEFGHIJDCGFLMZNZEOK
|
|
UADETCDJPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x th $. $d y ch $. $d y A $.
|
|
gencbval.1 $e |- A e. _V $.
|
|
gencbval.2 $e |- ( A = y -> ( ph <-> ps ) ) $.
|
|
gencbval.3 $e |- ( A = y -> ( ch <-> th ) ) $.
|
|
gencbval.4 $e |- ( th <-> E. x ( ch /\ A = y ) ) $.
|
|
$( Change of bound variable using implicit substitution. (Contributed by
|
|
NM, 17-May-1996.) $)
|
|
gencbval $p |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) ) $=
|
|
( wi wal wn wa wex cv wceq notbid exanali gencbvex 3bitr3i con4bii ) CALE
|
|
MZDBLFMZCANZOEPDBNZOFPUDNUENUFUGCDEFGHGFQRABISJKUACAETDBFTUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d x y $.
|
|
sbhypf.1 $e |- F/ x ps $.
|
|
sbhypf.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Introduce an explicit substitution into an implicit substitution
|
|
hypothesis. See also ~ csbhypf . (Contributed by Raph Levien,
|
|
10-Apr-2004.) $)
|
|
sbhypf $p |- ( y = A -> ( [ y / x ] ph <-> ps ) ) $=
|
|
( cv wceq wa wex wsb wb vex eqeq1 ceqsexv nfs1v nfbi sbequ12 bicomd
|
|
sylan9bb exlimi sylbir ) DHZEIZCHZUDIZUFEIZJZCKACDLZBMZUHUECUDDNUFUDEOPUI
|
|
UKCUJBCACDQFRUGUJAUHBUGAUJACDSTGUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d x z $.
|
|
$( Closed theorem form of ~ vtoclgf . (Contributed by NM, 17-Feb-2013.)
|
|
(Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
vtoclgft $p |- ( ( ( F/_ x A /\ F/ x ps )
|
|
/\ ( A. x ( x = A -> ( ph <-> ps ) )
|
|
/\ A. x ph ) /\ A e. V ) -> ps ) $=
|
|
( vz wcel wnfc wnf wa cv wceq wb wi wal cvv elex w3a wex mpbid elisset id
|
|
3ad2ant3 nfnfc1 nfcvd nfeqd eqeq1 a1i cbvexd 3adant3 bi1 imim2i com23 imp
|
|
ad2antrr alanimi 3ad2ant2 simp1r 19.23t syl mpd syl3an3 ) DEGCDHZBCIZJZCK
|
|
ZDLZABMZNZCOACOJZDPGZBDEQVEVJVKRZVGCSZBVLFKZDLZFSZVMVKVEVPVJFDPUAUCVEVJVP
|
|
VMMZVKVCVQVDVJVCVOVGFCCDUDVCCVNDVCCVNUEVCUBUFVNVFLVOVGMNVCVNVFDUGUHUIUOUJ
|
|
TVLVGBNZCOZVMBNZVJVEVSVKVIAVRCVIAVRVIVGABVHABNVGABUKULUMUNUPUQVLVDVSVTMVC
|
|
VDVJVKURVGBCUSUTTVAVB $.
|
|
$}
|
|
|
|
${
|
|
vtocld.1 $e |- ( ph -> A e. V ) $.
|
|
vtocld.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
vtocld.3 $e |- ( ph -> ps ) $.
|
|
${
|
|
vtocldf.4 $e |- F/ x ph $.
|
|
vtocldf.5 $e |- ( ph -> F/_ x A ) $.
|
|
vtocldf.6 $e |- ( ph -> F/ x ch ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed
|
|
by Mario Carneiro, 15-Oct-2016.) $)
|
|
vtocldf $p |- ( ph -> ch ) $=
|
|
( wnfc wnf cv wceq wb wi wal alrimi wcel ex vtoclgft syl221anc ) ADEMCD
|
|
NDOEPZBCQZRZDSBDSEFUACKLAUGDJAUEUFHUBTABDJITGBCDEFUCUD $.
|
|
$}
|
|
|
|
$d x A $. $d x ph $. $d x ch $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
Mario Carneiro, 15-Oct-2016.) $)
|
|
vtocld $p |- ( ph -> ch ) $=
|
|
( nfv nfcvd nfvd vtocldf ) ABCDEFGHIADJADEKACDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
vtoclf.1 $e |- F/ x ps $.
|
|
vtoclf.2 $e |- A e. _V $.
|
|
vtoclf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclf.4 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable. This is a
|
|
generalization of ~ chvar . (Contributed by NM, 30-Aug-1993.) $)
|
|
vtoclf $p |- ps $=
|
|
( cv wceq wex wi isseti biimpd eximi ax-mp 19.36i mpg ) ABCABCECIDJZCKABL
|
|
ZCKCDFMSTCSABGNOPQHR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
vtocl.1 $e |- A e. _V $.
|
|
vtocl.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl.3 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 30-Aug-1993.) $)
|
|
vtocl $p |- ps $=
|
|
( nfv vtoclf ) ABCDBCHEFGI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
vtocl2.1 $e |- A e. _V $.
|
|
vtocl2.2 $e |- B e. _V $.
|
|
vtocl2.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
vtocl2.4 $e |- ph $.
|
|
$( Implicit substitution of classes for setvar variables. (Contributed by
|
|
NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
vtocl2 $p |- ps $=
|
|
( wal wi wex cv wceq isseti wa eeanv biimpd 2eximi sylbir 19.36v 19.36aiv
|
|
mp2an exbii mpbi ax-gen mpg ) ADKZBCUIBCABLZDMZCMZUIBLZCMCNEOZCMZDNFOZDMZ
|
|
ULCEGPDFHPUOUQQUNUPQZDMCMULUNUPCDRURUJCDURABISTUAUDUKUMCABDUBUEUFUCADJUGU
|
|
H $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z ps $.
|
|
vtocl3.1 $e |- A e. _V $.
|
|
vtocl3.2 $e |- B e. _V $.
|
|
vtocl3.3 $e |- C e. _V $.
|
|
vtocl3.4 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
vtocl3.5 $e |- ph $.
|
|
$( Implicit substitution of classes for setvar variables. (Contributed by
|
|
NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
vtocl3 $p |- ps $=
|
|
( wal wi wex cv wceq isseti w3a eeeanv biimpd 2eximi sylbir 19.36v 2exbii
|
|
eximi mp3an mpbi exbii 19.36aiv gen2 mpg ) AENZDNZBCUOBCUNBOZDPZCPZUOBOZC
|
|
PABOZEPZDPCPZURCQFRZCPZDQGRZDPZEQHRZEPZVBCFISDGJSEHKSVDVFVHTVCVEVGTZEPZDP
|
|
CPVBVCVEVGCDEUAVJVACDVIUTEVIABLUBUGUCUDUHVAUPCDABEUEUFUIUQUSCUNBDUEUJUIUK
|
|
ADEMULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ch $. $d x th $.
|
|
vtoclb.1 $e |- A e. _V $.
|
|
vtoclb.2 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
vtoclb.3 $e |- ( x = A -> ( ps <-> th ) ) $.
|
|
vtoclb.4 $e |- ( ph <-> ps ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 23-Dec-1993.) $)
|
|
vtoclb $p |- ( ch <-> th ) $=
|
|
( wb cv wceq bibi12d vtocl ) ABKCDKEFGELFMACBDHINJO $.
|
|
$}
|
|
|
|
${
|
|
vtoclgf.1 $e |- F/_ x A $.
|
|
vtoclgf.2 $e |- F/ x ps $.
|
|
vtoclgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclgf.4 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable, with
|
|
bound-variable hypotheses in place of distinct variable restrictions.
|
|
(Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro,
|
|
10-Oct-2016.) $)
|
|
vtoclgf $p |- ( A e. V -> ps ) $=
|
|
( wcel cvv elex cv wceq wex issetf mpbii exlimi sylbi syl ) DEJDKJZBDELUA
|
|
CMDNZCOBCDFPUBBCGUBABIHQRST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
vtoclg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclg.2 $e |- ph $.
|
|
$( Implicit substitution of a class expression for a setvar variable.
|
|
(Contributed by NM, 17-Apr-1995.) $)
|
|
vtoclg $p |- ( A e. V -> ps ) $=
|
|
( nfcv nfv vtoclgf ) ABCDECDHBCIFGJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ch $. $d x th $.
|
|
vtoclbg.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
vtoclbg.2 $e |- ( x = A -> ( ps <-> th ) ) $.
|
|
vtoclbg.3 $e |- ( ph <-> ps ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 29-Apr-1994.) $)
|
|
vtoclbg $p |- ( A e. V -> ( ch <-> th ) ) $=
|
|
( wb cv wceq bibi12d vtoclg ) ABKCDKEFGELFMACBDHINJO $.
|
|
$}
|
|
|
|
${
|
|
vtocl2gf.1 $e |- F/_ x A $.
|
|
vtocl2gf.2 $e |- F/_ y A $.
|
|
vtocl2gf.3 $e |- F/_ y B $.
|
|
vtocl2gf.4 $e |- F/ x ps $.
|
|
vtocl2gf.5 $e |- F/ y ch $.
|
|
vtocl2gf.6 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl2gf.7 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl2gf.8 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 25-Apr-1995.) $)
|
|
vtocl2gf $p |- ( ( A e. V /\ B e. W ) -> ch ) $=
|
|
( wcel cvv wi elex nfel1 nfim cv wceq imbi2d vtoclgf mpan9 ) FHRFSRZGIRCF
|
|
HUAUIBTUICTEGILUICEEFSKUBNUCEUDGUEBCUIPUFABDFSJMOQUGUGUH $.
|
|
$}
|
|
|
|
${
|
|
vtocl3gf.a $e |- F/_ x A $.
|
|
vtocl3gf.b $e |- F/_ y A $.
|
|
vtocl3gf.c $e |- F/_ z A $.
|
|
vtocl3gf.d $e |- F/_ y B $.
|
|
vtocl3gf.e $e |- F/_ z B $.
|
|
vtocl3gf.f $e |- F/_ z C $.
|
|
vtocl3gf.1 $e |- F/ x ps $.
|
|
vtocl3gf.2 $e |- F/ y ch $.
|
|
vtocl3gf.3 $e |- F/ z th $.
|
|
vtocl3gf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl3gf.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl3gf.6 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
vtocl3gf.7 $e |- ph $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) $)
|
|
vtocl3gf $p |- ( ( A e. V /\ B e. W /\ C e. X ) -> th ) $=
|
|
( wcel cvv wa elex wi nfel1 nfim wceq imbi2d vtoclgf vtocl2gf mpan9 3impb
|
|
cv ) HKUGZILUGZJMUGZDVAHUHUGZVBVCUIDHKUJVDBUKVDCUKVDDUKFGIJLMQRSVDCFFHUHO
|
|
ULUAUMVDDGGHUHPULUBUMFUTIUNBCVDUDUOGUTJUNCDVDUEUOABEHUHNTUCUFUPUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d y B $. $d x ps $. $d y ch $.
|
|
vtocl2g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl2g.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl2g.3 $e |- ph $.
|
|
$( Implicit substitution of 2 classes for 2 setvar variables. (Contributed
|
|
by NM, 25-Apr-1995.) $)
|
|
vtocl2g $p |- ( ( A e. V /\ B e. W ) -> ch ) $=
|
|
( nfcv nfv vtocl2gf ) ABCDEFGHIDFMEFMEGMBDNCENJKLO $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
vtoclgaf.1 $e |- F/_ x A $.
|
|
vtoclgaf.2 $e |- F/ x ps $.
|
|
vtoclgaf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclgaf.4 $e |- ( x e. B -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.) $)
|
|
vtoclgaf $p |- ( A e. B -> ps ) $=
|
|
( wcel cv wi nfel1 nfim wceq eleq1 imbi12d vtoclgf pm2.43i ) DEJZBCKZEJZA
|
|
LTBLCDEFTBCCDEFMGNUADOUBTABUADEPHQIRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
vtoclga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclga.2 $e |- ( x e. B -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 20-Aug-1995.) $)
|
|
vtoclga $p |- ( A e. B -> ps ) $=
|
|
( nfcv nfv vtoclgaf ) ABCDECDHBCIFGJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d x y D $.
|
|
vtocl2gaf.a $e |- F/_ x A $.
|
|
vtocl2gaf.b $e |- F/_ y A $.
|
|
vtocl2gaf.c $e |- F/_ y B $.
|
|
vtocl2gaf.1 $e |- F/ x ps $.
|
|
vtocl2gaf.2 $e |- F/ y ch $.
|
|
vtocl2gaf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl2gaf.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl2gaf.5 $e |- ( ( x e. C /\ y e. D ) -> ph ) $.
|
|
$( Implicit substitution of 2 classes for 2 setvar variables. (Contributed
|
|
by NM, 10-Aug-2013.) $)
|
|
vtocl2gaf $p |- ( ( A e. C /\ B e. D ) -> ch ) $=
|
|
( wcel wa wi cv nfel1 nfan nfim wceq eleq1 anbi1d imbi12d anbi2d vtocl2gf
|
|
nfv pm2.43i ) FHRZGIRZSZCDUAZHRZEUAZIRZSZATUMUSSZBTUOCTDEFGHIJKLVABDUMUSD
|
|
DFHJUBUSDUKUCMUDUOCEUMUNEEFHKUBEGILUBUCNUDUPFUEZUTVAABVBUQUMUSUPFHUFUGOUH
|
|
URGUEZVAUOBCVCUSUNUMURGIUFUIPUHQUJUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y D $. $d x ps $. $d y ch $.
|
|
vtocl2ga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl2ga.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl2ga.3 $e |- ( ( x e. C /\ y e. D ) -> ph ) $.
|
|
$( Implicit substitution of 2 classes for 2 setvar variables. (Contributed
|
|
by NM, 20-Aug-1995.) $)
|
|
vtocl2ga $p |- ( ( A e. C /\ B e. D ) -> ch ) $=
|
|
( nfcv nfv vtocl2gaf ) ABCDEFGHIDFMEFMEGMBDNCENJKLO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z R $. $d x y z S $. $d x y z T $.
|
|
vtocl3gaf.a $e |- F/_ x A $.
|
|
vtocl3gaf.b $e |- F/_ y A $.
|
|
vtocl3gaf.c $e |- F/_ z A $.
|
|
vtocl3gaf.d $e |- F/_ y B $.
|
|
vtocl3gaf.e $e |- F/_ z B $.
|
|
vtocl3gaf.f $e |- F/_ z C $.
|
|
vtocl3gaf.1 $e |- F/ x ps $.
|
|
vtocl3gaf.2 $e |- F/ y ch $.
|
|
vtocl3gaf.3 $e |- F/ z th $.
|
|
vtocl3gaf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl3gaf.5 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl3gaf.6 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
vtocl3gaf.7 $e |- ( ( x e. R /\ y e. S /\ z e. T ) -> ph ) $.
|
|
$( Implicit substitution of 3 classes for 3 setvar variables. (Contributed
|
|
by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.) $)
|
|
vtocl3gaf $p |- ( ( A e. R /\ B e. S /\ C e. T ) -> th ) $=
|
|
( wcel w3a cv nfel1 nf3an nfim wceq eleq1 3anbi1d imbi12d 3anbi2d 3anbi3d
|
|
wi nfv vtocl3gf pm2.43i ) HKUGZILUGZJMUGZUHZDEUIZKUGZFUIZLUGZGUIZMUGZUHZA
|
|
USVCVJVLUHZBUSVCVDVLUHZCUSVFDUSEFGHIJKLMNOPQRSVNBEVCVJVLEEHKNUJVJEUTVLEUT
|
|
UKTULVOCFVCVDVLFFHKOUJFILQUJVLFUTUKUAULVFDGVCVDVEGGHKPUJGILRUJGJMSUJUKUBU
|
|
LVGHUMZVMVNABVPVHVCVJVLVGHKUNUOUCUPVIIUMZVNVOBCVQVJVDVCVLVIILUNUQUDUPVKJU
|
|
MZVOVFCDVRVLVEVCVDVKJMUNURUEUPUFVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d z C $. $d x y z D $. $d x y z R $.
|
|
$d x y z S $. $d x ps $. $d y ch $. $d z th $.
|
|
vtocl3ga.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtocl3ga.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
vtocl3ga.3 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
vtocl3ga.4 $e |- ( ( x e. D /\ y e. R /\ z e. S ) -> ph ) $.
|
|
$( Implicit substitution of 3 classes for 3 setvar variables. (Contributed
|
|
by NM, 20-Aug-1995.) $)
|
|
vtocl3ga $p |- ( ( A e. D /\ B e. R /\ C e. S ) -> th ) $=
|
|
( nfcv nfv vtocl3gaf ) ABCDEFGHIJKLMEHRFHRGHRFIRGIRGJRBESCFSDGSNOPQT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
vtocleg.1 $e |- ( x = A -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 10-Jan-2004.) $)
|
|
vtocleg $p |- ( A e. V -> ph ) $=
|
|
( wcel cv wceq wex elisset exlimiv syl ) CDFBGCHZBIABCDJMABEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Implicit substitution of a class for a setvar variable. (Closed theorem
|
|
version of ~ vtoclef .) (Contributed by NM, 7-Nov-2005.) (Revised by
|
|
Mario Carneiro, 11-Oct-2016.) $)
|
|
vtoclegft $p |- ( ( A e. B /\ F/ x ph /\
|
|
A. x ( x = A -> ph ) ) -> ph ) $=
|
|
( wcel wnf cv wceq wi wal w3a wex elisset mpan9 3adant2 wb 19.9t 3ad2ant2
|
|
exim mpbid ) CDEZABFZBGCHZAIBJZKABLZAUAUDUEUBUAUCBLUDUEBCDMUCABSNOUBUAUEA
|
|
PUDABQRT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
vtoclef.1 $e |- F/ x ph $.
|
|
vtoclef.2 $e |- A e. _V $.
|
|
vtoclef.3 $e |- ( x = A -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 18-Aug-1993.) $)
|
|
vtoclef $p |- ph $=
|
|
( cv wceq wex isseti exlimi ax-mp ) BGCHZBIABCEJMABDFKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
vtocle.1 $e |- A e. _V $.
|
|
vtocle.2 $e |- ( x = A -> ph ) $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 9-Sep-1993.) $)
|
|
vtocle $p |- ph $=
|
|
( cvv wcel vtocleg ax-mp ) CFGADABCFEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
vtoclri.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
vtoclri.2 $e |- A. x e. B ph $.
|
|
$( Implicit substitution of a class for a setvar variable. (Contributed by
|
|
NM, 21-Nov-1994.) $)
|
|
vtoclri $p |- ( A e. B -> ps ) $=
|
|
( rspec vtoclga ) ABCDEFACEGHI $.
|
|
$}
|
|
|
|
${
|
|
spcimgft.1 $e |- F/ x ps $.
|
|
spcimgft.2 $e |- F/_ x A $.
|
|
$( A closed version of ~ spcimgf . (Contributed by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
spcimgft $p |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B ->
|
|
( A. x ph -> ps ) ) ) $=
|
|
( wcel cvv cv wceq wi wal elex wex issetf exim syl5bi 19.36 syl6ib syl5 )
|
|
DEHDIHZCJDKZABLZLCMZACMBLZDENUEUBUDCOZUFUBUCCOUEUGCDGPUCUDCQRABCFSTUA $.
|
|
|
|
$( A closed version of ~ spcgf . (Contributed by Andrew Salmon,
|
|
6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcgft $p |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B ->
|
|
( A. x ph -> ps ) ) ) $=
|
|
( cv wceq wb wi wal wcel bi1 imim2i alimi spcimgft syl ) CHDIZABJZKZCLSAB
|
|
KZKZCLDEMACLBKKUAUCCTUBSABNOPABCDEFGQR $.
|
|
$}
|
|
|
|
${
|
|
spcimgf.1 $e |- F/_ x A $.
|
|
spcimgf.2 $e |- F/ x ps $.
|
|
${
|
|
spcimgf.3 $e |- ( x = A -> ( ph -> ps ) ) $.
|
|
$( Rule of specialization, using implicit substitution. Compare Theorem
|
|
7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcimgf $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
|
|
( cv wceq wi wcel wal spcimgft mpg ) CIDJABKKDELACMBKKCABCDEGFNHO $.
|
|
$}
|
|
|
|
spcimegf.3 $e |- ( x = A -> ( ps -> ph ) ) $.
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcimegf $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
|
|
( wcel wn wal wex nfn cv wceq con3d spcimgf con2d df-ex syl6ibr ) DEIZBAJ
|
|
ZCKZJACLUAUCBUBBJCDEFBCGMCNDOBAHPQRACST $.
|
|
$}
|
|
|
|
${
|
|
spcgf.1 $e |- F/_ x A $.
|
|
spcgf.2 $e |- F/ x ps $.
|
|
spcgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Rule of specialization, using implicit substitution. Compare Theorem
|
|
7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by
|
|
Andrew Salmon, 12-Aug-2011.) $)
|
|
spcgf $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
|
|
( cv wceq wb wi wcel wal spcgft mpg ) CIDJABKLDEMACNBLLCABCDEGFOHP $.
|
|
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by NM, 2-Feb-1997.) $)
|
|
spcegf $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
|
|
( wcel wn wal wex nfn cv wceq notbid spcgf con2d df-ex syl6ibr ) DEIZBAJZ
|
|
CKZJACLUAUCBUBBJCDEFBCGMCNDOABHPQRACST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $. $d x ch $.
|
|
spcimdv.1 $e |- ( ph -> A e. B ) $.
|
|
${
|
|
spcimdv.2 $e |- ( ( ph /\ x = A ) -> ( ps -> ch ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed
|
|
by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcimdv $p |- ( ph -> ( A. x ps -> ch ) ) $=
|
|
( cv wceq wi wal wcel ex alrimiv nfv nfcv spcimgft sylc ) ADIEJZBCKZKZD
|
|
LEFMBDLCKAUBDATUAHNOGBCDEFCDPDEQRS $.
|
|
$}
|
|
|
|
${
|
|
spcdv.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
$( Rule of specialization, using implicit substitution. Analogous to
|
|
~ rspcdv . (Contributed by David Moews, 1-May-2017.) $)
|
|
spcdv $p |- ( ph -> ( A. x ps -> ch ) ) $=
|
|
( cv wceq wa biimpd spcimdv ) ABCDEFGADIEJKBCHLM $.
|
|
$}
|
|
|
|
spcimedv.2 $e |- ( ( ph /\ x = A ) -> ( ch -> ps ) ) $.
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
spcimedv $p |- ( ph -> ( ch -> E. x ps ) ) $=
|
|
( wn wal wex cv wceq wa con3d spcimdv con2d df-ex syl6ibr ) ACBIZDJZIBDKA
|
|
UACATCIDEFGADLEMNCBHOPQBDRS $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
spcgv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Rule of specialization, using implicit substitution. Compare Theorem
|
|
7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) $)
|
|
spcgv $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
|
|
( nfcv nfv spcgf ) ABCDECDGBCHFI $.
|
|
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by NM, 14-Aug-1994.) $)
|
|
spcegv $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
|
|
( nfcv nfv spcegf ) ABCDECDGBCHFI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
spc2egv.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Existential specialization with 2 quantifiers, using implicit
|
|
substitution. (Contributed by NM, 3-Aug-1995.) $)
|
|
spc2egv $p |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ph ) ) $=
|
|
( wcel wa cv wceq wex elisset anim12i eeanv sylibr biimprcd 2eximdv
|
|
syl5com ) EGJZFHJZKZCLEMZDLFMZKZDNCNZBADNCNUDUECNZUFDNZKUHUBUIUCUJCEGODFH
|
|
OPUEUFCDQRBUGACDUGABISTUA $.
|
|
|
|
$( Specialization with 2 quantifiers, using implicit substitution.
|
|
(Contributed by NM, 27-Apr-2004.) $)
|
|
spc2gv $p |- ( ( A e. V /\ B e. W ) -> ( A. x A. y ph -> ps ) ) $=
|
|
( wcel wa wal wn wex cv wceq notbid spc2egv 2nalexn syl6ibr con4d ) EGJFH
|
|
JKZBADLCLZUBBMZAMZDNCNUCMUEUDCDEFGHCOEPDOFPKABIQRACDSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z ps $.
|
|
spc3egv.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
$( Existential specialization with 3 quantifiers, using implicit
|
|
substitution. (Contributed by NM, 12-May-2008.) $)
|
|
spc3egv $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( ps -> E. x E. y E. z ph ) ) $=
|
|
( wcel w3a cv wceq wex elisset 3anim123i eeeanv biimprcd 2eximdv syl5com
|
|
sylibr eximdv ) FIMZGJMZHKMZNZCOFPZDOGPZEOHPZNZEQZDQCQZBAEQZDQCQUIUJCQZUK
|
|
DQZULEQZNUOUFUQUGURUHUSCFIRDGJREHKRSUJUKULCDETUDBUNUPCDBUMAEUMABLUAUEUBUC
|
|
$.
|
|
|
|
$( Specialization with 3 quantifiers, using implicit substitution.
|
|
(Contributed by NM, 12-May-2008.) $)
|
|
spc3gv $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( A. x A. y A. z ph -> ps ) ) $=
|
|
( wcel w3a wal wn wex cv wceq exnal notbid spc3egv exbii bitr2i syl6ibr
|
|
bitri con4d ) FIMGJMHKMNZBAEOZDOZCOZUHBPZAPZEQZDQZCQZUKPZUMULCDEFGHIJKCRF
|
|
SDRGSERHSNABLUAUBUPUJPZCQUQUOURCUOUIPZDQURUNUSDAETUCUIDTUFUCUJCTUDUEUG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
spcv.1 $e |- A e. _V $.
|
|
spcv.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Rule of specialization, using implicit substitution. (Contributed by
|
|
NM, 22-Jun-1994.) $)
|
|
spcv $p |- ( A. x ph -> ps ) $=
|
|
( cvv wcel wal wi spcgv ax-mp ) DGHACIBJEABCDGFKL $.
|
|
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) $)
|
|
spcev $p |- ( ps -> E. x ph ) $=
|
|
( cvv wcel wex wi spcegv ax-mp ) DGHBACIJEABCDGFKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
spc2ev.1 $e |- A e. _V $.
|
|
spc2ev.2 $e |- B e. _V $.
|
|
spc2ev.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Existential specialization, using implicit substitution. (Contributed
|
|
by NM, 3-Aug-1995.) $)
|
|
spc2ev $p |- ( ps -> E. x E. y ph ) $=
|
|
( cvv wcel wex wi spc2egv mp2an ) EJKFJKBADLCLMGHABCDEFJJINO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
rspct.1 $e |- F/ x ps $.
|
|
$( A closed version of ~ rspc . (Contributed by Andrew Salmon,
|
|
6-Jun-2011.) $)
|
|
rspct $p |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B ->
|
|
( A. x e. B ph -> ps ) ) ) $=
|
|
( cv wceq wb wi wal wcel wral df-ral wa eleq1 adantr simpr imbi12d ex a2i
|
|
alimi nfv nfim nfcv spcgft syl syl7bi com34 pm2.43d ) CGZDHZABIZJZCKZDELZ
|
|
ACEMZBJUOUPUQUPBUQUKELZAJZCKZUOUPUPBJZACENUOULUSVAIZJZCKUPUTVAJJUNVCCULUM
|
|
VBULUMVBULUMOURUPABULURUPIUMUKDEPQULUMRSTUAUBUSVACDEUPBCUPCUCFUDCDUEUFUGU
|
|
HUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
rspc.1 $e |- F/ x ps $.
|
|
rspc.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) $)
|
|
rspc $p |- ( A e. B -> ( A. x e. B ph -> ps ) ) $=
|
|
( wral cv wcel wi wal df-ral nfcv nfv nfim wceq eleq1 imbi12d spcgf
|
|
pm2.43a syl5bi ) ACEHCIZEJZAKZCLZDEJZBACEMUFUGBUEUGBKCDECDNUGBCUGCOFPUCDQ
|
|
UDUGABUCDERGSTUAUB $.
|
|
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro,
|
|
11-Oct-2016.) $)
|
|
rspce $p |- ( ( A e. B /\ ps ) -> E. x e. B ph ) $=
|
|
( wcel wa cv wex wrex nfcv nfv nfan wceq eleq1 anbi12d spcegf anabsi5
|
|
df-rex sylibr ) DEHZBIZCJZEHZAIZCKZACELUCBUHUGUDCDECDMUCBCUCCNFOUEDPUFUCA
|
|
BUEDEQGRSTACEUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
rspcv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 26-May-1998.) $)
|
|
rspcv $p |- ( A e. B -> ( A. x e. B ph -> ps ) ) $=
|
|
( nfv rspc ) ABCDEBCGFH $.
|
|
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 2-Feb-2006.) $)
|
|
rspccv $p |- ( A. x e. B ph -> ( A e. B -> ps ) ) $=
|
|
( wcel wral rspcv com12 ) DEGACEHBABCDEFIJ $.
|
|
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 13-Sep-2005.) $)
|
|
rspcva $p |- ( ( A e. B /\ A. x e. B ph ) -> ps ) $=
|
|
( wcel wral rspcv imp ) DEGACEHBABCDEFIJ $.
|
|
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
rspccva $p |- ( ( A. x e. B ph /\ A e. B ) -> ps ) $=
|
|
( wcel wral rspcv impcom ) DEGACEHBABCDEFIJ $.
|
|
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by NM, 26-May-1998.) $)
|
|
rspcev $p |- ( ( A e. B /\ ps ) -> E. x e. B ph ) $=
|
|
( nfv rspce ) ABCDEBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $. $d x ch $.
|
|
rspcimdv.1 $e |- ( ph -> A e. B ) $.
|
|
${
|
|
rspcimdv.2 $e |- ( ( ph /\ x = A ) -> ( ps -> ch ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed
|
|
by Mario Carneiro, 4-Jan-2017.) $)
|
|
rspcimdv $p |- ( ph -> ( A. x e. B ps -> ch ) ) $=
|
|
( wral cv wcel wi wal df-ral wceq wa simpr eleq1d biimprd imim12d mpid
|
|
spcimdv syl5bi ) BDFIDJZFKZBLZDMZACBDFNAUGEFKZCGAUFUHCLDEFGAUDEOZPZUHUE
|
|
BCUJUEUHUJUDEFAUIQRSHTUBUAUC $.
|
|
$}
|
|
|
|
rspcimedv.2 $e |- ( ( ph /\ x = A ) -> ( ch -> ps ) ) $.
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by Mario Carneiro, 4-Jan-2017.) $)
|
|
rspcimedv $p |- ( ph -> ( ch -> E. x e. B ps ) ) $=
|
|
( wn wral wrex cv wceq wa con3d rspcimdv con2d dfrex2 syl6ibr ) ACBIZDFJZ
|
|
IBDFKAUACATCIDEFGADLEMNCBHOPQBDFRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $. $d x ch $.
|
|
rspcdv.1 $e |- ( ph -> A e. B ) $.
|
|
rspcdv.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
$( Restricted specialization, using implicit substitution. (Contributed by
|
|
NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.) $)
|
|
rspcdv $p |- ( ph -> ( A. x e. B ps -> ch ) ) $=
|
|
( cv wceq wa biimpd rspcimdv ) ABCDEFGADIEJKBCHLM $.
|
|
|
|
$( Restricted existential specialization, using implicit substitution.
|
|
(Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro,
|
|
4-Jan-2017.) $)
|
|
rspcedv $p |- ( ph -> ( ch -> E. x e. B ps ) ) $=
|
|
( cv wceq wa biimprd rspcimedv ) ABCDEFGADIEJKBCHLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x C $. $d x y D $.
|
|
rspc2.1 $e |- F/ x ch $.
|
|
rspc2.2 $e |- F/ y ps $.
|
|
rspc2.3 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
rspc2.4 $e |- ( y = B -> ( ch <-> ps ) ) $.
|
|
$( 2-variable restricted specialization, using implicit substitution.
|
|
(Contributed by NM, 9-Nov-2012.) $)
|
|
rspc2 $p |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph ->
|
|
ps ) ) $=
|
|
( wcel wral nfcv nfral cv wceq rspc ralbidv sylan9 ) FHNAEIOZDHOCEIOZGINB
|
|
UCUDDFHCDEIDIPJQDRFSACEILUATCBEGIKMTUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x C $. $d x y D $. $d x ch $. $d y ps $.
|
|
rspc2v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
rspc2v.2 $e |- ( y = B -> ( ch <-> ps ) ) $.
|
|
$( 2-variable restricted specialization, using implicit substitution.
|
|
(Contributed by NM, 13-Sep-1999.) $)
|
|
rspc2v $p |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph ->
|
|
ps ) ) $=
|
|
( nfv rspc2 ) ABCDEFGHICDLBELJKM $.
|
|
|
|
$( 2-variable restricted specialization, using implicit substitution.
|
|
(Contributed by NM, 18-Jun-2014.) $)
|
|
rspc2va $p |- ( ( ( A e. C /\ B e. D ) /\ A. x e. C A. y e. D ph ) ->
|
|
ps ) $=
|
|
( wcel wa wral rspc2v imp ) FHLGILMAEINDHNBABCDEFGHIJKOP $.
|
|
|
|
$( 2-variable restricted existential specialization, using implicit
|
|
substitution. (Contributed by NM, 16-Oct-1999.) $)
|
|
rspc2ev $p |- ( ( A e. C /\ B e. D /\ ps ) -> E. x e. C E. y e. D ph ) $=
|
|
( wcel w3a wrex wa rspcev anim2i 3impb cv wceq rexbidv syl ) FHLZGILZBMUC
|
|
CEINZOZAEINZDHNUCUDBUFUDBOUEUCCBEGIKPQRUGUEDFHDSFTACEIJUAPUB $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $. $d x ch $. $d y th $. $d x y z A $. $d y z B $. $d z C $.
|
|
$d x R $. $d x y S $. $d x y z T $.
|
|
rspc3v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
|
|
rspc3v.2 $e |- ( y = B -> ( ch <-> th ) ) $.
|
|
rspc3v.3 $e |- ( z = C -> ( th <-> ps ) ) $.
|
|
$( 3-variable restricted specialization, using implicit substitution.
|
|
(Contributed by NM, 10-May-2005.) $)
|
|
rspc3v $p |- ( ( A e. R /\ B e. S /\ C e. T ) ->
|
|
( A. x e. R A. y e. S A. z e. T ph -> ps ) ) $=
|
|
( wcel wral cv wceq wi wa ralbidv rspc2v rspcv sylan9 3impa ) HKQZILQZJMQ
|
|
ZAGMRZFLREKRZBUAUHUIUBULDGMRZUJBUKUMCGMREFHIKLESHTACGMNUCFSITCDGMOUCUDDBG
|
|
JMPUEUFUG $.
|
|
|
|
$( 3-variable restricted existentional specialization, using implicit
|
|
substitution. (Contributed by NM, 25-Jul-2012.) $)
|
|
rspc3ev $p |- ( ( ( A e. R /\ B e. S /\ C e. T ) /\ ps ) ->
|
|
E. x e. R E. y e. S E. z e. T ph ) $=
|
|
( wcel wrex cv wceq w3a wa simpl1 simpl2 rspcev 3ad2antl3 rexbidv rspc2ev
|
|
syl3anc ) HKQZILQZJMQZUABUBUJUKDGMRZAGMRZFLREKRUJUKULBUCUJUKULBUDULUJBUMU
|
|
KDBGJMPUEUFUNUMCGMREFHIKLESHTACGMNUGFSITCDGMOUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
eqvinc.1 $e |- A e. _V $.
|
|
$( A variable introduction law for class equality. (Contributed by NM,
|
|
14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
eqvinc $p |- ( A = B <-> E. x ( x = A /\ x = B ) ) $=
|
|
( wceq cv wa wex wi isseti ax-1 eqtr jca eximi pm3.43 mp2b 19.37aiv eqtr2
|
|
ex exlimiv impbii ) BCEZAFZBEZUCCEZGZAHUBUFAUDAHUBUDIZUBUEIZGZAHUBUFIZAHA
|
|
BDJUDUIAUDUGUHUDUBKUDUBUEUCBCLSMNUIUJAUBUDUEONPQUFUBAUCBCRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d x y $.
|
|
eqvincf.1 $e |- F/_ x A $.
|
|
eqvincf.2 $e |- F/_ x B $.
|
|
eqvincf.3 $e |- A e. _V $.
|
|
$( A variable introduction law for class equality, using bound-variable
|
|
hypotheses instead of distinct variable conditions. (Contributed by NM,
|
|
14-Sep-2003.) $)
|
|
eqvincf $p |- ( A = B <-> E. x ( x = A /\ x = B ) ) $=
|
|
( vy wceq cv wa wex eqvinc nfeq2 nfan nfv eqeq1 anbi12d cbvex bitri ) BCH
|
|
GIZBHZTCHZJZGKAIZBHZUDCHZJZAKGBCFLUCUGGAUAUBAATBDMATCEMNUGGOTUDHUAUEUBUFT
|
|
UDBPTUDCPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
alexeq.1 $e |- A e. _V $.
|
|
$( Two ways to express substitution of ` A ` for ` x ` in ` ph ` .
|
|
(Contributed by NM, 2-Mar-1995.) $)
|
|
alexeq $p |- ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) $=
|
|
( vy cv wceq wa wex wi wal anbi1d exbidv imbi1d albidv sb56 vtoclb bicomi
|
|
eqeq2 ) BFZCGZAHZBIZUAAJZBKZTEFZGZAHZBIUGAJZBKUCUEECDUFCGZUHUBBUJUGUAAUFC
|
|
TSZLMUJUIUDBUJUGUAAUKNOABEPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
$( Equality implies equivalence with substitution. (Contributed by NM,
|
|
2-Mar-1995.) $)
|
|
ceqex $p |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) $=
|
|
( vy cvv wcel cv wceq wa wex wb 19.8a isset sylibr wi eqeq2 anbi1d exbidv
|
|
bibi2d imbi12d ex wal vex alexeq sp com12 syl5bir impbid vtoclg mpcom ) C
|
|
EFZBGZCHZAUMAIZBJZKZUMUMBJUKUMBLBCMNULDGZHZAURAIZBJZKZOUMUPODCEUQCHZURUMV
|
|
AUPUQCULPZVBUTUOAVBUSUNBVBURUMAVCQRSTURAUTURAUTUSBLUAUTURAOZBUBZURAABUQDU
|
|
CUDVEURAVDBUEUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ceqsexg.1 $e |- F/ x ps $.
|
|
ceqsexg.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A representation of explicit substitution of a class for a variable,
|
|
inferred from an implicit substitution hypothesis. (Contributed by NM,
|
|
11-Oct-2004.) $)
|
|
ceqsexg $p |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) ) $=
|
|
( wb cv wceq wa wex nfcv nfe1 nfbi ceqex bibi12d biid vtoclgf ) AAHCIDJZA
|
|
KZCLZBHCDECDMUBBCUACNFOTAUBABACDPGQARS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
ceqsexgv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Elimination of an existential quantifier, using implicit substitution.
|
|
(Contributed by NM, 29-Dec-1996.) $)
|
|
ceqsexgv $p |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) ) $=
|
|
( nfv ceqsexg ) ABCDEBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
ceqsrexv.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Elimination of a restricted existential quantifier, using implicit
|
|
substitution. (Contributed by NM, 30-Apr-2004.) $)
|
|
ceqsrexv $p |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) ) $=
|
|
( cv wceq wa wrex wcel wex df-rex an12 exbii bitr4i eleq1 anbi12d bianabs
|
|
ceqsexgv syl5bb ) CGZDHZAIZCEJZUCUBEKZAIZIZCLZDEKZBUEUFUDIZCLUIUDCEMUHUKC
|
|
UCUFANOPUJUIBUGUJBICDEUCUFUJABUBDEQFRTSUA $.
|
|
|
|
$( Elimination of a restricted existential quantifier, using implicit
|
|
substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) $)
|
|
ceqsrexbv $p |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) ) $=
|
|
( wcel cv wceq wa wrex r19.42v eleq1 adantr pm5.32ri bicomi baib ceqsrexv
|
|
wb rexbiia pm5.32i 3bitr3i ) DEGZCHZDIZAJZJZCEKUCUFCEKZJUHUCBJUCUFCELUGUF
|
|
CEUGUDEGZUFUIUFJUGUFUIUCUEUIUCSAUDDEMNOPQTUCUHBABCDEFRUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x C $. $d x y D $. $d x ps $. $d y ch $.
|
|
ceqsrex2v.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ceqsrex2v.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( Elimination of a restricted existential quantifier, using implicit
|
|
substitution. (Contributed by NM, 29-Oct-2005.) $)
|
|
ceqsrex2v $p |- ( ( A e. C /\ B e. D ) ->
|
|
( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) ) $=
|
|
( wcel cv wceq wa wrex anass rexbii r19.42v ceqsrexv bitri anbi2d rexbidv
|
|
syl5bb sylan9bb ) FHLZDMFNZEMGNZOAOZEIPZDHPZUHBOZEIPZGILCUKUGUHAOZEIPZOZD
|
|
HPUFUMUJUPDHUJUGUNOZEIPUPUIUQEIUGUHAQRUGUNEISUARUOUMDFHUGUNULEIUGABUHJUBU
|
|
CTUDBCEGIKTUE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
clel2.1 $e |- A e. _V $.
|
|
$( An alternate definition of class membership when the class is a set.
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
clel2 $p |- ( A e. B <-> A. x ( x = A -> x e. B ) ) $=
|
|
( cv wceq wcel wi wal eleq1 ceqsalv bicomi ) AEZBFMCGZHAIBCGZNOABDMBCJKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An alternate definition of class membership when the class is a set.
|
|
(Contributed by NM, 13-Aug-2005.) $)
|
|
clel3g $p |- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) ) $=
|
|
( wcel cv wceq wa wex eleq2 ceqsexgv bicomd ) CDEAFZCGBMEZHAIBCEZNOACDMCB
|
|
JKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
clel3.1 $e |- B e. _V $.
|
|
$( An alternate definition of class membership when the class is a set.
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
clel3 $p |- ( A e. B <-> E. x ( x = B /\ A e. x ) ) $=
|
|
( cvv wcel cv wceq wa wex wb clel3g ax-mp ) CEFBCFAGZCHBNFIAJKDABCELM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
clel4.1 $e |- B e. _V $.
|
|
$( An alternate definition of class membership when the class is a set.
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
clel4 $p |- ( A e. B <-> A. x ( x = B -> A e. x ) ) $=
|
|
( cv wceq wcel wi wal eleq2 ceqsalv bicomi ) AEZCFBMGZHAIBCGZNOACDMCBJKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d y A z $. $d y B z $.
|
|
$( Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only ` A ` is
|
|
required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.) $)
|
|
pm13.183 $p |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) ) $=
|
|
( vy cv wceq wal eqeq1 eqeq2 bibi1d albidv alrimiv wsb stdpc4 sbbi bibi2i
|
|
wb eqsb3 sylbi equsb1 bi1 mpi syl impbii vtoclbg ) EFZCGZAFZUGGZUICGZRZAH
|
|
ZBCGUIBGZUKRZAHEBDUGBCIUGBGZULUOAUPUJUNUKUGBUIJKLUHUMUHULAUGCUIJMUMULAENZ
|
|
UHULAEOUQUJAENZUKAENZRZUHUJUKAEPUTURUHRZUHUSUHUREACSQVAURUHAEUAURUHUBUCTT
|
|
UDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y $. $d y ph $.
|
|
$( Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We
|
|
don't need the non-empty class condition of ~ r19.3rzv when there is an
|
|
outer quantifier. (Contributed by NM, 25-Oct-2012.) $)
|
|
rr19.3v $p |- ( A. x e. A A. y e. A ph <-> A. x e. A ph ) $=
|
|
( wral cv wceq biidd rspcv ralimia wcel ax-1 ralrimiv ralimi impbii ) ACD
|
|
EZBDEABDEPABDAACBFZDCFZQGAHIJAPBDAACDARDKLMNO $.
|
|
|
|
$( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We
|
|
don't need the non-empty class condition of ~ r19.28zv when there is an
|
|
outer quantifier. (Contributed by NM, 29-Oct-2012.) $)
|
|
rr19.28v $p |- ( A. x e. A A. y e. A ( ph /\ ps )
|
|
<-> A. x e. A ( ph /\ A. y e. A ps ) ) $=
|
|
( wa wral cv wcel simpl ralimi wceq biidd rspcv syl5 wi simpr a1i ralimia
|
|
jcad r19.28av impbii ) ABFZDEGZCEGABDEGZFZCEGUDUFCECHZEIZUDAUEUDADEGUHAUC
|
|
ADEABJKAADUGEDHUGLAMNOUDUEPUHUCBDEABQKRTSUFUDCEABDEUAKUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
$( Membership in a class abstraction, using implicit substitution. (Closed
|
|
theorem version of ~ elabg .) (Contributed by NM, 7-Nov-2005.) (Proof
|
|
shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
elabgt $p |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) ->
|
|
( A e. { x | ph } <-> ps ) ) $=
|
|
( cv wceq wb wal wcel cab abid eleq1 syl5bbr bibi1d biimpd a2i alimi nfcv
|
|
wi nfab1 nfel2 nfv nfbi pm5.5 spcgf imp sylan2 ) CFZDGZABHZTZCIDEJZUJDACK
|
|
ZJZBHZTZCIZUPULUQCUJUKUPUJUKUPUJAUOBAUIUNJUJUOACLUIDUNMNOPQRUMURUPUQUPCDE
|
|
CDSUOBCCDUNACUAUBBCUCUDUJUPUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
elabgf.1 $e |- F/_ x A $.
|
|
elabgf.2 $e |- F/ x ps $.
|
|
elabgf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, using implicit substitution. Compare
|
|
Theorem 6.13 of [Quine] p. 44. This version has bound-variable
|
|
hypotheses in place of distinct variable restrictions. (Contributed by
|
|
NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
elabgf $p |- ( A e. B -> ( A e. { x | ph } <-> ps ) ) $=
|
|
( cv cab wcel wb nfab1 nfel nfbi wceq eleq1 bibi12d abid vtoclgf ) CIZACJ
|
|
ZKZALDUBKZBLCDEFUDBCCDUBFACMNGOUADPUCUDABUADUBQHRACST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
elabf.1 $e |- F/ x ps $.
|
|
elabf.2 $e |- A e. _V $.
|
|
elabf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro,
|
|
12-Oct-2016.) $)
|
|
elabf $p |- ( A e. { x | ph } <-> ps ) $=
|
|
( cvv wcel cab wb nfcv elabgf ax-mp ) DHIDACJIBKFABCDHCDLEGMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab.1 $e |- A e. _V $.
|
|
elab.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, using implicit substitution. Compare
|
|
Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) $)
|
|
elab $p |- ( A e. { x | ph } <-> ps ) $=
|
|
( nfv elabf ) ABCDBCGEFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, using implicit substitution. Compare
|
|
Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) $)
|
|
elabg $p |- ( A e. V -> ( A e. { x | ph } <-> ps ) ) $=
|
|
( nfcv nfv elabgf ) ABCDECDGBCHFI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab2g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
elab2g.2 $e |- B = { x | ph } $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 13-Sep-1995.) $)
|
|
elab2g $p |- ( A e. V -> ( A e. B <-> ps ) ) $=
|
|
( wcel cab eleq2i elabg syl5bb ) DEIDACJZIDFIBENDHKABCDFGLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab2.1 $e |- A e. _V $.
|
|
elab2.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
elab2.3 $e |- B = { x | ph } $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 13-Sep-1995.) $)
|
|
elab2 $p |- ( A e. B <-> ps ) $=
|
|
( cvv wcel wb elab2g ax-mp ) DIJDEJBKFABCDEIGHLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab4g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
elab4g.2 $e |- B = { x | ph } $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 17-Oct-2012.) $)
|
|
elab4g $p |- ( A e. B <-> ( A e. _V /\ ps ) ) $=
|
|
( wcel cvv elex elab2g biadan2 ) DEHDIHBDEJABCDEIFGKL $.
|
|
$}
|
|
|
|
${
|
|
elab3gf.1 $e |- F/_ x A $.
|
|
elab3gf.2 $e |- F/ x ps $.
|
|
elab3gf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, with a weaker antecedent than
|
|
~ elabgf . (Contributed by NM, 6-Sep-2011.) $)
|
|
elab3gf $p |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) ) $=
|
|
( wcel cab wb wn elabgf ibi pm2.21 impbid2 ja ) BDEIDACJZIZBKBLSBSBABCDRF
|
|
GHMNBSOPABCDEFGHMQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab3g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction, with a weaker antecedent than
|
|
~ elabg . (Contributed by NM, 29-Aug-2006.) $)
|
|
elab3g $p |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) ) $=
|
|
( nfcv nfv elab3gf ) ABCDECDGBCHFI $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $.
|
|
elab3.1 $e |- ( ps -> A e. _V ) $.
|
|
elab3.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class abstraction using implicit substitution.
|
|
(Contributed by NM, 10-Nov-2000.) $)
|
|
elab3 $p |- ( A e. { x | ph } <-> ps ) $=
|
|
( cvv wcel wi cab wb elab3g ax-mp ) BDGHIDACJHBKEABCDGFLM $.
|
|
$}
|
|
|
|
${
|
|
elrabf.1 $e |- F/_ x A $.
|
|
elrabf.2 $e |- F/_ x B $.
|
|
elrabf.3 $e |- F/ x ps $.
|
|
elrabf.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a restricted class abstraction, using implicit
|
|
substitution. This version has bound-variable hypotheses in place of
|
|
distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) $)
|
|
elrabf $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
|
|
( crab wcel cvv wa elex adantr cv cab df-rab eleq2i nfel nfan wceq elabgf
|
|
eleq1 anbi12d syl5bb pm5.21nii ) DACEJZKZDLKZDEKZBMZDUHNUKUJBDENOUIDCPZEK
|
|
ZAMZCQZKUJULUHUPDACERSUOULCDLFUKBCCDEFGTHUAUMDUBUNUKABUMDEUDIUEUCUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $. $d x B $.
|
|
elrab.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Membership in a restricted class abstraction, using implicit
|
|
substitution. (Contributed by NM, 21-May-1999.) $)
|
|
elrab $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
|
|
( nfcv nfv elrabf ) ABCDECDGCEGBCHFI $.
|
|
|
|
$( Membership in a restricted class abstraction, using implicit
|
|
substitution. (Contributed by NM, 5-Oct-2006.) $)
|
|
elrab3 $p |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) ) $=
|
|
( crab wcel elrab baib ) DACEGHDEHBABCDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d x A $. $d x B $.
|
|
elrab2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
elrab2.2 $e |- C = { x e. B | ph } $.
|
|
$( Membership in a class abstraction, using implicit substitution.
|
|
(Contributed by NM, 2-Nov-2006.) $)
|
|
elrab2 $p |- ( A e. C <-> ( A e. B /\ ps ) ) $=
|
|
( wcel crab wa eleq2i elrab bitri ) DFIDACEJZIDEIBKFODHLABCDEGMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y ps $.
|
|
ralab.1 $e |- ( y = x -> ( ph <-> ps ) ) $.
|
|
$( Universal quantification over a class abstraction. (Contributed by Jeff
|
|
Madsen, 10-Jun-2010.) $)
|
|
ralab $p |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) ) $=
|
|
( cab wral cv wcel wi wal df-ral vex elab imbi1i albii bitri ) CDAEGZHDIZ
|
|
SJZCKZDLBCKZDLCDSMUBUCDUABCABETDNFOPQR $.
|
|
|
|
$( Universal quantification over a restricted class abstraction.
|
|
(Contributed by Jeff Madsen, 10-Jun-2010.) $)
|
|
ralrab $p |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) ) $=
|
|
( wi crab cv wcel wa elrab imbi1i impexp bitri ralbii2 ) CBCHZDAEFIZFDJZS
|
|
KZCHTFKZBLZCHUBRHUAUCCABETFGMNUBBCOPQ $.
|
|
|
|
$( Existential quantification over a class abstraction. (Contributed by
|
|
Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro,
|
|
3-Sep-2015.) $)
|
|
rexab $p |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) ) $=
|
|
( cab wrex cv wcel wa wex df-rex vex elab anbi1i exbii bitri ) CDAEGZHDIZ
|
|
SJZCKZDLBCKZDLCDSMUBUCDUABCABETDNFOPQR $.
|
|
|
|
$( Existential quantification over a class abstraction. (Contributed by
|
|
Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) $)
|
|
rexrab $p |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) ) $=
|
|
( wa crab cv wcel elrab anbi1i anass bitri rexbii2 ) CBCHZDAEFIZFDJZRKZCH
|
|
SFKZBHZCHUAQHTUBCABESFGLMUABCNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x A $. $d x ch $. $d x ph $. $d y ps $.
|
|
ralab2.1 $e |- ( x = y -> ( ps <-> ch ) ) $.
|
|
$( Universal quantification over a class abstraction. (Contributed by
|
|
Mario Carneiro, 3-Sep-2015.) $)
|
|
ralab2 $p |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) ) $=
|
|
( cab wral cv wcel wi wal df-ral nfsab1 nfv nfim wceq eleq1 abid syl6bb
|
|
imbi12d cbval bitri ) BDAEGZHDIZUDJZBKZDLACKZELBDUDMUGUHDEUFBEAEDNBEOPUHD
|
|
OUEEIZQZUFABCUJUFUIUDJAUEUIUDRAESTFUAUBUC $.
|
|
|
|
$( Universal quantification over a restricted class abstraction.
|
|
(Contributed by Mario Carneiro, 3-Sep-2015.) $)
|
|
ralrab2 $p |- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) ) $=
|
|
( crab wral cv wcel wa cab wi wal df-rab raleqi ralab2 impexp albii
|
|
df-ral bitr4i 3bitri ) BDAEFHZIBDEJFKZALZEMZIUFCNZEOZACNZEFIZBDUDUGAEFPQU
|
|
FBCDEGRUIUEUJNZEOUKUHULEUEACSTUJEFUAUBUC $.
|
|
|
|
$( Existential quantification over a class abstraction. (Contributed by
|
|
Mario Carneiro, 3-Sep-2015.) $)
|
|
rexab2 $p |- ( E. x e. { y | ph } ps <-> E. y ( ph /\ ch ) ) $=
|
|
( cab wrex cv wcel wa wex df-rex nfsab1 nfv nfan wceq eleq1 abid syl6bb
|
|
anbi12d cbvex bitri ) BDAEGZHDIZUDJZBKZDLACKZELBDUDMUGUHDEUFBEAEDNBEOPUHD
|
|
OUEEIZQZUFABCUJUFUIUDJAUEUIUDRAESTFUAUBUC $.
|
|
|
|
$( Existential quantification over a class abstraction. (Contributed by
|
|
Mario Carneiro, 3-Sep-2015.) $)
|
|
rexrab2 $p |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) ) $=
|
|
( crab wrex cv wcel wa cab wex df-rab rexeqi rexab2 anass exbii df-rex
|
|
bitr4i 3bitri ) BDAEFHZIBDEJFKZALZEMZIUECLZENZACLZEFIZBDUCUFAEFOPUEBCDEGQ
|
|
UHUDUILZENUJUGUKEUDACRSUIEFTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d A z $.
|
|
$( Identity used to create closed-form versions of bound-variable
|
|
hypothesis builders for class expressions. (Contributed by NM,
|
|
10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) $)
|
|
abidnf $p |- ( F/_ x A -> { z | A. x z e. A } = A ) $=
|
|
( wnfc cv wcel wal sp nfcr nfrd impbid2 abbi1dv ) ACDZBECFZAGZBCMONNAHMNA
|
|
ABCIJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d z A $.
|
|
dedhb.1 $e |- ( A = { z | A. x z e. A } -> ( ph <-> ps ) ) $.
|
|
dedhb.2 $e |- ps $.
|
|
$( A deduction theorem for converting the inference ` |- F/_ x A ` =>
|
|
` |- ph ` into a closed theorem. Use ~ nfa1 and ~ nfab to eliminate the
|
|
hypothesis of the substitution instance ` ps ` of the inference. For
|
|
converting the inference form into a deduction form, ~ abidnf is
|
|
useful. (Contributed by NM, 8-Dec-2006.) $)
|
|
dedhb $p |- ( F/_ x A -> ph ) $=
|
|
( wnfc cv wcel wal cab wceq wb abidnf eqcomd syl mpbiri ) CEHZABGSEDIEJCK
|
|
DLZMABNSTECDEOPFQR $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x y ps $. $d x y A $.
|
|
eqeu.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( A condition which implies existential uniqueness. (Contributed by Jeff
|
|
Hankins, 8-Sep-2009.) $)
|
|
eqeu $p |- ( ( A e. B /\ ps /\ A. x ( ph -> x = A ) ) -> E! x ph ) $=
|
|
( vy wcel cv wceq wi wal w3a wex weu spcegv imp 3adant3 eqeq2 imbi2d nfv
|
|
albidv 3adant2 eu3 sylanbrc ) DEHZBACIZDJZKZCLZMACNZAUGGIZJZKZCLZGNZACOUF
|
|
BUKUJUFBUKABCDEFPQRUFUJUPBUFUJUPUOUJGDEULDJZUNUICUQUMUHAULDUGSTUBPQUCACGA
|
|
GUAUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Equality has existential uniqueness. (Contributed by NM,
|
|
25-Nov-1994.) $)
|
|
eueq $p |- ( A e. _V <-> E! x x = A ) $=
|
|
( vy cv wceq wex wa wi wal cvv wcel weu eqtr3 biantru isset eqeq1 3bitr4i
|
|
gen2 eu4 ) ADZBEZAFZUBUACDZBEZGTUCEHZCIAIZGBJKUAALUFUBUEACTUCBMRNABOUAUDA
|
|
CTUCBPSQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
eueq1.1 $e |- A e. _V $.
|
|
$( Equality has existential uniqueness. (Contributed by NM,
|
|
5-Apr-1995.) $)
|
|
eueq1 $p |- E! x x = A $=
|
|
( cvv wcel cv wceq weu eueq mpbi ) BDEAFBGAHCABIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $.
|
|
eueq2.1 $e |- A e. _V $.
|
|
eueq2.2 $e |- B e. _V $.
|
|
$( Equality has existential uniqueness (split into 2 cases). (Contributed
|
|
by NM, 5-Apr-1995.) $)
|
|
eueq2 $p |- E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) $=
|
|
( cv wceq wa wn weu eueq1 euanv biimpri mpan2 euorv bianfd eubidv mpbid
|
|
wo notnot1 syl2anc orcom orbi2d syl5bb mpdan id orbi1d pm2.61i ) AABGZCHZ
|
|
IZAJZUJDHZIZTZBKZAUMULTZBKZUQAUMJULBKZUSAUAZAUKBKZUTBCELUTAVBIAUKBMNOUMUL
|
|
BPUBAURUPBURULUMTAUPUMULUCAUMUOULAUMUNVAQUDUERSUMAUOTZBKZUQUMUOBKZVDUMUNB
|
|
KZVEBDFLVEUMVFIUMUNBMNOAUOBPUFUMVCUPBUMAULUOUMAUKUMUGQUHRSUI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x ps $. $d x A $. $d x B $. $d x C $.
|
|
eueq3.1 $e |- A e. _V $.
|
|
eueq3.2 $e |- B e. _V $.
|
|
eueq3.3 $e |- C e. _V $.
|
|
eueq3.4 $e |- -. ( ph /\ ps ) $.
|
|
$( Equality has existential uniqueness (split into 3 cases). (Contributed
|
|
by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro,
|
|
28-Sep-2015.) $)
|
|
eueq3 $p |- E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B )
|
|
\/ ( ps /\ x = C ) ) $=
|
|
( wceq wa wo wn w3o weu eueq1 ibar wb con2i cv pm2.45 imnani jaoi orbi12d
|
|
bianfd mtbid biorf bitrd 3orrot df-3or bitri syl6bbr eubidv adantr pm2.46
|
|
syl mpbii simpl orim12i con3i 3orcomb ecase3 ) ABACUAZDKZLZABMZNZVDEKZLZB
|
|
VDFKZLZOZCPZAVECPVNCDGQAVEVMCAVEVJVLMZVFMZVMAVEVFVPAVERAVONVFVPSAVHBMZVOV
|
|
QAVHANBABUBZABABJUCZTUDTAVHVJBVLAVHVIVHAVRTUFABVKVSUFUEUGVOVFUHUQUIVMVJVL
|
|
VFOVPVFVJVLUJVJVLVFUKULUMUNURBVKCPVNCFIQBVKVMCBVKVFVJMZVLMZVMBVKVLWABVKRB
|
|
VTNVLWASVTBVFBNZVJAWBVEVSUOVHWBVIABUPUOUDTVTVLUHUQUIVFVJVLUKUMUNURVHVICPV
|
|
NCEHQVHVIVMCVHVIVFVLMZVJMZVMVHVIVJWDVHVIRVHWCNVJWDSWCVGVFAVLBAVEUSBVKUSUT
|
|
VAWCVJUHUQUIVMVFVLVJOWDVFVJVLVBVFVLVJUKULUMUNURVC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( There is at most one set equal to a class. (Contributed by NM,
|
|
8-Mar-1995.) $)
|
|
moeq $p |- E* x x = A $=
|
|
( cv wceq wmo wex weu wi cvv wcel isset eueq bitr3i biimpi df-mo mpbir )
|
|
ACBDZAEQAFZQAGZHRSRBIJSABKABLMNQAOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d x y ps $. $d x y A $. $d x y B $. $d x y C $.
|
|
moeq3.1 $e |- B e. _V $.
|
|
moeq3.2 $e |- C e. _V $.
|
|
moeq3.3 $e |- -. ( ph /\ ps ) $.
|
|
$( "At most one" property of equality (split into 3 cases). (The first 2
|
|
hypotheses could be eliminated with longer proof.) (Contributed by NM,
|
|
23-Apr-1995.) $)
|
|
moeq3 $p |- E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B )
|
|
\/ ( ps /\ x = C ) ) $=
|
|
( vy cvv wcel cv wceq wa wo wn w3o weu biidd eqeq2 anbi2d 3orbi123d eueq3
|
|
wmo eubidv vex vtoclg eumo syl wi eleq1 mpbii pm2.21 anim2d orim1d 3orass
|
|
wal syl5 3imtr4g alrimiv euimmo ee10 pm2.61i ) DKLZACMZDNZOZABPQVFENOZBVF
|
|
FNOZRZCUEZVEVKCSZVLAVFJMZNZOZVIVJRZCSZVMJDKVNDNZVQVKCVSVPVHVIVIVJVJVSVOVG
|
|
AVNDVFUAUBVSVITVSVJTUCUFABCVNEFJUGGHIUDZUHVKCUIUJVEQZVKVQUKZCURVRVLWAWBCW
|
|
AVHVIVJPZPVPWCPVKVQWAVHVPWCWAVGVOAVGVEWAVOVGVFKLVECUGVFDKULUMVEVOUNUSUOUP
|
|
VHVIVJUQVPVIVJUQUTVAVTVKVQCVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
mosub.1 $e |- E* x ph $.
|
|
$( "At most one" remains true after substitution. (Contributed by NM,
|
|
9-Mar-1995.) $)
|
|
mosub $p |- E* x E. y ( y = A /\ ph ) $=
|
|
( cv wceq wmo wal wa wex moeq ax-gen moexexv mp2an ) CFDGZCHABHZCIPAJCKBH
|
|
CDLQCEMPACBNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( Theorem for inferring "at most one." (Contributed by NM,
|
|
17-Oct-1996.) $)
|
|
mo2icl $p |- ( A. x ( ph -> x = A ) -> E* x ph ) $=
|
|
( vy cvv wcel cv wceq wi wal wmo eqeq2 imbi2d albidv imbi1d wex 19.8a nfv
|
|
mo2 wn sylibr vtoclg vex eleq1 mpbii imim2i con3rr3 alimdv alnex exmo ori
|
|
sylbi syl6 pm2.61i ) CEFZABGZCHZIZBJZABKZIZAUPDGZHZIZBJZUTIVADCEVBCHZVEUS
|
|
UTVFVDURBVFVCUQAVBCUPLMNOVEVEDPUTVEDQABDADRSUAUBUOTZUSATZBJZUTVGURVHBURAU
|
|
OUQUOAUQUPEFUOBUCUPCEUDUEUFUGUHVIABPZTUTABUIVJUTABUJUKULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $. $d x y ps $.
|
|
moi2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) $)
|
|
mob2 $p |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) ) $=
|
|
( vy wcel wmo w3a cv wceq simp3 syl5ibcom wi wa wsb wal nfs1v sbequ12 nfv
|
|
mo4f sylbi sbhypf anbi2d eqeq2 imbi12d spcgv syl5 imp exp3a 3impia impbid
|
|
sp ) DEHZACIZAJZCKZDLZBUQAUSBUOUPAMFNUOUPABUSOUOUPPABUSUOUPABPZUSOZUPAACG
|
|
QZPZURGKZLZOZGRZUOVAUPVGCRVGAVBCGACGSACGTUBVGCUNUCVFVAGDEVDDLZVCUTVEUSVHV
|
|
BBAABCGDBCUAFUDUEVDDURUFUGUHUIUJUKULUM $.
|
|
|
|
$( Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) $)
|
|
moi2 $p |- ( ( ( A e. B /\ E* x ph ) /\ ( ph /\ ps ) ) -> x = A ) $=
|
|
( wcel wmo wa cv wceq wb mob2 3expa biimprd impr ) DEGZACHZIZABCJDKZSAITB
|
|
QRATBLABCDEFMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ch $. $d x ps $.
|
|
moi.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
moi.2 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
$( Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) $)
|
|
mob $p |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) ->
|
|
( A = B <-> ch ) ) $=
|
|
( wcel wa wmo wceq wb wi cvv elex w3a nfv cv nfcv nfmo1 nf3an nfim bibi1d
|
|
3anbi3d eqeq1 imbi12d mob2 vtoclgf com12 3expib syl com3r imp 3impib ) EG
|
|
KZFHKZLADMZBEFNZCOZURUSUTBLZVBPUSVCURVBUSFQKZVCURVBPZPFHRVDUTBVEURVDUTBSZ
|
|
VBVDUTASZDUAZFNZCOZPVFVBPDEGDEUBVFVBDVDUTBDVDDTADUCBDTUDVBDTUEVHENZVGVFVJ
|
|
VBVKABVDUTIUGVKVIVACVHEFUHUFUIACDFQJUJUKULUMUNUOUPUQ $.
|
|
|
|
$( Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) $)
|
|
moi $p |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) ->
|
|
A = B ) $=
|
|
( wcel wa wmo wceq wi w3a mob biimprd 3expia imp3a 3impia ) EGKFHKLZADMZB
|
|
CLEFNZUBUCLBCUDUBUCBCUDOUBUCBPUDCABCDEFGHIJQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d B x $. $d A x $. $d ps x $.
|
|
morex.1 $e |- B e. _V $.
|
|
morex.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( Derive membership from uniqueness. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
morex $p |- ( ( E. x e. A ph /\ E* x ph ) -> ( ps -> B e. A ) ) $=
|
|
( wmo wrex wcel wi cv wa wex df-rex exancom bitri wal nfmo1 nfe1 nfan syl
|
|
mopick alrimi wceq eleq1 imbi12d spcv sylan2b ancoms ) ACHZACDIZBEDJZKZUL
|
|
UKACLZDJZMZCNZUNULUPAMCNURACDOUPACPQUKURMZAUPKZCRUNUSUTCUKURCACSUQCTUAAUP
|
|
CUCUDUTUNCEFUOEUEABUPUMGUOEDUFUGUHUBUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $.
|
|
euxfr2.1 $e |- A e. _V $.
|
|
euxfr2.2 $e |- E* y x = A $.
|
|
$( Transfer existential uniqueness from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by NM,
|
|
14-Nov-2004.) $)
|
|
euxfr2 $p |- ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) $=
|
|
( cv wceq wa wex weu wmo wi 2euswap moani ancom mobii mpbi mpg moeq biidd
|
|
impbii ceqsexv eubii bitri ) BGDHZAIZCJBKZUGBJZCKZACKUHUJUGCLZUHUJMBUGBCN
|
|
AUFIZCLUKUFACFOULUGCAUFPZQRSUGBLZUJUHMCUGCBNULBLUNUFABBDTOULUGBUMQRSUBUIA
|
|
CAABDEUFAUAUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x ps $. $d y ph $. $d x A $.
|
|
euxfr.1 $e |- A e. _V $.
|
|
euxfr.2 $e |- E! y x = A $.
|
|
euxfr.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Transfer existential uniqueness from a variable ` x ` to another
|
|
variable ` y ` contained in expression ` A ` . (Contributed by NM,
|
|
14-Nov-2004.) $)
|
|
euxfr $p |- ( E! x ph <-> E! y ps ) $=
|
|
( weu cv wceq wa wex euex ax-mp biantrur 19.41v pm5.32i exbii 3bitr2i
|
|
eubii eumoi euxfr2 bitri ) ACICJEKZBLZDMZCIBDIAUGCAUEDMZALUEALZDMUGUHAUED
|
|
IUHGUEDNOPUEADQUIUFDUEABHRSTUABCDEFUEDGUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d y z w ph $. $d x z ps $. $d y z w A $. $d x z B $. $d x y w $.
|
|
euind.1 $e |- B e. _V $.
|
|
euind.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
euind.3 $e |- ( x = y -> A = B ) $.
|
|
$( Existential uniqueness via an indirect equality. (Contributed by NM,
|
|
11-Oct-2010.) $)
|
|
euind $p |- ( ( A. x A. y ( ( ph /\ ps ) -> A = B ) /\ E. x ph )
|
|
-> E! z A. x ( ph -> z = A ) ) $=
|
|
( vw wa wceq wi wal wex cv exbii bitri imim2i cbvexv isseti 19.41v bitr3i
|
|
weu biantrur excom wb eqeq2 bi2 imbi1i impexp 3bitr3i sylib 2alimi 19.23v
|
|
an31 syl albii 19.21v eximdv syl5bi imp pm4.24 biimpi eqtr3 syl56 alanimi
|
|
prth com12 syl5 alrimivv adantl eqeq1 imbi2d albidv eu4 sylanbrc ) ABLZFG
|
|
MZNZDOCOZACPZLAEQZFMZNZCOZEPZWGAKQZFMZNZCOZLZWDWIMZNZKOEOZWGEUEWBWCWHWCWD
|
|
GMZBLZDPZEPZWBWHWCBDPZWTABCDIUAXAWQEPZBLZDPZWTBXCDXBBEGHUBUFRXDWREPZDPWTX
|
|
EXCDWQBEUCRWRDEUGUDSSWBWSWGEWBWRWFNZDOZCOZWSWGNZWAXFCDWAVSWEWQUHZNZXFVTXJ
|
|
VSFGWDUITXKVSWQWENZNZXFXJXLVSWEWQUJTVSWQLZWENWRALZWENXMXFXNXOWEABWQUQUKVS
|
|
WQWEULWRAWEULUMUNURUOXHWSWFNZCOXIXGXPCWRWFDUPUSWSWFCUTSUNVAVBVCWCWPWBWCWO
|
|
EKWMAWNNZCOZWCWNWFWKXQCAAALZWFWKLWEWJLWNAXSAVDVEAWEAWJVIWDWIFVFVGVHXRWCWN
|
|
XRWCWNNAWNCUPVEVJVKVLVMWGWLEKWNWFWKCWNWEWJAWDWIFVNVOVPVQVR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y ph $. $d x ps $.
|
|
$( A way to express restricted uniqueness. (Contributed by NM,
|
|
22-Nov-1994.) $)
|
|
reu2 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
|
|
A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
|
|
( cv wcel wa weu wex wsb wceq wi wal wreu nfv df-ral impexp albii 3bitr4i
|
|
wral wrex eu2 df-reu df-rex 19.21v nfs1v nfan sbequ12 anbi12d sbie anbi2i
|
|
eleq1 an4 bitri imbi1i 3bitri imbi2i bitr4i anbi12i ) BEZDFZAGZBHVBBIZVBV
|
|
BBCJZGZUTCEZKZLZCMZBMZGABDNABDUAZAABCJZGZVGLZCDTZBDTZGVBBCVBCOUBABDUCVKVC
|
|
VPVJABDUDVPVAVOLZBMVJVOBDPVIVQBVAVFDFZVNLZLZCMVAVSCMZLVIVQVAVSCUEVHVTCVHV
|
|
AVRGZVMGZVGLWBVNLVTVEWCVGVEVBVRVLGZGWCVDWDVBVBWDBCVRVLBVRBOABCUFUGVGVAVRA
|
|
VLUTVFDULABCUHUIUJUKVAAVRVLUMUNUOWBVMVGQVAVRVNQUPRVOWAVAVNCDPUQSRURUSS $.
|
|
|
|
$( A way to express restricted uniqueness. (Contributed by NM,
|
|
20-Oct-2006.) $)
|
|
reu6 $p |- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) ) $=
|
|
( cv wcel wa wceq wb wal wex wi bi1 bi2 adantr impbid imim2i imp 3bitr4i
|
|
ex wreu weu wral wrex df-reu wsb eleq1 sbequ12 anbi12d eqeq1 bibi12d eqid
|
|
19.28v tbt simpl sylbir syl6bi spimv expdimp simpr syl6 sps jca a5i imp3a
|
|
adantl eleq1a com23 adantll alimi impbii df-ral anbi2i exbii df-eu df-rex
|
|
jcai bitri ) ABDUABEZDFZAGZBUBZAVSCEZHZIZBDUCZCDUDZABDUEWAWDIZBJZCKWCDFZW
|
|
FGZCKWBWGWIWKCWJVTWELZGZBJZWJWLBJZGWIWKWJWLBUMWIWNWHWMBWIWJWLWHWJBCWDWHWJ
|
|
ABCUFZGZWCWCHZIZWJWDWAWQWDWRWDVTWJAWPVSWCDUGABCUHUIVSWCWCUJUKWSWQWJWRWQWC
|
|
ULUNWJWPUOUPUQURWHWLBWHVTWEWHVTGAWDWHVTAWDWAWDMUSWHWDALZVTWHWDWAAWAWDNVTA
|
|
UTVAOPTVBVCVDWMWHBWMWAWDWLWAWDLWJWLVTAWDWEAWDLVTAWDMQVEVFWMWDWAWMWDGVTAWM
|
|
WDVTWJWDVTLWLWCDVSVGORWLWDVTALZWJWLWDXAWLVTWDAWEWTVTAWDNQVHRVIVQTPVJVKWFW
|
|
OWJWEBDVLVMSVNWABCVOWFCDVPSVR $.
|
|
|
|
$( A way to express restricted uniqueness. (Contributed by NM,
|
|
24-Oct-2006.) $)
|
|
reu3 $p |- ( E! x e. A ph <->
|
|
( E. x e. A ph /\ E. y e. A A. x e. A ( ph -> x = y ) ) ) $=
|
|
( wreu wrex cv wceq wi wral wa reurex wb reu6 bi1 ralimi reximi sylbi wex
|
|
wal jca rexex anim2i weu nfv eu3 df-reu df-rex df-ral impexp albii bitr4i
|
|
wcel exbii anbi12i 3bitr4i sylibr impbii ) ABDEZABDFZABGZCGHZIZBDJZCDFZKZ
|
|
USUTVEABDLUSAVBMZBDJZCDFVEABCDNVHVDCDVGVCBDAVBOPQRUAVFUTVDCSZKZUSVEVIUTVD
|
|
CDUBUCVADUMZAKZBUDVLBSZVLVBIZBTZCSZKUSVJVLBCVLCUEUFABDUGUTVMVIVPABDUHVDVO
|
|
CVDVKVCIZBTVOVCBDUIVNVQBVKAVBUJUKULUNUOUPUQUR $.
|
|
|
|
$( A condition which implies existential uniqueness. (Contributed by Mario
|
|
Carneiro, 2-Oct-2015.) $)
|
|
reu6i $p |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph ) $=
|
|
( vy wcel cv wceq wb wral wa wrex wreu eqeq2 bibi2d ralbidv rspcev sylibr
|
|
reu6 ) DCFABGZDHZIZBCJZKATEGZHZIZBCJZECLABCMUGUCEDCUDDHZUFUBBCUHUEUAAUDDT
|
|
NOPQABECSR $.
|
|
|
|
eqreu.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( A condition which implies existential uniqueness. (Contributed by Mario
|
|
Carneiro, 2-Oct-2015.) $)
|
|
eqreu $p |- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) ->
|
|
E! x e. A ph ) $=
|
|
( wcel cv wceq wi wral wreu wa wb ralbiim ceqsralv anbi2d syl5bb reu6i ex
|
|
sylbird 3impib 3com23 ) EDGZACHEIZJCDKZBACDLZUDUFBUGUDUFBMZAUENCDKZUGUIUF
|
|
UEAJCDKZMUDUHAUECDOUDUJBUFABCEDFPQRUDUIUGACDESTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z ph $. $d x z ps $.
|
|
rmo4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Restricted "at most one" using implicit substitution. (Contributed by
|
|
NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) $)
|
|
rmo4 $p |- ( E* x e. A ph <->
|
|
A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) $=
|
|
( wrmo cv wcel wa wmo weq wi wral df-rmo wal bitri impexp albii df-ral
|
|
an4 ancom anbi1i imbi1i 3bitri r19.21v 3bitr2i eleq1 anbi12d mo4 3bitr4i
|
|
) ACEGCHZEIZAJZCKZABJZCDLZMZDENZCENZACEOUNDHZEIZBJZJZUQMZDPZCPUMUSMZCPUOU
|
|
TVFVGCVFVBUMURMZMZDPVHDENVGVEVIDVEVBUMJZUPJZUQMVJURMVIVDVKUQVDUMVBJZUPJVK
|
|
UMAVBBUAVLVJUPUMVBUBUCQUDVJUPUQRVBUMURRUESVHDETUMURDEUFUGSUNVCCDUQUMVBABU
|
|
LVAEUHFUIUJUSCETUKQ $.
|
|
|
|
$( Restricted uniqueness using implicit substitution. (Contributed by NM,
|
|
23-Nov-1994.) $)
|
|
reu4 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
|
|
A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) ) $=
|
|
( wreu wrex wrmo wa weq wi wral reu5 rmo4 anbi2i bitri ) ACEGACEHZACEIZJR
|
|
ABJCDKLDEMCEMZJACENSTRABCDEFOPQ $.
|
|
|
|
$( Restricted uniqueness using implicit substitution. (Contributed by NM,
|
|
24-Oct-2006.) $)
|
|
reu7 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\
|
|
E. x e. A A. y e. A ( ps -> x = y ) ) ) $=
|
|
( vz wreu wrex cv wceq wi wral wa reu3 eqeq1 eqcom syl6bb imbi12d bitri
|
|
cbvralv rexbii imbi2d ralbidv cbvrexv anbi2i ) ACEHACEIZACJZGJZKZLZCEMZGE
|
|
IZNUGBUHDJZKZLZDEMZCEIZNACGEOUMURUGUMBUIUNKZLZDEMZGEIURULVAGEUKUTCDEUOABU
|
|
JUSFUOUJUNUIKUSUHUNUIPUNUIQRSUAUBVAUQGCEUIUHKZUTUPDEVBUSUOBUIUHUNPUCUDUET
|
|
UFT $.
|
|
|
|
$( Restricted uniqueness using implicit substitution. (Contributed by NM,
|
|
24-Oct-2006.) $)
|
|
reu8 $p |- ( E! x e. A ph <-> E. x e. A ( ph /\
|
|
A. y e. A ( ps -> x = y ) ) ) $=
|
|
( wreu cv wceq wb wral wrex wi wa cbvreuv reu6 wcel ralbii wal syl5bb a1i
|
|
dfbi2 ancom equcom imbi2i biimt df-ral bi2.04 albii eleq1 imbi12d equcoms
|
|
vex bicomd ceqsalv 3bitrri syl6bb anbi12d r19.26 syl6rbbr rexbiia 3bitri
|
|
) ACEGBDEGBDHZCHZIZJZDEKZCELABVDVCIZMZDEKZNZCELABCDEFOBDCEPVGVKCEVGBVEMZV
|
|
EBMZNZDEKZVDEQZVKVFVNDEBVEUBRVPVKVLDEKZVMDEKZNZVOVKVJANVPVSAVJUCVPVJVQAVR
|
|
VJVQJVPVIVLDEVHVEBCDUDUERUAVPAVPAMZVRVPAUFVRVCEQZVMMZDSVEWABMZMZDSVTVMDEU
|
|
GWBWDDWAVEBUHUIWCVTDVDCUMWCVTJCDVHVTWCVHVPWAABVDVCEUJFUKUNULUOUPUQURTVLVM
|
|
DEUSUTTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality has existential uniqueness. (Contributed by Mario Carneiro,
|
|
1-Sep-2015.) $)
|
|
reueq $p |- ( B e. A <-> E! x e. A x = B ) $=
|
|
( wcel cv wceq wrex wreu risset wrmo wmo moeq mormo ax-mp mpbiran2 bitr4i
|
|
reu5 ) CBDAECFZABGZRABHZACBITSRABJZRAKUAACLRABMNRABQOP $.
|
|
$}
|
|
|
|
$( Restricted "at most one" still holds when a conjunct is added.
|
|
(Contributed by NM, 16-Jun-2017.) $)
|
|
rmoan $p |- ( E* x e. A ph -> E* x e. A ( ps /\ ph ) ) $=
|
|
( cv wcel wa wmo wrmo moan an12 mobii sylib df-rmo 3imtr4i ) CEDFZAGZCHZPBA
|
|
GZGZCHZACDISCDIRBQGZCHUAQBCJUBTCBPAKLMACDNSCDNO $.
|
|
|
|
$( Restricted "at most one" is preserved through implication (note wff
|
|
reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoim $p |- ( A. x e. A ( ph -> ps )
|
|
-> ( E* x e. A ps -> E* x e. A ph ) ) $=
|
|
( wi wral cv wcel wa wal wrmo df-ral imdistan albii wmo moim df-rmo 3imtr4g
|
|
bitri sylbi ) ABEZCDFZCGDHZAIZUCBIZEZCJZBCDKZACDKZEUBUCUAEZCJUGUACDLUJUFCUC
|
|
ABMNSUGUECOUDCOUHUIUDUECPBCDQACDQRT $.
|
|
|
|
${
|
|
rmoimia.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Restricted "at most one" is preserved through implication (note wff
|
|
reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoimia $p |- ( E* x e. A ps -> E* x e. A ph ) $=
|
|
( wi wrmo rmoim mprg ) ABFBCDGACDGFCDABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
rmoimi2.1 $e |- A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ps ) ) $.
|
|
$( Restricted "at most one" is preserved through implication (note wff
|
|
reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
rmoimi2 $p |- ( E* x e. B ps -> E* x e. A ph ) $=
|
|
( cv wcel wa wmo wrmo wi wal moim ax-mp df-rmo 3imtr4i ) CGZEHBIZCJZRDHAI
|
|
ZCJZBCEKACDKUASLCMTUBLFUASCNOBCEPACDPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x B $.
|
|
$( A condition allowing swap of uniqueness and existential quantifiers.
|
|
(Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM,
|
|
16-Jun-2017.) $)
|
|
2reuswap $p |- ( A. x e. A E* y e. B ph ->
|
|
( E! x e. A E. y e. B ph -> E! y e. B E. x e. A ph ) ) $=
|
|
( wral cv wcel wa wmo wrex wreu wal wex weu df-reu r19.42v df-rex bitri
|
|
wi wrmo df-rmo ralbii df-ral moanimv albii bitr4i bitr3i an12 exbii eubii
|
|
2euswap 3imtr4g sylbi ) ACEUAZBDFCGEHZAIZCJZBDFZACEKZBDLZABDKZCELZTZUOURB
|
|
DACEUBUCUSBGDHZUQIZCJZBMZVDUSVEURTZBMVHURBDUDVGVIBVEUQCUEUFUGVHVFCNZBOZVF
|
|
BNZCOZVAVCVFBCULVAVEUTIZBOVKUTBDPVNVJBVNUPVEAIZIZCNZVJVNVOCEKVQVEACEQVOCE
|
|
RUHVPVFCUPVEAUIUJSUKSVCUPVBIZCOVMVBCEPVRVLCVRUQBDKVLUPABDQUQBDRUHUKSUMUNU
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d w y z A $. $d x z B $. $d w x y z C $. $d w y z ph $. $d x z ps $.
|
|
reuind.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
reuind.2 $e |- ( x = y -> A = B ) $.
|
|
$( Existential uniqueness via an indirect equality. (Contributed by NM,
|
|
16-Oct-2010.) $)
|
|
reuind $p |- ( ( A. x A. y ( ( ( A e. C /\ ph ) /\ ( B e. C /\ ps ) )
|
|
-> A = B ) /\ E. x ( A e. C /\ ph ) )
|
|
-> E! z e. C A. x ( ( A e. C /\ ph ) -> z = A ) ) $=
|
|
( vw wcel wa wceq wi wal wex cv wrex bitri wral wreu eleq1d anbi12d exbii
|
|
cbvexv r19.41v rexcom4 risset anbi1i 3bitr4ri wb eqeq2 imim2i an31 imbi1i
|
|
bi2 impexp 3bitr3i sylib 2alimi 19.23v an12 adantr pm5.32ri bitr4i 19.42v
|
|
syl eleq1 albii 19.21v exp3a reximdvai syl5bi imp pm4.24 prth eqtr3 syl56
|
|
biimpi alanimi com12 syl5 a1d ralrimivv adantl eqeq1 imbi2d reu4 sylanbrc
|
|
albidv ) FHLZAMZGHLZBMZMZFGNZOZDPCPZWMCQZMWMERZFNZOZCPZEHSZXDWMKRZFNZOZCP
|
|
ZMZXAXFNZOZKHUAEHUAZXDEHUBWSWTXEWTXAGNZBMZDQZEHSZWSXEWTWODQZXQWMWOCDCRDRN
|
|
ZWLWNABXSFGHJUCIUDUFXOEHSZDQXNEHSZBMZDQXQXRXTYBDXNBEHUGUEXOEDHUHWOYBDWNYA
|
|
BEGHUIUJUEUKTWSXPXDEHWSXAHLZXPXDWSXNWOMZXCOZDPZCPZYCXPMZXDOZWRYECDWRWPXBX
|
|
NULZOZYEWQYJWPFGXAUMUNYKWPXNXBOZOZYEYJYLWPXBXNUQUNWPXNMZXBOYDWMMZXBOYMYEY
|
|
NYOXBWMWOXNUOUPWPXNXBURYDWMXBURUSUTVHVAYGYHXCOZCPYIYFYPCYFYDDQZXCOYPYDXCD
|
|
VBYQYHXCYQYCXOMZDQYHYDYRDYDWNXOMYRXNWNBVCXOYCWNXNYCWNULBXAGHVIVDVEVFUEYCX
|
|
ODVGTUPTVJYHXCCVKTUTVLVMVNVOWTXMWSWTXLEKHHWTXLYCXFHLMXJWMXKOZCPZWTXKXCXHY
|
|
SCWMWMWMMZXCXHMXBXGMXKWMUUAWMVPVTWMXBWMXGVQXAXFFVRVSWAYTWTXKYTWTXKOWMXKCV
|
|
BVTWBWCWDWEWFXDXIEKHXKXCXHCXKXBXGWMXAXFFWGWHWKWIWJ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x B $. $d x y $.
|
|
$( Double restricted quantification with "at most one," analogous to
|
|
~ 2moex . (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
|
|
2rmorex $p |- ( E* x e. A E. y e. B ph -> A. y e. B E* x e. A ph ) $=
|
|
( wrex wrmo nfcv nfre1 nfrmo cv wcel wi wral ex ralrimivw rmoim syl com12
|
|
rspe ralrimi ) ACEFZBDGZABDGZCEUBCBDCDHACEIJCKELZUCUDUEAUBMZBDNUCUDMUEUFB
|
|
DUEAUBACETOPAUBBDQRSUA $.
|
|
|
|
$( Lemma for ~ 2reu5 . Note that ` E! x e. A E! y e. B ph ` does not mean
|
|
"there is exactly one ` x ` in ` A ` and exactly one ` y ` in ` B ` such
|
|
that ` ph ` holds;" see comment for ~ 2eu5 . (Contributed by Alexander
|
|
van der Vekens, 17-Jun-2017.) $)
|
|
2reu5lem1 $p |- ( E! x e. A E! y e. B ph <->
|
|
E! x E! y ( x e. A /\ y e. B /\ ph ) ) $=
|
|
( wreu cv wcel wa weu w3a df-reu reubii euanv bicomi 3anass eubii bitri )
|
|
ACEFZBDFCGEHZAIZCJZBDFZBGDHZTAKZCJZBJZSUBBDACELMUCUDUBIZBJUGUBBDLUHUFBUHU
|
|
DUAIZCJZUFUJUHUDUACNOUIUECUEUIUDTAPOQRQRR $.
|
|
|
|
$( Lemma for ~ 2reu5 . (Contributed by Alexander van der Vekens,
|
|
17-Jun-2017.) $)
|
|
2reu5lem2 $p |- ( A. x e. A E* y e. B ph <->
|
|
A. x E* y ( x e. A /\ y e. B /\ ph ) ) $=
|
|
( wrmo wral cv wcel wa wmo w3a wal df-rmo ralbii wi df-ral moanimv bicomi
|
|
bitri 3anass mobii albii ) ACEFZBDGCHEIZAJZCKZBDGZBHDIZUEALZCKZBMZUDUGBDA
|
|
CENOUHUIUGPZBMULUGBDQUMUKBUMUIUFJZCKZUKUOUMUIUFCRSUNUJCUJUNUIUEAUASUBTUCT
|
|
T $.
|
|
$}
|
|
|
|
${
|
|
$d w y z A $. $d w x z B $. $d x y $. $d ph w $. $d ph z $.
|
|
$( Lemma for ~ 2reu5 . This lemma is interesting in its own right, showing
|
|
that existential restriction in the last conjunct (the "at most one"
|
|
part) is optional; compare ~ rmo2 . (Contributed by Alexander van der
|
|
Vekens, 17-Jun-2017.) $)
|
|
2reu5lem3 $p |- ( ( E! x e. A E! y e. B ph /\ A. x e. A E* y e. B ph )
|
|
<-> ( E. x e. A E. y e. B ph
|
|
/\ E. z E. w A. x e. A A. y e. B
|
|
( ph -> ( x = z /\ y = w ) ) ) ) $=
|
|
( wreu wral wa cv wcel weu wal wex weq wi wrex exbii 3bitri w3a 2reu5lem1
|
|
wrmo wmo 2reu5lem2 anbi12i 2eu5 3anass 19.42v df-rex bicomi anbi2i bitr4i
|
|
3anan12 imbi1i impexp imbi2i albii df-ral r19.21v 3bitr2i ) ACGHBFHZACGUC
|
|
BFIZJBKFLZCKGLZAUAZCMBMZVFCUDBNZJVFCOZBOZVFBDPCEPJZQZCNZBNZEOZDOZJACGRZBF
|
|
RZAVKQZCGIZBFIZEOZDOZJVBVGVCVHABCFGUBABCFGUEUFVFBCDEUGVJVRVPWCVJVDVQJZBOV
|
|
RVIWDBVIVDVEAJZJZCOVDWECOZJWDVFWFCVDVEAUHSVDWECUIWGVQVDVQWGACGUJUKULTSVQB
|
|
FUJUMVOWBDVNWAEVNVDVTQZBNWAVMWHBVMVEVDVSQZQZCNWICGIWHVLWJCVLVEVDAJZJZVKQV
|
|
EWKVKQZQWJVFWLVKVDVEAUNUOVEWKVKUPWMWIVEVDAVKUPUQTURWICGUSVDVSCGUTVAURVTBF
|
|
USUMSSUFT $.
|
|
|
|
$d x A $. $d y B $.
|
|
$( Double restricted existential uniqueness in terms of restricted
|
|
existential quantification and restricted universal quantification,
|
|
analogous to ~ 2eu5 and ~ reu3 . (Contributed by Alexander van der
|
|
Vekens, 17-Jun-2017.) $)
|
|
2reu5 $p |- ( ( E! x e. A E! y e. B ph /\ A. x e. A E* y e. B ph )
|
|
<-> ( E. x e. A E. y e. B ph
|
|
/\ E. z e. A E. w e. B A. x e. A A. y e. B
|
|
( ph -> ( x = z /\ y = w ) ) ) ) $=
|
|
( wrex weq wa wral wex cv wcel wreu r19.29r reximi eleq1 ex df-rex pm3.35
|
|
wi wrmo bi2anan9 biimpac ancomd rexlimivv syl 3syl pm4.71rd anass 2exbidv
|
|
syl6bb pm5.32i 2reu5lem3 r19.42v bitr3i exbii bitri anbi2i 3bitr4i ) ACGH
|
|
ZBFHZABDIZCEIZJZUBZCGKZBFKZELDLZJVCEMZGNZDMZFNZVIJZJZELZDLZJACGOBFOACGUCB
|
|
FKJVCVIEGHZDFHZJVCVJVRVCVIVPDEVCVIVLVNJZVIJVPVCVIWAVCVIWAVCVIJVBVHJZBFHAV
|
|
GJZCGHZBFHZWAVBVHBFPWBWDBFAVGCGPQWEVFCGHZBFHWAWDWFBFWCVFCGAVFUAQQVFWABCFG
|
|
BMZFNZCMZGNZJZVFWAWKVFJVNVLVFWKVNVLJVDWHVNVEWJVLWGVMFRWIVKGRUDUEUFSUGUHUI
|
|
SUJVLVNVIUKUMULUNABCDEFGUOVTVRVCVTVNVSJZDLVRVSDFTWLVQDWLVOEGHVQVNVIEGUPVO
|
|
EGTUQURUSUTVA $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Russell's Paradox
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y $.
|
|
$( Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.
|
|
|
|
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension,
|
|
expressed in our notation as ` A e. _V ` , asserted that any collection
|
|
of sets ` A ` is a set i.e. belongs to the universe ` _V ` of all sets.
|
|
In particular, by substituting ` { x | x e/ x } ` (the "Russell class")
|
|
for ` A ` , it asserted ` { x | x e/ x } e. _V ` , meaning that the
|
|
"collection of all sets which are not members of themselves" is a set.
|
|
However, here we prove ` { x | x e/ x } e/ _V ` . This contradiction
|
|
was discovered by Russell in 1901 (published in 1903), invalidating the
|
|
Comprehension Axiom and leading to the collapse of Frege's system.
|
|
|
|
In 1908, Zermelo rectified this fatal flaw by replacing Comprehension
|
|
with a weaker Subset (or Separation) Axiom ssex in set.mm asserting that
|
|
` A ` is a set only when it is smaller than some other set ` B ` .
|
|
However, Zermelo was then faced with a "chicken and egg" problem of how
|
|
to show ` B ` is a set, leading him to introduce the set-building axioms
|
|
of Null Set ~ 0ex , Pairing ~ prex , Union ~ uniex , Power Set ~ pwex ,
|
|
and Infinity omex in set.mm to give him some starting sets to work with
|
|
(all of which, before Russell's Paradox, were immediate consequences of
|
|
Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom
|
|
with our present Replacement Axiom funimaex in set.mm (whose modern
|
|
formalization is due to Skolem, also in 1922). Thus, in a very real
|
|
sense Russell's Paradox spawned the invention of ZF set theory and
|
|
completely revised the foundations of mathematics!
|
|
|
|
Another mainstream formalization of set theory, devised by von Neumann,
|
|
Bernays, and Goedel, uses class variables rather than setvar variables
|
|
as its primitives. The axiom system NBG in [Mendelson] p. 225 is
|
|
suitable for a Metamath encoding. NBG is a conservative extension of ZF
|
|
in that it proves exactly the same theorems as ZF that are expressible
|
|
in the language of ZF. An advantage of NBG is that it is finitely
|
|
axiomatizable - the Axiom of Replacement can be broken down into a
|
|
finite set of formulas that eliminate its wff metavariable. Finite
|
|
axiomatizability is required by some proof languages (although not by
|
|
Metamath). There is a stronger version of NBG called Morse-Kelley
|
|
(axiom system MK in [Mendelson] p. 287).
|
|
|
|
Russell himself continued in a different direction, avoiding the paradox
|
|
with his "theory of types." Quine extended Russell's ideas to formulate
|
|
his New Foundations set theory (axiom system NF of [Quine] p. 331). In
|
|
NF, the collection of all sets is a set, contradicting ZF and NBG set
|
|
theories, and it has other bizarre consequences: when sets become too
|
|
huge (beyond the size of those used in standard mathematics), the Axiom
|
|
of Choice ac4 in set.mm and Cantor's Theorem canth in set.mm are
|
|
provably false! (See ncanth in set.mm for some intuition behind the
|
|
latter.) Recent results (as of 2014) seem to show that NF is
|
|
equiconsistent to Z (ZF in which ax-sep in set.mm replaces ax-rep in
|
|
set.mm) with ax-sep restricted to only bounded quantifiers. NF is
|
|
finitely axiomatizable and can be encoded in Metamath using the axioms
|
|
from T. Hailperin, "A set of axioms for logic," _J. Symb. Logic_ 9:1-19
|
|
(1944).
|
|
|
|
Under ZF set theory, every set is a member of the Russell class by
|
|
elirrv in set.mm (derived from the Axiom of Regularity), so for us the
|
|
Russell class equals the universe ` _V ` (theorem ruv in set.mm). See
|
|
ruALT in set.mm for an alternate proof of ~ ru derived from that fact.
|
|
(Contributed by NM, 7-Aug-1994.) $)
|
|
ru $p |- { x | x e/ x } e/ _V $=
|
|
( vy cv wnel cab cvv wcel wn wceq wex wb wal pm5.19 df-nel eleq12d notbid
|
|
eleq1 id syl5bb mtbir bibi12d spv mto abeq2 nex isset mpbir ) ACZUHDZAEZF
|
|
DUJFGZHUKBCZUJIZBJUMBUMUHULGZUIKZALZUPULULGZUQHZKZUQMUOUSABUHULIZUNUQUIUR
|
|
UHULULQUIUHUHGZHUTURUHUHNUTVAUQUTUHULUHULUTRZVBOPSUAUBUCUIAULUDTUEBUJUFTU
|
|
JFNUG $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Proper substitution of classes for sets
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c [. $.
|
|
$c ]. $.
|
|
|
|
$( Extend wff notation to include the proper substitution of a class for a
|
|
set. Read this notation as "the proper substitution of class ` A ` for
|
|
setvar variable ` x ` in wff ` ph ` ." $)
|
|
wsbc $a wff [. A / x ]. ph $.
|
|
|
|
$( Define the proper substitution of a class for a set.
|
|
|
|
When ` A ` is a proper class, our definition evaluates to false. This is
|
|
somewhat arbitrary: we could have, instead, chosen the conclusion of
|
|
~ sbc6 for our definition, which always evaluates to true for proper
|
|
classes.
|
|
|
|
Our definition also does not produce the same results as discussed in the
|
|
proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does
|
|
hold, as shown by ~ dfsbcq below). For example, if ` A ` is a proper
|
|
class, Quine's substitution of ` A ` for ` y ` in 0 ` e. y ` evaluates to
|
|
0 ` e. A ` rather than our falsehood. (This can be seen by substituting
|
|
` A ` , ` y ` , and 0 for alpha, beta, and gamma in Subcase 1 of Quine's
|
|
discussion on p. 42.) Unfortunately, Quine's definition requires a
|
|
recursive syntactical breakdown of ` ph ` , and it does not seem possible
|
|
to express it with a single closed formula.
|
|
|
|
If we did not want to commit to any specific proper class behavior, we
|
|
could use this definition _only_ to prove theorem ~ dfsbcq , which holds
|
|
for both our definition and Quine's, and from which we can derive a weaker
|
|
version of ~ df-sbc in the form of ~ sbc8g . However, the behavior of
|
|
Quine's definition at proper classes is similarly arbitrary, and for
|
|
practical reasons (to avoid having to prove sethood of ` A ` in every use
|
|
of this definition) we allow direct reference to ~ df-sbc and assert that
|
|
` [. A / x ]. ph ` is always false when ` A ` is a proper class.
|
|
|
|
The theorem ~ sbc2or shows the apparently "strongest" statement we can
|
|
make regarding behavior at proper classes if we start from ~ dfsbcq .
|
|
|
|
The related definition ~ df-csb defines proper substitution into a class
|
|
variable (as opposed to a wff variable). (Contributed by NM,
|
|
14-Apr-1995.) (Revised by NM, 25-Dec-2016.) $)
|
|
df-sbc $a |- ( [. A / x ]. ph <-> A e. { x | ph } ) $.
|
|
|
|
$( --- Start of old code before overloading prevention patch. $)
|
|
$(
|
|
@( Extend wff notation to include the proper substitution of a class for a
|
|
set. This definition "overloads" the previously defined variable
|
|
substitution ~ wsb (where the first argument is a setvar variable rather
|
|
than a class variable). We take care to ensure that this new definition
|
|
is a conservative extension. Read this notation as "the proper
|
|
substitution of class ` A ` for setvar variable ` x ` in wff ` ph ` ." @)
|
|
wsbcSBC @a wff [ A / x ] ph @.
|
|
$)
|
|
$( --- End of old code before overloading prevention patch. $)
|
|
|
|
$( This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
|
|
under both our definition and Quine's, provides us with a weak definition
|
|
of the proper substitution of a class for a set. Since our ~ df-sbc does
|
|
not result in the same behavior as Quine's for proper classes, if we
|
|
wished to avoid conflict with Quine's definition we could start with this
|
|
theorem and ~ dfsbcq2 instead of ~ df-sbc . ( ~ dfsbcq2 is needed because
|
|
unlike Quine we do not overload the ~ df-sb syntax.) As a consequence of
|
|
these theorems, we can derive ~ sbc8g , which is a weaker version of
|
|
~ df-sbc that leaves substitution undefined when ` A ` is a proper class.
|
|
|
|
However, it is often a nuisance to have to prove the sethood hypothesis of
|
|
~ sbc8g , so we will allow direct use of ~ df-sbc after theorem ~ sbc2or
|
|
below. Proper substiution with a proper class is rarely needed, and when
|
|
it is, we can simply use the expansion of Quine's definition.
|
|
(Contributed by NM, 14-Apr-1995.) $)
|
|
dfsbcq $p |- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) ) $=
|
|
( wceq cab wcel wsbc eleq1 df-sbc 3bitr4g ) CDECABFZGDLGABCHABDHCDLIABCJABD
|
|
JK $.
|
|
|
|
$( This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
|
|
under both our definition and Quine's, relates logic substitution ~ df-sb
|
|
and substitution for class variables ~ df-sbc . Unlike Quine, we use a
|
|
different syntax for each in order to avoid overloading it. See remarks
|
|
in ~ dfsbcq . (Contributed by NM, 31-Dec-2016.) $)
|
|
dfsbcq2 $p |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) ) $=
|
|
( cv wceq cab wcel wsb wsbc eleq1 df-clab df-sbc bicomi 3bitr3g ) CEZDFPABG
|
|
ZHDQHZABCIABDJZPDQKACBLSRABDMNO $.
|
|
|
|
$( Show that ~ df-sb and ~ df-sbc are equivalent when the class term ` A ` in
|
|
~ df-sbc is a setvar variable. This theorem lets us reuse theorems based
|
|
on ~ df-sb for proofs involving ~ df-sbc . (Contributed by NM,
|
|
31-Dec-2016.) (Proof modification is discouraged.) $)
|
|
sbsbc $p |- ( [ y / x ] ph <-> [. y / x ]. ph ) $=
|
|
( weq wsb cv wsbc wb eqid dfsbcq2 ax-mp ) CCDABCEABCFZGHLIABCLJK $.
|
|
|
|
${
|
|
sbceq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for class substitution. (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) $)
|
|
sbceq1d $p |- ( ph -> ( [. A / x ]. ph <-> [. B / x ]. ph ) ) $=
|
|
( wceq wsbc wb dfsbcq syl ) ACDFABCGABDGHEABCDIJ $.
|
|
|
|
sbceq1dd.2 $e |- ( ph -> [. A / x ]. ph ) $.
|
|
$( Equality theorem for class substitution. (Contributed by Mario
|
|
Carneiro, 9-Feb-2017.) $)
|
|
sbceq1dd $p |- ( ph -> [. B / x ]. ph ) $=
|
|
( wsbc sbceq1d mpbid ) AABCGABDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y ph $. $d x y $.
|
|
$( This is the closest we can get to ~ df-sbc if we start from ~ dfsbcq
|
|
(see its comments) and ~ dfsbcq2 . (Contributed by NM, 18-Nov-2008.)
|
|
(Proof shortened by Andrew Salmon, 29-Jun-2011.)
|
|
(Proof modification is discouraged.) $)
|
|
sbc8g $p |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) ) $=
|
|
( vy cv wsbc cab wcel dfsbcq eleq1 wsb df-clab weq wb equid dfsbcq2 ax-mp
|
|
bitr2i vtoclbg ) ABEFZGZUAABHZIZABCGCUCIECDABUACJUACUCKUDABELZUBAEBMEENUE
|
|
UBOEPABEUAQRST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( The disjunction of two equivalences for class substitution does not
|
|
require a class existence hypothesis. This theorem tells us that there
|
|
are only 2 possibilities for ` [ A / x ] ph ` behavior at proper
|
|
classes, matching the ~ sbc5 (false) and ~ sbc6 (true) conclusions.
|
|
This is interesting since ~ dfsbcq and ~ dfsbcq2 (from which it is
|
|
derived) do not appear to say anything obvious about proper class
|
|
behavior. Note that this theorem doesn't tell us that it is always one
|
|
or the other at proper classes; it could "flip" between false (the first
|
|
disjunct) and true (the second disjunct) as a function of some other
|
|
variable ` y ` that ` ph ` or ` A ` may contain. (Contributed by NM,
|
|
11-Oct-2004.) (Proof modification is discouraged.) $)
|
|
sbc2or $p |- ( ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) \/
|
|
( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) ) $=
|
|
( vy cvv wcel wsbc cv wceq wa wex wb wi wal wo wsb weq wn mpbii con3i sb5
|
|
dfsbcq2 eqeq2 anbi1d exbidv vtoclbg pm5.15 vex eleq1 adantr nexdv pm2.21d
|
|
orcd alrimiv 2thd bibi2d orbi2d pm2.61i ) CEFZABCGZBHZCIZAJZBKZLZUTVBAMZB
|
|
NZLZOZUSVEVHABDPBDQZAJZBKUTVDDCEABDCUBDHZCIZVKVCBVMVJVBAVLCVAUCUDUEABDUAU
|
|
FUMUSRZVEUTVDRZLZOVIUTVDUGVNVPVHVEVNVOVGUTVNVOVGVNVCBVCUSVBUSAVBVAEFUSBUH
|
|
VACEUISZUJTUKVNVFBVNVBAVBUSVQTULUNUOUPUQSUR $.
|
|
$}
|
|
|
|
$( By our definition of proper substitution, it can only be true if the
|
|
substituted expression is a set. (Contributed by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
sbcex $p |- ( [. A / x ]. ph -> A e. _V ) $=
|
|
( wsbc cab wcel cvv df-sbc elex sylbi ) ABCDCABEZFCGFABCHCKIJ $.
|
|
|
|
$( Equality theorem for class substitution. Class version of ~ sbequ12 .
|
|
(Contributed by NM, 26-Sep-2003.) $)
|
|
sbceq1a $p |- ( x = A -> ( ph <-> [. A / x ]. ph ) ) $=
|
|
( wsb cv wceq wsbc sbid dfsbcq2 syl5bbr ) AABBDBECFABCGABHABBCIJ $.
|
|
|
|
$( Equality theorem for class substitution. Class version of ~ sbequ12r .
|
|
(Contributed by NM, 4-Jan-2017.) $)
|
|
sbceq2a $p |- ( A = x -> ( [. A / x ]. ph <-> ph ) ) $=
|
|
( cv wceq wsbc wb sbceq1a eqcoms bicomd ) CBDZEAABCFZALGKCABCHIJ $.
|
|
|
|
${
|
|
$d ph y $. $d A y $. $d x y $.
|
|
$( Specialization: if a formula is true for all sets, it is true for any
|
|
class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See
|
|
also ~ stdpc4 and ~ rspsbc . (Contributed by NM, 16-Jan-2004.) $)
|
|
spsbc $p |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) ) $=
|
|
( vy wal wsbc wi cv wceq wsb stdpc4 sbsbc sylib dfsbcq syl5ib vtocleg ) A
|
|
BFZABCGZHECDRABEIZGZTCJSRABEKUAABELABEMNABTCOPQ $.
|
|
|
|
spsbcd.1 $e |- ( ph -> A e. V ) $.
|
|
spsbcd.2 $e |- ( ph -> A. x ps ) $.
|
|
$( Specialization: if a formula is true for all sets, it is true for any
|
|
class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See
|
|
also ~ stdpc4 and ~ rspsbc . (Contributed by Mario Carneiro,
|
|
9-Feb-2017.) $)
|
|
spsbcd $p |- ( ph -> [. A / x ]. ps ) $=
|
|
( wcel wal wsbc spsbc sylc ) ADEHBCIBCDJFGBCDEKL $.
|
|
$}
|
|
|
|
${
|
|
sbcth.1 $e |- ph $.
|
|
$( A substitution into a theorem remains true (when ` A ` is a set).
|
|
(Contributed by NM, 5-Nov-2005.) $)
|
|
sbcth $p |- ( A e. V -> [. A / x ]. ph ) $=
|
|
( wcel wal wsbc ax-gen spsbc mpi ) CDFABGABCHABEIABCDJK $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
sbcthdv.1 $e |- ( ph -> ps ) $.
|
|
$( Deduction version of ~ sbcth . (Contributed by NM, 30-Nov-2005.)
|
|
(Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
sbcthdv $p |- ( ( ph /\ A e. V ) -> [. A / x ]. ps ) $=
|
|
( wal wcel wsbc alrimiv spsbc mpan9 ) ABCGDEHBCDIABCFJBCDEKL $.
|
|
$}
|
|
|
|
$( An identity theorem for substitution. See ~ sbid . (Contributed by Mario
|
|
Carneiro, 18-Feb-2017.) $)
|
|
sbcid $p |- ( [. x / x ]. ph <-> ph )
|
|
$=
|
|
( cv wsbc wsb sbsbc sbid bitr3i ) ABBCDABBEAABBFABGH $.
|
|
|
|
${
|
|
nfsbc1d.2 $e |- ( ph -> F/_ x A ) $.
|
|
$( Deduction version of ~ nfsbc1 . (Contributed by NM, 23-May-2006.)
|
|
(Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbc1d $p |- ( ph -> F/ x [. A / x ]. ps ) $=
|
|
( wsbc cab wcel df-sbc wnfc nfab1 a1i nfeld nfxfrd ) BCDFDBCGZHACBCDIACDO
|
|
ECOJABCKLMN $.
|
|
$}
|
|
|
|
${
|
|
nfsbc1.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for class substitution. (Contributed
|
|
by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbc1 $p |- F/ x [. A / x ]. ph $=
|
|
( wsbc wnf wtru wnfc a1i nfsbc1d trud ) ABCEBFGABCBCHGDIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Bound-variable hypothesis builder for class substitution. (Contributed
|
|
by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbc1v $p |- F/ x [. A / x ]. ph $=
|
|
( nfcv nfsbc1 ) ABCBCDE $.
|
|
$}
|
|
|
|
${
|
|
nfsbcd.1 $e |- F/ y ph $.
|
|
nfsbcd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfsbcd.3 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfsbc . (Contributed by NM, 23-Nov-2005.)
|
|
(Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbcd $p |- ( ph -> F/ x [. A / y ]. ps ) $=
|
|
( wsbc cab wcel df-sbc nfabd nfeld nfxfrd ) BDEIEBDJZKACBDELACEPGABCDFHMN
|
|
O $.
|
|
$}
|
|
|
|
${
|
|
nfsbc.1 $e |- F/_ x A $.
|
|
nfsbc.2 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for class substitution. (Contributed
|
|
by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfsbc $p |- F/ x [. A / y ]. ph $=
|
|
( wsbc wnf wtru nftru wnfc a1i nfsbcd trud ) ACDGBHIABCDCJBDKIELABHIFLMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d z A $. $d y z ph $.
|
|
$( A composition law for class substitution. (Contributed by NM,
|
|
26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
sbcco $p |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) $=
|
|
( vz cv wsbc cvv wcel sbcex dfsbcq wsb sbsbc sbbii sbco2 3bitr3ri vtoclbg
|
|
nfv bitri pm5.21nii ) ABCFGZCDGZDHIABDGZUACDJABDJUACEFZGZABUDGZUBUCEDHUAC
|
|
UDDKABUDDKUEABELZUFABCLZCELUACELUGUEUHUACEABCMNABECACROUACEMPABEMSQT $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d A y $.
|
|
sbcco2.1 $e |- ( x = y -> A = B ) $.
|
|
$( A composition law for class substitution. Importantly, ` x ` may occur
|
|
free in the class expression substituted for ` A ` . (Contributed by
|
|
NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
sbcco2 $p |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph ) $=
|
|
( wsbc cv wsb sbsbc nfv weq wceq wb eqcoms dfsbcq bicomd syl sbie bitr3i
|
|
) ABEGZCBHZGUACBIABDGZUACBJUAUCCBUCCKCBLDEMZUAUCNUDUBCHFOUDUCUAABDEPQRST
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( An equivalence for class substitution. (Contributed by NM,
|
|
23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
sbc5 $p |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) $=
|
|
( vy wsbc cvv wcel cv wceq wex sbcex exsimpl isset sylibr wsb weq dfsbcq2
|
|
wa eqeq2 anbi1d exbidv sb5 vtoclbg pm5.21nii ) ABCEZCFGZBHZCIZARZBJZABCKU
|
|
JUHBJUFUHABLBCMNABDOBDPZARZBJUEUJDCFABDCQDHZCIZULUIBUNUKUHAUMCUGSTUAABDUB
|
|
UCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( An equivalence for class substitution. (Contributed by NM,
|
|
11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) $)
|
|
sbc6g $p |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) ) $=
|
|
( wcel cv wceq wi wal wa wex wsbc nfe1 ceqex ceqsalg sbc5 syl6rbbr ) CDEB
|
|
FCGZAHBIRAJZBKZABCLATBCDSBMABCNOABCPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
sbc6.1 $e |- A e. _V $.
|
|
$( An equivalence for class substitution. (Contributed by NM,
|
|
23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) $)
|
|
sbc6 $p |- ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) $=
|
|
( cvv wcel wsbc cv wceq wi wal wb sbc6g ax-mp ) CEFABCGBHCIAJBKLDABCEMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y ph $. $d x y $.
|
|
$( An equivalence for class substitution in the spirit of ~ df-clab . Note
|
|
that ` x ` and ` A ` don't have to be distinct. (Contributed by NM,
|
|
18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
sbc7 $p |- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) ) $=
|
|
( wsbc cv wceq wa wex sbcco sbc5 bitr3i ) ABDEABCFZEZCDEMDGNHCIABCDJNCDKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
cbvsbc.1 $e |- F/ y ph $.
|
|
cbvsbc.2 $e |- F/ x ps $.
|
|
cbvsbc.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variables in a wff substitution. (Contributed by Jeff
|
|
Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon,
|
|
8-Jun-2011.) $)
|
|
cbvsbc $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
|
|
( cab wcel wsbc cbvab eleq2i df-sbc 3bitr4i ) EACIZJEBDIZJACEKBDEKPQEABCD
|
|
FGHLMACENBDENO $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $.
|
|
cbvsbcv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change the bound variable of a class substitution using implicit
|
|
substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario
|
|
Carneiro, 13-Oct-2016.) $)
|
|
cbvsbcv $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
|
|
( nfv cbvsbc ) ABCDEADGBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
using a bound-variable hypothesis instead of distinct variables.
|
|
(Closed theorem version of ~ sbciegf .) (Contributed by NM,
|
|
10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
sbciegft $p |- ( ( A e. V /\ F/ x ps /\
|
|
A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) ) $=
|
|
( wcel wnf cv wceq wb wi wal w3a wsbc imim2i alimi biimpa sylan2 3adant1
|
|
wa wex sbc5 bi1 imp3a 19.23t syl5bi bi2 com23 19.21t sbc6g sylibrd impbid
|
|
3ad2ant1 ) DEFZBCGZCHDIZABJZKZCLZMZACDNZBVAUPATZCUAZUTBACDUBUOUSVCBKZUNUS
|
|
UOVBBKZCLZVDURVECURUPABUQABKUPABUCOUDPUOVFVDVBBCUEQRSUFUTBUPAKZCLZVAUOUSB
|
|
VHKZUNUSUOBVGKZCLZVIURVJCURUPBAUQBAKUPABUGOUHPUOVKVIBVGCUIQRSUNUOVAVHJUSA
|
|
CDEUJUMUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
sbciegf.1 $e |- F/ x ps $.
|
|
sbciegf.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
sbciegf $p |- ( A e. V -> ( [. A / x ]. ph <-> ps ) ) $=
|
|
( wcel wnf cv wceq wb wi wal wsbc ax-gen sbciegft mp3an23 ) DEHBCICJDKABL
|
|
MZCNACDOBLFSCGPABCDEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
sbcieg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 10-Nov-2005.) $)
|
|
sbcieg $p |- ( A e. V -> ( [. A / x ]. ph <-> ps ) ) $=
|
|
( wcel cvv wsbc wb elex nfv sbciegf syl ) DEGDHGACDIBJDEKABCDHBCLFMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $. $d ch y $. $d ph y $. $d ps x $.
|
|
sbcie2g.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
sbcie2g.2 $e |- ( y = A -> ( ps <-> ch ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
This version of ~ sbcie avoids a disjointness condition on ` x , A ` by
|
|
substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) $)
|
|
sbcie2g $p |- ( A e. V -> ( [. A / x ]. ph <-> ch ) ) $=
|
|
( cv wsbc dfsbcq wsb sbsbc nfv sbie bitr3i vtoclbg ) ADEJZKZBADFKCEFGADSF
|
|
LITADEMBADENABDEBDOHPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
sbcie.1 $e |- A e. _V $.
|
|
sbcie.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 4-Sep-2004.) $)
|
|
sbcie $p |- ( [. A / x ]. ph <-> ps ) $=
|
|
( cvv wcel wsbc wb sbcieg ax-mp ) DGHACDIBJEABCDGFKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
sbcied.1 $e |- ( ph -> A e. V ) $.
|
|
sbcied.2 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
|
|
${
|
|
sbciedf.3 $e |- F/ x ph $.
|
|
sbciedf.4 $e |- ( ph -> F/ x ch ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
deduction form. (Contributed by NM, 29-Dec-2014.) $)
|
|
sbciedf $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
|
|
( wcel wnf cv wceq wb wi wal wsbc ex alrimi sbciegft syl3anc ) AEFKCDLD
|
|
MENZBCOZPZDQBDERCOGJAUEDIAUCUDHSTBCDEFUAUB $.
|
|
$}
|
|
|
|
$d x ph $. $d x ch $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
deduction form. (Contributed by NM, 13-Dec-2014.) $)
|
|
sbcied $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
|
|
( nfv nfvd sbciedf ) ABCDEFGHADIACDJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $. $d x ch $.
|
|
sbcied2.1 $e |- ( ph -> A e. V ) $.
|
|
sbcied2.2 $e |- ( ph -> A = B ) $.
|
|
sbcied2.3 $e |- ( ( ph /\ x = B ) -> ( ps <-> ch ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
deduction form. (Contributed by NM, 13-Dec-2014.) $)
|
|
sbcied2 $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
|
|
( cv wceq wb id sylan9eqr syldan sbcied ) ABCDEGHADKZELZRFLBCMSAREFSNIOJP
|
|
Q $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y ph $. $d x y $.
|
|
elrabsf.1 $e |- F/_ x B $.
|
|
$( Membership in a restricted class abstraction, expressed with explicit
|
|
class substitution. (The variation ~ elrabf has implicit
|
|
substitution). The hypothesis specifies that ` x ` must not be a free
|
|
variable in ` B ` . (Contributed by NM, 30-Sep-2003.) (Proof shortened
|
|
by Mario Carneiro, 13-Oct-2016.) $)
|
|
elrabsf $p |- ( A e. { x e. B | ph }
|
|
<-> ( A e. B /\ [. A / x ]. ph ) ) $=
|
|
( vy cv wsbc crab dfsbcq nfcv nfv nfsbc1v sbceq1a cbvrab elrab2 ) ABFGZHZ
|
|
ABCHFCDABDIABQCJARBFDEFDKAFLABQMABQNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y B $. $d y A $.
|
|
$( Substitution applied to an atomic wff. Set theory version of ~ eqsb3 .
|
|
(Contributed by Andrew Salmon, 29-Jun-2011.) $)
|
|
eqsbc3 $p |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) ) $=
|
|
( vy cv wceq wsbc dfsbcq eqeq1 wsb sbsbc eqsb3 bitr3i vtoclbg ) AFCGZAEFZ
|
|
HZQCGZPABHBCGEBDPAQBIQBCJRPAEKSPAELEACMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y ph $. $d y ps $.
|
|
$( Move negation in and out of class substitution. (Contributed by NM,
|
|
16-Jan-2004.) $)
|
|
sbcng $p |- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) ) $=
|
|
( vy wn wsb wsbc dfsbcq2 cv wceq notbid sbn vtoclbg ) AFZBEGABEGZFOBCHABC
|
|
HZFECDOBECIEJCKPQABECILABEMN $.
|
|
|
|
$( Distribution of class substitution over implication. (Contributed by
|
|
NM, 16-Jan-2004.) $)
|
|
sbcimg $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) ) $=
|
|
( vy wi wsb wsbc dfsbcq2 cv wceq imbi12d sbim vtoclbg ) ABGZCFHACFHZBCFHZ
|
|
GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
|
|
|
|
$( Distribution of class substitution over conjunction. (Contributed by
|
|
NM, 31-Dec-2016.) $)
|
|
sbcan $p |- ( [. A / x ]. ( ph /\ ps )
|
|
<-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) $=
|
|
( vy wa wsbc cvv wcel sbcex adantl dfsbcq2 cv wceq anbi12d sban pm5.21nii
|
|
wsb vtoclbg ) ABFZCDGZDHIZACDGZBCDGZFZTCDJUDUBUCBCDJKTCERACERZBCERZFUAUEE
|
|
DHTCEDLEMDNUFUCUGUDACEDLBCEDLOABCEPSQ $.
|
|
|
|
$( Distribution of class substitution over conjunction. (Contributed by
|
|
NM, 21-May-2004.) $)
|
|
sbcang $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) ) $=
|
|
( vy wa wsb wsbc dfsbcq2 cv wceq anbi12d sban vtoclbg ) ABGZCFHACFHZBCFHZ
|
|
GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
|
|
|
|
$( Distribution of class substitution over disjunction. (Contributed by
|
|
NM, 31-Dec-2016.) $)
|
|
sbcor $p |- ( [. A / x ]. ( ph \/ ps )
|
|
<-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) $=
|
|
( vy wo wsbc cvv wcel sbcex jaoi wsb dfsbcq2 cv wceq orbi12d sbor vtoclbg
|
|
pm5.21nii ) ABFZCDGZDHIZACDGZBCDGZFZTCDJUCUBUDACDJBCDJKTCELACELZBCELZFUAU
|
|
EEDHTCEDMENDOUFUCUGUDACEDMBCEDMPABCEQRS $.
|
|
|
|
$( Distribution of class substitution over disjunction. (Contributed by
|
|
NM, 21-May-2004.) $)
|
|
sbcorg $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) ) $=
|
|
( vy wo wsb wsbc dfsbcq2 cv wceq orbi12d sbor vtoclbg ) ABGZCFHACFHZBCFHZ
|
|
GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
|
|
|
|
$( Distribution of class substitution over biconditional. (Contributed by
|
|
Raph Levien, 10-Apr-2004.) $)
|
|
sbcbig $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) ) $=
|
|
( vy wb wsb wsbc dfsbcq2 cv wceq bibi12d sbbi vtoclbg ) ABGZCFHACFHZBCFHZ
|
|
GPCDIACDIZBCDIZGFDEPCFDJFKDLQSRTACFDJBCFDJMABCFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d x y z $. $d z ph $.
|
|
$( Move universal quantifier in and out of class substitution.
|
|
(Contributed by NM, 31-Dec-2016.) $)
|
|
sbcal $p |- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) $=
|
|
( vz wal wsbc cvv wcel sbcex sps dfsbcq2 cv wceq albidv vtoclbg pm5.21nii
|
|
wsb sbal ) ABFZCDGZDHIZACDGZBFZTCDJUCUBBACDJKTCERACERZBFUAUDEDHTCEDLEMDNU
|
|
EUCBACEDLOABCESPQ $.
|
|
|
|
$( Move universal quantifier in and out of class substitution.
|
|
(Contributed by NM, 16-Jan-2004.) $)
|
|
sbcalg $p |- ( A e. V
|
|
-> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) ) $=
|
|
( vz wal wsb wsbc dfsbcq2 cv wceq albidv sbal vtoclbg ) ABGZCFHACFHZBGPCD
|
|
IACDIZBGFDEPCFDJFKDLQRBACFDJMABCFNO $.
|
|
|
|
$( Move existential quantifier in and out of class substitution.
|
|
(Contributed by NM, 21-May-2004.) $)
|
|
sbcex2 $p |- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) $=
|
|
( vz wex wsbc cvv wcel sbcex exlimiv wsb dfsbcq2 wceq exbidv sbex vtoclbg
|
|
cv pm5.21nii ) ABFZCDGZDHIZACDGZBFZTCDJUCUBBACDJKTCELACELZBFUAUDEDHTCEDME
|
|
RDNUEUCBACEDMOABCEPQS $.
|
|
|
|
$( Move existential quantifier in and out of class substitution.
|
|
(Contributed by NM, 21-May-2004.) $)
|
|
sbcexg $p |- ( A e. V
|
|
-> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) ) $=
|
|
( vz wex wsb wsbc dfsbcq2 cv wceq exbidv sbex vtoclbg ) ABGZCFHACFHZBGPCD
|
|
IACDIZBGFDEPCFDJFKDLQRBACFDJMABCFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x A $.
|
|
$( Set theory version of sbeqal1 in set.mm. (Contributed by Andrew Salmon,
|
|
28-Jun-2011.) $)
|
|
sbceqal $p |- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) ) $=
|
|
( wcel cv wceq wi wal wsbc spsbc sbcimg wb eqsbc3 mpbiri pm5.5 syl 3bitrd
|
|
eqid sylibd ) BDEZAFZBGZUBCGZHZAIUEABJZBCGZUEABDKUAUFUCABJZUDABJZHZUIUGUC
|
|
UDABDLUAUHUJUIMUAUHBBGBSABBDNOUHUIPQABCDNRT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew
|
|
Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) $)
|
|
sbeqalb $p |- ( A e. V -> ( ( A. x ( ph <-> x = A ) /\ A. x ( ph <->
|
|
x = B ) ) -> A = B ) ) $=
|
|
( cv wceq wb wal wa wi wcel bibi1 biimpa biimpd alanimi sbceqal syl5 ) AB
|
|
FZCGZHZBIASDGZHZBIJTUBKZBICELCDGUAUCUDBUAUCJTUBUAUCTUBHATUBMNOPBCDEQR $.
|
|
$}
|
|
|
|
${
|
|
sbcbid.1 $e |- F/ x ph $.
|
|
sbcbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building deduction rule for class substitution. (Contributed by
|
|
NM, 29-Dec-2014.) $)
|
|
sbcbid $p |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) ) $=
|
|
( cab wcel wsbc abbid eleq2d df-sbc 3bitr4g ) AEBDHZIECDHZIBDEJCDEJAOPEAB
|
|
CDFGKLBDEMCDEMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
sbcbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building deduction rule for class substitution. (Contributed by
|
|
NM, 29-Dec-2014.) $)
|
|
sbcbidv $p |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) ) $=
|
|
( nfv sbcbid ) ABCDEADGFH $.
|
|
$}
|
|
|
|
${
|
|
sbcbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Formula-building inference rule for class substitution. (Contributed by
|
|
NM, 11-Nov-2005.) $)
|
|
sbcbii $p |- ( [. A / x ]. ph <-> [. A / x ]. ps ) $=
|
|
( wsbc wb wtru a1i sbcbidv trud ) ACDFBCDFGHABCDABGHEIJK $.
|
|
|
|
$( Formula-building inference rule for class substitution. (Contributed by
|
|
NM, 11-Nov-2005.) (New usage is discouraged.) $)
|
|
sbcbiiOLD $p |- ( A e. V -> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) $=
|
|
( wsbc wb wcel sbcbii a1i ) ACDGBCDGHDEIABCDFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x C $. $d x A $.
|
|
$( ~ eqsbc3 with setvar variable on right side of equals sign. This proof
|
|
was automatically generated from the virtual deduction proof eqsbc3rVD
|
|
in set.mm using a translation program. (Contributed by Alan Sare,
|
|
24-Oct-2011.) $)
|
|
eqsbc3r $p |- ( A e. B -> ( [. A / x ]. C = x <-> C = A ) ) $=
|
|
( wcel wceq wsbc eqcom sbcbii biimpi eqsbc3 syl5ib syl6ib syl6ibr sylibrd
|
|
cv idd impbid ) BCEZDAPZFZABGZDBFZSUBBDFZUCUBTDFZABGZSUDUBUFUAUEABDTHIZJA
|
|
BDCKZLBDHZMSUCUFUBSUCUDUFSUCUCUDSUCQUINUHOUGNR $.
|
|
$}
|
|
|
|
${
|
|
$d y ch $. $d y ps $. $d y ph $. $d y A $. $d x y $.
|
|
$( Distribution of class substitution over triple conjunction.
|
|
(Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbc3ang $p |- ( A e. V -> ( [. A / x ]. ( ph /\ ps /\ ch ) <->
|
|
( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) ) ) $=
|
|
( vy w3a wsb wsbc dfsbcq2 cv wceq 3anbi123d sb3an vtoclbg ) ABCHZDGIADGIZ
|
|
BDGIZCDGIZHQDEJADEJZBDEJZCDEJZHGEFQDGEKGLEMRUASUBTUCADGEKBDGEKCDGEKNABCDG
|
|
OP $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y B $.
|
|
$( Class substitution into a membership relation. (Contributed by NM,
|
|
17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
sbcel1gv $p |- ( A e. V -> ( [. A / x ]. x e. B <-> A e. B ) ) $=
|
|
( vy cv wcel wsb wsbc dfsbcq2 eleq1 clelsb3 vtoclbg ) AFCGZAEHEFZCGNABIBC
|
|
GEBDNAEBJOBCKEACLM $.
|
|
$}
|
|
|
|
${
|
|
$d y B $. $d x y A $.
|
|
$( Class substitution into a membership relation. (Contributed by NM,
|
|
17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
sbcel2gv $p |- ( B e. V -> ( [. B / x ]. A e. x <-> A e. B ) ) $=
|
|
( vy cv wcel wsb wsbc dfsbcq2 eleq2 nfv sbie vtoclbg ) BAFZGZAEHBEFZGZPAC
|
|
IBCGECDPAECJQCBKPRAERALOQBKMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
sbcimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed
|
|
by NM, 11-Nov-2005.) $)
|
|
sbcimdv $p |- ( ( ph /\ A e. V ) ->
|
|
( [. A / x ]. ps -> [. A / x ]. ch ) ) $=
|
|
( wcel wsbc wi wal alrimiv spsbc syl5 sbcimg sylibd impcom ) EFHZABDEICDE
|
|
IJZRABCJZDEIZSATDKRUAATDGLTDEFMNBCDEFOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y ph $.
|
|
$( Substitution for a variable not free in a wff does not affect it.
|
|
(Contributed by Mario Carneiro, 14-Oct-2016.) $)
|
|
sbctt $p |- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) ) $=
|
|
( vy wcel wnf wsbc wb wsb wi wceq dfsbcq2 bibi1d imbi2d sbft vtoclg imp
|
|
cv ) CDFABGZABCHZAIZTABEJZAIZKTUBKECDESCLZUDUBTUEUCUAAABECMNOABEPQR $.
|
|
$}
|
|
|
|
${
|
|
sbcgf.1 $e |- F/ x ph $.
|
|
$( Substitution for a variable not free in a wff does not affect it.
|
|
(Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbcgf $p |- ( A e. V -> ( [. A / x ]. ph <-> ph ) ) $=
|
|
( wcel wnf wsbc wb sbctt mpan2 ) CDFABGABCHAIEABCDJK $.
|
|
|
|
$( Substitution for a variable not free in antecedent affects only the
|
|
consequent. (Contributed by NM, 11-Oct-2004.) $)
|
|
sbc19.21g $p |- ( A e. V ->
|
|
( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) ) $=
|
|
( wcel wi wsbc sbcimg sbcgf imbi1d bitrd ) DEGZABHCDIACDIZBCDIZHAPHABCDEJ
|
|
NOAPACDEFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( Substitution for a variable not occurring in a wff does not affect it.
|
|
Distinct variable form of ~ sbcgf . (Contributed by Alan Sare,
|
|
10-Nov-2012.) $)
|
|
sbcg $p |- ( A e. V -> ( [. A / x ]. ph <-> ph ) ) $=
|
|
( nfv sbcgf ) ABCDABEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x V $. $d y W $.
|
|
sbc2iegf.1 $e |- F/ x ps $.
|
|
sbc2iegf.2 $e |- F/ y ps $.
|
|
sbc2iegf.3 $e |- F/ x B e. W $.
|
|
sbc2iegf.4 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by Mario Carneiro, 19-Dec-2013.) $)
|
|
sbc2iegf $p |- ( ( A e. V /\ B e. W ) ->
|
|
( [. A / x ]. [. B / y ]. ph <-> ps ) ) $=
|
|
( wcel wa simpl cv wceq wb adantll nfv wsbc wnf a1i sbciedf nfan ) EGMZFH
|
|
MZNZADFUAZBCEGUFUGOUGCPEQZUIBRUFUGUJNZABDFHUGUJOUJDPFQABRUGLSUKDTBDUBUKJU
|
|
CUDSUFUGCUFCTKUEBCUBUHIUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y ps $.
|
|
sbc2ie.1 $e |- A e. _V $.
|
|
sbc2ie.2 $e |- B e. _V $.
|
|
sbc2ie.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro,
|
|
19-Dec-2013.) $)
|
|
sbc2ie $p |- ( [. A / x ]. [. B / y ]. ph <-> ps ) $=
|
|
( cvv wcel wsbc wb nfv nfth sbc2iegf mp2an ) EJKFJKZADFLCELBMGHABCDEFJJBC
|
|
NBDNRCHOIPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y ph $. $d x y ch $.
|
|
sbc2iedv.1 $e |- A e. _V $.
|
|
sbc2iedv.2 $e |- B e. _V $.
|
|
sbc2iedv.3 $e |- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro,
|
|
18-Oct-2016.) $)
|
|
sbc2iedv $p |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) ) $=
|
|
( wsbc cvv wcel a1i cv wceq wa wb impl sbcied ) ABEGKCDFLFLMAHNADOFPZQZBC
|
|
EGLGLMUBINAUAEOGPBCRJSTT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d z C $. $d x y z ps $.
|
|
sbc3ie.1 $e |- A e. _V $.
|
|
sbc3ie.2 $e |- B e. _V $.
|
|
sbc3ie.3 $e |- C e. _V $.
|
|
sbc3ie.4 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
(Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario
|
|
Carneiro, 29-Dec-2014.) $)
|
|
sbc3ie $p |- ( [. A / x ]. [. B / y ]. [. C / z ]. ph <-> ps ) $=
|
|
( wsbc cv wceq wa cvv wcel a1i wb 3expa sbcied sbc2ie ) AEHMBCDFGIJCNFOZD
|
|
NGOZPZABEHQHQRUFKSUDUEENHOABTLUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Lemma for ~ sbccom . (Contributed by NM, 14-Nov-2005.) (Revised by
|
|
Mario Carneiro, 18-Oct-2016.) $)
|
|
sbccomlem $p |- ( [. A / x ]. [. B / y ]. ph
|
|
<-> [. B / y ]. [. A / x ]. ph ) $=
|
|
( cv wceq wa wex wsbc excom exdistr an12 exbii bitri 3bitr3i sbc5 3bitr4i
|
|
19.42v sbcbii ) CFEGZAHZCIZBDJZBFDGZAHZBIZCEJZACEJZBDJABDJZCEJUEUCHBIZUAU
|
|
GHZCIZUDUHUEUBHZCIBIUNBIZCIUKUMUNBCKUEUBBCLUOULCUOUAUFHZBIULUNUPBUEUAAMNU
|
|
AUFBSONPUCBDQUGCEQRUIUCBDACEQTUJUGCEABDQTR $.
|
|
$}
|
|
|
|
${
|
|
$d w y z A $. $d w x z B $. $d w z ph $. $d x y $.
|
|
$( Commutative law for double class substitution. (Contributed by NM,
|
|
15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) $)
|
|
sbccom $p |- ( [. A / x ]. [. B / y ]. ph
|
|
<-> [. B / y ]. [. A / x ]. ph ) $=
|
|
( vw vz cv wsbc sbccomlem sbcbii bitri 3bitr3i sbcco ) ACFHZIZFEIZBDIZABG
|
|
HZIZGDIZCEIZACEIZBDIABDIZCEIQBSIZGDIZUACOIZFEIZRUBTCOIZFEIZGDIUIGDIZFEIUF
|
|
UHUIGFDEJUJUEGDUJPBSIZFEIUEUIULFEACBOSJKPFBESJLKUKUGFETGCDOJKMQBGDNUACFEN
|
|
MQUCBDACFENKUAUDCEABGDNKM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d A z $. $d B x z $. $d V z $. $d ph z $.
|
|
$( Interchange class substitution and restricted quantifier. (Contributed
|
|
by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.) $)
|
|
sbcralt $p |- ( ( A e. V /\ F/_ y A ) ->
|
|
( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) ) $=
|
|
( vz wral wsbc cv wcel wnfc wa sbcco simpl wsb wceq sbsbc nfcv wb sbequ12
|
|
nfs1v nfral weq ralbidv bitr3i nfnfc1 nfcvd id nfeqd nfan1 dfsbcq2 adantl
|
|
sbie ralbid adantll syl5bb sbcied syl5bbr ) ACEHZBDIUTBGJZIZGDIDFKZCDLZMZ
|
|
ABDIZCEHZUTBGDNVEVBVGGDFVCVDOVBABGPZCEHZVEVADQZMVGVBUTBGPVIUTBGRUTVIBGVHB
|
|
CEBESABGUBUCBGUDAVHCEABGUAUEUNUFVDVJVIVGTVCVDVJMVHVFCEVDVJCCDUGVDCVADVDCV
|
|
AUHVDUIUJUKVJVHVFTVDABGDULUMUOUPUQURUS $.
|
|
|
|
$( Interchange class substitution and restricted existential quantifier.
|
|
(Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
sbcrext $p |- ( ( A e. V /\ F/_ y A ) ->
|
|
( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) ) $=
|
|
( wcel cvv wnfc wrex wsbc wb elex wa wn wral sbcng adantr bitrd dfrex2 id
|
|
sbcralt nfnfc1 nfcvd nfeld nfan1 adantl ralbid ancoms notbid sbcbii sylan
|
|
3bitr4g ) DFGDHGZCDIZACEJZBDKZABDKZCEJZLDFMUNUONZAOZCEPZOZBDKZUROZCEPZOZU
|
|
QUSUTVDVBBDKZOZVGUNVDVILUOVBBDHQRUTVHVFUTVHVABDKZCEPZVFVABCDEHUBUOUNVKVFL
|
|
UOUNNVJVECEUOUNCCDUCUOCDHUOUAUOCHUDUEUFUNVJVELUOABDHQUGUHUISUJSUPVCBDACET
|
|
UKURCETUMUL $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x B $. $d x y z $. $d ph z $. $d B z $.
|
|
$( Interchange class substitution and restricted quantifier. (Contributed
|
|
by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbcralg $p |- ( A e. V ->
|
|
( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) ) $=
|
|
( vz wral wsb wsbc dfsbcq2 wceq ralbidv nfcv nfs1v nfral weq sbequ12 sbie
|
|
cv vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGTDLUCUECEABGDKMUBUDB
|
|
GUCBCEBENABGOPBGQAUCCEABGRMSUA $.
|
|
|
|
$( Interchange class substitution and restricted existential quantifier.
|
|
(Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbcrexg $p |- ( A e. V ->
|
|
( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) ) $=
|
|
( vz wrex wsb wsbc dfsbcq2 wceq rexbidv nfcv nfs1v nfrex weq sbequ12 sbie
|
|
cv vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGTDLUCUECEABGDKMUBUDB
|
|
GUCBCEBENABGOPBGQAUCCEABGRMSUA $.
|
|
|
|
$( Interchange class substitution and restricted uniqueness quantifier.
|
|
(Contributed by NM, 24-Feb-2013.) $)
|
|
sbcreug $p |- ( A e. V ->
|
|
( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) ) $=
|
|
( vz wreu wsb wsbc dfsbcq2 wceq reubidv nfcv nfs1v nfreu weq sbequ12 sbie
|
|
cv vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGTDLUCUECEABGDKMUBUDB
|
|
GUCBCEBENABGOPBGQAUCCEABGRMSUA $.
|
|
$}
|
|
|
|
${
|
|
$d y w A $. $d w B $. $d w ph $. $d x y $. $d w x $.
|
|
sbcabel.1 $e |- F/_ x B $.
|
|
$( Interchange class substitution and class abstraction. (Contributed by
|
|
NM, 5-Nov-2005.) $)
|
|
sbcabel $p |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <->
|
|
{ y | [. A / x ]. ph } e. B ) ) $=
|
|
( vw wcel cvv cab wsbc wb cv wceq wa wex wal abeq2 bitrd elex sbcexg sbcg
|
|
sbcang sbcbii sbcalg sbcbig bibi1d albidv syl6bbr anbi12d df-clel 3bitr4g
|
|
syl5bb nfcri sbcgf exbidv syl ) DFIDJIZACKZEIZBDLZABDLZCKZEIZMDFUAUSHNZUT
|
|
OZVFEIZPZHQZBDLZVFVDOZVHPZHQZVBVEUSVKVIBDLZHQVNVIHBDJUBUSVOVMHUSVOVGBDLZV
|
|
HBDLZPVMVGVHBDJUDUSVPVLVQVHUSVPCNVFIZVCMZCRZVLVPVRAMZCRZBDLZUSVTVGWBBDACV
|
|
FSUEUSWCWABDLZCRVTWACBDJUFUSWDVSCUSWDVRBDLZVCMVSVRABDJUGUSWEVRVCVRBDJUCUH
|
|
TUITUNVCCVFSUJVHBDJBHEGUOUPUKTUQTVAVJBDHUTEULUEHVDEULUMUR $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y B $. $d y ph $.
|
|
$( Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This
|
|
provides an axiom for a predicate calculus for a restricted domain.
|
|
This theorem generalizes the unrestricted ~ stdpc4 and ~ spsbc . See
|
|
also ~ rspsbca and ~ rspcsbela . (Contributed by NM, 17-Nov-2006.)
|
|
(Proof shortened by Mario Carneiro, 13-Oct-2016.) $)
|
|
rspsbc $p |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) ) $=
|
|
( vy wral wsb wcel wsbc cbvralsv dfsbcq2 rspcv syl5bi ) ABDFABEGZEDFCDHAB
|
|
CIZABEDJNOECDABECKLM $.
|
|
|
|
$( Restricted quantifier version of Axiom 4 of [Mendelson] p. 69.
|
|
(Contributed by NM, 14-Dec-2005.) $)
|
|
rspsbca $p |- ( ( A e. B /\ A. x e. B ph ) -> [. A / x ]. ph ) $=
|
|
( wcel wral wsbc rspsbc imp ) CDEABDFABCGABCDHI $.
|
|
|
|
$( Existence form of ~ rspsbca . (Contributed by NM, 29-Feb-2008.) (Proof
|
|
shortened by Mario Carneiro, 13-Oct-2016.) $)
|
|
rspesbca $p |- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph ) $=
|
|
( vy wcel wsbc wa wsb wrex dfsbcq2 rspcev cbvrexsv sylibr ) CDFABCGZHABEI
|
|
ZEDJABDJPOECDABECKLABEDMN $.
|
|
|
|
$( Existence form of ~ spsbc . (Contributed by Mario Carneiro,
|
|
18-Nov-2016.) $)
|
|
spesbc $p |- ( [. A / x ]. ph -> E. x ph ) $=
|
|
( wsbc cvv wrex wex wcel sbcex rspesbca mpancom rexv sylib ) ABCDZABEFZAB
|
|
GCEHNOABCIABCEJKABLM $.
|
|
|
|
spesbcd.1 $e |- ( ph -> [. A / x ]. ps ) $.
|
|
$( form of ~ spsbc . (Contributed by Mario Carneiro, 9-Feb-2017.) $)
|
|
spesbcd $p |- ( ph -> E. x ps ) $=
|
|
( wsbc wex spesbc syl ) ABCDFBCGEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
sbcth2.1 $e |- ( x e. B -> ph ) $.
|
|
$( A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof
|
|
shortened by Mario Carneiro, 13-Oct-2016.) $)
|
|
sbcth2 $p |- ( A e. B -> [. A / x ]. ph ) $=
|
|
( wcel wral wsbc rgen rspsbc mpi ) CDFABDGABCHABDEIABCDJK $.
|
|
$}
|
|
|
|
${
|
|
ra5.1 $e |- F/ x ph $.
|
|
$( Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is
|
|
an axiom of a predicate calculus for a restricted domain. Compare the
|
|
unrestricted ~ stdpc5 . (Contributed by NM, 16-Jan-2004.) $)
|
|
ra5 $p |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) ) $=
|
|
( wi wral cv wcel wal df-ral bi2.04 albii bitri stdpc5 sylbi syl6ibr ) AB
|
|
FZCDGZACHDIZBFZCJZBCDGSAUAFZCJZAUBFSTRFZCJUDRCDKUEUCCTABLMNAUACEOPBCDKQ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
rmo2.1 $e |- F/ y ph $.
|
|
$( Alternate definition of restricted "at most one." Note that
|
|
` E* x e. A ph ` is not equivalent to
|
|
` E. y e. A A. x e. A ( ph -> x = y ) ` (in analogy to ~ reu6 ); to see
|
|
this, let ` A ` be the empty set. However, one direction of this
|
|
pattern holds; see ~ rmo2i . (Contributed by NM, 17-Jun-2017.) $)
|
|
rmo2 $p |- ( E* x e. A ph <-> E. y A. x e. A ( ph -> x = y ) ) $=
|
|
( wrmo cv wcel wa wmo weq wi wal wex wral df-rmo nfv nfan mo2 impexp
|
|
albii df-ral bitr4i exbii 3bitri ) ABDFBGDHZAIZBJUGBCKZLZBMZCNAUHLZBDOZCN
|
|
ABDPUGBCUFACUFCQERSUJULCUJUFUKLZBMULUIUMBUFAUHTUAUKBDUBUCUDUE $.
|
|
|
|
$( Condition implying restricted "at most one." (Contributed by NM,
|
|
17-Jun-2017.) $)
|
|
rmo2i $p |- ( E. y e. A A. x e. A ( ph -> x = y ) -> E* x e. A ph ) $=
|
|
( weq wi wral wrex wex wrmo rexex rmo2 sylibr ) ABCFGBDHZCDIOCJABDKOCDLAB
|
|
CDEMN $.
|
|
|
|
$( Restricted "at most one" using explicit substitution. (Contributed by
|
|
NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) $)
|
|
rmo3 $p |- ( E* x e. A ph <->
|
|
A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
|
|
( wrmo cv wcel wa wmo wsb wi wral anbi1i bitri 3bitri impexp albii df-ral
|
|
wal weq df-rmo sban clelsb3 anbi2i an4 ancom r19.21v 3bitr2i nfv nfan mo3
|
|
imbi1i 3bitr4i ) ABDFBGDHZAIZBJZAABCKZIZBCUAZLZCDMZBDMZABDUBUPUPBCKZIZUTL
|
|
ZCTZBTUOVBLZBTUQVCVGVHBVGCGDHZUOVALZLZCTVJCDMVHVFVKCVFVIUOIZUSIZUTLVLVALV
|
|
KVEVMUTVEUPVIURIZIUOVIIZUSIVMVDVNUPVDUOBCKZURIVNUOABCUCVPVIURCBDUDNOUEUOA
|
|
VIURUFVOVLUSUOVIUGNPUMVLUSUTQVIUOVAQPRVJCDSUOVACDUHUIRUPBCUOACUOCUJEUKULV
|
|
BBDSUNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ps $. $d x ch $.
|
|
rmoi.b $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
rmoi.c $e |- ( x = C -> ( ph <-> ch ) ) $.
|
|
$( Consequence of "at most one", using implicit substitution. (Contributed
|
|
by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.) $)
|
|
rmob $p |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) ->
|
|
( B = C <-> ( C e. A /\ ch ) ) ) $=
|
|
( wrmo cv wcel wa wmo wceq wb df-rmo simprl eleq1 anbi12d syl5ibcom simpl
|
|
wi a1i simplrl simpr simpll simplrr mob syl212anc ex pm5.21ndd sylanb ) A
|
|
DEJDKZELZAMZDNZFELZBMZFGOZGELZCMZPZADEQUQUSMZVAUTVBVDURUTVAUQURBRFGESUAVB
|
|
VAUCVDVACUBUDVDVAVCVDVAMURVAUQURBVCUQURBVAUEZVDVAUFUQUSVAUGVEUQURBVAUHUPU
|
|
SVBDFGEEUNFOUOURABUNFESHTUNGOUOVAACUNGESITUIUJUKULUM $.
|
|
|
|
$( Consequence of "at most one", using implicit substitution. (Contributed
|
|
by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) $)
|
|
rmoi $p |- ( ( E* x e. A ph
|
|
/\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C ) $=
|
|
( wrmo wcel wa wceq rmob biimp3ar ) ADEJFEKBLFGMGEKCLABCDEFGHINO $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Proper substitution of classes for sets into classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c [_ $. $( Underlined left bracket $)
|
|
$c ]_ $. $( Underlined right bracket $)
|
|
|
|
$( Extend class notation to include the proper substitution of a class for a
|
|
set into another class. $)
|
|
csb $a class [_ A / x ]_ B $.
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( Define the proper substitution of a class for a set into another class.
|
|
The underlined brackets distinguish it from the substitution into a wff,
|
|
~ wsbc , to prevent ambiguity. Theorem ~ sbcel1g shows an example of
|
|
how ambiguity could arise if we didn't use distinguished brackets.
|
|
Theorem ~ sbccsbg recreates substitution into a wff from this
|
|
definition. (Contributed by NM, 10-Nov-2005.) $)
|
|
df-csb $a |- [_ A / x ]_ B = { y | [. A / x ]. y e. B } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y $.
|
|
$( Alternate expression for the proper substitution into a class, without
|
|
referencing substitution into a wff. Note that ` x ` can be free in
|
|
` B ` but cannot occur in ` A ` . (Contributed by NM, 2-Dec-2013.) $)
|
|
csb2 $p |- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) } $=
|
|
( csb cv wcel wsbc cab wceq wa wex df-csb sbc5 abbii eqtri ) ACDEBFDGZACH
|
|
ZBIAFCJQKALZBIABCDMRSBQACNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $.
|
|
$( Analog of ~ dfsbcq for proper substitution into a class. (Contributed
|
|
by NM, 10-Nov-2005.) $)
|
|
csbeq1 $p |- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C ) $=
|
|
( vy wceq cv wcel wsbc cab csb dfsbcq abbidv df-csb 3eqtr4g ) BCFZEGDHZAB
|
|
IZEJQACIZEJABDKACDKPRSEQABCLMAEBDNAECDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z C $. $d z D $.
|
|
cbvcsb.1 $e |- F/_ y C $.
|
|
cbvcsb.2 $e |- F/_ x D $.
|
|
cbvcsb.3 $e |- ( x = y -> C = D ) $.
|
|
$( Change bound variables in a class substitution. Interestingly, this
|
|
does not require any bound variable conditions on ` A ` . (Contributed
|
|
by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro,
|
|
11-Dec-2016.) $)
|
|
cbvcsb $p |- [_ A / x ]_ C = [_ A / y ]_ D $=
|
|
( vz cv wcel wsbc cab csb nfcri wceq eleq2d cbvsbc abbii df-csb 3eqtr4i )
|
|
IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOAJBJPDEUBHQRSAICDTBI
|
|
CETUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y B $. $d x C $.
|
|
cbvcsbv.1 $e |- ( x = y -> B = C ) $.
|
|
$( Change the bound variable of a proper substitution into a class using
|
|
implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by
|
|
Mario Carneiro, 13-Oct-2016.) $)
|
|
cbvcsbv $p |- [_ A / x ]_ B = [_ A / y ]_ C $=
|
|
( nfcv cbvcsb ) ABCDEBDGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
csbeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for proper substitution into a class. (Contributed
|
|
by NM, 3-Dec-2005.) $)
|
|
csbeq1d $p |- ( ph -> [_ A / x ]_ C = [_ B / x ]_ C ) $=
|
|
( wceq csb csbeq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( Analog of ~ sbid for proper substitution into a class. (Contributed by
|
|
NM, 10-Nov-2005.) $)
|
|
csbid $p |- [_ x / x ]_ A = A $=
|
|
( vy cv csb wcel wsbc cab df-csb wsb sbsbc sbid bitr3i abbii abid2 3eqtri
|
|
) AADZBECDBFZAQGZCHRCHBACQBISRCSRAAJRRAAKRALMNCBOP $.
|
|
$}
|
|
|
|
$( Equality theorem for proper substitution into a class. (Contributed by
|
|
NM, 10-Nov-2005.) $)
|
|
csbeq1a $p |- ( x = A -> B = [_ A / x ]_ B ) $=
|
|
( cv wceq csb csbid csbeq1 syl5eqr ) ADZBECAJCFABCFACGAJBCHI $.
|
|
|
|
${
|
|
$d z A $. $d y z B $. $d x z $.
|
|
$( Composition law for chained substitutions into a class. (Contributed by
|
|
NM, 10-Nov-2005.) $)
|
|
csbco $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $=
|
|
( vz cv csb wcel wsbc cab df-csb abeq2i sbcbii sbcco bitri abbii 3eqtr4i
|
|
) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM
|
|
UCABCNOPBECTKAECDKQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( The existence of proper substitution into a class. (Contributed by NM,
|
|
10-Nov-2005.) $)
|
|
csbexg $p |- ( ( A e. V /\ A. x B e. W ) -> [_ A / x ]_ B e. _V ) $=
|
|
( vy wcel wal wa csb wsbc cab cvv df-csb abid2 elex syl5eqel alimi spsbc
|
|
cv syl5 imp wb nfcv sbcabel adantr mpbid ) BDGZCEGZAHZIZABCJFTCGZABKFLZMA
|
|
FBCNUKULFLZMGZABKZUMMGZUHUJUPUJUOAHUHUPUIUOAUIUNCMFCOCEPQRUOABDSUAUBUHUPU
|
|
QUCUJULAFBMDAMUDUEUFUGQ $.
|
|
$}
|
|
|
|
${
|
|
csbex.1 $e |- A e. _V $.
|
|
csbex.2 $e |- B e. _V $.
|
|
$( The existence of proper substitution into a class. (Contributed by NM,
|
|
7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
csbex $p |- [_ A / x ]_ B e. _V $=
|
|
( cvv wcel csb wal csbexg mpan mpg ) CFGZABCHFGZABFGMAINDABCFFJKEL $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y V $. $d x y $.
|
|
$( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
|
|
free). (Contributed by Mario Carneiro, 14-Oct-2016.) $)
|
|
csbtt $p |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B ) $=
|
|
( vy wcel wnfc wa csb cv wsbc cab df-csb wnf wb nfcr sbctt sylan2 abbi1dv
|
|
syl5eq ) BDFZACGZHZABCIEJCFZABKZELCAEBCMUCUEECUBUAUDANUEUDOAECPUDABDQRST
|
|
$.
|
|
$}
|
|
|
|
${
|
|
csbconstgf.1 $e |- F/_ x B $.
|
|
$( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
|
|
free). (Contributed by NM, 10-Nov-2005.) $)
|
|
csbconstgf $p |- ( A e. V -> [_ A / x ]_ B = B ) $=
|
|
( wcel wnfc csb wceq csbtt mpan2 ) BDFACGABCHCIEABCDJK $.
|
|
$}
|
|
|
|
${
|
|
$d B x $.
|
|
$( Substitution doesn't affect a constant ` B ` (in which ` x ` is not
|
|
free). ~ csbconstgf with distinct variable requirement. (Contributed by
|
|
Alan Sare, 22-Jul-2012.) $)
|
|
csbconstg $p |- ( A e. V -> [_ A / x ]_ B = B ) $=
|
|
( nfcv csbconstgf ) ABCDACEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d y z A $. $d y z B $. $d y z C $.
|
|
$( Distribute proper substitution through a membership relation.
|
|
(Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbcel12g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
|
|
[_ A / x ]_ B e. [_ A / x ]_ C ) ) $=
|
|
( vy vz wcel wsbc cv cab csb wsb dfsbcq2 abbidv eleq12d nfs1v nfab df-csb
|
|
sbab wceq nfel weq sbie vtoclbg eleq12i syl6bbr ) BEHCDHZABIZFJZCHZABIZFK
|
|
ZUJDHZABIZFKZHZABCLZABDLZHUHAGMUKAGMZFKZUNAGMZFKZHZUIUQGBEUHAGBNGJBUAZVAU
|
|
MVCUPVEUTULFUKAGBNOVEVBUOFUNAGBNOPUHVDAGAVAVCUTAFUKAGQRVBAFUNAGQRUBAGUCCV
|
|
ADVCAGFCTAGFDTPUDUEURUMUSUPAFBCSAFBDSUFUG $.
|
|
|
|
$( Distribute proper substitution through an equality relation.
|
|
(Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
sbceqg $p |- ( A e. V -> ( [. A / x ]. B = C <->
|
|
[_ A / x ]_ B = [_ A / x ]_ C ) ) $=
|
|
( vy vz wcel wceq wsbc cab csb wsb dfsbcq2 abbidv eqeq12d nfs1v nfab sbab
|
|
cv nfeq weq sbie vtoclbg df-csb eqeq12i syl6bbr ) BEHCDIZABJZFTZCHZABJZFK
|
|
ZUJDHZABJZFKZIZABCLZABDLZIUHAGMUKAGMZFKZUNAGMZFKZIZUIUQGBEUHAGBNGTBIZVAUM
|
|
VCUPVEUTULFUKAGBNOVEVBUOFUNAGBNOPUHVDAGAVAVCUTAFUKAGQRVBAFUNAGQRUAAGUBCVA
|
|
DVCAGFCSAGFDSPUCUDURUMUSUPAFBCUEAFBDUEUFUG $.
|
|
$}
|
|
|
|
$( Distribute proper substitution through negated membership. (Contributed
|
|
by Andrew Salmon, 18-Jun-2011.) $)
|
|
sbcnel12g $p |- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/
|
|
[_ A / x ]_ C ) ) $=
|
|
( wcel wnel wsbc wn csb wb df-nel sbcbii a1i sbcel12g notbid syl6bbr 3bitrd
|
|
sbcng ) BEFZCDGZABHZCDFZIZABHZUCABHZIZABCJZABDJZGZUBUEKTUAUDABCDLMNUCABESTU
|
|
GUHUIFZIUJTUFUKABCDEOPUHUILQR $.
|
|
|
|
$( Distribute proper substitution through an inequality. (Contributed by
|
|
Andrew Salmon, 18-Jun-2011.) $)
|
|
sbcne12g $p |- ( A e. V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/=
|
|
[_ A / x ]_ C ) ) $=
|
|
( wcel wne wsbc csb wn wceq wb nne sbcbii a1i sbcng syl6bbr 3bitr3d con4bid
|
|
sbceqg ) BEFZCDGZABHZABCIZABDIZGZUAUBJZABHZCDKZABHZUCJUFJZUHUJLUAUGUIABCDMN
|
|
OUBABEPUAUJUDUEKUKABCDETUDUEMQRS $.
|
|
|
|
${
|
|
$d x C $.
|
|
$( Move proper substitution in and out of a membership relation. Note that
|
|
the scope of ` [. A / x ]. ` is the wff ` B e. C ` , whereas the scope
|
|
of ` [_ A / x ]_ ` is the class ` B ` . (Contributed by NM,
|
|
10-Nov-2005.) $)
|
|
sbcel1g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
|
|
[_ A / x ]_ B e. C ) ) $=
|
|
( wcel wsbc csb sbcel12g csbconstg eleq2d bitrd ) BEFZCDFABGABCHZABDHZFND
|
|
FABCDEIMODNABDEJKL $.
|
|
|
|
$( Move proper substitution to first argument of an equality. (Contributed
|
|
by NM, 30-Nov-2005.) $)
|
|
sbceq1g $p |- ( A e. V -> ( [. A / x ]. B = C <->
|
|
[_ A / x ]_ B = C ) ) $=
|
|
( wcel wceq wsbc csb sbceqg csbconstg eqeq2d bitrd ) BEFZCDGABHABCIZABDIZ
|
|
GODGABCDEJNPDOABDEKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x B $.
|
|
$( Move proper substitution in and out of a membership relation.
|
|
(Contributed by NM, 14-Nov-2005.) $)
|
|
sbcel2g $p |- ( A e. V -> ( [. A / x ]. B e. C <->
|
|
B e. [_ A / x ]_ C ) ) $=
|
|
( wcel wsbc csb sbcel12g csbconstg eleq1d bitrd ) BEFZCDFABGABCHZABDHZFCO
|
|
FABCDEIMNCOABCEJKL $.
|
|
|
|
$( Move proper substitution to second argument of an equality.
|
|
(Contributed by NM, 30-Nov-2005.) $)
|
|
sbceq2g $p |- ( A e. V -> ( [. A / x ]. B = C <->
|
|
B = [_ A / x ]_ C ) ) $=
|
|
( wcel wceq wsbc csb sbceqg csbconstg eqeq1d bitrd ) BEFZCDGABHABCIZABDIZ
|
|
GCPGABCDEJNOCPABCEKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z B $. $d z C $. $d x y $.
|
|
$( Commutative law for double substitution into a class. (Contributed by
|
|
NM, 14-Nov-2005.) $)
|
|
csbcomg $p |- ( ( A e. V /\ B e. W ) ->
|
|
[_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C ) $=
|
|
( vz wcel cvv csb wceq elex wsbc wb sbcel2g sbcbidv adantl adantr 3bitr3d
|
|
wa cv sbccom a1i eqrdv syl2an ) CFICJIZDJIZACBDEKZKZBDACEKZKZLDGICFMDGMUG
|
|
UHUAZHUJULUMHUBZUIIZACNZUNUKIZBDNZUNUJIZUNULIZUMUNEIZBDNZACNZVAACNZBDNZUP
|
|
URVCVEOUMVAABCDUCUDUHVCUPOUGUHVBUOACBDUNEJPQRUGVEUROUHUGVDUQBDACUNEJPQSTU
|
|
GUPUSOUHACUNUIJPSUHURUTOUGBDUNUKJPRTUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $. $d y ph $.
|
|
csbeq2d.1 $e |- F/ x ph $.
|
|
csbeq2d.2 $e |- ( ph -> B = C ) $.
|
|
$( Formula-building deduction rule for class substitution. (Contributed by
|
|
NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
csbeq2d $p |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C ) $=
|
|
( vy cv wcel wsbc cab csb eleq2d sbcbid abbidv df-csb 3eqtr4g ) AHIZDJZBC
|
|
KZHLSEJZBCKZHLBCDMBCEMAUAUCHATUBBCFADESGNOPBHCDQBHCEQR $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
csbeq2dv.1 $e |- ( ph -> B = C ) $.
|
|
$( Formula-building deduction rule for class substitution. (Contributed by
|
|
NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
csbeq2dv $p |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C ) $=
|
|
( nfv csbeq2d ) ABCDEABGFH $.
|
|
$}
|
|
|
|
${
|
|
csbeq2i.1 $e |- B = C $.
|
|
$( Formula-building inference rule for class substitution. (Contributed by
|
|
NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
csbeq2i $p |- [_ A / x ]_ B = [_ A / x ]_ C $=
|
|
( csb wceq wtru a1i csbeq2dv trud ) ABCFABDFGHABCDCDGHEIJK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $.
|
|
$( The proper substitution of a class for setvar variable results in the
|
|
class (if the class exists). (Contributed by NM, 10-Nov-2005.) $)
|
|
csbvarg $p |- ( A e. V -> [_ A / x ]_ x = A ) $=
|
|
( vz vy wcel cvv cv csb wceq elex wsbc cab df-csb sbcel2gv abbi1dv syl5eq
|
|
vex ax-mp csbeq2i csbco 3eqtr3i syl ) BCFBGFZABAHZIZBJBCKUDUFDHZEHZFEBLZD
|
|
MZBEBAUHUEIZIEBUHIUFUJEBUKUHUHGFZUKUHJERULUKUGUEFAUHLZDMUHADUHUENULUMDUHA
|
|
UGUHGOPQSTAEBUEUAEDBUHNUBUDUIDBEUGBGOPQUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Substitution into a wff expressed in terms of substitution into a
|
|
class. (Contributed by NM, 15-Aug-2007.) $)
|
|
sbccsbg $p |- ( A e. V ->
|
|
( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } ) ) $=
|
|
( wsbc cv cab wcel csb abid sbcbii sbcel2g syl5bbr ) ABDFCGZACHZIZBDFDEIO
|
|
BDPJIQABDACKLBDOPEMN $.
|
|
$}
|
|
|
|
$( Substitution into a wff expressed in using substitution into a class.
|
|
(Contributed by NM, 27-Nov-2005.) $)
|
|
sbccsb2g $p |- ( A e. V ->
|
|
( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) ) $=
|
|
( wsbc cv cab wcel csb abid sbcbii sbcel12g csbvarg eleq1d bitrd syl5bbr )
|
|
ABCEBFZABGZHZBCEZCDHZCBCRIZHZSABCABJKUATBCQIZUBHUCBCQRDLUAUDCUBBCDMNOP $.
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y ph $.
|
|
nfcsb1d.1 $e |- ( ph -> F/_ x A ) $.
|
|
$( Bound-variable hypothesis builder for substitution into a class.
|
|
(Contributed by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfcsb1d $p |- ( ph -> F/_ x [_ A / x ]_ B ) $=
|
|
( vy csb cv wcel wsbc cab df-csb nfv nfsbc1d nfabd nfcxfrd ) ABBCDGFHDIZB
|
|
CJZFKBFCDLARBFAFMAQBCENOP $.
|
|
$}
|
|
|
|
${
|
|
nfcsb1.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for substitution into a class.
|
|
(Contributed by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfcsb1 $p |- F/_ x [_ A / x ]_ B $=
|
|
( csb wnfc wtru a1i nfcsb1d trud ) AABCEFGABCABFGDHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Bound-variable hypothesis builder for substitution into a class.
|
|
(Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro,
|
|
12-Oct-2016.) $)
|
|
nfcsb1v $p |- F/_ x [_ A / x ]_ B $=
|
|
( nfcv nfcsb1 ) ABCABDE $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z B $. $d z ph $.
|
|
nfcsbd.1 $e |- F/ y ph $.
|
|
nfcsbd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfcsbd.3 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of ~ nfcsb . (Contributed by NM, 21-Nov-2005.)
|
|
(Revised by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfcsbd $p |- ( ph -> F/_ x [_ A / y ]_ B ) $=
|
|
( vz csb cv wcel wsbc cab df-csb nfv nfcrd nfsbcd nfabd nfcxfrd ) ABCDEJI
|
|
KELZCDMZINCIDEOAUBBIAIPAUABCDFGABIEHQRST $.
|
|
$}
|
|
|
|
${
|
|
nfcsb.1 $e |- F/_ x A $.
|
|
nfcsb.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for substitution into a class.
|
|
(Contributed by Mario Carneiro, 12-Oct-2016.) $)
|
|
nfcsb $p |- F/_ x [_ A / y ]_ B $=
|
|
( csb wnfc wtru nftru a1i nfcsbd trud ) ABCDGHIABCDBJACHIEKADHIFKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
csbhypf.1 $e |- F/_ x A $.
|
|
csbhypf.2 $e |- F/_ x C $.
|
|
csbhypf.3 $e |- ( x = A -> B = C ) $.
|
|
$( Introduce an explicit substitution into an implicit substitution
|
|
hypothesis. See ~ sbhypf for class substitution version. (Contributed
|
|
by NM, 19-Dec-2008.) $)
|
|
csbhypf $p |- ( y = A -> [_ y / x ]_ B = C ) $=
|
|
( cv wceq wi csb nfeq2 nfcsb1v nfeq nfim eqeq1 csbeq1a eqeq1d imbi12d
|
|
chvar ) AIZCJZDEJZKBIZCJZAUEDLZEJZKABUFUHAAUECFMAUGEAUEDNGOPUBUEJZUCUFUDU
|
|
HUBUECQUIDUGEAUEDRSTHUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Closed theorem version of ~ csbiegf .) (Contributed by NM,
|
|
11-Nov-2005.) $)
|
|
csbiebt $p |- ( ( A e. V /\ F/_ x C ) ->
|
|
( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) ) $=
|
|
( wcel cvv wnfc cv wceq wi wal csb wb elex wa wsbc adantl a1i nfeqd spsbc
|
|
adantr simpl biimt csbeq1a eqeq1d bitr3d nfv nfnfc1 nfcsb1v simpr sbciedf
|
|
nfan sylibd id nfan1 biimprcd alrimi ex impbid sylan ) BEFBGFZADHZAIBJZCD
|
|
JZKZALZABCMZDJZNBEOVBVCPZVGVIVJVGVFABQZVIVBVGVKKVCVFABGUAUBVJVFVIABGVBVCU
|
|
CVDVFVINVJVDVEVFVIVDVEUDVDCVHDABCUEUFZUGRVBVCAVBAUHADUIZUMVJAVHDAVHHZVJAB
|
|
CUJZSVBVCUKTULUNVCVIVGKVBVCVIVGVCVIPVFAVCVIAVMVCAVHDVNVCVOSVCUOTUPVIVFVCV
|
|
DVEVIVLUQRURUSRUTVA $.
|
|
|
|
csbiedf.1 $e |- F/ x ph $.
|
|
csbiedf.2 $e |- ( ph -> F/_ x C ) $.
|
|
csbiedf.3 $e |- ( ph -> A e. V ) $.
|
|
csbiedf.4 $e |- ( ( ph /\ x = A ) -> B = C ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by Mario Carneiro, 13-Oct-2016.) $)
|
|
csbiedf $p |- ( ph -> [_ A / x ]_ B = C ) $=
|
|
( cv wceq wi wal csb ex alrimi wcel wnfc wb csbiebt syl2anc mpbid ) ABKCL
|
|
ZDELZMZBNZBCDOELZAUFBGAUDUEJPQACFRBESUGUHTIHBCDEFUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
csbieb.1 $e |- A e. _V $.
|
|
csbieb.2 $e |- F/_ x C $.
|
|
$( Bidirectional conversion between an implicit class substitution
|
|
hypothesis ` x = A -> B = C ` and its explicit substitution equivalent.
|
|
(Contributed by NM, 2-Mar-2008.) $)
|
|
csbieb $p |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) $=
|
|
( cvv wcel wnfc cv wceq wi wal csb wb csbiebt mp2an ) BGHADIAJBKCDKLAMABC
|
|
NDKOEFABCDGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d a x A $. $d a B $. $d a C $.
|
|
csbiebg.2 $e |- F/_ x C $.
|
|
$( Bidirectional conversion between an implicit class substitution
|
|
hypothesis ` x = A -> B = C ` and its explicit substitution equivalent.
|
|
(Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro,
|
|
11-Dec-2016.) $)
|
|
csbiebg $p |- ( A e. V ->
|
|
( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) ) $=
|
|
( va cv wceq wal csb eqeq2 imbi1d albidv csbeq1 eqeq1d vex csbieb vtoclbg
|
|
wi ) AHZGHZIZCDIZTZAJAUBCKZDIUABIZUDTZAJABCKZDIGBEUBBIZUEUHAUJUCUGUDUBBUA
|
|
LMNUJUFUIDAUBBCOPAUBCDGQFRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x V $.
|
|
csbiegf.1 $e |- ( A e. V -> F/_ x C ) $.
|
|
csbiegf.2 $e |- ( x = A -> B = C ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
csbiegf $p |- ( A e. V -> [_ A / x ]_ B = C ) $=
|
|
( wcel cv wceq wi wal csb ax-gen wnfc wb csbiebt mpdan mpbii ) BEHZAIBJCD
|
|
JKZALZABCMDJZUAAGNTADOUBUCPFABCDEQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
csbief.1 $e |- A e. _V $.
|
|
csbief.2 $e |- F/_ x C $.
|
|
csbief.3 $e |- ( x = A -> B = C ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro,
|
|
13-Oct-2016.) $)
|
|
csbief $p |- [_ A / x ]_ B = C $=
|
|
( cvv wcel csb wceq wnfc a1i csbiegf ax-mp ) BHIZABCJDKEABCDHADLPFMGNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x ph $.
|
|
csbied.1 $e |- ( ph -> A e. V ) $.
|
|
csbied.2 $e |- ( ( ph /\ x = A ) -> B = C ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario
|
|
Carneiro, 13-Oct-2016.) $)
|
|
csbied $p |- ( ph -> [_ A / x ]_ B = C ) $=
|
|
( nfv nfcvd csbiedf ) ABCDEFABIABEJGHK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $. $d x D $.
|
|
csbied2.1 $e |- ( ph -> A e. V ) $.
|
|
csbied2.2 $e |- ( ph -> A = B ) $.
|
|
csbied2.3 $e |- ( ( ph /\ x = B ) -> C = D ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution,
|
|
deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.) $)
|
|
csbied2 $p |- ( ph -> [_ A / x ]_ C = D ) $=
|
|
( cv wceq id sylan9eqr syldan csbied ) ABCEFGHABKZCLZQDLEFLRAQCDRMINJOP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y D $.
|
|
csbie2t.1 $e |- A e. _V $.
|
|
csbie2t.2 $e |- B e. _V $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class (closed form of ~ csbie2 ). (Contributed by NM, 3-Sep-2007.)
|
|
(Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
csbie2t $p |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) ->
|
|
[_ A / x ]_ [_ B / y ]_ C = D ) $=
|
|
( cv wceq wa wi wal csb cvv nfa1 nfcvd wcel a1i csbiedf nfa2 nfv nfan sps
|
|
sp impl ) AICJZBIDJZKEFJZLZBMZAMZACBDENFOUKAPULAFQCORULGSULUGKZBDEFOULUGB
|
|
UJBAUAUGBUBUCUMBFQDORUMHSULUGUHUIUKUJAUJBUEUDUFTT $.
|
|
|
|
csbie2.3 $e |- ( ( x = A /\ y = B ) -> C = D ) $.
|
|
$( Conversion of implicit substitution to explicit substitution into a
|
|
class. (Contributed by NM, 27-Aug-2007.) $)
|
|
csbie2 $p |- [_ A / x ]_ [_ B / y ]_ C = D $=
|
|
( cv wceq wa wi wal csb gen2 csbie2t ax-mp ) AJCKBJDKLEFKMZBNANACBDEOOFKS
|
|
ABIPABCDEFGHQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d A y z $. $d B y z $. $d C x $. $d D y z $. $d V z $.
|
|
csbie2g.1 $e |- ( x = y -> B = C ) $.
|
|
csbie2g.2 $e |- ( y = A -> C = D ) $.
|
|
$( Conversion of implicit substitution to explicit class substitution.
|
|
This version of ~ sbcie avoids a disjointness condition on ` x , A ` by
|
|
substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.) $)
|
|
csbie2g $p |- ( A e. V -> [_ A / x ]_ B = D ) $=
|
|
( vz wcel csb cv wsbc cab df-csb wceq eleq2d sbcie2g abbi1dv syl5eq ) CGK
|
|
ZACDLJMZDKZACNZJOFAJCDPUBUEJFUDUCEKUCFKABCGAMBMZQDEUCHRUFCQEFUCIRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z B $. $d z C $. $d z ph $.
|
|
$( Nest the composition of two substitutions. (Contributed by Mario
|
|
Carneiro, 11-Nov-2016.) $)
|
|
sbcnestgf $p |- ( ( A e. V /\ A. y F/ x ph ) ->
|
|
( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $=
|
|
( vz wcel wnf wal wsbc csb wb cv wi wceq dfsbcq syl cvv a1i csbeq1 imbi2d
|
|
bibi12d vex csbeq1a adantl nfnf1 nfal nfa1 wnfc nfcsb1v sp nfsbcd sbciedf
|
|
vtoclg imp ) DFHABIZCJZACEKZBDKZACBDELZKZMZURUSBGNZKZACBVDELZKZMZOURVCOGD
|
|
FVDDPZVHVCURVIVEUTVGVBUSBVDDQVIVFVAPVGVBMBVDDEUAACVFVAQRUCUBURUSVGBVDSVDS
|
|
HURGUDTBNVDPZUSVGMZURVJEVFPVKBVDEUEACEVFQRUFUQBCABUGUHURABCVFUQCUIBVFUJUR
|
|
BVDEUKTUQCULUMUNUOUP $.
|
|
|
|
$( Nest the composition of two substitutions. (Contributed by NM,
|
|
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) $)
|
|
csbnestgf $p |- ( ( A e. V /\ A. y F/_ x C ) ->
|
|
[_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
|
|
( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb abeq2i wb nfcr
|
|
sbcbii wnf alimi sbcnestgf sylan2 syl5bb abbidv sylan 3eqtr4g ) CFHZAEIZB
|
|
JZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVBG
|
|
URUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUDZBJVGVBUAUMVHBAGEUBUEUTABCDPU
|
|
FUGUHUIUJAGCUPSBGVAESUK $.
|
|
|
|
$d x ph $.
|
|
$( Nest the composition of two substitutions. (Contributed by NM,
|
|
27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) $)
|
|
sbcnestg $p |- ( A e. V ->
|
|
( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $=
|
|
( wcel wnf wal wsbc csb wb nfv ax-gen sbcnestgf mpan2 ) DFGABHZCIACEJBDJA
|
|
CBDEKJLQCABMNABCDEFOP $.
|
|
|
|
$d x C $.
|
|
$( Nest the composition of two substitutions. (Contributed by NM,
|
|
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) $)
|
|
csbnestg $p |- ( A e. V ->
|
|
[_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
|
|
( wcel wnfc wal csb wceq nfcv ax-gen csbnestgf mpan2 ) CFGAEHZBIACBDEJJBA
|
|
CDJEJKPBAELMABCDEFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x C $.
|
|
$( Nest the composition of two substitutions. (New usage is discouraged.)
|
|
(Contributed by NM, 23-Nov-2005.) $)
|
|
csbnestgOLD $p |- ( ( A e. V /\ A. x B e. W ) ->
|
|
[_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
|
|
( wcel csb wceq wal csbnestg adantr ) CFHACBDEIIBACDIEIJDGHAKABCDEFLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y C $.
|
|
$( Nest the composition of two substitutions. (Contributed by NM,
|
|
23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) $)
|
|
csbnest1g $p |- ( A e. V ->
|
|
[_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C ) $=
|
|
( vy wcel cv csb wnfc wceq nfcsb1v ax-gen csbnestgf mpan2 csbeq2i 3eqtr3g
|
|
wal csbco ) BEGZABFCAFHZDIZIZIZFABCIZUBIZABACDIZIAUEDITAUBJZFRUDUFKUHFAUA
|
|
DLMAFBCUBENOABUCUGAFCDSPAFUEDSQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Nest the composition of two substitutions. Obsolete as of 11-Nov-2016.
|
|
(Contributed by NM, 23-May-2006.) (New usage is discouraged.) $)
|
|
csbnest1gOLD $p |- ( ( A e. V /\ A. x B e. W ) ->
|
|
[_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C ) $=
|
|
( wcel csb wceq wal csbnest1g adantr ) BEGABACDHHAABCHDHICFGAJABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Idempotent law for class substitutions. (Contributed by NM,
|
|
1-Mar-2008.) $)
|
|
csbidmg $p |- ( A e. V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B ) $=
|
|
( wcel cvv csb wceq elex csbnest1g csbconstg csbeq1d eqtrd syl ) BDEBFEZA
|
|
BABCGZGZPHBDIOQAABBGZCGPABBCFJOARBCABBFKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $. $d x C $. $d x D $.
|
|
sbcco3g.1 $e |- ( x = A -> B = C ) $.
|
|
$( Composition of two substitutions. (Contributed by NM, 27-Nov-2005.)
|
|
(Revised by Mario Carneiro, 11-Nov-2016.) $)
|
|
sbcco3g $p |- ( A e. V ->
|
|
( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) ) $=
|
|
( wcel wsbc csb sbcnestg cvv wceq wb elex nfcvd csbiegf dfsbcq 3syl bitrd
|
|
) DGIZACEJBDJACBDEKZJZACFJZABCDEGLUBDMIZUCFNUDUEODGPBDEFMUFBFQHRACUCFSTUA
|
|
$.
|
|
|
|
$( Composition of two substitutions. (Contributed by NM, 27-Nov-2005.)
|
|
(New usage is discouraged.) $)
|
|
sbcco3gOLD $p |- ( ( A e. V /\ A. x B e. W ) ->
|
|
( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) ) $=
|
|
( wcel wsbc wb wal sbcco3g adantr ) DGJACEKBDKACFKLEHJBMABCDEFGINO $.
|
|
|
|
$( Composition of two class substitutions. (Contributed by NM,
|
|
27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) $)
|
|
csbco3g $p |- ( A e. V ->
|
|
[_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D ) $=
|
|
( wcel csb csbnestg cvv wceq elex nfcvd csbiegf syl csbeq1d eqtrd ) CGIZA
|
|
CBDFJJBACDJZFJBEFJABCDFGKTBUAEFTCLIZUAEMCGNACDELUBAEOHPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $. $d x y $.
|
|
csbco3g.1 $e |- ( x = A -> B = D ) $.
|
|
$( Composition of two class substitutions. Obsolete as of 11-Nov-2016.
|
|
(Contributed by NM, 27-Nov-2005.) (New usage is discouraged.) $)
|
|
csbco3gOLD $p |- ( ( A e. V /\ A. x B e. W ) ->
|
|
[_ A / x ]_ [_ B / y ]_ C = [_ D / y ]_ C ) $=
|
|
( wcel csb wceq wal csbco3g adantr ) CGJACBDEKKBFEKLDHJAMABCDFEGINO $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x D $.
|
|
$( Special case related to ~ rspsbc . (Contributed by NM, 10-Dec-2005.)
|
|
(Proof shortened by Eric Schmidt, 17-Jan-2007.) $)
|
|
rspcsbela $p |- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D ) $=
|
|
( wcel wral csb wsbc rspsbc sbcel1g sylibd imp ) BCFZDEFZACGZABDHEFZNPOAB
|
|
IQOABCJABDECKLM $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z $. $d w y z A $.
|
|
$( Two ways of expressing " ` x ` is (effectively) not free in ` A ` ."
|
|
(Contributed by Mario Carneiro, 14-Oct-2016.) $)
|
|
sbnfc2 $p |- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A ) $=
|
|
( vw cv csb wceq wal cvv wcel vex csbtt mpan wsb wsbc sbsbc sbcel2g ax-mp
|
|
wb eqtr4d alrimivv nfv wnf eleq2 bitri 3bitr4g 2alimi sbnf2 sylibr impbii
|
|
wnfc nfcd ) ADULZABFZDGZACFZDGZHZCIBIZUNUSBCUNUPDURUOJKZUNUPDHBLZAUODJMNU
|
|
QJKZUNURDHCLZAUQDJMNUAUBUTAEDUTEUCUTEFZDKZABOZVFACOZTZCIBIVFAUDUSVIBCUSVE
|
|
UPKZVEURKZVGVHUPURVEUEVGVFAUOPZVJVFABQVAVLVJTVBAUOVEDJRSUFVHVFAUQPZVKVFAC
|
|
QVCVMVKTVDAUQVEDJRSUFUGUHVFABCUIUJUMUK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d z ph $. $d x y z $. $d V z $.
|
|
$( Move substitution into a class abstraction. (Contributed by NM,
|
|
13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
csbabg $p |- ( A e. V ->
|
|
[_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } ) $=
|
|
( vz wcel cab csb wsbc cv sbccom wsb df-clab sbsbc sbcbii 3bitr4i sbcel2g
|
|
bitri syl5rbb eqrdv ) DEGZFBDACHZIZABDJZCHZFKZUFGZUGUCGZBDJZUBUGUDGUECUGJ
|
|
ZACUGJZBDJUHUJACBUGDLUHUECFMUKUEFCNUECFOSUIULBDUIACFMULAFCNACFOSPQBDUGUCE
|
|
RTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x v z $. $d y v z $. $d A v z $. $d B v z $. $d ph v z $.
|
|
$d ps v z $.
|
|
cbvralcsf.1 $e |- F/_ y A $.
|
|
cbvralcsf.2 $e |- F/_ x B $.
|
|
cbvralcsf.3 $e |- F/ y ph $.
|
|
cbvralcsf.4 $e |- F/ x ps $.
|
|
cbvralcsf.5 $e |- ( x = y -> A = B ) $.
|
|
cbvralcsf.6 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( A more general version of ~ cbvralf that doesn't require ` A ` and ` B `
|
|
to be distinct from ` x ` or ` y ` . Changes bound variables using
|
|
implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) $)
|
|
cbvralcsf $p |- ( A. x e. A ph <-> A. y e. B ps ) $=
|
|
( vz vv cv wcel wi wal wsbc nfcri wral csb nfv nfcsb1v nfsbc1v id csbeq1a
|
|
nfim wceq eleq12d sbceq1a imbi12d cbval nfcsb nfsbc csbeq1 cab df-csb wsb
|
|
nfcv eleq2d sbsbc bitr3i abbi2i eqtr4i syl6eq dfsbcq syl6bb bitri 3bitr4i
|
|
weq sbie df-ral ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACWBSZQ
|
|
ZMRWAVPWFCMVPMUCWDWECCMWCCWBEUDTACWBUEUHVNWBUIZVOWDAWEWGVNWBEWCWGUFCWBEUG
|
|
UJACWBUKULUMWFVTMDWDWEDDMWCDCWBEDWBUTZGUNTADCWBWHIUOUHVTMUCWBVRUIZWDVSWEB
|
|
WIWBVRWCFWIUFWIWCCVREUBZFCWBVREUPWJNOZEPZCVRSZNUQFCNVREURWMNFWKFPZWLCDUSW
|
|
MWLWNCDCNFHTCDVKEFWKKVAVLWLCDVBVCVDVEVFUJWIWEACVRSZBACWBVRVGWOACDUSBACDVB
|
|
ABCDJLVLVCVHULUMVIACEVMBDFVMVJ $.
|
|
|
|
$( A more general version of ~ cbvrexf that has no distinct variable
|
|
restrictions. Changes bound variables using implicit substitution.
|
|
(Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario
|
|
Carneiro, 7-Dec-2014.) $)
|
|
cbvrexcsf $p |- ( E. x e. A ph <-> E. y e. B ps ) $=
|
|
( wn wral wrex nfn cv wceq notbid dfrex2 cbvralcsf notbii 3bitr4i ) AMZCE
|
|
NZMBMZDFNZMACEOBDFOUEUGUDUFCDEFGHADIPBCJPKCQDQRABLSUAUBACETBDFTUC $.
|
|
|
|
$( A more general version of ~ cbvreuv that has no distinct variable
|
|
rextrictions. Changes bound variables using implicit substitution.
|
|
(Contributed by Andrew Salmon, 13-Jul-2011.) $)
|
|
cbvreucsf $p |- ( E! x e. A ph <-> E! y e. B ps ) $=
|
|
( vz vv cv wcel wa weu wsb nfcri wreu csb nfcsb1v nfs1v nfan wceq csbeq1a
|
|
nfv eleq12d sbequ12 anbi12d cbveu nfcv nfcsb nfsb csbeq1 wsbc sbsbc abbii
|
|
cab eleq2d bicomi abbi2i df-csb 3eqtr4ri syl6eq sbequ syl6bb bitri df-reu
|
|
id sbie 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACMSZQZ
|
|
MRWAVPWFCMVPMUHWDWECCMWCCWBEUCTACMUDUEVNWBUFZVOWDAWEWGVNWBEWCWGVKCWBEUGUI
|
|
ACMUJUKULWFVTMDWDWEDDMWCDCWBEDWBUMGUNTACMDIUOUEVTMUHWBVRUFZWDVSWEBWHWBVRW
|
|
CFWHVKWHWCCVREUBZFCWBVREUPNOZEPZCDSZNUTWKCVRUQZNUTFWIWLWMNWKCDURUSWLNFWLW
|
|
JFPZWKWNCDCNFHTVNVRUFEFWJKVAVLVBVCCNVREVDVEVFUIWHWEACDSBAMDCVGABCDJLVLVHU
|
|
KULVIACEVJBDFVJVM $.
|
|
|
|
$( A more general version of ~ cbvrab with no distinct variable
|
|
restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.) $)
|
|
cbvrabcsf $p |- { x e. A | ph } = { y e. B | ps } $=
|
|
( vz vv cv wcel wa cab wsb nfcri crab csb nfcsb1v nfs1v nfan wceq csbeq1a
|
|
nfv id eleq12d sbequ12 anbi12d cbvab nfcv nfcsb nfsb csbeq1 df-csb eleq2d
|
|
wsbc weq sbie sbsbc bitr3i abbi2i eqtr4i syl6eq sbequ syl6bb eqtri df-rab
|
|
3eqtr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACMSZQZMRVTVOWE
|
|
CMVOMUHWCWDCCMWBCWAEUCTACMUDUEVMWAUFZVNWCAWDWFVMWAEWBWFUICWAEUGUJACMUKULU
|
|
MWEVSMDWCWDDDMWBDCWAEDWAUNGUOTACMDIUPUEVSMUHWAVQUFZWCVRWDBWGWAVQWBFWGUIWG
|
|
WBCVQEUBZFCWAVQEUQWHNOZEPZCVQUTZNRFCNVQEURWKNFWIFPZWJCDSWKWJWLCDCNFHTCDVA
|
|
EFWIKUSVBWJCDVCVDVEVFVGUJWGWDACDSBAMDCVHABCDJLVBVIULUMVJACEVKBDFVKVL $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d ps y $. $d B x $. $d ch x $.
|
|
cbvralv2.1 $e |- ( x = y -> ( ps <-> ch ) ) $.
|
|
cbvralv2.2 $e |- ( x = y -> A = B ) $.
|
|
$( Rule used to change the bound variable in a restricted universal
|
|
quantifier with implicit substitution which also changes the quantifier
|
|
domain. (Contributed by David Moews, 1-May-2017.) $)
|
|
cbvralv2 $p |- ( A. x e. A ps <-> A. y e. B ch ) $=
|
|
( nfcv nfv cbvralcsf ) ABCDEFDEICFIADJBCJHGK $.
|
|
|
|
$( Rule used to change the bound variable in a restricted existential
|
|
quantifier with implicit substitution which also changes the quantifier
|
|
domain. (Contributed by David Moews, 1-May-2017.) $)
|
|
cbvrexv2 $p |- ( E. x e. A ps <-> E. y e. B ch ) $=
|
|
( nfcv nfv cbvrexcsf ) ABCDEFDEICFIADJBCJHGK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Define boolean set operations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new symbols. $)
|
|
$c -i^i $. $( Lined cap (anti-intersection) $)
|
|
$c ~ $. $( Tilde (complement) $)
|
|
$c \ $. $( Backslash (difference) $)
|
|
$c u. $. $( Cup (union) $)
|
|
$c i^i $. $( Cap (intersection) $)
|
|
$c (+) $. $( Circled plus (symmetric difference) $)
|
|
|
|
$( Extend class notation to include anti-intersection (read: "the
|
|
anti-intersection of ` A ` and ` B ` "). $)
|
|
cnin $a class ( A -i^i B ) $.
|
|
|
|
$( Extend class notation to include complement. (read: "the complement of
|
|
` A ` " ). $)
|
|
ccompl $a class ~ A $.
|
|
|
|
$( Extend class notation to include class difference (read: " ` A ` minus
|
|
` B ` "). $)
|
|
cdif $a class ( A \ B ) $.
|
|
|
|
$( Extend class notation to include union of two classes (read: " ` A `
|
|
union ` B ` "). $)
|
|
cun $a class ( A u. B ) $.
|
|
|
|
$( Extend class notation to include the intersection of two classes
|
|
(read: " ` A ` intersect ` B ` "). $)
|
|
cin $a class ( A i^i B ) $.
|
|
|
|
$( Extend class notation to include the symmetric difference of two
|
|
classes. $)
|
|
csymdif $a class ( A (+) B ) $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Soundness theorem for ~ df-nin . (Contributed by SF, 10-Jan-2015.) $)
|
|
ninjust $p |- { x | ( x e. A -/\ x e. B ) } =
|
|
{ y | ( y e. A -/\ y e. B ) } $=
|
|
( cv wcel wnan weq eleq1 nanbi12d cbvabv ) AEZCFZLDFZGBEZCFZODFZGABABHMPN
|
|
QLOCILODIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
|
|
$( Define the anti-intersection of two classes. This operation is used
|
|
implicitly after Axiom P1 of [Hailperin] p. 6, though there does not
|
|
seem to be any notation for it in the literature. (Contributed by SF,
|
|
10-Jan-2015.) $)
|
|
df-nin $a |- ( A -i^i B ) = { x | ( x e. A -/\ x e. B ) } $.
|
|
$}
|
|
|
|
$( Define the complement of a class. Compare ~ nic-dfneg . (Contributed by
|
|
SF, 10-Jan-2015.) $)
|
|
df-compl $a |- ~ A = ( A -i^i A ) $.
|
|
|
|
$( Define the intersection of two classes. See ~ elin for membership.
|
|
(Contributed by SF, 10-Jan-2015.) $)
|
|
df-in $a |- ( A i^i B ) = ~ ( A -i^i B ) $.
|
|
|
|
$( Define the union of two classes. See ~ elun for membership. (Contributed
|
|
by SF, 10-Jan-2015.) $)
|
|
df-un $a |- ( A u. B ) = ( ~ A -i^i ~ B ) $.
|
|
|
|
$( Define the difference of two classes. See ~ eldif for membership.
|
|
(Contributed by SF, 10-Jan-2015.) $)
|
|
df-dif $a |- ( A \ B ) = ( A i^i ~ B ) $.
|
|
|
|
$( Define the symmetric difference of two classes. Definition IX.9.10,
|
|
[Rosser] p. 238. (Contributed by SF, 10-Jan-2015.) $)
|
|
df-symdif $a |- ( A (+) B ) = ( ( A \ B ) u. ( B \ A ) ) $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Membership in anti-intersection. (Contributed by SF, 10-Jan-2015.) $)
|
|
elning $p |- ( A e. V ->
|
|
( A e. ( B -i^i C ) <-> ( A e. B -/\ A e. C ) ) ) $=
|
|
( vx cv wcel wnan cnin wceq eleq1 nanbi12d df-nin elab2g ) EFZBGZOCGZHABG
|
|
ZACGZHEABCIDOAJPRQSOABKOACKLEBCMN $.
|
|
$}
|
|
|
|
$( Membership in class complement. (Contributed by SF, 10-Jan-2015.) $)
|
|
elcomplg $p |- ( A e. V -> ( A e. ~ B <-> -. A e. B ) ) $=
|
|
( ccompl wcel cnin wn df-compl eleq2i elning wa df-nan anidm xchbinx syl6bb
|
|
wnan syl5bb ) ABDZEABBFZEZACEZABEZGZRSABHIUATUBUBPZUCABBCJUDUBUBKUBUBUBLUBM
|
|
NOQ $.
|
|
|
|
$( Membership in intersection. (Contributed by SF, 10-Jan-2015.) $)
|
|
elin $p |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) ) $=
|
|
( cin wcel cvv wa elex adantr cnin ccompl wnan elcomplg elning notbid bitrd
|
|
wn df-in eleq2i df-nan con2bii 3bitr4g pm5.21nii ) ABCDZEZAFEZABEZACEZGZAUD
|
|
HUGUFUHABHIUFABCJZKZEZUGUHLZQZUEUIUFULAUJEZQUNAUJFMUFUOUMABCFNOPUDUKABCRSUM
|
|
UIUGUHTUAUBUC $.
|
|
|
|
$( Membership in union. (Contributed by SF, 10-Jan-2015.) $)
|
|
elun $p |- ( A e. ( B u. C ) <-> ( A e. B \/ A e. C ) ) $=
|
|
( cun wcel cvv wo elex jaoi ccompl cnin wnan elning elcomplg nanbi12d bitrd
|
|
wn df-un eleq2i wa oran df-nan bitr4i 3bitr4g pm5.21nii ) ABCDZEZAFEZABEZAC
|
|
EZGZAUFHUIUHUJABHACHIUHABJZCJZKZEZUIQZUJQZLZUGUKUHUOAULEZAUMEZLURAULUMFMUHU
|
|
SUPUTUQABFNACFNOPUFUNABCRSUKUPUQTQURUIUJUAUPUQUBUCUDUE $.
|
|
|
|
$( Membership in difference. (Contributed by SF, 10-Jan-2015.) $)
|
|
eldif $p |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) ) $=
|
|
( cdif wcel ccompl cin wa wn df-dif eleq2i elin elcomplg pm5.32i 3bitri ) A
|
|
BCDZEABCFZGZEABEZAQEZHSACEIZHPRABCJKABQLSTUAACBMNO $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Alternate definition of class difference. (Contributed by NM,
|
|
25-Mar-2004.) $)
|
|
dfdif2 $p |- ( A \ B ) = { x e. A | -. x e. B } $=
|
|
( cdif cv wcel wn wa cab crab eldif abbi2i df-rab eqtr4i ) BCDZAEZBFPCFGZ
|
|
HZAIQABJRAOPBCKLQABMN $.
|
|
$}
|
|
|
|
$( Membership in symmetric difference. (Contributed by SF, 10-Jan-2015.) $)
|
|
elsymdif $p |- ( A e. ( B (+) C ) <-> -. ( A e. B <-> A e. C ) ) $=
|
|
( cdif cun wcel wn wa wo csymdif wb elun eldif orbi12i df-symdif eleq2i xor
|
|
bitri 3bitr4i ) ABCDZCBDZEZFZABFZACFZGHZUEUDGHZIZABCJZFUDUEKGUCATFZAUAFZIUH
|
|
ATUALUJUFUKUGABCMACBMNRUIUBABCOPUDUEQS $.
|
|
|
|
${
|
|
elbool.1 $e |- A e. _V $.
|
|
$( Membership in anti-intersection. (Contributed by SF, 10-Jan-2015.) $)
|
|
elnin $p |- ( A e. ( B -i^i C ) <-> ( A e. B -/\ A e. C ) ) $=
|
|
( cvv wcel cnin wnan wb elning ax-mp ) AEFABCGFABFACFHIDABCEJK $.
|
|
|
|
$( Membership in complement. (Contributed by SF, 10-Jan-2015.) $)
|
|
elcompl $p |- ( A e. ~ B <-> -. A e. B ) $=
|
|
( cvv wcel ccompl wn wb elcomplg ax-mp ) ADEABFEABEGHCABDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Anti-intersection commutes. (Contributed by SF, 10-Jan-2015.) $)
|
|
nincom $p |- ( A -i^i B ) = ( B -i^i A ) $=
|
|
( vx cnin cv wcel wnan nancom vex elnin 3bitr4i eqriv ) CABDZBADZCEZAFZOB
|
|
FZGQPGOMFONFPQHOABCIZJOBARJKL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Double complement law. (Contributed by SF, 10-Jan-2015.) $)
|
|
dblcompl $p |- ~ ~ A = A $=
|
|
( vx ccompl cv wcel wn vex elcompl con2bii bitr4i eqriv ) BACZCZABDZMENLE
|
|
ZFNAEZNLBGZHOPNAQHIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d A y $. $d B y $.
|
|
nfnin.1 $e |- F/_ x A $.
|
|
nfnin.2 $e |- F/_ x B $.
|
|
$( Hypothesis builder for anti-intersection. (Contributed by SF,
|
|
2-Jan-2018.) $)
|
|
nfnin $p |- F/_ x ( A -i^i B ) $=
|
|
( vy cnin cv wcel wnan cab df-nin nfel2 nfnan nfab nfcxfr ) ABCGFHZBIZQCI
|
|
ZJZFKFBCLTAFRSAAQBDMAQCEMNOP $.
|
|
$}
|
|
|
|
${
|
|
nfbool.1 $e |- F/_ x A $.
|
|
$( Hypothesis builder for complement. (Contributed by SF, 2-Jan-2018.) $)
|
|
nfcompl $p |- F/_ x ~ A $=
|
|
( ccompl cnin df-compl nfnin nfcxfr ) ABDBBEBFABBCCGH $.
|
|
|
|
nfbool.2 $e |- F/_ x B $.
|
|
$( Hypothesis builder for intersection. (Contributed by SF,
|
|
2-Jan-2018.) $)
|
|
nfin $p |- F/_ x ( A i^i B ) $=
|
|
( cin cnin ccompl df-in nfnin nfcompl nfcxfr ) ABCFBCGZHBCIAMABCDEJKL $.
|
|
|
|
$( Hypothesis builder for union. (Contributed by SF, 2-Jan-2018.) $)
|
|
nfun $p |- F/_ x ( A u. B ) $=
|
|
( cun ccompl cnin df-un nfcompl nfnin nfcxfr ) ABCFBGZCGZHBCIAMNABDJACEJK
|
|
L $.
|
|
|
|
$( Hypothesis builder for difference. (Contributed by SF, 2-Jan-2018.) $)
|
|
nfdif $p |- F/_ x ( A \ B ) $=
|
|
( cdif ccompl cin df-dif nfcompl nfin nfcxfr ) ABCFBCGZHBCIABMDACEJKL $.
|
|
|
|
$( Hypothesis builder for symmetric difference. (Contributed by SF,
|
|
2-Jan-2018.) $)
|
|
nfsymdif $p |- F/_ x ( A (+) B ) $=
|
|
( csymdif cdif cun df-symdif nfdif nfun nfcxfr ) ABCFBCGZCBGZHBCIAMNABCDE
|
|
JACBEDJKL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Equality law for anti-intersection. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
nineq1 $p |- ( A = B -> ( A -i^i C ) = ( B -i^i C ) ) $=
|
|
( vx wceq cv wcel wnan cab cnin eleq2 nanbi1d abbidv df-nin 3eqtr4g ) ABE
|
|
ZDFZAGZQCGZHZDIQBGZSHZDIACJBCJPTUBDPRUASABQKLMDACNDBCNO $.
|
|
$}
|
|
|
|
$( Equality law for anti-intersection. (Contributed by SF, 11-Jan-2015.) $)
|
|
nineq2 $p |- ( A = B -> ( C -i^i A ) = ( C -i^i B ) ) $=
|
|
( wceq cnin nineq1 nincom 3eqtr3g ) ABDACEBCECAECBEABCFACGBCGH $.
|
|
|
|
$( Equality law for anti-intersection. (Contributed by SF, 11-Jan-2015.) $)
|
|
nineq12 $p |- ( ( A = B /\ C = D ) -> ( A -i^i C ) = ( B -i^i D ) ) $=
|
|
( wceq cnin nineq1 nineq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
nineqi.1 $e |- A = B $.
|
|
$( Equality inference for anti-intersection. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
nineq1i $p |- ( A -i^i C ) = ( B -i^i C ) $=
|
|
( wceq cnin nineq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for anti-intersection. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
nineq2i $p |- ( C -i^i A ) = ( C -i^i B ) $=
|
|
( wceq cnin nineq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
nineq12i.2 $e |- C = D $.
|
|
$( Equality inference for anti-intersection. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
nineq12i $p |- ( A -i^i C ) = ( B -i^i D ) $=
|
|
( wceq cnin nineq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
nineqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for anti-intersection. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
nineq1d $p |- ( ph -> ( A -i^i C ) = ( B -i^i C ) ) $=
|
|
( wceq cnin nineq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for anti-intersection. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
nineq2d $p |- ( ph -> ( C -i^i A ) = ( C -i^i B ) ) $=
|
|
( wceq cnin nineq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
nineq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality inference for anti-intersection. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
nineq12d $p |- ( ph -> ( A -i^i C ) = ( B -i^i D ) ) $=
|
|
( wceq cnin nineq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
$( Equality law for complement. (Contributed by SF, 11-Jan-2015.) $)
|
|
compleq $p |- ( A = B -> ~ A = ~ B ) $=
|
|
( wceq cnin ccompl nineq12 anidms df-compl 3eqtr4g ) ABCZAADZBBDZAEBEJKLCAB
|
|
ABFGAHBHI $.
|
|
|
|
${
|
|
compleqi.1 $e |- A = B $.
|
|
$( Equality inference for complement. (Contributed by SF, 11-Jan-2015.) $)
|
|
compleqi $p |- ~ A = ~ B $=
|
|
( wceq ccompl compleq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
compleqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for complement. (Contributed by SF, 11-Jan-2015.) $)
|
|
compleqd $p |- ( ph -> ~ A = ~ B ) $=
|
|
( wceq ccompl compleq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
difeq1 $p |- ( A = B -> ( A \ C ) = ( B \ C ) ) $=
|
|
( wceq ccompl cnin cdif nineq1 compleqd cin df-dif df-in eqtri 3eqtr4g ) AB
|
|
DZACEZFZEZBPFZEZACGZBCGZOQSABPHIUAAPJRACKAPLMUBBPJTBCKBPLMN $.
|
|
|
|
$( Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
difeq2 $p |- ( A = B -> ( C \ A ) = ( C \ B ) ) $=
|
|
( wceq ccompl cnin cdif compleq nineq2d compleqd df-dif df-in eqtri 3eqtr4g
|
|
cin ) ABDZCAEZFZEZCBEZFZEZCAGZCBGZPRUAPQTCABHIJUCCQOSCAKCQLMUDCTOUBCBKCTLMN
|
|
$.
|
|
|
|
$( Equality law for intersection. (Contributed by SF, 11-Jan-2015.) $)
|
|
symdifeq1 $p |- ( A = B -> ( A (+) C ) = ( B (+) C ) ) $=
|
|
( wceq cdif ccompl cnin difeq1 compleqd difeq2 nineq12d cun df-symdif df-un
|
|
csymdif eqtri 3eqtr4g ) ABDZACEZFZCAEZFZGZBCEZFZCBEZFZGZACOZBCOZRTUEUBUGRSU
|
|
DABCHIRUAUFABCJIKUISUALUCACMSUANPUJUDUFLUHBCMUDUFNPQ $.
|
|
|
|
$( Equality law for intersection. (Contributed by SF, 11-Jan-2015.) $)
|
|
symdifeq2 $p |- ( A = B -> ( C (+) A ) = ( C (+) B ) ) $=
|
|
( wceq cdif ccompl cnin difeq2 compleqd difeq1 nineq12d cun df-symdif df-un
|
|
csymdif eqtri 3eqtr4g ) ABDZCAEZFZACEZFZGZCBEZFZBCEZFZGZCAOZCBOZRTUEUBUGRSU
|
|
DABCHIRUAUFABCJIKUISUALUCCAMSUANPUJUDUFLUHCBMUDUFNPQ $.
|
|
|
|
$( Equality law for intersection. (Contributed by SF, 11-Jan-2015.) $)
|
|
symdifeq12 $p |- ( ( A = B /\ C = D ) -> ( A (+) C ) = ( B (+) D ) ) $=
|
|
( wceq csymdif symdifeq1 symdifeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
symdifeqi.1 $e |- A = B $.
|
|
$( Equality inference for symmetric difference. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
symdifeq1i $p |- ( A (+) C ) = ( B (+) C ) $=
|
|
( wceq csymdif symdifeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for symmetric difference. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
symdifeq2i $p |- ( C (+) A ) = ( C (+) B ) $=
|
|
( wceq csymdif symdifeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
symdifeq12i.2 $e |- C = D $.
|
|
$( Equality inference for symmetric difference. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
symdifeq12i $p |- ( A (+) C ) = ( B (+) D ) $=
|
|
( wceq csymdif symdifeq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
symdifeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for symmetric difference. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
symdifeq1d $p |- ( ph -> ( A (+) C ) = ( B (+) C ) ) $=
|
|
( wceq csymdif symdifeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for symmetric difference. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
symdifeq2d $p |- ( ph -> ( C (+) A ) = ( C (+) B ) ) $=
|
|
( wceq csymdif symdifeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
symdifeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality inference for symmetric difference. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
symdifeq12d $p |- ( ph -> ( A (+) C ) = ( B (+) D ) ) $=
|
|
( wceq csymdif symdifeq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Subclasses and subsets
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c C_ $. $( Subclass or subset symbol $)
|
|
$c C. $. $( Proper subclass or subset symbol $)
|
|
|
|
$( Extend wff notation to include the subclass relation. This is
|
|
read " ` A ` is a subclass of ` B ` " or " ` B ` includes ` A ` ." When
|
|
` A ` exists as a set, it is also read " ` A ` is a subset of ` B ` ." $)
|
|
wss $a wff A C_ B $.
|
|
|
|
$( Extend wff notation with proper subclass relation. $)
|
|
wpss $a wff A C. B $.
|
|
|
|
$( Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18.
|
|
For example, ` { ` 1 , 2 ` } C_ { ` 1 , 2 , 3 ` } ` (ex-ss in set.mm).
|
|
Note that ` A C_ A ` (proved in ~ ssid ). Contrast this relationship with
|
|
the relationship ` A C. B ` (as will be defined in ~ df-pss ). For a more
|
|
traditional definition, but requiring a dummy variable, see ~ dfss2 .
|
|
Other possible definitions are given by ~ dfss3 , ~ dfss4 , ~ sspss ,
|
|
~ ssequn1 , ~ ssequn2 , ~ sseqin2 , and ~ ssdif0 . (Contributed by NM,
|
|
27-Apr-1994.) $)
|
|
df-ss $a |- ( A C_ B <-> ( A i^i B ) = A ) $.
|
|
|
|
$( Variant of subclass definition ~ df-ss . (Contributed by NM,
|
|
3-Sep-2004.) $)
|
|
dfss $p |- ( A C_ B <-> A = ( A i^i B ) ) $=
|
|
( wss cin wceq df-ss eqcom bitri ) ABCABDZAEAIEABFIAGH $.
|
|
|
|
$( Define proper subclass relationship between two classes. Definition 5.9
|
|
of [TakeutiZaring] p. 17. For example, ` { ` 1 , 2 ` } C. { ` 1 , 2 , 3
|
|
` } ` (ex-pss in set.mm). Note that ` -. A C. A ` (proved in ~ pssirr ).
|
|
Contrast this relationship with the relationship ` A C_ B ` (as defined in
|
|
~ df-ss ). Other possible definitions are given by ~ dfpss2 and
|
|
~ dfpss3 . (Contributed by NM, 7-Feb-1996.) $)
|
|
df-pss $a |- ( A C. B <-> ( A C_ B /\ A =/= B ) ) $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Alternate definition of the subclass relationship between two classes.
|
|
Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM,
|
|
8-Jan-2002.) $)
|
|
dfss2 $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
|
|
( wss cv wcel wa wb wal wi cin wceq dfss dfcleq bibi2i albii bitri pm4.71
|
|
elin bitr4i ) BCDZAEZBFZUCUBCFZGZHZAIZUCUDJZAIUABBCKZLZUGBCMUJUCUBUIFZHZA
|
|
IUGABUINULUFAUKUEUCUBBCSOPQQUHUFAUCUDRPT $.
|
|
|
|
$( Alternate definition of subclass relationship. (Contributed by NM,
|
|
14-Oct-1999.) $)
|
|
dfss3 $p |- ( A C_ B <-> A. x e. A x e. B ) $=
|
|
( wss cv wcel wi wal wral dfss2 df-ral bitr4i ) BCDAEZBFMCFZGAHNABIABCJNA
|
|
BKL $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d x z $.
|
|
dfss2f.1 $e |- F/_ x A $.
|
|
dfss2f.2 $e |- F/_ x B $.
|
|
$( Equivalence for subclass relation, using bound-variable hypotheses
|
|
instead of distinct variable conditions. (Contributed by NM,
|
|
3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.) $)
|
|
dfss2f $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
|
|
( vz wss cv wcel wal dfss2 nfcri nfim nfv wceq eleq1 imbi12d cbval bitri
|
|
wi ) BCGFHZBIZUACIZTZFJAHZBIZUECIZTZAJFBCKUDUHFAUBUCAAFBDLAFCELMUHFNUAUEO
|
|
UBUFUCUGUAUEBPUAUECPQRS $.
|
|
|
|
$( Equivalence for subclass relation, using bound-variable hypotheses
|
|
instead of distinct variable conditions. (Contributed by NM,
|
|
20-Mar-2004.) $)
|
|
dfss3f $p |- ( A C_ B <-> A. x e. A x e. B ) $=
|
|
( wss cv wcel wi wal wral dfss2f df-ral bitr4i ) BCFAGZBHOCHZIAJPABKABCDE
|
|
LPABMN $.
|
|
|
|
$( If ` x ` is not free in ` A ` and ` B ` , it is not free in
|
|
` A C_ B ` . (Contributed by NM, 27-Dec-1996.) $)
|
|
nfss $p |- F/ x A C_ B $=
|
|
( wss cv wcel wral dfss3f nfra1 nfxfr ) BCFAGCHZABIAABCDEJMABKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Membership relationships follow from a subclass relationship.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
ssel $p |- ( A C_ B -> ( C e. A -> C e. B ) ) $=
|
|
( vx wss cv wceq wa wex wi wal dfss2 biimpi 19.21bi anim2d eximdv df-clel
|
|
wcel 3imtr4g ) ABEZDFZCGZUAARZHZDIUBUABRZHZDICARCBRTUDUFDTUCUEUBTUCUEJZDT
|
|
UGDKDABLMNOPDCAQDCBQS $.
|
|
$}
|
|
|
|
$( Membership relationships follow from a subclass relationship.
|
|
(Contributed by NM, 7-Jun-2004.) $)
|
|
ssel2 $p |- ( ( A C_ B /\ C e. A ) -> C e. B ) $=
|
|
( wss wcel ssel imp ) ABDCAECBEABCFG $.
|
|
|
|
${
|
|
sseli.1 $e |- A C_ B $.
|
|
$( Membership inference from subclass relationship. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
sseli $p |- ( C e. A -> C e. B ) $=
|
|
( wss wcel wi ssel ax-mp ) ABECAFCBFGDABCHI $.
|
|
|
|
${
|
|
sselii.2 $e |- C e. A $.
|
|
$( Membership inference from subclass relationship. (Contributed by NM,
|
|
31-May-1999.) $)
|
|
sselii $p |- C e. B $=
|
|
( wcel sseli ax-mp ) CAFCBFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
sseldi.2 $e |- ( ph -> C e. A ) $.
|
|
$( Membership inference from subclass relationship. (Contributed by NM,
|
|
25-Jun-2014.) $)
|
|
sseldi $p |- ( ph -> C e. B ) $=
|
|
( wcel sseli syl ) ADBGDCGFBCDEHI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
sseld.1 $e |- ( ph -> A C_ B ) $.
|
|
$( Membership deduction from subclass relationship. (Contributed by NM,
|
|
15-Nov-1995.) $)
|
|
sseld $p |- ( ph -> ( C e. A -> C e. B ) ) $=
|
|
( wss wcel wi ssel syl ) ABCFDBGDCGHEBCDIJ $.
|
|
|
|
$( Membership deduction from subclass relationship. (Contributed by NM,
|
|
26-Jun-2014.) $)
|
|
sselda $p |- ( ( ph /\ C e. A ) -> C e. B ) $=
|
|
( wcel sseld imp ) ADBFDCFABCDEGH $.
|
|
|
|
${
|
|
sseldd.2 $e |- ( ph -> C e. A ) $.
|
|
$( Membership inference from subclass relationship. (Contributed by NM,
|
|
14-Dec-2004.) $)
|
|
sseldd $p |- ( ph -> C e. B ) $=
|
|
( wcel sseld mpd ) ADBGDCGFABCDEHI $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ssneld.1 $e |- ( ph -> A C_ B ) $.
|
|
$( If a class is not in another class, it is also not in a subclass of that
|
|
class. Deduction form. (Contributed by David Moews, 1-May-2017.) $)
|
|
ssneld $p |- ( ph -> ( -. C e. B -> -. C e. A ) ) $=
|
|
( wcel sseld con3d ) ADBFDCFABCDEGH $.
|
|
|
|
ssneldd.2 $e |- ( ph -> -. C e. B ) $.
|
|
$( If an element is not in a class, it is also not in a subclass of that
|
|
class. Deduction form. (Contributed by David Moews, 1-May-2017.) $)
|
|
ssneldd $p |- ( ph -> -. C e. A ) $=
|
|
( wcel wn ssneld mpd ) ADCGHDBGHFABCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
ssriv.1 $e |- ( x e. A -> x e. B ) $.
|
|
$( Inference rule based on subclass definition. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ssriv $p |- A C_ B $=
|
|
( wss cv wcel wi dfss2 mpgbir ) BCEAFZBGKCGHAABCIDJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
ssrdv.1 $e |- ( ph -> ( x e. A -> x e. B ) ) $.
|
|
$( Deduction rule based on subclass definition. (Contributed by NM,
|
|
15-Nov-1995.) $)
|
|
ssrdv $p |- ( ph -> A C_ B ) $=
|
|
( cv wcel wi wal wss alrimiv dfss2 sylibr ) ABFZCGNDGHZBICDJAOBEKBCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
|
|
14-Jun-2011.) $)
|
|
sstr2 $p |- ( A C_ B -> ( B C_ C -> A C_ C ) ) $=
|
|
( vx wss cv wcel wi wal ssel imim1d alimdv dfss2 3imtr4g ) ABEZDFZBGZPCGZ
|
|
HZDIPAGZRHZDIBCEACEOSUADOTQRABPJKLDBCMDACMN $.
|
|
$}
|
|
|
|
$( Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by
|
|
NM, 5-Sep-2003.) $)
|
|
sstr $p |- ( ( A C_ B /\ B C_ C ) -> A C_ C ) $=
|
|
( wss sstr2 imp ) ABDBCDACDABCEF $.
|
|
|
|
${
|
|
sstri.1 $e |- A C_ B $.
|
|
sstri.2 $e |- B C_ C $.
|
|
$( Subclass transitivity inference. (Contributed by NM, 5-May-2000.) $)
|
|
sstri $p |- A C_ C $=
|
|
( wss sstr2 mp2 ) ABFBCFACFDEABCGH $.
|
|
$}
|
|
|
|
${
|
|
sstrd.1 $e |- ( ph -> A C_ B ) $.
|
|
sstrd.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.) $)
|
|
sstrd $p |- ( ph -> A C_ C ) $=
|
|
( wss sstr syl2anc ) ABCGCDGBDGEFBCDHI $.
|
|
$}
|
|
|
|
${
|
|
syl5ss.1 $e |- A C_ B $.
|
|
syl5ss.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.) $)
|
|
syl5ss $p |- ( ph -> A C_ C ) $=
|
|
( wss a1i sstrd ) ABCDBCGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl6ss.1 $e |- ( ph -> A C_ B ) $.
|
|
syl6ss.2 $e |- B C_ C $.
|
|
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
syl6ss $p |- ( ph -> A C_ C ) $=
|
|
( wss a1i sstrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
sylan9ss.1 $e |- ( ph -> A C_ B ) $.
|
|
sylan9ss.2 $e |- ( ps -> B C_ C ) $.
|
|
$( A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
|
|
(Proof shortened by Andrew Salmon, 14-Jun-2011.) $)
|
|
sylan9ss $p |- ( ( ph /\ ps ) -> A C_ C ) $=
|
|
( wss sstr syl2an ) ACDHDEHCEHBFGCDEIJ $.
|
|
$}
|
|
|
|
${
|
|
sylan9ssr.1 $e |- ( ph -> A C_ B ) $.
|
|
sylan9ssr.2 $e |- ( ps -> B C_ C ) $.
|
|
$( A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) $)
|
|
sylan9ssr $p |- ( ( ps /\ ph ) -> A C_ C ) $=
|
|
( wss sylan9ss ancoms ) ABCEHABCDEFGIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The subclass relationship is antisymmetric. Compare Theorem 4 of
|
|
[Suppes] p. 22. (Contributed by NM, 5-Aug-1993.) $)
|
|
eqss $p |- ( A = B <-> ( A C_ B /\ B C_ A ) ) $=
|
|
( vx cv wcel wb wal wi wa wceq wss albiim dfcleq dfss2 anbi12i 3bitr4i )
|
|
CDZAEZQBEZFCGRSHCGZSRHCGZIABJABKZBAKZIRSCLCABMUBTUCUACABNCBANOP $.
|
|
$}
|
|
|
|
${
|
|
eqssi.1 $e |- A C_ B $.
|
|
eqssi.2 $e |- B C_ A $.
|
|
$( Infer equality from two subclass relationships. Compare Theorem 4 of
|
|
[Suppes] p. 22. (Contributed by NM, 9-Sep-1993.) $)
|
|
eqssi $p |- A = B $=
|
|
( wceq wss eqss mpbir2an ) ABEABFBAFCDABGH $.
|
|
$}
|
|
|
|
${
|
|
eqssd.1 $e |- ( ph -> A C_ B ) $.
|
|
eqssd.2 $e |- ( ph -> B C_ A ) $.
|
|
$( Equality deduction from two subclass relationships. Compare Theorem 4
|
|
of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.) $)
|
|
eqssd $p |- ( ph -> A = B ) $=
|
|
( wss wceq eqss sylanbrc ) ABCFCBFBCGDEBCHI $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Any class is a subclass of itself. Exercise 10 of [TakeutiZaring]
|
|
p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 14-Jun-2011.) $)
|
|
ssid $p |- A C_ A $=
|
|
( vx cv wcel id ssriv ) BAABCADEF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Any class is a subclass of the universal class. (Contributed by NM,
|
|
31-Oct-1995.) $)
|
|
ssv $p |- A C_ _V $=
|
|
( vx cvv cv elex ssriv ) BACBDAEF $.
|
|
$}
|
|
|
|
$( Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof
|
|
shortened by Andrew Salmon, 21-Jun-2011.) $)
|
|
sseq1 $p |- ( A = B -> ( A C_ C <-> B C_ C ) ) $=
|
|
( wceq wss wa wb eqss wi sstr2 adantl adantr impbid sylbi ) ABDABEZBAEZFZAC
|
|
EZBCEZGABHQRSPRSIOBACJKOSRIPABCJLMN $.
|
|
|
|
$( Equality theorem for the subclass relationship. (Contributed by NM,
|
|
25-Jun-1998.) $)
|
|
sseq2 $p |- ( A = B -> ( C C_ A <-> C C_ B ) ) $=
|
|
( wss wa wi wceq wb sstr2 com12 anim12i eqss dfbi2 3imtr4i ) ABDZBADZECADZC
|
|
BDZFZRQFZEABGQRHOSPTQORCABIJRPQCBAIJKABLQRMN $.
|
|
|
|
$( Equality theorem for the subclass relationship. (Contributed by NM,
|
|
31-May-1999.) $)
|
|
sseq12 $p |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) ) $=
|
|
( wceq wss sseq1 sseq2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.
|
|
|
|
${
|
|
sseq1i.1 $e |- A = B $.
|
|
$( An equality inference for the subclass relationship. (Contributed by
|
|
NM, 18-Aug-1993.) $)
|
|
sseq1i $p |- ( A C_ C <-> B C_ C ) $=
|
|
( wceq wss wb sseq1 ax-mp ) ABEACFBCFGDABCHI $.
|
|
|
|
$( An equality inference for the subclass relationship. (Contributed by
|
|
NM, 30-Aug-1993.) $)
|
|
sseq2i $p |- ( C C_ A <-> C C_ B ) $=
|
|
( wceq wss wb sseq2 ax-mp ) ABECAFCBFGDABCHI $.
|
|
|
|
${
|
|
sseq12i.2 $e |- C = D $.
|
|
$( An equality inference for the subclass relationship. (Contributed by
|
|
NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
|
|
sseq12i $p |- ( A C_ C <-> B C_ D ) $=
|
|
( wceq wss wb sseq12 mp2an ) ABGCDGACHBDHIEFABCDJK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
sseq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( An equality deduction for the subclass relationship. (Contributed by
|
|
NM, 14-Aug-1994.) $)
|
|
sseq1d $p |- ( ph -> ( A C_ C <-> B C_ C ) ) $=
|
|
( wceq wss wb sseq1 syl ) ABCFBDGCDGHEBCDIJ $.
|
|
|
|
$( An equality deduction for the subclass relationship. (Contributed by
|
|
NM, 14-Aug-1994.) $)
|
|
sseq2d $p |- ( ph -> ( C C_ A <-> C C_ B ) ) $=
|
|
( wceq wss wb sseq2 syl ) ABCFDBGDCGHEBCDIJ $.
|
|
|
|
${
|
|
sseq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( An equality deduction for the subclass relationship. (Contributed by
|
|
NM, 31-May-1999.) $)
|
|
sseq12d $p |- ( ph -> ( A C_ C <-> B C_ D ) ) $=
|
|
( wss sseq1d sseq2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
eqsstr.1 $e |- A = B $.
|
|
eqsstr.2 $e |- B C_ C $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 16-Jul-1995.) $)
|
|
eqsstri $p |- A C_ C $=
|
|
( wss sseq1i mpbir ) ACFBCFEABCDGH $.
|
|
$}
|
|
|
|
${
|
|
eqsstr3.1 $e |- B = A $.
|
|
eqsstr3.2 $e |- B C_ C $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 19-Oct-1999.) $)
|
|
eqsstr3i $p |- A C_ C $=
|
|
( eqcomi eqsstri ) ABCBADFEG $.
|
|
$}
|
|
|
|
${
|
|
sseqtr.1 $e |- A C_ B $.
|
|
sseqtr.2 $e |- B = C $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 28-Jul-1995.) $)
|
|
sseqtri $p |- A C_ C $=
|
|
( wss sseq2i mpbi ) ABFACFDBCAEGH $.
|
|
$}
|
|
|
|
${
|
|
sseqtr4.1 $e |- A C_ B $.
|
|
sseqtr4.2 $e |- C = B $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 4-Apr-1995.) $)
|
|
sseqtr4i $p |- A C_ C $=
|
|
( eqcomi sseqtri ) ABCDCBEFG $.
|
|
$}
|
|
|
|
${
|
|
eqsstrd.1 $e |- ( ph -> A = B ) $.
|
|
eqsstrd.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 25-Apr-2004.) $)
|
|
eqsstrd $p |- ( ph -> A C_ C ) $=
|
|
( wss sseq1d mpbird ) ABDGCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqsstr3d.1 $e |- ( ph -> B = A ) $.
|
|
eqsstr3d.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 25-Apr-2004.) $)
|
|
eqsstr3d $p |- ( ph -> A C_ C ) $=
|
|
( eqcomd eqsstrd ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
${
|
|
sseqtrd.1 $e |- ( ph -> A C_ B ) $.
|
|
sseqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 25-Apr-2004.) $)
|
|
sseqtrd $p |- ( ph -> A C_ C ) $=
|
|
( wss sseq2d mpbid ) ABCGBDGEACDBFHI $.
|
|
$}
|
|
|
|
${
|
|
sseqtr4d.1 $e |- ( ph -> A C_ B ) $.
|
|
sseqtr4d.2 $e |- ( ph -> C = B ) $.
|
|
$( Substitution of equality into a subclass relationship. (Contributed by
|
|
NM, 25-Apr-2004.) $)
|
|
sseqtr4d $p |- ( ph -> A C_ C ) $=
|
|
( eqcomd sseqtrd ) ABCDEADCFGH $.
|
|
$}
|
|
|
|
${
|
|
3sstr3.1 $e |- A C_ B $.
|
|
3sstr3.2 $e |- A = C $.
|
|
3sstr3.3 $e |- B = D $.
|
|
$( Substitution of equality in both sides of a subclass relationship.
|
|
(Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt,
|
|
26-Jan-2007.) $)
|
|
3sstr3i $p |- C C_ D $=
|
|
( wss sseq12i mpbi ) ABHCDHEACBDFGIJ $.
|
|
$}
|
|
|
|
${
|
|
3sstr4.1 $e |- A C_ B $.
|
|
3sstr4.2 $e |- C = A $.
|
|
3sstr4.3 $e |- D = B $.
|
|
$( Substitution of equality in both sides of a subclass relationship.
|
|
(Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt,
|
|
26-Jan-2007.) $)
|
|
3sstr4i $p |- C C_ D $=
|
|
( wss sseq12i mpbir ) CDHABHECADBFGIJ $.
|
|
$}
|
|
|
|
${
|
|
3sstr3g.1 $e |- ( ph -> A C_ B ) $.
|
|
3sstr3g.2 $e |- A = C $.
|
|
3sstr3g.3 $e |- B = D $.
|
|
$( Substitution of equality into both sides of a subclass relationship.
|
|
(Contributed by NM, 1-Oct-2000.) $)
|
|
3sstr3g $p |- ( ph -> C C_ D ) $=
|
|
( wss sseq12i sylib ) ABCIDEIFBDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3sstr4g.1 $e |- ( ph -> A C_ B ) $.
|
|
3sstr4g.2 $e |- C = A $.
|
|
3sstr4g.3 $e |- D = B $.
|
|
$( Substitution of equality into both sides of a subclass relationship.
|
|
(Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt,
|
|
26-Jan-2007.) $)
|
|
3sstr4g $p |- ( ph -> C C_ D ) $=
|
|
( wss sseq12i sylibr ) ABCIDEIFDBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
3sstr3d.1 $e |- ( ph -> A C_ B ) $.
|
|
3sstr3d.2 $e |- ( ph -> A = C ) $.
|
|
3sstr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( Substitution of equality into both sides of a subclass relationship.
|
|
(Contributed by NM, 1-Oct-2000.) $)
|
|
3sstr3d $p |- ( ph -> C C_ D ) $=
|
|
( wss sseq12d mpbid ) ABCIDEIFABDCEGHJK $.
|
|
$}
|
|
|
|
${
|
|
3sstr4d.1 $e |- ( ph -> A C_ B ) $.
|
|
3sstr4d.2 $e |- ( ph -> C = A ) $.
|
|
3sstr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( Substitution of equality into both sides of a subclass relationship.
|
|
(Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt,
|
|
26-Jan-2007.) $)
|
|
3sstr4d $p |- ( ph -> C C_ D ) $=
|
|
( wss sseq12d mpbird ) ADEIBCIFADBECGHJK $.
|
|
$}
|
|
|
|
${
|
|
syl5eqss.1 $e |- A = B $.
|
|
syl5eqss.2 $e |- ( ph -> B C_ C ) $.
|
|
$( B chained subclass and equality deduction. (Contributed by NM,
|
|
25-Apr-2004.) $)
|
|
syl5eqss $p |- ( ph -> A C_ C ) $=
|
|
( wss sseq1i sylibr ) ACDGBDGFBCDEHI $.
|
|
$}
|
|
|
|
${
|
|
syl5eqssr.1 $e |- B = A $.
|
|
syl5eqssr.2 $e |- ( ph -> B C_ C ) $.
|
|
$( B chained subclass and equality deduction. (Contributed by NM,
|
|
25-Apr-2004.) $)
|
|
syl5eqssr $p |- ( ph -> A C_ C ) $=
|
|
( eqcomi syl5eqss ) ABCDCBEGFH $.
|
|
$}
|
|
|
|
${
|
|
syl6sseq.1 $e |- ( ph -> A C_ B ) $.
|
|
syl6sseq.2 $e |- B = C $.
|
|
$( A chained subclass and equality deduction. (Contributed by NM,
|
|
25-Apr-2004.) $)
|
|
syl6sseq $p |- ( ph -> A C_ C ) $=
|
|
( wss sseq2i sylib ) ABCGBDGECDBFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6ssr.1 $e |- ( ph -> A C_ B ) $.
|
|
syl6ssr.2 $e |- C = B $.
|
|
$( A chained subclass and equality deduction. (Contributed by NM,
|
|
25-Apr-2004.) $)
|
|
syl6sseqr $p |- ( ph -> A C_ C ) $=
|
|
( eqcomi syl6sseq ) ABCDEDCFGH $.
|
|
$}
|
|
|
|
${
|
|
syl5sseq.1 $e |- B C_ A $.
|
|
syl5sseq.2 $e |- ( ph -> A = C ) $.
|
|
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
syl5sseq $p |- ( ph -> B C_ C ) $=
|
|
( wceq wss sseq2 biimpa sylancl ) ABDGZCBHZCDHZFELMNBDCIJK $.
|
|
$}
|
|
|
|
${
|
|
syl5sseqr.1 $e |- B C_ A $.
|
|
syl5sseqr.2 $e |- ( ph -> C = A ) $.
|
|
$( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
|
|
3-Jun-2011.) $)
|
|
syl5sseqr $p |- ( ph -> B C_ C ) $=
|
|
( wss a1i sseqtr4d ) ACBDCBGAEHFI $.
|
|
$}
|
|
|
|
${
|
|
syl6eqss.1 $e |- ( ph -> A = B ) $.
|
|
syl6eqss.2 $e |- B C_ C $.
|
|
$( A chained subclass and equality deduction. (Contributed by Mario
|
|
Carneiro, 2-Jan-2017.) $)
|
|
syl6eqss $p |- ( ph -> A C_ C ) $=
|
|
( wss a1i eqsstrd ) ABCDECDGAFHI $.
|
|
$}
|
|
|
|
${
|
|
syl6eqssr.1 $e |- ( ph -> B = A ) $.
|
|
syl6eqssr.2 $e |- B C_ C $.
|
|
$( A chained subclass and equality deduction. (Contributed by Mario
|
|
Carneiro, 2-Jan-2017.) $)
|
|
syl6eqssr $p |- ( ph -> A C_ C ) $=
|
|
( eqcomd syl6eqss ) ABCDACBEGFH $.
|
|
$}
|
|
|
|
$( Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.)
|
|
(Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
|
|
eqimss $p |- ( A = B -> A C_ B ) $=
|
|
( wceq wss eqss simplbi ) ABCABDBADABEF $.
|
|
|
|
$( Equality implies the subclass relation. (Contributed by NM,
|
|
23-Nov-2003.) $)
|
|
eqimss2 $p |- ( B = A -> A C_ B ) $=
|
|
( wss eqimss eqcoms ) ABCABABDE $.
|
|
|
|
${
|
|
eqimssi.1 $e |- A = B $.
|
|
$( Infer subclass relationship from equality. (Contributed by NM,
|
|
6-Jan-2007.) $)
|
|
eqimssi $p |- A C_ B $=
|
|
( ssid sseqtri ) AABADCE $.
|
|
|
|
$( Infer subclass relationship from equality. (Contributed by NM,
|
|
7-Jan-2007.) $)
|
|
eqimss2i $p |- B C_ A $=
|
|
( ssid sseqtr4i ) BBABDCE $.
|
|
$}
|
|
|
|
$( Two classes are different if they don't include the same class.
|
|
(Contributed by NM, 23-Apr-2015.) $)
|
|
nssne1 $p |- ( ( A C_ B /\ -. A C_ C ) -> B =/= C ) $=
|
|
( wss wn wne wceq sseq2 biimpcd necon3bd imp ) ABDZACDZEBCFLMBCBCGLMBCAHIJK
|
|
$.
|
|
|
|
$( Two classes are different if they are not subclasses of the same class.
|
|
(Contributed by NM, 23-Apr-2015.) $)
|
|
nssne2 $p |- ( ( A C_ C /\ -. B C_ C ) -> A =/= B ) $=
|
|
( wss wn wne wceq sseq1 biimpcd necon3bd imp ) ACDZBCDZEABFLMABABGLMABCHIJK
|
|
$.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Negation of subclass relationship. Exercise 13 of [TakeutiZaring]
|
|
p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew
|
|
Salmon, 21-Jun-2011.) $)
|
|
nss $p |- ( -. A C_ B <-> E. x ( x e. A /\ -. x e. B ) ) $=
|
|
( cv wcel wn wa wex wss wi wal exanali dfss2 xchbinxr bicomi ) ADZBEZPCEZ
|
|
FGAHZBCIZFSQRJAKTQRALABCMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Quantification restricted to a subclass. (Contributed by NM,
|
|
11-Mar-2006.) $)
|
|
ssralv $p |- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) ) $=
|
|
( wss cv wcel ssel imim1d ralimdv2 ) CDEZAABDCKBFZCGLDGACDLHIJ $.
|
|
|
|
$( Existential quantification restricted to a subclass. (Contributed by
|
|
NM, 11-Jan-2007.) $)
|
|
ssrexv $p |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) ) $=
|
|
( wss cv wcel ssel anim1d reximdv2 ) CDEZAABCDKBFZCGLDGACDLHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Restricted universal quantification on a subset in terms of superset.
|
|
(Contributed by Stefan O'Rear, 3-Apr-2015.) $)
|
|
ralss $p |- ( A C_ B -> ( A. x e. A ph <->
|
|
A. x e. B ( x e. A -> ph ) ) ) $=
|
|
( wss cv wcel wi wa ssel pm4.71rd imbi1d impexp syl6bb ralbidv2 ) CDEZABF
|
|
ZCGZAHZBCDPSQDGZRIZAHTSHPRUAAPRTCDQJKLTRAMNO $.
|
|
|
|
$( Restricted existential quantification on a subset in terms of superset.
|
|
(Contributed by Stefan O'Rear, 3-Apr-2015.) $)
|
|
rexss $p |- ( A C_ B -> ( E. x e. A ph <->
|
|
E. x e. B ( x e. A /\ ph ) ) ) $=
|
|
( wss cv wcel wa ssel pm4.71rd anbi1d anass syl6bb rexbidv2 ) CDEZABFZCGZ
|
|
AHZBCDORPDGZQHZAHSRHOQTAOQSCDPIJKSQALMN $.
|
|
$}
|
|
|
|
$( Class abstractions in a subclass relationship. (Contributed by NM,
|
|
3-Jul-1994.) $)
|
|
ss2ab $p |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) ) $=
|
|
( cab wss cv wcel wi wal nfab1 dfss2f abid imbi12i albii bitri ) ACDZBCDZEC
|
|
FZPGZRQGZHZCIABHZCICPQACJBCJKUAUBCSATBACLBCLMNO $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Class abstraction in a subclass relationship. (Contributed by NM,
|
|
16-Aug-2006.) $)
|
|
abss $p |- ( { x | ph } C_ A <-> A. x ( ph -> x e. A ) ) $=
|
|
( cab wss cv wcel wi wal abid2 sseq2i ss2ab bitr3i ) ABDZCENBFCGZBDZEAOHB
|
|
IPCNBCJKAOBLM $.
|
|
|
|
$( Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.) $)
|
|
ssab $p |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) ) $=
|
|
( cab wss cv wcel wi wal abid2 sseq1i ss2ab bitr3i ) CABDZEBFCGZBDZNEOAHB
|
|
IPCNBCJKOABLM $.
|
|
|
|
$( The relation for a subclass of a class abstraction is equivalent to
|
|
restricted quantification. (Contributed by NM, 6-Sep-2006.) $)
|
|
ssabral $p |- ( A C_ { x | ph } <-> A. x e. A ph ) $=
|
|
( cab wss cv wcel wi wal wral ssab df-ral bitr4i ) CABDEBFCGAHBIABCJABCKA
|
|
BCLM $.
|
|
$}
|
|
|
|
${
|
|
ss2abi.1 $e |- ( ph -> ps ) $.
|
|
$( Inference of abstraction subclass from implication. (Contributed by NM,
|
|
31-Mar-1995.) $)
|
|
ss2abi $p |- { x | ph } C_ { x | ps } $=
|
|
( cab wss wi ss2ab mpgbir ) ACEBCEFABGCABCHDI $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ss2abdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Deduction of abstraction subclass from implication. (Contributed by NM,
|
|
29-Jul-2011.) $)
|
|
ss2abdv $p |- ( ph -> { x | ps } C_ { x | ch } ) $=
|
|
( wi wal cab wss alrimiv ss2ab sylibr ) ABCFZDGBDHCDHIAMDEJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $.
|
|
abssdv.1 $e |- ( ph -> ( ps -> x e. A ) ) $.
|
|
$( Deduction of abstraction subclass from implication. (Contributed by NM,
|
|
20-Jan-2006.) $)
|
|
abssdv $p |- ( ph -> { x | ps } C_ A ) $=
|
|
( cv wcel wi wal cab wss alrimiv abss sylibr ) ABCFDGHZCIBCJDKAOCELBCDMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
abssi.1 $e |- ( ph -> x e. A ) $.
|
|
$( Inference of abstraction subclass from implication. (Contributed by NM,
|
|
20-Jan-2006.) $)
|
|
abssi $p |- { x | ph } C_ A $=
|
|
( cab cv wcel ss2abi abid2 sseqtri ) ABEBFCGZBECAKBDHBCIJ $.
|
|
$}
|
|
|
|
$( Restricted abstraction classes in a subclass relationship. (Contributed
|
|
by NM, 30-May-1999.) $)
|
|
ss2rab $p |- ( { x e. A | ph } C_ { x e. A | ps } <->
|
|
A. x e. A ( ph -> ps ) ) $=
|
|
( crab wss cv wcel wa cab wi wal df-rab sseq12i ss2ab df-ral imdistan albii
|
|
wral bitr2i 3bitri ) ACDEZBCDEZFCGDHZAIZCJZUDBIZCJZFUEUGKZCLZABKZCDSZUBUFUC
|
|
UHACDMBCDMNUEUGCOULUDUKKZCLUJUKCDPUMUICUDABQRTUA $.
|
|
|
|
${
|
|
$d x B $.
|
|
$( Restricted class abstraction in a subclass relationship. (Contributed
|
|
by NM, 16-Aug-2006.) $)
|
|
rabss $p |- ( { x e. A | ph } C_ B <-> A. x e. A ( ph -> x e. B ) ) $=
|
|
( crab wss cv wcel wa cab wal wral df-rab sseq1i abss impexp albii df-ral
|
|
wi bitr4i 3bitri ) ABCEZDFBGZCHZAIZBJZDFUEUCDHZSZBKZAUGSZBCLZUBUFDABCMNUE
|
|
BDOUIUDUJSZBKUKUHULBUDAUGPQUJBCRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass of a restricted class abstraction. (Contributed by NM,
|
|
16-Aug-2006.) $)
|
|
ssrab $p |- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) ) $=
|
|
( crab wss cv wcel wa cab wal wral df-rab sseq2i ssab dfss3 anbi1i r19.26
|
|
wi df-ral 3bitr2ri 3bitri ) DABCEZFDBGZCHZAIZBJZFUDDHUFSBKZDCFZABDLZIZUCU
|
|
GDABCMNUFBDOUKUEBDLZUJIUFBDLUHUIULUJBDCPQUEABDRUFBDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ph $.
|
|
ssrabdv.1 $e |- ( ph -> B C_ A ) $.
|
|
ssrabdv.2 $e |- ( ( ph /\ x e. B ) -> ps ) $.
|
|
$( Subclass of a restricted class abstraction (deduction rule).
|
|
(Contributed by NM, 31-Aug-2006.) $)
|
|
ssrabdv $p |- ( ph -> B C_ { x e. A | ps } ) $=
|
|
( wss wral crab ralrimiva ssrab sylanbrc ) AEDHBCEIEBCDJHFABCEGKBCDELM $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x ph $.
|
|
rabssdv.1 $e |- ( ( ph /\ x e. A /\ ps ) -> x e. B ) $.
|
|
$( Subclass of a restricted class abstraction (deduction rule).
|
|
(Contributed by NM, 2-Feb-2015.) $)
|
|
rabssdv $p |- ( ph -> { x e. A | ps } C_ B ) $=
|
|
( cv wcel wi wral crab wss 3exp ralrimiv rabss sylibr ) ABCGZEHZIZCDJBCDK
|
|
ELASCDAQDHBRFMNBCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
ss2rabdv.1 $e |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) $.
|
|
$( Deduction of restricted abstraction subclass from implication.
|
|
(Contributed by NM, 30-May-2006.) $)
|
|
ss2rabdv $p |- ( ph -> { x e. A | ps } C_ { x e. A | ch } ) $=
|
|
( wi wral crab wss ralrimiva ss2rab sylibr ) ABCGZDEHBDEICDEIJANDEFKBCDEL
|
|
M $.
|
|
$}
|
|
|
|
${
|
|
ss2rabi.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
|
|
$( Inference of restricted abstraction subclass from implication.
|
|
(Contributed by NM, 14-Oct-1999.) $)
|
|
ss2rabi $p |- { x e. A | ph } C_ { x e. A | ps } $=
|
|
( crab wss wi ss2rab mprgbir ) ACDFBCDFGABHCDABCDIEJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass law for restricted abstraction. (Contributed by NM,
|
|
18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
rabss2 $p |- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } ) $=
|
|
( wss cv wcel wa cab crab wi wal pm3.45 alimi dfss2 ss2ab 3imtr4i 3sstr4g
|
|
df-rab ) CDEZBFZCGZAHZBIZUADGZAHZBIZABCJABDJUBUEKZBLUCUFKZBLTUDUGEUHUIBUB
|
|
UEAMNBCDOUCUFBPQABCSABDSR $.
|
|
|
|
$( Subclass relation for the restriction of a class abstraction.
|
|
(Contributed by NM, 31-Mar-1995.) $)
|
|
ssab2 $p |- { x | ( x e. A /\ ph ) } C_ A $=
|
|
( cv wcel wa simpl abssi ) BDCEZAFBCIAGH $.
|
|
|
|
$( Subclass relation for a restricted class. (Contributed by NM,
|
|
19-Mar-1997.) $)
|
|
ssrab2 $p |- { x e. A | ph } C_ A $=
|
|
( crab cv wcel wa cab df-rab ssab2 eqsstri ) ABCDBECFAGBHCABCIABCJK $.
|
|
$}
|
|
|
|
$( A restricted class is a subclass of the corresponding unrestricted class.
|
|
(Contributed by Mario Carneiro, 23-Dec-2016.) $)
|
|
rabssab $p |- { x e. A | ph } C_ { x | ph } $=
|
|
( crab cv wcel wa cab df-rab simpr ss2abi eqsstri ) ABCDBECFZAGZBHABHABCINA
|
|
BMAJKL $.
|
|
|
|
${
|
|
$d x y $. $d y z A $. $d y z B $. $d x z C $.
|
|
$( A subset relationship useful for converting union to indexed union using
|
|
~ dfiun2 or ~ dfiun2g and intersection to indexed intersection using
|
|
~ dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario
|
|
Carneiro, 26-Sep-2015.) $)
|
|
uniiunlem $p |- ( A. x e. A B e. D ->
|
|
( A. x e. A B e. C <-> { y | E. x e. A y = B } C_ C ) ) $=
|
|
( vz cv wceq wrex cab wss wcel wi wal wral eqeq1 rexbidv cbvabv wb sseq1i
|
|
r19.23v albii ralcom4 abss 3bitr4i bitr4i nfv eleq1 ceqsalg ralbi syl5rbb
|
|
ralimi syl ) BHZDIZACJZBKZELZGHZDIZUTEMZNZGOZACPZDFMZACPZDEMZACPZUSVAACJZ
|
|
GKZELZVEURVKEUQVJBGUOUTIUPVAACUOUTDQRSUAVCACPZGOVJVBNZGOVEVLVMVNGVAVBACUB
|
|
UCVCAGCUDVJGEUEUFUGVGVDVHTZACPVEVITVFVOACVBVHGDFVHGUHUTDEUIUJUMVDVHACUKUN
|
|
UL $.
|
|
$}
|
|
|
|
$( Alternate definition of proper subclass. (Contributed by NM,
|
|
7-Feb-1996.) $)
|
|
dfpss2 $p |- ( A C. B <-> ( A C_ B /\ -. A = B ) ) $=
|
|
( wpss wss wne wa wceq wn df-pss df-ne anbi2i bitri ) ABCABDZABEZFMABGHZFAB
|
|
INOMABJKL $.
|
|
|
|
$( Alternate definition of proper subclass. (Contributed by NM,
|
|
7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
dfpss3 $p |- ( A C. B <-> ( A C_ B /\ -. B C_ A ) ) $=
|
|
( wpss wss wceq wn wa dfpss2 eqss baib notbid pm5.32i bitri ) ABCABDZABEZFZ
|
|
GNBADZFZGABHNPRNOQONQABIJKLM $.
|
|
|
|
$( Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.) $)
|
|
psseq1 $p |- ( A = B -> ( A C. C <-> B C. C ) ) $=
|
|
( wceq wss wne wa wpss sseq1 neeq1 anbi12d df-pss 3bitr4g ) ABDZACEZACFZGBC
|
|
EZBCFZGACHBCHNOQPRABCIABCJKACLBCLM $.
|
|
|
|
$( Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.) $)
|
|
psseq2 $p |- ( A = B -> ( C C. A <-> C C. B ) ) $=
|
|
( wceq wss wne wa wpss sseq2 neeq2 anbi12d df-pss 3bitr4g ) ABDZCAEZCAFZGCB
|
|
EZCBFZGCAHCBHNOQPRABCIABCJKCALCBLM $.
|
|
|
|
${
|
|
psseq1i.1 $e |- A = B $.
|
|
$( An equality inference for the proper subclass relationship.
|
|
(Contributed by NM, 9-Jun-2004.) $)
|
|
psseq1i $p |- ( A C. C <-> B C. C ) $=
|
|
( wceq wpss wb psseq1 ax-mp ) ABEACFBCFGDABCHI $.
|
|
|
|
$( An equality inference for the proper subclass relationship.
|
|
(Contributed by NM, 9-Jun-2004.) $)
|
|
psseq2i $p |- ( C C. A <-> C C. B ) $=
|
|
( wceq wpss wb psseq2 ax-mp ) ABECAFCBFGDABCHI $.
|
|
|
|
${
|
|
psseq12i.2 $e |- C = D $.
|
|
$( An equality inference for the proper subclass relationship.
|
|
(Contributed by NM, 9-Jun-2004.) $)
|
|
psseq12i $p |- ( A C. C <-> B C. D ) $=
|
|
( wpss psseq1i psseq2i bitri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
psseq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( An equality deduction for the proper subclass relationship.
|
|
(Contributed by NM, 9-Jun-2004.) $)
|
|
psseq1d $p |- ( ph -> ( A C. C <-> B C. C ) ) $=
|
|
( wceq wpss wb psseq1 syl ) ABCFBDGCDGHEBCDIJ $.
|
|
|
|
$( An equality deduction for the proper subclass relationship.
|
|
(Contributed by NM, 9-Jun-2004.) $)
|
|
psseq2d $p |- ( ph -> ( C C. A <-> C C. B ) ) $=
|
|
( wceq wpss wb psseq2 syl ) ABCFDBGDCGHEBCDIJ $.
|
|
|
|
${
|
|
psseq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( An equality deduction for the proper subclass relationship.
|
|
(Contributed by NM, 9-Jun-2004.) $)
|
|
psseq12d $p |- ( ph -> ( A C. C <-> B C. D ) ) $=
|
|
( wpss psseq1d psseq2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
$}
|
|
|
|
$( A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23.
|
|
(Contributed by NM, 7-Feb-1996.) $)
|
|
pssss $p |- ( A C. B -> A C_ B ) $=
|
|
( wpss wss wne df-pss simplbi ) ABCABDABEABFG $.
|
|
|
|
$( Two classes in a proper subclass relationship are not equal. (Contributed
|
|
by NM, 16-Feb-2015.) $)
|
|
pssne $p |- ( A C. B -> A =/= B ) $=
|
|
( wpss wss wne df-pss simprbi ) ABCABDABEABFG $.
|
|
|
|
${
|
|
pssssd.1 $e |- ( ph -> A C. B ) $.
|
|
$( Deduce subclass from proper subclass. (Contributed by NM,
|
|
29-Feb-1996.) $)
|
|
pssssd $p |- ( ph -> A C_ B ) $=
|
|
( wpss wss pssss syl ) ABCEBCFDBCGH $.
|
|
|
|
$( Proper subclasses are unequal. Deduction form of ~ pssne .
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
pssned $p |- ( ph -> A =/= B ) $=
|
|
( wpss wne pssne syl ) ABCEBCFDBCGH $.
|
|
$}
|
|
|
|
$( Subclass in terms of proper subclass. (Contributed by NM,
|
|
25-Feb-1996.) $)
|
|
sspss $p |- ( A C_ B <-> ( A C. B \/ A = B ) ) $=
|
|
( wss wpss wceq wo wn dfpss2 simplbi2 con1d orrd pssss eqimss jaoi impbii )
|
|
ABCZABDZABEZFPQRPRQQPRGABHIJKQPRABLABMNO $.
|
|
|
|
$( Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23.
|
|
(Contributed by NM, 7-Feb-1996.) $)
|
|
pssirr $p |- -. A C. A $=
|
|
( wpss wss wn wa pm3.24 dfpss3 mtbir ) AABAACZIDEIFAAGH $.
|
|
|
|
$( Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes]
|
|
p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
pssn2lp $p |- -. ( A C. B /\ B C. A ) $=
|
|
( wpss wn wi wa wss dfpss3 simprbi pssss nsyl imnan mpbi ) ABCZBACZDENOFDNB
|
|
AGZONABGPDABHIBAJKNOLM $.
|
|
|
|
$( Two ways of stating trichotomy with respect to inclusion. (Contributed by
|
|
NM, 12-Aug-2004.) $)
|
|
sspsstri $p |- ( ( A C_ B \/ B C_ A ) <-> ( A C. B \/ A = B \/ B C. A ) ) $=
|
|
( wpss wo wceq wss w3o or32 sspss eqcom orbi2i bitri orbi12i orordir bitr4i
|
|
df-3or 3bitr4i ) ABCZBACZDABEZDZRTDZSDABFZBAFZDZRTSGRSTHUEUBSTDZDUAUCUBUDUF
|
|
ABIUDSBAEZDUFBAIUGTSBAJKLMRSTNORTSPQ $.
|
|
|
|
$( Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
ssnpss $p |- ( A C_ B -> -. B C. A ) $=
|
|
( wpss wss wn dfpss3 simprbi con2i ) BACZABDZIBADJEBAFGH $.
|
|
|
|
$( Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23.
|
|
(Contributed by NM, 7-Feb-1996.) $)
|
|
psstr $p |- ( ( A C. B /\ B C. C ) -> A C. C ) $=
|
|
( wpss wa wss wceq pssss sylan9ss pssn2lp psseq1 anbi1d mtbiri con2i dfpss2
|
|
wn sylanbrc ) ABDZBCDZEZACFACGZPACDRSABCABHBCHIUATUATCBDZSECBJUARUBSACBKLMN
|
|
ACOQ $.
|
|
|
|
$( Transitive law for subclass and proper subclass. (Contributed by NM,
|
|
3-Apr-1996.) $)
|
|
sspsstr $p |- ( ( A C_ B /\ B C. C ) -> A C. C ) $=
|
|
( wss wpss wceq wo sspss wi psstr ex psseq1 biimprd jaoi imp sylanb ) ABDAB
|
|
EZABFZGZBCEZACEZABHSTUAQTUAIRQTUAABCJKRUATABCLMNOP $.
|
|
|
|
$( Transitive law for subclass and proper subclass. (Contributed by NM,
|
|
3-Apr-1996.) $)
|
|
psssstr $p |- ( ( A C. B /\ B C_ C ) -> A C. C ) $=
|
|
( wss wpss wceq wo sspss psstr ex psseq2 biimpcd jaod imp sylan2b ) BCDABEZ
|
|
BCEZBCFZGZACEZBCHPSTPQTRPQTABCIJRPTBCAKLMNO $.
|
|
|
|
${
|
|
psstrd.1 $e |- ( ph -> A C. B ) $.
|
|
psstrd.2 $e |- ( ph -> B C. C ) $.
|
|
$( Proper subclass inclusion is transitive. Deduction form of ~ psstr .
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
psstrd $p |- ( ph -> A C. C ) $=
|
|
( wpss psstr syl2anc ) ABCGCDGBDGEFBCDHI $.
|
|
$}
|
|
|
|
${
|
|
sspsstrd.1 $e |- ( ph -> A C_ B ) $.
|
|
sspsstrd.2 $e |- ( ph -> B C. C ) $.
|
|
$( Transitivity involving subclass and proper subclass inclusion.
|
|
Deduction form of ~ sspsstr . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
sspsstrd $p |- ( ph -> A C. C ) $=
|
|
( wss wpss sspsstr syl2anc ) ABCGCDHBDHEFBCDIJ $.
|
|
$}
|
|
|
|
${
|
|
psssstrd.1 $e |- ( ph -> A C. B ) $.
|
|
psssstrd.2 $e |- ( ph -> B C_ C ) $.
|
|
$( Transitivity involving subclass and proper subclass inclusion.
|
|
Deduction form of ~ psssstr . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
psssstrd $p |- ( ph -> A C. C ) $=
|
|
( wpss wss psssstr syl2anc ) ABCGCDHBDGEFBCDIJ $.
|
|
$}
|
|
|
|
$( A class is not a proper subclass of another iff it satisfies a
|
|
one-directional form of ~ eqss . (Contributed by Mario Carneiro,
|
|
15-May-2015.) $)
|
|
npss $p |- ( -. A C. B <-> ( A C_ B -> A = B ) ) $=
|
|
( wss wceq wi wpss wn wa pm4.61 dfpss2 bitr4i con1bii ) ABCZABDZEZABFZOGMNG
|
|
HPMNIABJKL $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The difference, union, and intersection of two classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Equality theorem for class difference. (Contributed by FL,
|
|
31-Aug-2009.) $)
|
|
difeq12 $p |- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) ) $=
|
|
( wceq cdif difeq1 difeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
difeq1i.1 $e |- A = B $.
|
|
$( Inference adding difference to the right in a class equality.
|
|
(Contributed by NM, 15-Nov-2002.) $)
|
|
difeq1i $p |- ( A \ C ) = ( B \ C ) $=
|
|
( wceq cdif difeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Inference adding difference to the left in a class equality.
|
|
(Contributed by NM, 15-Nov-2002.) $)
|
|
difeq2i $p |- ( C \ A ) = ( C \ B ) $=
|
|
( wceq cdif difeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
difeq12i.2 $e |- C = D $.
|
|
$( Equality inference for class difference. (Contributed by NM,
|
|
29-Aug-2004.) $)
|
|
difeq12i $p |- ( A \ C ) = ( B \ D ) $=
|
|
( cdif difeq1i difeq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
difeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction adding difference to the right in a class equality.
|
|
(Contributed by NM, 15-Nov-2002.) $)
|
|
difeq1d $p |- ( ph -> ( A \ C ) = ( B \ C ) ) $=
|
|
( wceq cdif difeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Deduction adding difference to the left in a class equality.
|
|
(Contributed by NM, 15-Nov-2002.) $)
|
|
difeq2d $p |- ( ph -> ( C \ A ) = ( C \ B ) ) $=
|
|
( wceq cdif difeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
difeq12d.1 $e |- ( ph -> A = B ) $.
|
|
difeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for class difference. (Contributed by FL,
|
|
29-May-2014.) $)
|
|
difeq12d $p |- ( ph -> ( A \ C ) = ( B \ D ) ) $=
|
|
( cdif difeq1d difeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
difeqri.1 $e |- ( ( x e. A /\ -. x e. B ) <-> x e. C ) $.
|
|
$( Inference from membership to difference. (Contributed by NM,
|
|
17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
difeqri $p |- ( A \ B ) = C $=
|
|
( cdif cv wcel wn wa eldif bitri eqriv ) ABCFZDAGZNHOBHOCHIJODHOBCKELM $.
|
|
$}
|
|
|
|
$( Implication of membership in a class difference. (Contributed by NM,
|
|
29-Apr-1994.) $)
|
|
eldifi $p |- ( A e. ( B \ C ) -> A e. B ) $=
|
|
( cdif wcel wn eldif simplbi ) ABCDEABEACEFABCGH $.
|
|
|
|
$( Implication of membership in a class difference. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
eldifn $p |- ( A e. ( B \ C ) -> -. A e. C ) $=
|
|
( cdif wcel wn eldif simprbi ) ABCDEABEACEFABCGH $.
|
|
|
|
$( A set does not belong to a class excluding it. (Contributed by NM,
|
|
27-Jun-1994.) $)
|
|
elndif $p |- ( A e. B -> -. A e. ( C \ B ) ) $=
|
|
( cdif wcel eldifn con2i ) ACBDEABEACBFG $.
|
|
|
|
$( Implication of membership in a class difference. (Contributed by NM,
|
|
28-Jun-1994.) $)
|
|
neldif $p |- ( ( A e. B /\ -. A e. ( B \ C ) ) -> A e. C ) $=
|
|
( wcel cdif wn eldif simplbi2 con1d imp ) ABDZABCEDZFACDZKMLLKMFABCGHIJ $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
|
|
(Contributed by NM, 17-May-1998.) $)
|
|
difdif $p |- ( A \ ( B \ A ) ) = A $=
|
|
( vx cdif cv wcel wi wa wn pm4.45im eldif xchbinxr anbi2i bitr2i difeqri
|
|
iman ) CABADZACEZAFZSRBFZSGZHSRQFZIZHSTJUAUCSUATSIHUBTSPRBAKLMNO $.
|
|
|
|
$( Subclass relationship for class difference. Exercise 14 of
|
|
[TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.) $)
|
|
difss $p |- ( A \ B ) C_ A $=
|
|
( vx cdif cv eldifi ssriv ) CABDACEABFG $.
|
|
$}
|
|
|
|
$( A difference of two classes is contained in the minuend. Deduction form
|
|
of ~ difss . (Contributed by David Moews, 1-May-2017.) $)
|
|
difssd $p |- ( ph -> ( A \ B ) C_ A ) $=
|
|
( cdif wss difss a1i ) BCDBEABCFG $.
|
|
|
|
$( If a class is contained in a difference, it is contained in the minuend.
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
difss2 $p |- ( A C_ ( B \ C ) -> A C_ B ) $=
|
|
( cdif wss id difss syl6ss ) ABCDZEZAIBJFBCGH $.
|
|
|
|
${
|
|
difss2d.1 $e |- ( ph -> A C_ ( B \ C ) ) $.
|
|
$( If a class is contained in a difference, it is contained in the
|
|
minuend. Deduction form of ~ difss2 . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
difss2d $p |- ( ph -> A C_ B ) $=
|
|
( cdif wss difss2 syl ) ABCDFGBCGEBCDHI $.
|
|
$}
|
|
|
|
$( Preservation of a subclass relationship by class difference. (Contributed
|
|
by NM, 15-Feb-2007.) $)
|
|
ssdifss $p |- ( A C_ B -> ( A \ C ) C_ B ) $=
|
|
( cdif wss difss sstr mpan ) ACDZAEABEIBEACFIABGH $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Double complement under universal class. Exercise 4.10(s) of
|
|
[Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) $)
|
|
ddif $p |- ( _V \ ( _V \ A ) ) = A $=
|
|
( vx cvv cdif cv wcel wn wa eldif mpbiran con2bii biantrur bitr2i difeqri
|
|
vex ) BCCADZABEZAFZQPFZGZQCFZTHSRSUARGBOZQCAIJKUATUBLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.) $)
|
|
ssconb $p |- ( ( A C_ C /\ B C_ C ) ->
|
|
( A C_ ( C \ B ) <-> B C_ ( C \ A ) ) ) $=
|
|
( vx wss wa cv wcel cdif wi wal wn wb ssel pm5.1 jcab 3bitr4g eldif dfss2
|
|
imbi2i syl2an con2b a1i anbi12d albidv ) ACEZBCEZFZDGZAHZUICBIZHZJZDKUIBH
|
|
ZUICAIZHZJZDKAUKEBUOEUHUMUQDUHUJUICHZUNLZFZJZUNURUJLZFZJZUMUQUHUJURJZUJUS
|
|
JZFUNURJZUNVBJZFVAVDUHVEVGVFVHUFVEVGVEVGMUGACUINBCUINVEVGOUAVFVHMUHUJUNUB
|
|
UCUDUJURUSPUNURVBPQULUTUJUICBRTUPVCUNUICARTQUEDAUKSDBUOSQ $.
|
|
|
|
$( Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
|
|
(Contributed by NM, 22-Mar-1998.) $)
|
|
sscon $p |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) ) $=
|
|
( vx wss cdif cv wcel wn wa ssel con3d anim2d eldif 3imtr4g ssrdv ) ABEZD
|
|
CBFZCAFZQDGZCHZTBHZIZJUATAHZIZJTRHTSHQUCUEUAQUDUBABTKLMTCBNTCANOP $.
|
|
|
|
$( Difference law for subsets. (Contributed by NM, 28-May-1998.) $)
|
|
ssdif $p |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) ) $=
|
|
( vx wss cdif cv wcel wn wa ssel anim1d eldif 3imtr4g ssrdv ) ABEZDACFZBC
|
|
FZPDGZAHZSCHIZJSBHZUAJSQHSRHPTUBUAABSKLSACMSBCMNO $.
|
|
$}
|
|
|
|
${
|
|
ssdifd.1 $e |- ( ph -> A C_ B ) $.
|
|
$( If ` A ` is contained in ` B ` , then ` ( A \ C ) ` is contained in
|
|
` ( B \ C ) ` . Deduction form of ~ ssdif . (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
ssdifd $p |- ( ph -> ( A \ C ) C_ ( B \ C ) ) $=
|
|
( wss cdif ssdif syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( If ` A ` is contained in ` B ` , then ` ( C \ B ) ` is contained in
|
|
` ( C \ A ) ` . Deduction form of ~ sscon . (Contributed by David
|
|
Moews, 1-May-2017.) $)
|
|
sscond $p |- ( ph -> ( C \ B ) C_ ( C \ A ) ) $=
|
|
( wss cdif sscon syl ) ABCFDCGDBGFEBCDHI $.
|
|
|
|
$( If ` A ` is contained in ` B ` , then ` ( A \ C ) ` is also contained in
|
|
` B ` . Deduction form of ~ ssdifss . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
ssdifssd $p |- ( ph -> ( A \ C ) C_ B ) $=
|
|
( wss cdif ssdifss syl ) ABCFBDGCFEBCDHI $.
|
|
|
|
ssdif2d.2 $e |- ( ph -> C C_ D ) $.
|
|
$( If ` A ` is contained in ` B ` and ` C ` is contained in ` D ` , then
|
|
` ( A \ D ) ` is contained in ` ( B \ C ) ` . Deduction form.
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
ssdif2d $p |- ( ph -> ( A \ D ) C_ ( B \ C ) ) $=
|
|
( cdif sscond ssdifd sstrd ) ABEHBDHCDHADEBGIABCDFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
uneqri.1 $e |- ( ( x e. A \/ x e. B ) <-> x e. C ) $.
|
|
$( Inference from membership to union. (Contributed by NM, 5-Aug-1993.) $)
|
|
uneqri $p |- ( A u. B ) = C $=
|
|
( cun cv wcel wo elun bitri eqriv ) ABCFZDAGZMHNBHNCHINDHNBCJEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
unidm $p |- ( A u. A ) = A $=
|
|
( vx cv wcel oridm uneqri ) BAAABCADEF $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Commutative law for union of classes. Exercise 6 of [TakeutiZaring]
|
|
p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
uncom $p |- ( A u. B ) = ( B u. A ) $=
|
|
( vx cun cv wcel wo orcom elun bitr4i uneqri ) CABBADZCEZAFZMBFZGONGMLFNO
|
|
HMBAIJK $.
|
|
$}
|
|
|
|
$( If a class equals the union of two other classes, then it equals the union
|
|
of those two classes commuted. ~ equncom was automatically derived from
|
|
equncomVD in set.mm using the tools program
|
|
translate_without_overwriting.cmd and minimizing. (Contributed by Alan
|
|
Sare, 18-Feb-2012.) $)
|
|
equncom $p |- ( A = ( B u. C ) <-> A = ( C u. B ) ) $=
|
|
( cun uncom eqeq2i ) BCDCBDABCEF $.
|
|
|
|
${
|
|
equncomi.1 $e |- A = ( B u. C ) $.
|
|
$( Inference form of ~ equncom . ~ equncomi was automatically derived from
|
|
equncomiVD in set.mm using the tools program
|
|
translate_without_overwriting.cmd and minimizing. (Contributed by Alan
|
|
Sare, 18-Feb-2012.) $)
|
|
equncomi $p |- A = ( C u. B ) $=
|
|
( cun wceq equncom mpbi ) ABCEFACBEFDABCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality theorem for union of two classes. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
uneq1 $p |- ( A = B -> ( A u. C ) = ( B u. C ) ) $=
|
|
( vx wceq cun cv wcel wo eleq2 orbi1d elun 3bitr4g eqrdv ) ABEZDACFZBCFZO
|
|
DGZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
|
|
$}
|
|
|
|
$( Equality theorem for the union of two classes. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
uneq2 $p |- ( A = B -> ( C u. A ) = ( C u. B ) ) $=
|
|
( wceq cun uneq1 uncom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Equality theorem for union of two classes. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
uneq12 $p |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) ) $=
|
|
( wceq cun uneq1 uneq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
uneq1i.1 $e |- A = B $.
|
|
$( Inference adding union to the right in a class equality. (Contributed
|
|
by NM, 30-Aug-1993.) $)
|
|
uneq1i $p |- ( A u. C ) = ( B u. C ) $=
|
|
( wceq cun uneq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Inference adding union to the left in a class equality. (Contributed by
|
|
NM, 30-Aug-1993.) $)
|
|
uneq2i $p |- ( C u. A ) = ( C u. B ) $=
|
|
( wceq cun uneq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
uneq12i.2 $e |- C = D $.
|
|
$( Equality inference for union of two classes. (Contributed by NM,
|
|
12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
|
|
uneq12i $p |- ( A u. C ) = ( B u. D ) $=
|
|
( wceq cun uneq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
uneq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction adding union to the right in a class equality. (Contributed
|
|
by NM, 29-Mar-1998.) $)
|
|
uneq1d $p |- ( ph -> ( A u. C ) = ( B u. C ) ) $=
|
|
( wceq cun uneq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Deduction adding union to the left in a class equality. (Contributed by
|
|
NM, 29-Mar-1998.) $)
|
|
uneq2d $p |- ( ph -> ( C u. A ) = ( C u. B ) ) $=
|
|
( wceq cun uneq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
${
|
|
uneq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for union of two classes. (Contributed by NM,
|
|
29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
uneq12d $p |- ( ph -> ( A u. C ) = ( B u. D ) ) $=
|
|
( wceq cun uneq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Associative law for union of classes. Exercise 8 of [TakeutiZaring]
|
|
p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
unass $p |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) ) $=
|
|
( vx cun cv wcel wo elun orbi2i orbi1i orass bitr2i 3bitrri uneqri ) DABE
|
|
ZCABCEZEZDFZRGSAGZSQGZHTSBGZSCGZHZHZSPGZUCHZSAQIUAUDTSBCIJUGTUBHZUCHUEUFU
|
|
HUCSABIKTUBUCLMNO $.
|
|
$}
|
|
|
|
$( A rearrangement of union. (Contributed by NM, 12-Aug-2004.) $)
|
|
un12 $p |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) ) $=
|
|
( cun uncom uneq1i unass 3eqtr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH
|
|
$.
|
|
|
|
$( A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
un23 $p |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B ) $=
|
|
( cun unass un12 uncom 3eqtri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.
|
|
|
|
$( A rearrangement of the union of 4 classes. (Contributed by NM,
|
|
12-Aug-2004.) $)
|
|
un4 $p |- ( ( A u. B ) u. ( C u. D ) ) =
|
|
( ( A u. C ) u. ( B u. D ) ) $=
|
|
( cun un12 uneq2i unass 3eqtr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHA
|
|
CLHI $.
|
|
|
|
$( Union distributes over itself. (Contributed by NM, 17-Aug-2004.) $)
|
|
unundi $p |- ( A u. ( B u. C ) ) = ( ( A u. B ) u. ( A u. C ) ) $=
|
|
( cun unidm uneq1i un4 eqtr3i ) AADZBCDZDAJDABDACDDIAJAEFAABCGH $.
|
|
|
|
$( Union distributes over itself. (Contributed by NM, 17-Aug-2004.) $)
|
|
unundir $p |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. ( B u. C ) ) $=
|
|
( cun unidm uneq2i un4 eqtr3i ) ABDZCCDZDICDACDBCDDJCICEFABCCGH $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass relationship for union of classes. Theorem 25 of [Suppes]
|
|
p. 27. (Contributed by NM, 5-Aug-1993.) $)
|
|
ssun1 $p |- A C_ ( A u. B ) $=
|
|
( vx cun cv wcel wo orc elun sylibr ssriv ) CAABDZCEZAFZNMBFZGMLFNOHMABIJ
|
|
K $.
|
|
$}
|
|
|
|
$( Subclass relationship for union of classes. (Contributed by NM,
|
|
30-Aug-1993.) $)
|
|
ssun2 $p |- A C_ ( B u. A ) $=
|
|
( cun ssun1 uncom sseqtri ) AABCBACABDABEF $.
|
|
|
|
$( Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.) $)
|
|
ssun3 $p |- ( A C_ B -> A C_ ( B u. C ) ) $=
|
|
( wss cun ssun1 sstr2 mpi ) ABDBBCEZDAIDBCFABIGH $.
|
|
|
|
$( Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.) $)
|
|
ssun4 $p |- ( A C_ B -> A C_ ( C u. B ) ) $=
|
|
( wss cun ssun2 sstr2 mpi ) ABDBCBEZDAIDBCFABIGH $.
|
|
|
|
$( Membership law for union of classes. (Contributed by NM, 5-Aug-1993.) $)
|
|
elun1 $p |- ( A e. B -> A e. ( B u. C ) ) $=
|
|
( cun ssun1 sseli ) BBCDABCEF $.
|
|
|
|
$( Membership law for union of classes. (Contributed by NM, 30-Aug-1993.) $)
|
|
elun2 $p |- ( A e. B -> A e. ( C u. B ) ) $=
|
|
( cun ssun2 sseli ) BCBDABCEF $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
unss1 $p |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) ) $=
|
|
( vx wss cun cv wcel wo ssel orim1d elun 3imtr4g ssrdv ) ABEZDACFZBCFZODG
|
|
ZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
|
|
|
|
$( A relationship between subclass and union. Theorem 26 of [Suppes]
|
|
p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
ssequn1 $p |- ( A C_ B <-> ( A u. B ) = B ) $=
|
|
( vx cv wcel wi wal cun wb wceq wo bicom pm4.72 elun bibi1i 3bitr4i albii
|
|
wss dfss2 dfcleq ) CDZAEZUABEZFZCGUAABHZEZUCIZCGABRUEBJUDUGCUCUBUCKZIUHUC
|
|
IUDUGUCUHLUBUCMUFUHUCUAABNOPQCABSCUEBTP $.
|
|
$}
|
|
|
|
$( Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.
|
|
(Contributed by NM, 14-Oct-1999.) $)
|
|
unss2 $p |- ( A C_ B -> ( C u. A ) C_ ( C u. B ) ) $=
|
|
( wss cun unss1 uncom 3sstr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.) $)
|
|
unss12 $p |- ( ( A C_ B /\ C C_ D ) -> ( A u. C ) C_ ( B u. D ) ) $=
|
|
( wss cun unss1 unss2 sylan9ss ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
$( A relationship between subclass and union. (Contributed by NM,
|
|
13-Jun-1994.) $)
|
|
ssequn2 $p |- ( A C_ B <-> ( B u. A ) = B ) $=
|
|
( wss cun wceq ssequn1 uncom eqeq1i bitri ) ABCABDZBEBADZBEABFJKBABGHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27
|
|
and its converse. (Contributed by NM, 11-Jun-2004.) $)
|
|
unss $p |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C ) $=
|
|
( vx cun wss cv wcel wi wal wa dfss2 19.26 elun imbi1i jaob bitri anbi12i
|
|
wo albii 3bitr4i bitr2i ) ABEZCFDGZUCHZUDCHZIZDJZACFZBCFZKZDUCCLUDAHZUFIZ
|
|
UDBHZUFIZKZDJUMDJZUODJZKUHUKUMUODMUGUPDUGULUNSZUFIUPUEUSUFUDABNOULUFUNPQT
|
|
UIUQUJURDACLDBCLRUAUB $.
|
|
$}
|
|
|
|
${
|
|
unssi.1 $e |- A C_ C $.
|
|
unssi.2 $e |- B C_ C $.
|
|
$( An inference showing the union of two subclasses is a subclass.
|
|
(Contributed by Raph Levien, 10-Dec-2002.) $)
|
|
unssi $p |- ( A u. B ) C_ C $=
|
|
( wss wa cun pm3.2i unss mpbi ) ACFZBCFZGABHCFLMDEIABCJK $.
|
|
$}
|
|
|
|
${
|
|
unssd.1 $e |- ( ph -> A C_ C ) $.
|
|
unssd.2 $e |- ( ph -> B C_ C ) $.
|
|
$( A deduction showing the union of two subclasses is a subclass.
|
|
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
unssd $p |- ( ph -> ( A u. B ) C_ C ) $=
|
|
( wss cun wa unss biimpi syl2anc ) ABDGZCDGZBCHDGZEFMNIOBCDJKL $.
|
|
$}
|
|
|
|
${
|
|
unssad.1 $e |- ( ph -> ( A u. B ) C_ C ) $.
|
|
$( If ` ( A u. B ) ` is contained in ` C ` , so is ` A ` . One-way
|
|
deduction form of ~ unss . Partial converse of ~ unssd . (Contributed
|
|
by David Moews, 1-May-2017.) $)
|
|
unssad $p |- ( ph -> A C_ C ) $=
|
|
( wss cun wa unss sylibr simpld ) ABDFZCDFZABCGDFLMHEBCDIJK $.
|
|
|
|
$( If ` ( A u. B ) ` is contained in ` C ` , so is ` B ` . One-way
|
|
deduction form of ~ unss . Partial converse of ~ unssd . (Contributed
|
|
by David Moews, 1-May-2017.) $)
|
|
unssbd $p |- ( ph -> B C_ C ) $=
|
|
( wss cun wa unss sylibr simprd ) ABDFZCDFZABCGDFLMHEBCDIJK $.
|
|
$}
|
|
|
|
$( A condition that implies inclusion in the union of two classes.
|
|
(Contributed by NM, 23-Nov-2003.) $)
|
|
ssun $p |- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) ) $=
|
|
( wss cun ssun3 ssun4 jaoi ) ABDABCEDACDABCFACBGH $.
|
|
|
|
$( Restricted existential quantification over union. (Contributed by Jeff
|
|
Madsen, 5-Jan-2011.) $)
|
|
rexun $p |- ( E. x e. ( A u. B ) ph <->
|
|
( E. x e. A ph \/ E. x e. B ph ) ) $=
|
|
( cun wrex cv wcel wa wo df-rex 19.43 elun anbi1i andir bitri exbii orbi12i
|
|
wex 3bitr4i ) ABCDEZFBGZUAHZAIZBSZABCFZABDFZJZABUAKUBCHZAIZUBDHZAIZJZBSUJBS
|
|
ZULBSZJUEUHUJULBLUDUMBUDUIUKJZAIUMUCUPAUBCDMNUIUKAOPQUFUNUGUOABCKABDKRTP $.
|
|
|
|
$( Restricted quantification over a union. (Contributed by Scott Fenton,
|
|
12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
ralunb $p |- ( A. x e. ( A u. B ) ph <->
|
|
( A. x e. A ph /\ A. x e. B ph ) ) $=
|
|
( cv cun wcel wi wal wa wral wo elun imbi1i jaob bitri albii df-ral anbi12i
|
|
19.26 3bitr4i ) BEZCDFZGZAHZBIZUBCGZAHZBIZUBDGZAHZBIZJZABUCKABCKZABDKZJUFUH
|
|
UKJZBIUMUEUPBUEUGUJLZAHUPUDUQAUBCDMNUGAUJOPQUHUKBTPABUCRUNUIUOULABCRABDRSUA
|
|
$.
|
|
|
|
$( Restricted quantification over union. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
ralun $p |- ( ( A. x e. A ph /\ A. x e. B ph ) -> A. x e. ( A u. B ) ph ) $=
|
|
( cun wral wa ralunb biimpri ) ABCDEFABCFABDFGABCDHI $.
|
|
|
|
${
|
|
elin2.x $e |- X = ( B i^i C ) $.
|
|
$( Membership in a class defined as an intersection. (Contributed by
|
|
Stefan O'Rear, 29-Mar-2015.) $)
|
|
elin2 $p |- ( A e. X <-> ( A e. B /\ A e. C ) ) $=
|
|
( wcel cin wa eleq2i elin bitri ) ADFABCGZFABFACFHDLAEIABCJK $.
|
|
$}
|
|
|
|
${
|
|
elin3.x $e |- X = ( ( B i^i C ) i^i D ) $.
|
|
$( Membership in a class defined as a ternary intersection. (Contributed
|
|
by Stefan O'Rear, 29-Mar-2015.) $)
|
|
elin3 $p |- ( A e. X <-> ( A e. B /\ A e. C /\ A e. D ) ) $=
|
|
( cin wcel wa w3a elin anbi1i elin2 df-3an 3bitr4i ) ABCGZHZADHZIABHZACHZ
|
|
IZRIAEHSTRJQUARABCKLAPDEFMSTRNO $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Commutative law for intersection of classes. Exercise 7 of
|
|
[TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) $)
|
|
incom $p |- ( A i^i B ) = ( B i^i A ) $=
|
|
( vx cin cv wcel wa ancom elin 3bitr4i eqriv ) CABDZBADZCEZAFZNBFZGPOGNLF
|
|
NMFOPHNABINBAIJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
ineqri.1 $e |- ( ( x e. A /\ x e. B ) <-> x e. C ) $.
|
|
$( Inference from membership to intersection. (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
ineqri $p |- ( A i^i B ) = C $=
|
|
( cin cv wcel wa elin bitri eqriv ) ABCFZDAGZMHNBHNCHINDHNBCJEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Equality theorem for intersection of two classes. (Contributed by NM,
|
|
14-Dec-1993.) $)
|
|
ineq1 $p |- ( A = B -> ( A i^i C ) = ( B i^i C ) ) $=
|
|
( vx wceq cin cv wcel wa eleq2 anbi1d elin 3bitr4g eqrdv ) ABEZDACFZBCFZO
|
|
DGZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
|
|
$}
|
|
|
|
$( Equality theorem for intersection of two classes. (Contributed by NM,
|
|
26-Dec-1993.) $)
|
|
ineq2 $p |- ( A = B -> ( C i^i A ) = ( C i^i B ) ) $=
|
|
( wceq cin ineq1 incom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Equality theorem for intersection of two classes. (Contributed by NM,
|
|
8-May-1994.) $)
|
|
ineq12 $p |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) ) $=
|
|
( wceq cin ineq1 ineq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
ineq1i.1 $e |- A = B $.
|
|
$( Equality inference for intersection of two classes. (Contributed by NM,
|
|
26-Dec-1993.) $)
|
|
ineq1i $p |- ( A i^i C ) = ( B i^i C ) $=
|
|
( wceq cin ineq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for intersection of two classes. (Contributed by NM,
|
|
26-Dec-1993.) $)
|
|
ineq2i $p |- ( C i^i A ) = ( C i^i B ) $=
|
|
( wceq cin ineq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
ineq12i.2 $e |- C = D $.
|
|
$( Equality inference for intersection of two classes. (Contributed by
|
|
NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) $)
|
|
ineq12i $p |- ( A i^i C ) = ( B i^i D ) $=
|
|
( wceq cin ineq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ineq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for intersection of two classes. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
ineq1d $p |- ( ph -> ( A i^i C ) = ( B i^i C ) ) $=
|
|
( wceq cin ineq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for intersection of two classes. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
ineq2d $p |- ( ph -> ( C i^i A ) = ( C i^i B ) ) $=
|
|
( wceq cin ineq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
${
|
|
ineq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for intersection of two classes. (Contributed by
|
|
NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
ineq12d $p |- ( ph -> ( A i^i C ) = ( B i^i D ) ) $=
|
|
( wceq cin ineq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
ineqan12d.2 $e |- ( ps -> C = D ) $.
|
|
$( Equality deduction for intersection of two classes. (Contributed by
|
|
NM, 7-Feb-2007.) $)
|
|
ineqan12d $p |- ( ( ph /\ ps ) -> ( A i^i C ) = ( B i^i D ) ) $=
|
|
( wceq cin ineq12 syl2an ) ACDIEFICEJDFJIBGHCDEFKL $.
|
|
$}
|
|
$}
|
|
|
|
$( A frequently-used variant of subclass definition ~ df-ss . (Contributed
|
|
by NM, 10-Jan-2015.) $)
|
|
dfss1 $p |- ( A C_ B <-> ( B i^i A ) = A ) $=
|
|
( wss cin wceq df-ss incom eqeq1i bitri ) ABCABDZAEBADZAEABFJKAABGHI $.
|
|
|
|
$( Another definition of subclasshood. Similar to ~ df-ss , ~ dfss , and
|
|
~ dfss1 . (Contributed by David Moews, 1-May-2017.) $)
|
|
dfss5 $p |- ( A C_ B <-> A = ( B i^i A ) ) $=
|
|
( wss cin wceq dfss1 eqcom bitri ) ABCBADZAEAIEABFIAGH $.
|
|
|
|
${
|
|
$d A y $. $d C y $. $d D y $. $d x y $.
|
|
$( Distribute proper substitution through an intersection relation.
|
|
(Contributed by Alan Sare, 22-Jul-2012.) $)
|
|
csbing $p |- ( A e. B -> [_ A / x ]_ ( C i^i D ) =
|
|
( [_ A / x ]_ C i^i [_ A / x ]_ D ) ) $=
|
|
( vy cv cin csb wceq csbeq1 ineq12d eqeq12d nfcsb1v csbeq1a csbief vtoclg
|
|
vex nfin ) AFGZDEHZIZATDIZATEIZHZJABUAIZABDIZABEIZHZJFBCTBJZUBUFUEUIATBUA
|
|
KUJUCUGUDUHATBDKATBEKLMATUAUEFRAUCUDATDNATENSAGTJDUCEUDATDOATEOLPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $.
|
|
rabbi2dva.1 $e |- ( ( ph /\ x e. A ) -> ( x e. B <-> ps ) ) $.
|
|
$( Deduction from a wff to a restricted class abstraction. (Contributed by
|
|
NM, 14-Jan-2014.) $)
|
|
rabbi2dva $p |- ( ph -> ( A i^i B ) = { x e. A | ps } ) $=
|
|
( cin cv wcel crab wa cab elin abbi2i df-rab eqtr4i rabbidva syl5eq ) ADE
|
|
GZCHZEIZCDJZBCDJSTDIUAKZCLUBUCCSTDEMNUACDOPAUABCDFQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Idempotent law for intersection of classes. Theorem 15 of [Suppes]
|
|
p. 26. (Contributed by NM, 5-Aug-1993.) $)
|
|
inidm $p |- ( A i^i A ) = A $=
|
|
( vx cv wcel anidm ineqri ) BAAABCADEF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Associative law for intersection of classes. Exercise 9 of
|
|
[TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) $)
|
|
inass $p |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) ) $=
|
|
( vx cin cv wcel wa anass elin anbi2i bitr4i anbi1i 3bitr4i ineqri ) DABE
|
|
ZCABCEZEZDFZAGZSBGZHZSCGZHZTSQGZHZSPGZUCHSRGUDTUAUCHZHUFTUAUCIUEUHTSBCJKL
|
|
UGUBUCSABJMSAQJNO $.
|
|
$}
|
|
|
|
$( A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) $)
|
|
in12 $p |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) ) $=
|
|
( cin incom ineq1i inass 3eqtr3i ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH
|
|
$.
|
|
|
|
$( A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
in32 $p |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i B ) $=
|
|
( cin inass in12 incom 3eqtri ) ABDCDABCDDBACDZDIBDABCEABCFBIGH $.
|
|
|
|
$( A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.) $)
|
|
in13 $p |- ( A i^i ( B i^i C ) ) = ( C i^i ( B i^i A ) ) $=
|
|
( cin in32 incom 3eqtr4i ) BCDZADBADZCDAHDCIDBCAEAHFCIFG $.
|
|
|
|
$( A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.) $)
|
|
in31 $p |- ( ( A i^i B ) i^i C ) = ( ( C i^i B ) i^i A ) $=
|
|
( cin in12 incom 3eqtr4i ) CABDZDACBDZDHCDIADCABEHCFIAFG $.
|
|
|
|
$( Rotate the intersection of 3 classes. (Contributed by NM,
|
|
27-Aug-2012.) $)
|
|
inrot $p |- ( ( A i^i B ) i^i C ) = ( ( C i^i A ) i^i B ) $=
|
|
( cin in31 in32 eqtri ) ABDCDCBDADCADBDABCECBAFG $.
|
|
|
|
$( Rearrangement of intersection of 4 classes. (Contributed by NM,
|
|
21-Apr-2001.) $)
|
|
in4 $p |- ( ( A i^i B ) i^i ( C i^i D ) ) =
|
|
( ( A i^i C ) i^i ( B i^i D ) ) $=
|
|
( cin in12 ineq2i inass 3eqtr4i ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHA
|
|
CLHI $.
|
|
|
|
$( Intersection distributes over itself. (Contributed by NM, 6-May-1994.) $)
|
|
inindi $p |- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) ) $=
|
|
( cin inidm ineq1i in4 eqtr3i ) AADZBCDZDAJDABDACDDIAJAEFAABCGH $.
|
|
|
|
$( Intersection distributes over itself. (Contributed by NM,
|
|
17-Aug-2004.) $)
|
|
inindir $p |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i ( B i^i C ) ) $=
|
|
( cin inidm ineq2i in4 eqtr3i ) ABDZCCDZDICDACDBCDDJCICEFABCCGH $.
|
|
|
|
$( A relationship between subclass and intersection. Similar to Exercise 9
|
|
of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.) $)
|
|
sseqin2 $p |- ( A C_ B <-> ( B i^i A ) = A ) $=
|
|
( dfss1 ) ABC $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The intersection of two classes is a subset of one of them. Part of
|
|
Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM,
|
|
27-Apr-1994.) $)
|
|
inss1 $p |- ( A i^i B ) C_ A $=
|
|
( vx cin cv wcel elin simplbi ssriv ) CABDZACEZJFKAFKBFKABGHI $.
|
|
$}
|
|
|
|
$( The intersection of two classes is a subset of one of them. Part of
|
|
Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM,
|
|
27-Apr-1994.) $)
|
|
inss2 $p |- ( A i^i B ) C_ B $=
|
|
( cin incom inss1 eqsstr3i ) ABCBACBBADBAEF $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.
|
|
(Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
ssin $p |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) ) $=
|
|
( vx cv wcel wi wal wa cin wss elin imbi2i albii jcab 19.26 3bitrri dfss2
|
|
anbi12i 3bitr4i ) DEZAFZUABFZGZDHZUBUACFZGZDHZIZUBUABCJZFZGZDHZABKZACKZIA
|
|
UJKUMUBUCUFIZGZDHUDUGIZDHUIULUQDUKUPUBUABCLMNUQURDUBUCUFONUDUGDPQUNUEUOUH
|
|
DABRDACRSDAUJRT $.
|
|
$}
|
|
|
|
${
|
|
ssini.1 $e |- A C_ B $.
|
|
ssini.2 $e |- A C_ C $.
|
|
$( An inference showing that a subclass of two classes is a subclass of
|
|
their intersection. (Contributed by NM, 24-Nov-2003.) $)
|
|
ssini $p |- A C_ ( B i^i C ) $=
|
|
( wss wa cin pm3.2i ssin mpbi ) ABFZACFZGABCHFLMDEIABCJK $.
|
|
$}
|
|
|
|
${
|
|
ssind.1 $e |- ( ph -> A C_ B ) $.
|
|
ssind.2 $e |- ( ph -> A C_ C ) $.
|
|
$( A deduction showing that a subclass of two classes is a subclass of
|
|
their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
ssind $p |- ( ph -> A C_ ( B i^i C ) ) $=
|
|
( wss cin wa ssin biimpi syl2anc ) ABCGZBDGZBCDHGZEFMNIOBCDJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Add right intersection to subclass relation. (Contributed by NM,
|
|
16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
ssrin $p |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) ) $=
|
|
( vx wss cin cv wcel wa ssel anim1d elin 3imtr4g ssrdv ) ABEZDACFZBCFZODG
|
|
ZAHZRCHZIRBHZTIRPHRQHOSUATABRJKRACLRBCLMN $.
|
|
|
|
$( Add left intersection to subclass relation. (Contributed by NM,
|
|
19-Oct-1999.) $)
|
|
sslin $p |- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) ) $=
|
|
( wss cin ssrin incom 3sstr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
$}
|
|
|
|
$( Intersection of subclasses. (Contributed by NM, 5-May-2000.) $)
|
|
ss2in $p |- ( ( A C_ B /\ C C_ D ) -> ( A i^i C ) C_ ( B i^i D ) ) $=
|
|
( wss cin ssrin sslin sylan9ss ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
$( Intersection preserves subclass relationship. (Contributed by NM,
|
|
14-Sep-1999.) $)
|
|
ssinss1 $p |- ( A C_ C -> ( A i^i B ) C_ C ) $=
|
|
( cin wss wi inss1 sstr2 ax-mp ) ABDZAEACEJCEFABGJACHI $.
|
|
|
|
$( Inclusion of an intersection of two classes. (Contributed by NM,
|
|
30-Oct-2014.) $)
|
|
inss $p |- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C ) $=
|
|
( wss cin ssinss1 incom syl5eqss jaoi ) ACDABEZCDBCDZABCFKJBAECABGBACFHI $.
|
|
|
|
$( Absorption law for union. (Contributed by NM, 16-Apr-2006.) $)
|
|
unabs $p |- ( A u. ( A i^i B ) ) = A $=
|
|
( cin wss cun wceq inss1 ssequn2 mpbi ) ABCZADAJEAFABGJAHI $.
|
|
|
|
$( Absorption law for intersection. (Contributed by NM, 16-Apr-2006.) $)
|
|
inabs $p |- ( A i^i ( A u. B ) ) = A $=
|
|
( cun wss cin wceq ssun1 df-ss mpbi ) AABCZDAJEAFABGAJHI $.
|
|
|
|
$( Negation of subclass expressed in terms of intersection and proper
|
|
subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
nssinpss $p |- ( -. A C_ B <-> ( A i^i B ) C. A ) $=
|
|
( cin wne wss wa wn wpss inss1 biantrur df-ss necon3bbii df-pss 3bitr4i ) A
|
|
BCZADZOAEZPFABEZGOAHQPABIJROAABKLOAMN $.
|
|
|
|
$( Negation of subclass expressed in terms of proper subclass and union.
|
|
(Contributed by NM, 15-Sep-2004.) $)
|
|
nsspssun $p |- ( -. A C_ B <-> B C. ( A u. B ) ) $=
|
|
( wss wn cun wa wpss ssun2 biantrur ssid biantru unss bitri xchnxbir dfpss3
|
|
bitr4i ) ABCZDBABEZCZRBCZDZFZBRGTUBQSUABAHIQQBBCZFTUCQBJKABBLMNBROP $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass defined in terms of class difference. See comments under
|
|
~ dfun2 . (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew
|
|
Salmon, 26-Jun-2011.) $)
|
|
dfss4 $p |- ( A C_ B <-> ( B \ ( B \ A ) ) = A ) $=
|
|
( vx wss wceq cdif sseqin2 cv wcel wn wa eldif notbii anbi2i wi elin abai
|
|
cin iman bitr4i 3bitri difeqri eqeq1i ) ABDBARZAEBBAFZFZAEABGUFUDACBUEUDC
|
|
HZBIZUGUEIZJZKUHUHUGAIZJKZJZKZUGUDIZUJUMUHUIULUGBALMNUOUHUKKUHUHUKOZKUNUG
|
|
BAPUHUKQUPUMUHUHUKSNUATUBUCT $.
|
|
|
|
$( An alternate definition of the union of two classes in terms of class
|
|
difference, requiring no dummy variables. Along with ~ dfin2 and
|
|
~ dfss4 it shows we can express union, intersection, and subset directly
|
|
in terms of the single "primitive" operation ` \ ` (class difference).
|
|
(Contributed by NM, 10-Jun-2004.) $)
|
|
dfun2 $p |- ( A u. B ) = ( _V \ ( ( _V \ A ) \ B ) ) $=
|
|
( vx cvv cdif cv wcel wo wn wa eldif mpbiran anbi1i ioran 3bitr4i con2bii
|
|
vex bitr4i uneqri ) CABDDAEZBEZEZCFZAGZUCBGZHZUCUAGZIZUCUBGZUGUFUCTGZUEIZ
|
|
JUDIZUKJUGUFIUJULUKUJUCDGZULCQZUCDAKLMUCTBKUDUENOPUIUMUHUNUCDUAKLRS $.
|
|
|
|
$( An alternate definition of the intersection of two classes in terms of
|
|
class difference, requiring no dummy variables. See comments under
|
|
~ dfun2 . Another version is given by ~ dfin4 . (Contributed by NM,
|
|
10-Jun-2004.) $)
|
|
dfin2 $p |- ( A i^i B ) = ( A \ ( _V \ B ) ) $=
|
|
( vx cvv cdif cv wcel wa vex eldif mpbiran con2bii anbi2i bitr4i ineqri
|
|
wn ) CABADBEZEZCFZAGZSBGZHTSQGZPZHSRGUAUCTUBUAUBSDGUAPCISDBJKLMSAQJNO $.
|
|
|
|
$( Difference with intersection. Theorem 33 of [Suppes] p. 29.
|
|
(Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
difin $p |- ( A \ ( A i^i B ) ) = ( A \ B ) $=
|
|
( vx cin cdif cv wcel wi wn pm4.61 anclb elin imbi2i iman 3bitr2i con2bii
|
|
wa eldif 3bitr4i difeqri ) CAABDZABEZCFZAGZUCBGZHZIUDUEIQUDUCUAGZIQZUCUBG
|
|
UDUEJUFUHUFUDUDUEQZHUDUGHUHIUDUEKUGUIUDUCABLMUDUGNOPUCABRST $.
|
|
$}
|
|
|
|
$( Union defined in terms of intersection (De Morgan's law). Definition of
|
|
union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) $)
|
|
dfun3 $p |- ( A u. B ) = ( _V \ ( ( _V \ A ) i^i ( _V \ B ) ) ) $=
|
|
( cun cvv cdif cin dfun2 dfin2 ddif difeq2i eqtr2i eqtri ) ABCDDAEZBEZEDMDB
|
|
EZFZEABGNPDPMDOEZENMOHQBMBIJKJL $.
|
|
|
|
$( Intersection defined in terms of union (De Morgan's law. Similar to
|
|
Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM,
|
|
8-Jan-2002.) $)
|
|
dfin3 $p |- ( A i^i B ) = ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) $=
|
|
( cvv cdif cun cin ddif dfun2 difeq1i difeq2i eqtri dfin2 3eqtr4ri ) CCACBD
|
|
ZDZDZDOCCADZNEZDABFOGRPCRCCQDZNDZDPQNHTOCSANAGIJKJABLM $.
|
|
|
|
$( Alternate definition of the intersection of two classes. Exercise 4.10(q)
|
|
of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.) $)
|
|
dfin4 $p |- ( A i^i B ) = ( A \ ( A \ B ) ) $=
|
|
( cin cdif wss wceq inss1 dfss4 mpbi difin difeq2i eqtr3i ) AAABCZDZDZMAABD
|
|
ZDMAEOMFABGMAHINPAABJKL $.
|
|
|
|
$( Intersection with universal complement. Remark in [Stoll] p. 20.
|
|
(Contributed by NM, 17-Aug-2004.) $)
|
|
invdif $p |- ( A i^i ( _V \ B ) ) = ( A \ B ) $=
|
|
( cvv cdif cin dfin2 ddif difeq2i eqtri ) ACBDZEACJDZDABDAJFKBABGHI $.
|
|
|
|
$( Intersection with class difference. Theorem 34 of [Suppes] p. 29.
|
|
(Contributed by NM, 17-Aug-2004.) $)
|
|
indif $p |- ( A i^i ( A \ B ) ) = ( A \ B ) $=
|
|
( cdif cin dfin4 difeq2i difin 3eqtr2i ) AABCZDAAICZCAABDZCIAIEKJAABEFABGH
|
|
$.
|
|
|
|
$( Bring an intersection in and out of a class difference. (Contributed by
|
|
Jeff Hankins, 15-Jul-2009.) $)
|
|
indif2 $p |- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C ) $=
|
|
( cin cvv cdif inass invdif ineq2i 3eqtr3ri ) ABDZECFZDABLDZDKCFABCFZDABLGK
|
|
CHMNABCHIJ $.
|
|
|
|
$( Bring an intersection in and out of a class difference. (Contributed by
|
|
Mario Carneiro, 15-May-2015.) $)
|
|
indif1 $p |- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C ) $=
|
|
( cdif cin indif2 incom difeq1i 3eqtr3i ) BACDZEBAEZCDJBEABEZCDBACFBJGKLCBA
|
|
GHI $.
|
|
|
|
$( Commutation law for intersection and difference. (Contributed by Scott
|
|
Fenton, 18-Feb-2013.) $)
|
|
indifcom $p |- ( A i^i ( B \ C ) ) = ( B i^i ( A \ C ) ) $=
|
|
( cin cdif incom difeq1i indif2 3eqtr4i ) ABDZCEBADZCEABCEDBACEDJKCABFGABCH
|
|
BACHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Distributive law for intersection over union. Exercise 10 of
|
|
[TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
indi $p |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) ) $=
|
|
( vx cun cin cv wcel wo wa andi elin orbi12i bitr4i anbi2i 3bitr4i ineqri
|
|
elun ) DABCEZABFZACFZEZDGZAHZUCBHZUCCHZIZJZUCTHZUCUAHZIZUDUCSHZJUCUBHUHUD
|
|
UEJZUDUFJZIUKUDUEUFKUIUMUJUNUCABLUCACLMNULUGUDUCBCROUCTUARPQ $.
|
|
|
|
$( Distributive law for union over intersection. Exercise 11 of
|
|
[TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
undi $p |- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) ) $=
|
|
( vx cin cv wcel wo wa elin orbi2i ordi elun anbi12i bitr2i 3bitri uneqri
|
|
cun ) DABCEZABRZACRZEZDFZAGZUCSGZHUDUCBGZUCCGZIZHUDUFHZUDUGHZIZUCUBGZUEUH
|
|
UDUCBCJKUDUFUGLULUCTGZUCUAGZIUKUCTUAJUMUIUNUJUCABMUCACMNOPQ $.
|
|
$}
|
|
|
|
$( Distributive law for intersection over union. Theorem 28 of [Suppes]
|
|
p. 27. (Contributed by NM, 30-Sep-2002.) $)
|
|
indir $p |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) ) $=
|
|
( cun cin indi incom uneq12i 3eqtr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMK
|
|
NLACGBCGHI $.
|
|
|
|
$( Distributive law for union over intersection. Theorem 29 of [Suppes]
|
|
p. 27. (Contributed by NM, 30-Sep-2002.) $)
|
|
undir $p |- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) ) $=
|
|
( cin cun undi uncom ineq12i 3eqtr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABFJCGMK
|
|
NLACGBCGHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Infer equality from equalities of union and intersection. Exercise 20
|
|
of [Enderton] p. 32 and its converse. (Contributed by NM,
|
|
10-Aug-2004.) $)
|
|
unineq $p |- ( ( ( A u. C ) = ( B u. C ) /\ ( A i^i C ) = ( B i^i C ) )
|
|
<-> A = B ) $=
|
|
( vx cun wceq cin wa wcel wb eleq2 elin 3bitr3g iba bibi12d syl5ibr uncom
|
|
wo elun biorf cv wi adantld eqeq12i sylbi adantrd pm2.61i eqrdv uneq1 jca
|
|
wn ineq1 impbii ) ACEZBCEZFZACGZBCGZFZHZABFZUTDABDUAZCIZUTVBAIZVBBIZJZUBV
|
|
CUSVFUPUSVFVCVDVCHZVEVCHZJUSVBUQIVBURIVGVHUQURVBKVBACLVBBCLMVCVDVGVEVHVCV
|
|
DNVCVENOPUCVCUKZUPVFUSUPVFVIVCVDRZVCVERZJUPVBCAEZIZVBCBEZIZVJVKUPVLVNFVMV
|
|
OJUNVLUOVNACQBCQUDVLVNVBKUEVBCASVBCBSMVIVDVJVEVKVCVDTVCVETOPUFUGUHVAUPUSA
|
|
BCUIABCULUJUM $.
|
|
$}
|
|
|
|
$( Equality of union and intersection implies equality of their arguments.
|
|
(Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
uneqin $p |- ( ( A u. B ) = ( A i^i B ) <-> A = B ) $=
|
|
( cun cin wceq wss wa eqimss unss ssin sstr sylbir simpl anim12i syl sylibr
|
|
eqss unidm inidm eqtr4i uneq2 ineq2 3eqtr3a impbii ) ABCZABDZEZABEZUGABFZBA
|
|
FZGZUHUGUEUFFZUKUEUFHULAUFFZBUFFZGUKABUFIUMUIUNUJUMAAFUIGUIAABJAABKLUNUJBBF
|
|
ZGUJBABJUJUOMLNLOABQPUHAACZAADZUEUFUPAUQARASTABAUAABAUBUCUD $.
|
|
|
|
$( Distributive law for class difference. Theorem 39 of [Suppes] p. 29.
|
|
(Contributed by NM, 17-Aug-2004.) $)
|
|
difundi $p |- ( A \ ( B u. C ) ) = ( ( A \ B ) i^i ( A \ C ) ) $=
|
|
( cun cdif cvv cin dfun3 difeq2i inindi dfin2 invdif ineq12i 3eqtr3i eqtri
|
|
) ABCDZEAFFBEZFCEZGZEZEZABEZACEZGZPTABCHIASGAQGZARGZGUAUDAQRJASKUEUBUFUCABL
|
|
ACLMNO $.
|
|
|
|
$( Distributive law for class difference. (Contributed by NM,
|
|
17-Aug-2004.) $)
|
|
difundir $p |- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) ) $=
|
|
( cun cvv cdif cin indir invdif uneq12i 3eqtr3i ) ABDZECFZGAMGZBMGZDLCFACFZ
|
|
BCFZDABMHLCINPOQACIBCIJK $.
|
|
|
|
$( Distributive law for class difference. Theorem 40 of [Suppes] p. 29.
|
|
(Contributed by NM, 17-Aug-2004.) $)
|
|
difindi $p |- ( A \ ( B i^i C ) ) = ( ( A \ B ) u. ( A \ C ) ) $=
|
|
( cin cdif cvv cun dfin3 difeq2i indi dfin2 invdif uneq12i 3eqtr3i eqtri )
|
|
ABCDZEAFFBEZFCEZGZEZEZABEZACEZGZPTABCHIASDAQDZARDZGUAUDAQRJASKUEUBUFUCABLAC
|
|
LMNO $.
|
|
|
|
$( Distributive law for class difference. (Contributed by NM,
|
|
17-Aug-2004.) $)
|
|
difindir $p |- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) ) $=
|
|
( cin cvv cdif inindir invdif ineq12i 3eqtr3i ) ABDZECFZDALDZBLDZDKCFACFZBC
|
|
FZDABLGKCHMONPACHBCHIJ $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Distribute intersection over difference. (Contributed by Scott Fenton,
|
|
14-Apr-2011.) $)
|
|
indifdir $p |- ( ( A \ B ) i^i C ) = ( ( A i^i C ) \ ( B i^i C ) ) $=
|
|
( vx cdif cin cv wcel wn wa pm3.24 intnan anass mtbir biorfi 3bitr4i elin
|
|
wo eldif bitri an32 andi ianor anbi2i bitr4i anbi1i notbii anbi12i eqriv
|
|
) DABEZCFZACFZBCFZEZDGZAHZUOBHZIZJZUOCHZJZUPUTJZUQUTJZIZJZUOUKHZUOUNHZVAV
|
|
BURUTIZRZJZVEVBURJZVKVBVHJZRVAVJVLVKVLUPUTVHJZJVMUPUTKLUPUTVHMNOUPURUTUAV
|
|
BURVHUBPVDVIVBUQUTUCUDUEVFUOUJHZUTJVAUOUJCQVNUSUTUOABSUFTVGUOULHZUOUMHZIZ
|
|
JVEUOULUMSVOVBVQVDUOACQVPVCUOBCQUGUHTPUI $.
|
|
$}
|
|
|
|
$( De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19.
|
|
(Contributed by NM, 18-Aug-2004.) $)
|
|
undm $p |- ( _V \ ( A u. B ) ) = ( ( _V \ A ) i^i ( _V \ B ) ) $=
|
|
( cvv difundi ) CABD $.
|
|
|
|
$( De Morgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19.
|
|
(Contributed by NM, 18-Aug-2004.) $)
|
|
indm $p |- ( _V \ ( A i^i B ) ) = ( ( _V \ A ) u. ( _V \ B ) ) $=
|
|
( cvv difindi ) CABD $.
|
|
|
|
$( A relationship involving double difference and union. (Contributed by NM,
|
|
29-Aug-2004.) $)
|
|
difun1 $p |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C ) $=
|
|
( cvv cdif cin cun inass invdif eqtr3i undm ineq2i difeq1i ) ADBEZFZCEZABCG
|
|
ZEZABEZCEANDCEZFZFZPROTFUBPANTHOCIJADQEZFUBRUCUAABCKLAQIJJOSCABIMJ $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( An equality involving class union and class difference. The first
|
|
equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan
|
|
Sare, 17-Apr-2012.) $)
|
|
undif3 $p |- ( A u. ( B \ C ) ) = ( ( A u. B ) \ ( C \ A ) ) $=
|
|
( vx cdif cun cv wcel wn wa elun pm4.53 eldif xchnxbir anbi12i orbi2i orc
|
|
wo olc jca anim12i simpl adantr adantl ccase impbii 3bitri 3bitr4ri eqriv
|
|
jaoi orcd ) DABCEZFZABFZCAEZEZDGZUNHZUQUOHZIZJUQAHZUQBHZRZUQCHZIZVARZJZUQ
|
|
UPHUQUMHZURVCUTVFUQABKVDVAIJVFUSVDVALUQCAMNOUQUNUOMVHVAUQULHZRVAVBVEJZRZV
|
|
GUQAULKVIVJVAUQBCMPVKVGVAVGVJVAVCVFVAVBQVAVESTVBVCVEVFVBVASVEVAQUAUJVAVEV
|
|
BVAVKVAVEJVAVJVAVEUBUKVJVASVAVKVAVAVJQZUCVAVKVBVLUDUEUFUGUHUI $.
|
|
|
|
$( Represent a set difference as an intersection with a larger difference.
|
|
(Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
difin2 $p |- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) ) $=
|
|
( vx wss cdif cin cv wcel wn ssel pm4.71d anbi1d eldif anbi1i ancom anass
|
|
wa elin bitr4i 3bitri 3bitr4g eqrdv ) ACEZDABFZCBFZAGZUDDHZAIZUHBIJZRUIUH
|
|
CIZRZUJRZUHUEIUHUGIZUDUIULUJUDUIUKACUHKLMUHABNUNUHUFIZUIRUKUJRZUIRZUMUHUF
|
|
ASUOUPUIUHCBNOUQUIUPRUMUPUIPUIUKUJQTUAUBUC $.
|
|
$}
|
|
|
|
$( Swap second and third argument of double difference. (Contributed by NM,
|
|
18-Aug-2004.) $)
|
|
dif32 $p |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B ) $=
|
|
( cun cdif uncom difeq2i difun1 3eqtr3i ) ABCDZEACBDZEABECEACEBEJKABCFGABCH
|
|
ACBHI $.
|
|
|
|
$( Absorption-like law for class difference: you can remove a class only
|
|
once. (Contributed by FL, 2-Aug-2009.) $)
|
|
difabs $p |- ( ( A \ B ) \ B ) = ( A \ B ) $=
|
|
( cun cdif difun1 unidm difeq2i eqtr3i ) ABBCZDABDZBDJABBEIBABFGH $.
|
|
|
|
$( Two ways to express symmetric difference. This theorem shows the
|
|
equivalence of the definition of symmetric difference in [Stoll] p. 13 and
|
|
the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by
|
|
NM, 17-Aug-2004.) $)
|
|
symdif1 $p |- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) ) $=
|
|
( cun cin cdif difundir difin incom difeq2i eqtri uneq12i eqtr2i ) ABCABDZE
|
|
AMEZBMEZCABEZBAEZCABMFNPOQABGOBBADZEQMRBABHIBAGJKL $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Two ways to express symmetric difference. (Contributed by NM,
|
|
17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
symdif2 $p |- ( ( A \ B ) u. ( B \ A ) ) =
|
|
{ x | -. ( x e. A <-> x e. B ) } $=
|
|
( cv wcel wb wn cdif cun wo wa eldif orbi12i elun xor 3bitr4i abbi2i ) AD
|
|
ZBEZRCEZFGZABCHZCBHZIZRUBEZRUCEZJSTGKZTSGKZJRUDEUAUEUGUFUHRBCLRCBLMRUBUCN
|
|
STOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d ph y $. $d ps y $.
|
|
$( Union of two class abstractions. (Contributed by NM, 29-Sep-2002.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
unab $p |- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) } $=
|
|
( vy cab wo wsb cv wcel sbor df-clab orbi12i 3bitr4ri uneqri ) DACEZBCEZA
|
|
BFZCEZQCDGACDGZBCDGZFDHZRIUAOIZUAPIZFABCDJQDCKUBSUCTADCKBDCKLMN $.
|
|
|
|
$( Intersection of two class abstractions. (Contributed by NM,
|
|
29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
inab $p |- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) } $=
|
|
( vy cab wa wsb cv wcel sban df-clab anbi12i 3bitr4ri ineqri ) DACEZBCEZA
|
|
BFZCEZQCDGACDGZBCDGZFDHZRIUAOIZUAPIZFABCDJQDCKUBSUCTADCKBDCKLMN $.
|
|
|
|
$( Difference of two class abstractions. (Contributed by NM,
|
|
23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
difab $p |- ( { x | ph } \ { x | ps } ) = { x | ( ph /\ -. ps ) } $=
|
|
( vy cab wn wcel wsb df-clab sban bicomi xchbinxr anbi12i 3bitrri difeqri
|
|
wa cv sbn ) DACEZBCEZABFZPZCEZDQZUCGUBCDHACDHZUACDHZPUDSGZUDTGZFZPUBDCIAU
|
|
ACDJUEUGUFUIUGUEADCIKUFBCDHUHBCDRBDCILMNO $.
|
|
|
|
$( Complement of a class abstraction. (Contributed by SF, 12-Jan-2015.) $)
|
|
complab $p |- ~ { x | ph } = { x | -. ph } $=
|
|
( vy cab ccompl wn wcel wsb df-clab notbii sbn bitr4i vex elcompl 3bitr4i
|
|
cv eqriv ) CABDZEZAFZBDZCPZRGZFZTBCHZUBSGUBUAGUDABCHZFUEUCUFACBIJABCKLUBR
|
|
CMNTCBIOQ $.
|
|
$}
|
|
|
|
$( A class builder defined by a negation. (Contributed by FL,
|
|
18-Sep-2010.) $)
|
|
notab $p |- { x | -. ph } = ( _V \ { x | ph } ) $=
|
|
( cv cvv wcel wn wa cab cdif crab df-rab rabab eqtr3i difab abid2 difeq1i )
|
|
BCDEZAFZGBHZRBHZDABHZIZRBDJSTRBDKRBLMQBHZUAISUBQABNUCDUABDOPMM $.
|
|
|
|
$( Union of two restricted class abstractions. (Contributed by NM,
|
|
25-Mar-2004.) $)
|
|
unrab $p |- ( { x e. A | ph } u. { x e. A | ps } ) =
|
|
{ x e. A | ( ph \/ ps ) } $=
|
|
( crab cun cv wcel wa cab wo df-rab uneq12i unab andi abbii eqtr4i ) ACDEZB
|
|
CDEZFCGDHZAIZCJZTBIZCJZFZABKZCDEZRUBSUDACDLBCDLMUGTUFIZCJZUEUFCDLUEUAUCKZCJ
|
|
UIUAUCCNUHUJCTABOPQQQ $.
|
|
|
|
$( Intersection of two restricted class abstractions. (Contributed by NM,
|
|
1-Sep-2006.) $)
|
|
inrab $p |- ( { x e. A | ph } i^i { x e. A | ps } ) =
|
|
{ x e. A | ( ph /\ ps ) } $=
|
|
( crab cin cv wcel wa cab df-rab ineq12i inab anandi abbii eqtr4i ) ACDEZBC
|
|
DEZFCGDHZAIZCJZSBIZCJZFZABIZCDEZQUARUCACDKBCDKLUFSUEIZCJZUDUECDKUDTUBIZCJUH
|
|
TUBCMUGUICSABNOPPP $.
|
|
|
|
${
|
|
$d x B $.
|
|
$( Intersection with a restricted class abstraction. (Contributed by NM,
|
|
19-Nov-2007.) $)
|
|
inrab2 $p |- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph } $=
|
|
( crab cin cv wcel cab df-rab abid2 eqcomi ineq12i inab elin anbi1i bitri
|
|
wa an32 eqtr4i abbii ) ABCEZDFBGZCHZARZBIZUCDHZBIZFZABCDFZEZUBUFDUHABCJUH
|
|
DBDKLMUKUCUJHZARZBIZUIABUJJUIUEUGRZBIUNUEUGBNUMUOBUMUDUGRZARUOULUPAUCCDOP
|
|
UDUGASQUATTT $.
|
|
$}
|
|
|
|
$( Difference of two restricted class abstractions. (Contributed by NM,
|
|
23-Oct-2004.) $)
|
|
difrab $p |- ( { x e. A | ph } \ { x e. A | ps } ) =
|
|
{ x e. A | ( ph /\ -. ps ) } $=
|
|
( crab cdif cv wcel wa wn df-rab difeq12i difab anass simpr con3i anim2i wi
|
|
cab eqtr4i pm3.2 adantr con3d imdistani impbii bitr3i abbii ) ACDEZBCDEZFCG
|
|
DHZAIZCSZUJBIZCSZFZABJZIZCDEZUHULUIUNACDKBCDKLURUJUQIZCSZUOUQCDKUOUKUMJZIZC
|
|
SUTUKUMCMUSVBCUSUKUPIZVBUJAUPNVCVBUPVAUKUMBUJBOPQUKVAUPUKBUMUJBUMRAUJBUAUBU
|
|
CUDUEUFUGTTT $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Alternate definition of restricted class abstraction. (Contributed by
|
|
NM, 20-Sep-2003.) $)
|
|
dfrab2 $p |- { x e. A | ph } = ( { x | ph } i^i A ) $=
|
|
( crab cv wcel wa cab cin df-rab inab abid2 ineq1i eqtr3i incom 3eqtri )
|
|
ABCDBECFZAGBHZCABHZIZSCIABCJQBHZSIRTQABKUACSBCLMNCSOP $.
|
|
|
|
$( Alternate definition of restricted class abstraction. (Contributed by
|
|
Mario Carneiro, 8-Sep-2013.) $)
|
|
dfrab3 $p |- { x e. A | ph } = ( A i^i { x | ph } ) $=
|
|
( crab cv wcel wa cab cin df-rab inab abid2 ineq1i 3eqtr2i ) ABCDBECFZAGB
|
|
HOBHZABHZICQIABCJOABKPCQBCLMN $.
|
|
|
|
$( Complementation of restricted class abstractions. (Contributed by Mario
|
|
Carneiro, 3-Sep-2015.) $)
|
|
notrab $p |- ( A \ { x e. A | ph } ) = { x e. A | -. ph } $=
|
|
( cv wcel cab cdif wn crab difab cin difin dfrab3 difeq2i difeq1i 3eqtr4i
|
|
wa abid2 df-rab ) BDCEZBFZABFZGZTAHZQBFCABCIZGZUDBCITABJCCUBKZGCUBGUFUCCU
|
|
BLUEUGCABCMNUACUBBCROPUDBCSP $.
|
|
|
|
$( Restricted class abstraction with a common superset. (Contributed by
|
|
Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro,
|
|
8-Nov-2015.) $)
|
|
dfrab3ss $p |- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) ) $=
|
|
( wss cab crab wceq df-ss ineq1 eqcomd sylbi dfrab3 ineq2i eqtr4i 3eqtr4g
|
|
cin inass ) CDEZCABFZQZCDQZTQZABCGCABDGZQZSUBCHZUAUCHCDIUFUCUAUBCTJKLABCM
|
|
UECDTQZQUCUDUGCABDMNCDTROP $.
|
|
$}
|
|
|
|
$( Abstraction restricted to a union. (Contributed by Stefan O'Rear,
|
|
5-Feb-2015.) $)
|
|
rabun2 $p |- { x e. ( A u. B ) | ph } =
|
|
( { x e. A | ph } u. { x e. B | ph } ) $=
|
|
( cun crab cv wcel wa cab df-rab uneq12i elun anbi1i andir bitri abbii unab
|
|
wo eqtr4i ) ABCDEZFBGZUAHZAIZBJZABCFZABDFZEZABUAKUHUBCHZAIZBJZUBDHZAIZBJZEZ
|
|
UEUFUKUGUNABCKABDKLUEUJUMSZBJUOUDUPBUDUIULSZAIUPUCUQAUBCDMNUIULAOPQUJUMBRTT
|
|
T $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Transfer uniqueness to a smaller subclass. (Contributed by NM,
|
|
20-Oct-2005.) $)
|
|
reuss2 $p |- ( ( ( A C_ B /\ A. x e. A ( ph -> ps ) ) /\
|
|
( E. x e. A ph /\ E! x e. B ps ) ) -> E! x e. A ph ) $=
|
|
( wrex wreu wa wss wi wral wcel wex weu df-rex df-reu anbi12i wal sylan2b
|
|
cv wmo df-ral ssel prth sylan exp4b com23 a2d imp4a alimdv imp euimmo syl
|
|
eu5 simplbi2 syl9 imp32 sylibr ) ACDFZBCEGZHDEIZABJZCDKZHZCTZDLZAHZCMZVEE
|
|
LZBHZCNZHZACDGZUSVHUTVKACDOBCEPQVDVLHVGCNZVMVDVHVKVNVDVKVGCUAZVHVNVDVGVJJ
|
|
ZCRZVKVOJVCVAVFVBJZCRZVQVBCDUBVAVSVQVAVRVPCVAVRVFAVJVAVFVBAVJJZVAVBVFVTVA
|
|
VBVFAVJVAVFVIJVBVPDEVEUCVFVIABUDUEUFUGUHUIUJUKSVGVJCULUMVNVHVOVGCUNUOUPUQ
|
|
ACDPURS $.
|
|
|
|
$( Transfer uniqueness to a smaller subclass. (Contributed by NM,
|
|
21-Aug-1999.) $)
|
|
reuss $p |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) ->
|
|
E! x e. A ph ) $=
|
|
( wss wrex wreu wi wral wa cv wcel idd rgen reuss2 mpanl2 3impb ) CDEZABC
|
|
FZABDGZABCGZRAAHZBCISTJUAUBBCBKCLAMNAABCDOPQ $.
|
|
|
|
$( Transfer uniqueness to a smaller class. (Contributed by NM,
|
|
21-Oct-2005.) $)
|
|
reuun1 $p |- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) )
|
|
-> E! x e. A ph ) $=
|
|
( cun wss wo wi wral wrex wreu wa ssun1 orc rgenw reuss2 mpanl12 ) DDEFZG
|
|
AABHZIZCDJACDKTCSLMACDLDENUACDABOPATCDSQR $.
|
|
|
|
$( Transfer uniqueness to a smaller or larger class. (Contributed by NM,
|
|
21-Oct-2005.) $)
|
|
reuun2 $p |- ( -. E. x e. B ph ->
|
|
( E! x e. ( A u. B ) ph <-> E! x e. A ph ) ) $=
|
|
( wrex wn cv wcel wa wo weu cun wreu wex df-rex euor2 sylnbi df-reu bitri
|
|
wb elun anbi1i andir orcom eubii 3bitr4g ) ABDEZFBGZDHZAIZUHCHZAIZJZBKZUL
|
|
BKZABCDLZMZABCMUGUJBNUNUOTABDOUJULBPQUQUHUPHZAIZBKUNABUPRUSUMBUSUKUIJZAIZ
|
|
UMURUTAUHCDUAUBVAULUJJUMUKUIAUCULUJUDSSUESABCRUF $.
|
|
|
|
$( Restricted uniqueness "picks" a member of a subclass. (Contributed by
|
|
NM, 21-Aug-1999.) $)
|
|
reupick $p |- ( ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) /\ ph ) ->
|
|
( x e. A <-> x e. B ) ) $=
|
|
( wss wrex wreu wa cv wcel wi ssel ad2antrr wex weu df-rex df-reu anbi12i
|
|
ancrd anim1d an32 syl6ib eximdv eupick ex syl9 com23 imp32 exp3acom23 imp
|
|
sylan2b impbid ) CDEZABCFZABDGZHZHZAHBIZCJZURDJZUMUSUTKUPACDURLZMUQAUTUSK
|
|
UQUTAUSUPUMUSAHZBNZUTAHZBOZHVDUSKZUNVCUOVEABCPABDQRUMVCVEVFUMVEVCVFUMVCVD
|
|
USHZBNZVEVFUMVBVGBUMVBUTUSHZAHVGUMUSVIAUMUSUTVASTUTUSAUAUBUCVEVHVFVDUSBUD
|
|
UEUFUGUHUKUIUJUL $.
|
|
|
|
$( Restricted uniqueness "picks" a member of a subclass. (Contributed by
|
|
Mario Carneiro, 19-Nov-2016.) $)
|
|
reupick3 $p |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) ->
|
|
( ph -> ps ) ) $=
|
|
( wreu wa wrex cv wcel wi weu wex df-reu df-rex anass exbii bitr4i eupick
|
|
syl2anb exp3a 3impia ) ACDEZABFZCDGZCHDIZABJUBUDFUEABUBUEAFZCKUFBFZCLZUFB
|
|
JUDACDMUDUEUCFZCLUHUCCDNUGUICUEABOPQUFBCRSTUA $.
|
|
|
|
$( Restricted uniqueness "picks" a member of a subclass. (Contributed by
|
|
Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro,
|
|
19-Nov-2016.) $)
|
|
reupick2 $p |- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\
|
|
E! x e. A ph ) /\ x e. A ) -> ( ph <-> ps ) ) $=
|
|
( wi wral wrex wreu w3a cv wcel ancr ralimi rexim syl reupick3 3exp com12
|
|
wa syl6 3imp1 rsp 3ad2ant1 imp impbid ) BAEZCDFZBCDGZACDHZIZCJDKZSABUGUHU
|
|
IUKABEZUGUHABSZCDGZUIUKULEZEUGBUMEZCDFUHUNEUFUPCDBALMBUMCDNOUIUNUOUIUNUKU
|
|
LABCDPQRTUAUJUKUFUGUHUKUFEUIUFCDUBUCUDUE $.
|
|
$}
|
|
|
|
$( Symmetric difference commutes. (Contributed by SF, 11-Jan-2015.) $)
|
|
symdifcom $p |- ( A (+) B ) = ( B (+) A ) $=
|
|
( cdif cun csymdif uncom df-symdif 3eqtr4i ) ABCZBACZDJIDABEBAEIJFABGBAGH
|
|
$.
|
|
|
|
$( Two classes are equal iff their complements are equal. (Contributed by
|
|
SF, 11-Jan-2015.) $)
|
|
compleqb $p |- ( A = B <-> ~ A = ~ B ) $=
|
|
( wceq ccompl compleq dblcompl 3eqtr3g impbii ) ABCADZBDZCZABEKIDJDABIJEAFB
|
|
FGH $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( A class is not equal to its complement. (Contributed by SF,
|
|
11-Jan-2015.) $)
|
|
necompl $p |- ~ A =/= A $=
|
|
( vx ccompl wne cv wcel wb wn wex pm5.19 elcompl bibi2i mtbir 19.8a ax-mp
|
|
vex wal dfcleq necon3abii exnal bitr4i mpbir necomi ) AACZAUDDZBEZAFZUFUD
|
|
FZGZHZBIZUJUKUIUGUGHZGUGJUHULUGUFABPKLMUJBNOUEUIBQZHUKUMAUDBAUDRSUIBTUAUB
|
|
UC $.
|
|
$}
|
|
|
|
$( Definition of intersection in terms of union. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
dfin5 $p |- ( A i^i B ) = ~ ( ~ A u. ~ B ) $=
|
|
( ccompl cnin cun cin dblcompl nineq12i compleqi df-un df-in 3eqtr4ri ) ACZ
|
|
CZBCZCZDZCABDZCMOEZCABFQRNAPBAGBGHISQMOJIABKL $.
|
|
|
|
$( Definition of union in terms of intersection. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
dfun4 $p |- ( A u. B ) = ~ ( ~ A i^i ~ B ) $=
|
|
( ccompl cin cun dfin5 compleqi dblcompl uneq12i 3eqtrri ) ACZBCZDZCKCZLCZE
|
|
ZCZCPABEMQKLFGPHNAOBAHBHIJ $.
|
|
|
|
$( Intersection of two complements is equal to the complement of a union.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
iunin $p |- ~ ( A u. B ) = ( ~ A i^i ~ B ) $=
|
|
( ccompl cin cun dfin5 dblcompl uneq12i compleqi eqtr2i ) ACZBCZDKCZLCZEZCA
|
|
BEZCKLFOPMANBAGBGHIJ $.
|
|
|
|
$( Complement of intersection is equal to union of complements. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
iinun $p |- ~ ( A i^i B ) = ( ~ A u. ~ B ) $=
|
|
( ccompl cun cin dfun4 dblcompl ineq12i compleqi eqtr2i ) ACZBCZDKCZLCZEZCA
|
|
BEZCKLFOPMANBAGBGHIJ $.
|
|
|
|
$( A difference is a subset of the complement of its second argument.
|
|
(Contributed by SF, 10-Mar-2015.) $)
|
|
difsscompl $p |- ( A \ B ) C_ ~ B $=
|
|
( cdif ccompl cin df-dif inss2 eqsstri ) ABCABDZEIABFAIGH $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The empty set
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the symbol for the empty or null set. $)
|
|
$c (/) $. $( null set $)
|
|
|
|
$( Extend class notation to include the empty set. $)
|
|
c0 $a class (/) $.
|
|
|
|
$( Define the empty set. Special case of Exercise 4.10(o) of [Mendelson]
|
|
p. 231. For a more traditional definition, but requiring a dummy
|
|
variable, see ~ dfnul2 . (Contributed by NM, 5-Aug-1993.) $)
|
|
df-nul $a |- (/) = ( _V \ _V ) $.
|
|
|
|
$( Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring]
|
|
p. 20. (Contributed by NM, 26-Dec-1996.) $)
|
|
dfnul2 $p |- (/) = { x | -. x = x } $=
|
|
( cv wceq wn c0 wcel cvv cdif wa df-nul eleq2i eldif eqid pm3.24 2th 3bitri
|
|
con2bii abbi2i ) ABZSCZDZAESEFSGGHZFSGFZUCDIZUAEUBSJKSGGLTUDTUDDSMUCNOQPR
|
|
$.
|
|
|
|
$( Alternate definition of the empty set. (Contributed by NM,
|
|
25-Mar-2004.) $)
|
|
dfnul3 $p |- (/) = { x e. A | -. x e. A } $=
|
|
( cv wceq wn cab wcel wa crab pm3.24 eqid 2th con1bii dfnul2 df-rab 3eqtr4i
|
|
c0 abbii ) ACZSDZEZAFSBGZUBEZHZAFQUCABIUAUDAUDTUDETUBJSKLMRANUCABOP $.
|
|
|
|
$( The empty set has no elements. Theorem 6.14 of [Quine] p. 44.
|
|
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro,
|
|
1-Sep-2015.) $)
|
|
noel $p |- -. A e. (/) $=
|
|
( c0 wcel cvv cdif eldifi eldifn pm2.65i df-nul eleq2i mtbir ) ABCADDEZCZMA
|
|
DCADDFADDGHBLAIJK $.
|
|
|
|
$( If a set has elements, it is not empty. (Contributed by NM,
|
|
31-Dec-1993.) $)
|
|
n0i $p |- ( B e. A -> -. A = (/) ) $=
|
|
( c0 wceq wcel noel eleq2 mtbiri con2i ) ACDZBAEZJKBCEBFACBGHI $.
|
|
|
|
$( If a set has elements, it is not empty. (Contributed by NM,
|
|
31-Dec-1993.) $)
|
|
ne0i $p |- ( B e. A -> A =/= (/) ) $=
|
|
( wcel c0 wceq wn wne n0i df-ne sylibr ) BACADEFADGABHADIJ $.
|
|
|
|
$( The universal class is not equal to the empty set. (Contributed by NM,
|
|
11-Sep-2008.) $)
|
|
vn0 $p |- _V =/= (/) $=
|
|
( vx cv cvv wcel c0 wne vex ne0i ax-mp ) ABZCDCEFAGCJHI $.
|
|
|
|
${
|
|
n0f.1 $e |- F/_ x A $.
|
|
$( A nonempty class has at least one element. Proposition 5.17(1) of
|
|
[TakeutiZaring] p. 20. This version of ~ n0 requires only that ` x `
|
|
not be free in, rather than not occur in, ` A ` . (Contributed by NM,
|
|
17-Oct-2003.) $)
|
|
n0f $p |- ( A =/= (/) <-> E. x x e. A ) $=
|
|
( c0 wne cv wcel wn wal wex wceq wb nfcv cleqf noel nbn bitr4i necon3abii
|
|
albii df-ex ) BDEAFZBGZHZAIZHUBAJUDBDBDKUBUADGZLZAIUDABDCADMNUCUFAUEUBUAO
|
|
PSQRUBATQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A nonempty class has at least one element. Proposition 5.17(1) of
|
|
[TakeutiZaring] p. 20. (Contributed by NM, 29-Sep-2006.) $)
|
|
n0 $p |- ( A =/= (/) <-> E. x x e. A ) $=
|
|
( nfcv n0f ) ABABCD $.
|
|
|
|
$( A nonempty class has at least one element. Proposition 5.17(1) of
|
|
[TakeutiZaring] p. 20. (Contributed by NM, 5-Aug-1993.) $)
|
|
neq0 $p |- ( -. A = (/) <-> E. x x e. A ) $=
|
|
( c0 wceq wn wne cv wcel wex df-ne n0 bitr3i ) BCDEBCFAGBHAIBCJABKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
reximdva0.1 $e |- ( ( ph /\ x e. A ) -> ps ) $.
|
|
$( Restricted existence deduced from non-empty class. (Contributed by NM,
|
|
1-Feb-2012.) $)
|
|
reximdva0 $p |- ( ( ph /\ A =/= (/) ) -> E. x e. A ps ) $=
|
|
( c0 wne wa cv wcel wex wrex n0 ex ancld eximdv imp sylan2b df-rex sylibr
|
|
) ADFGZHCIDJZBHZCKZBCDLUAAUBCKZUDCDMAUEUDAUBUCCAUBBAUBBENOPQRBCDST $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( A case of equivalence of "at most one" and "only one". (Contributed by
|
|
FL, 6-Dec-2010.) $)
|
|
n0moeu $p |- ( A =/= (/) -> ( E* x x e. A <-> E! x x e. A ) ) $=
|
|
( c0 wne cv wcel wmo wex wa weu n0 biimpi biantrurd eu5 syl6bbr ) BCDZAEB
|
|
FZAGZQAHZRIQAJPSRPSABKLMQANO $.
|
|
$}
|
|
|
|
$( Vacuous existential quantification is false. (Contributed by NM,
|
|
15-Oct-2003.) $)
|
|
rex0 $p |- -. E. x e. (/) ph $=
|
|
( c0 cv wcel wn noel pm2.21i nrex ) ABCBDZCEAFJGHI $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The empty set has no elements. Theorem 2 of [Suppes] p. 22.
|
|
(Contributed by NM, 29-Aug-1993.) $)
|
|
eq0 $p |- ( A = (/) <-> A. x -. x e. A ) $=
|
|
( c0 wceq cv wcel wn wal wex neq0 df-ex bitri con4bii ) BCDZAEBFZGAHZNGOA
|
|
IPGABJOAKLM $.
|
|
|
|
$( The universe contains every set. (Contributed by NM, 11-Sep-2006.) $)
|
|
eqv $p |- ( A = _V <-> A. x x e. A ) $=
|
|
( cvv wceq cv wcel wb wal dfcleq vex tbt albii bitr4i ) BCDAEZBFZNCFZGZAH
|
|
OAHABCIOQAPOAJKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y $.
|
|
$( Membership of the empty set in another class. (Contributed by NM,
|
|
29-Jun-2004.) $)
|
|
0el $p |- ( (/) e. A <-> E. x e. A A. y -. y e. x ) $=
|
|
( c0 wcel cv wceq wrex wn wal risset eq0 rexbii bitri ) DCEAFZDGZACHBFOEI
|
|
BJZACHADCKPQACBOLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
$( The class builder of a wff not containing the abstraction variable is
|
|
either the universal class or the empty set. (Contributed by Mario
|
|
Carneiro, 29-Aug-2013.) $)
|
|
abvor0 $p |- ( { x | ph } = _V \/ { x | ph } = (/) ) $=
|
|
( cab cvv wceq c0 wn cv wcel vex a1i 2thd abbi1dv con3i noel 2falsed orri
|
|
id syl ) ABCZDEZTFEZUAGAGZUBAUAAABDAABHZDIZARUEABJKLMNUCABFUCAUDFIZUCRUFG
|
|
UCUDOKPMSQ $.
|
|
$}
|
|
|
|
$( Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof
|
|
shortened by Mario Carneiro, 11-Nov-2016.) $)
|
|
abn0 $p |- ( { x | ph } =/= (/) <-> E. x ph ) $=
|
|
( cab c0 wne cv wcel wex nfab1 n0f abid exbii bitri ) ABCZDEBFNGZBHABHBNABI
|
|
JOABABKLM $.
|
|
|
|
$( Empty class abstraction. (Contributed by SF, 5-Jan-2018.) $)
|
|
ab0 $p |- ( { x | ph } = (/) <-> A. x -. ph ) $=
|
|
( cab c0 wceq wn wal wne wex abn0 df-ne df-ex 3bitr3i con4bii ) ABCZDEZAFBG
|
|
ZODHABIPFQFABJODKABLMN $.
|
|
|
|
$( Non-empty restricted class abstraction. (Contributed by NM,
|
|
29-Aug-1999.) $)
|
|
rabn0 $p |- ( { x e. A | ph } =/= (/) <-> E. x e. A ph ) $=
|
|
( cv wcel wa cab c0 wne wex crab wrex abn0 df-rab neeq1i df-rex 3bitr4i ) B
|
|
DCEAFZBGZHIRBJABCKZHIABCLRBMTSHABCNOABCPQ $.
|
|
|
|
$( Any restricted class abstraction restricted to the empty set is empty.
|
|
(Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
rab0 $p |- { x e. (/) | ph } = (/) $=
|
|
( cv c0 wcel wa cab wceq wn crab equid noel intnanr 2th abbii df-rab dfnul2
|
|
con2bii 3eqtr4i ) BCZDEZAFZBGTTHZIZBGABDJDUBUDBUCUBUCUBIBKUAATLMNROABDPBQS
|
|
$.
|
|
|
|
$( Condition for a restricted class abstraction to be empty. (Contributed by
|
|
Jeff Madsen, 7-Jun-2010.) $)
|
|
rabeq0 $p |- ( { x e. A | ph } = (/) <-> A. x e. A -. ph ) $=
|
|
( wn wral wrex crab c0 wceq ralnex rabn0 necon1bbii bitr2i ) ADBCEABCFZDABC
|
|
GZHIABCJNOHABCKLM $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( Law of excluded middle, in terms of restricted class abstractions.
|
|
(Contributed by Jeff Madsen, 20-Jun-2011.) $)
|
|
rabxm $p |- A = ( { x e. A | ph } u. { x e. A | -. ph } ) $=
|
|
( wn wo crab cun wceq rabid2 cv wcel exmid a1i mprgbir unrab eqtr4i ) CAA
|
|
DZEZBCFZABCFQBCFGCSHRBCRBCIRBJCKALMNAQBCOP $.
|
|
|
|
$( Law of noncontradiction, in terms of restricted class abstractions.
|
|
(Contributed by Jeff Madsen, 20-Jun-2011.) $)
|
|
rabnc $p |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/) $=
|
|
( crab wn cin wa c0 inrab wceq rabeq0 cv wcel pm3.24 a1i mprgbir eqtri )
|
|
ABCDAEZBCDFARGZBCDZHARBCITHJSEZBCSBCKUABLCMANOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The union of a class with the empty set is itself. Theorem 24 of
|
|
[Suppes] p. 27. (Contributed by NM, 5-Aug-1993.) $)
|
|
un0 $p |- ( A u. (/) ) = A $=
|
|
( vx c0 cv wcel wo noel biorfi bicomi uneqri ) BACABDZAEZLKCEZFMLKGHIJ $.
|
|
|
|
$( The intersection of a class with the empty set is the empty set.
|
|
Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.) $)
|
|
in0 $p |- ( A i^i (/) ) = (/) $=
|
|
( vx c0 cv wcel wa noel bianfi bicomi ineqri ) BACCBDZCEZKAEZLFLMKGHIJ $.
|
|
$}
|
|
|
|
$( The intersection of a class with the universal class is itself. Exercise
|
|
4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.) $)
|
|
inv1 $p |- ( A i^i _V ) = A $=
|
|
( cvv cin inss1 ssid ssv ssini eqssi ) ABCAABDAABAEAFGH $.
|
|
|
|
$( The union of a class with the universal class is the universal class.
|
|
Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM,
|
|
17-May-1998.) $)
|
|
unv $p |- ( A u. _V ) = _V $=
|
|
( cvv cun ssv ssun2 eqssi ) ABCZBGDBAEF $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( The null set is a subset of any class. Part of Exercise 1 of
|
|
[TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.) $)
|
|
0ss $p |- (/) C_ A $=
|
|
( vx c0 cv wcel noel pm2.21i ssriv ) BCABDZCEIAEIFGH $.
|
|
$}
|
|
|
|
$( Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its
|
|
converse. (Contributed by NM, 17-Sep-2003.) $)
|
|
ss0b $p |- ( A C_ (/) <-> A = (/) ) $=
|
|
( c0 wceq wss 0ss eqss mpbiran2 bicomi ) ABCZABDZIJBADAEABFGH $.
|
|
|
|
$( Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23.
|
|
(Contributed by NM, 13-Aug-1994.) $)
|
|
ss0 $p |- ( A C_ (/) -> A = (/) ) $=
|
|
( c0 wss wceq ss0b biimpi ) ABCABDAEF $.
|
|
|
|
$( A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
sseq0 $p |- ( ( A C_ B /\ B = (/) ) -> A = (/) ) $=
|
|
( c0 wceq wss sseq2 ss0 syl6bi impcom ) BCDZABEZACDZJKACELBCAFAGHI $.
|
|
|
|
$( A class with a nonempty subclass is nonempty. (Contributed by NM,
|
|
17-Feb-2007.) $)
|
|
ssn0 $p |- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) ) $=
|
|
( wss c0 wne wceq sseq0 ex necon3d imp ) ABCZADEBDEKBDADKBDFADFABGHIJ $.
|
|
|
|
${
|
|
abf.1 $e |- -. ph $.
|
|
$( A class builder with a false argument is empty. (Contributed by NM,
|
|
20-Jan-2012.) $)
|
|
abf $p |- { x | ph } = (/) $=
|
|
( cab c0 wss wceq cv wcel pm2.21i abssi ss0 ax-mp ) ABDZEFNEGABEABHEICJKN
|
|
LM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
eq0rdv.1 $e |- ( ph -> -. x e. A ) $.
|
|
$( Deduction rule for equality to the empty set. (Contributed by NM,
|
|
11-Jul-2014.) $)
|
|
eq0rdv $p |- ( ph -> A = (/) ) $=
|
|
( c0 wss wceq cv wcel pm2.21d ssrdv ss0 syl ) ACEFCEGABCEABHZCINEIDJKCLM
|
|
$.
|
|
$}
|
|
|
|
$( Two classes are empty iff their union is empty. (Contributed by NM,
|
|
11-Aug-2004.) $)
|
|
un00 $p |- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) ) $=
|
|
( c0 wceq wa cun uneq12 un0 syl6eq wss ssun1 sseq2 mpbii sylib ssun2 impbii
|
|
ss0b jca ) ACDZBCDZEZABFZCDZUAUBCCFCACBCGCHIUCSTUCACJZSUCAUBJUDABKUBCALMAQN
|
|
UCBCJZTUCBUBJUEBAOUBCBLMBQNRP $.
|
|
|
|
$( Only the universal class has the universal class as a subclass.
|
|
(Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
vss $p |- ( _V C_ A <-> A = _V ) $=
|
|
( cvv wss wa wceq ssv biantrur eqss bitr4i ) BACZABCZJDABEKJAFGABHI $.
|
|
|
|
$( The null set is a proper subset of any non-empty set. (Contributed by NM,
|
|
27-Feb-1996.) $)
|
|
0pss $p |- ( (/) C. A <-> A =/= (/) ) $=
|
|
( c0 wpss wne wss 0ss df-pss mpbiran necom bitri ) BACZBADZABDKBAELAFBAGHBA
|
|
IJ $.
|
|
|
|
$( No set is a proper subset of the empty set. (Contributed by NM,
|
|
17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
npss0 $p |- -. A C. (/) $=
|
|
( c0 wpss wss wn wa wi 0ss a1i iman mpbi dfpss3 mtbir ) ABCABDZBADZEFZNOGPE
|
|
ONAHINOJKABLM $.
|
|
|
|
$( Any non-universal class is a proper subclass of the universal class.
|
|
(Contributed by NM, 17-May-1998.) $)
|
|
pssv $p |- ( A C. _V <-> -. A = _V ) $=
|
|
( cvv wpss wss wceq wn ssv dfpss2 mpbiran ) ABCABDABEFAGABHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Two ways of saying that two classes are disjoint (have no members in
|
|
common). (Contributed by NM, 17-Feb-2004.) $)
|
|
disj $p |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B ) $=
|
|
( cv cin wcel wn wal wi c0 wceq wral wa df-an bitr2i con1bii albii df-ral
|
|
elin eq0 3bitr4i ) ADZBCEZFZGZAHUBBFZUBCFZGZIZAHUCJKUHABLUEUIAUIUDUDUFUGM
|
|
UIGUBBCSUFUGNOPQAUCTUHABRUA $.
|
|
|
|
$( Two ways of saying that two classes are disjoint. (Contributed by Jeff
|
|
Madsen, 19-Jun-2011.) $)
|
|
disjr $p |- ( ( A i^i B ) = (/) <-> A. x e. B -. x e. A ) $=
|
|
( cin c0 wceq cv wcel wn wral incom eqeq1i disj bitri ) BCDZEFCBDZEFAGBHI
|
|
ACJOPEBCKLACBMN $.
|
|
|
|
$( Two ways of saying that two classes are disjoint (have no members in
|
|
common). (Contributed by NM, 19-Aug-1993.) $)
|
|
disj1 $p |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) ) $=
|
|
( cin c0 wceq cv wcel wn wral wi wal disj df-ral bitri ) BCDEFAGZCHIZABJP
|
|
BHQKALABCMQABNO $.
|
|
|
|
$( Two ways of saying that two classes are disjoint, using the complement
|
|
of ` B ` relative to a universe ` C ` . (Contributed by NM,
|
|
15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
reldisj $p |- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) ) $=
|
|
( vx wss cv wcel wn wi wal cdif cin c0 wceq wb dfss2 pm5.44 eldif imbi2i
|
|
wa syl6bbr sps sylbi albidv disj1 3bitr4g ) ACEZDFZAGZUHBGHZIZDJUIUHCBKZG
|
|
ZIZDJABLMNAULEUGUKUNDUGUIUHCGZIZDJUKUNOZDACPUPUQDUPUKUIUOUJTZIUNUIUOUJQUM
|
|
URUIUHCBRSUAUBUCUDDABUEDAULPUF $.
|
|
|
|
$( Two ways of saying that two classes are disjoint. (Contributed by NM,
|
|
19-May-1998.) $)
|
|
disj3 $p |- ( ( A i^i B ) = (/) <-> A = ( A \ B ) ) $=
|
|
( vx cv wcel wn wi wal cdif wb cin c0 wa pm4.71 eldif bibi2i bitr4i albii
|
|
wceq disj1 dfcleq 3bitr4i ) CDZAEZUCBEFZGZCHUDUCABIZEZJZCHABKLSAUGSUFUICU
|
|
FUDUDUEMZJUIUDUENUHUJUDUCABOPQRCABTCAUGUAUB $.
|
|
|
|
$( Members of disjoint sets are not equal. (Contributed by NM,
|
|
28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
disjne $p |- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D ) $=
|
|
( vx cin c0 wceq wcel wne cv wn wral wi disj eleq1 notbid rspccva eleq1a
|
|
wa necon3bd syl5com sylanb 3impia ) ABFGHZCAIZDBIZCDJZUEEKZBIZLZEAMZUFUGU
|
|
HNEABOULUFTCBIZLZUGUHUKUNECAUICHUJUMUICBPQRUGUMCDDBCSUAUBUCUD $.
|
|
$}
|
|
|
|
$( A set can't belong to both members of disjoint classes. (Contributed by
|
|
NM, 28-Feb-2015.) $)
|
|
disjel $p |- ( ( ( A i^i B ) = (/) /\ C e. A ) -> -. C e. B ) $=
|
|
( cin c0 wceq wcel wn cdif wi disj3 eleq2 eldifn syl6bi sylbi imp ) ABDEFZC
|
|
AGZCBGHZQAABIZFZRSJABKUARCTGSATCLCABMNOP $.
|
|
|
|
$( Two ways of saying that two classes are disjoint. (Contributed by NM,
|
|
17-May-1998.) $)
|
|
disj2 $p |- ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) ) $=
|
|
( cvv wss cin c0 wceq cdif wb ssv reldisj ax-mp ) ACDABEFGACBHDIAJABCKL $.
|
|
|
|
$( Two ways of saying that two classes are disjoint. (Contributed by NM,
|
|
21-Mar-2004.) $)
|
|
disj4 $p |- ( ( A i^i B ) = (/) <-> -. ( A \ B ) C. A ) $=
|
|
( cin c0 wceq cdif wpss disj3 eqcom wss difss dfpss2 mpbiran con2bii 3bitri
|
|
wn ) ABCDEAABFZEQAEZQAGZPABHAQISRSQAJRPABKQALMNO $.
|
|
|
|
$( Intersection with a subclass of a disjoint class. (Contributed by FL,
|
|
24-Jan-2007.) $)
|
|
ssdisj $p |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) = (/) ) $=
|
|
( wss cin c0 wceq wa ss0b wi ssrin sstr2 syl syl5bir imp ss0 ) ABDZBCEZFGZH
|
|
ACEZFDZTFGQSUASRFDZQUARIQTRDUBUAJABCKTRFLMNOTPM $.
|
|
|
|
$( A class is a proper subset of its union with a disjoint nonempty class.
|
|
(Contributed by NM, 15-Sep-2004.) $)
|
|
disjpss $p |- ( ( ( A i^i B ) = (/) /\ B =/= (/) ) -> A C. ( A u. B ) ) $=
|
|
( cin c0 wceq wne wa wss wn cun wpss ssid biantru bitri sseq2 syl5bb syl6bi
|
|
ssin ss0 necon3ad imp nsspssun uncom psseq2i sylib ) ABCZDEZBDFZGBAHZIZAABJ
|
|
ZKZUGUHUJUGUIBDUGUIBDHZBDEUIBUFHZUGUMUIUIBBHZGUNUOUIBLMBABRNUFDBOPBSQTUAUJA
|
|
BAJZKULBAUBUPUKABAUCUDNUE $.
|
|
|
|
$( The union of disjoint classes is disjoint. (Contributed by NM,
|
|
26-Sep-2004.) $)
|
|
undisj1 $p |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <->
|
|
( ( A u. B ) i^i C ) = (/) ) $=
|
|
( cin c0 wceq wa cun un00 indir eqeq1i bitr4i ) ACDZEFBCDZEFGMNHZEFABHCDZEF
|
|
MNIPOEABCJKL $.
|
|
|
|
$( The union of disjoint classes is disjoint. (Contributed by NM,
|
|
13-Sep-2004.) $)
|
|
undisj2 $p |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <->
|
|
( A i^i ( B u. C ) ) = (/) ) $=
|
|
( cin c0 wceq wa cun un00 indi eqeq1i bitr4i ) ABDZEFACDZEFGMNHZEFABCHDZEFM
|
|
NIPOEABCJKL $.
|
|
|
|
$( Subclass expressed in terms of intersection with difference from the
|
|
universal class. (Contributed by NM, 17-Sep-2003.) $)
|
|
ssindif0 $p |- ( A C_ B <-> ( A i^i ( _V \ B ) ) = (/) ) $=
|
|
( cvv cdif cin c0 wceq wss disj2 ddif sseq2i bitr2i ) ACBDZEFGACMDZHABHAMIN
|
|
BABJKL $.
|
|
|
|
$( The intersection of classes with a common member is nonempty.
|
|
(Contributed by NM, 7-Apr-1994.) $)
|
|
inelcm $p |- ( ( A e. B /\ A e. C ) -> ( B i^i C ) =/= (/) ) $=
|
|
( wcel wa cin c0 wne elin ne0i sylbir ) ABDACDEABCFZDLGHABCILAJK $.
|
|
|
|
$( A minimum element of a class has no elements in common with the class.
|
|
(Contributed by NM, 22-Jun-1994.) $)
|
|
minel $p |- ( ( A e. B /\ ( C i^i B ) = (/) ) -> -. A e. C ) $=
|
|
( cin c0 wceq wcel wn wa wi inelcm necon2bi imnan sylibr con2d impcom ) CBD
|
|
ZEFZABGZACGZHRTSRTSIZHTSHJUAQEACBKLTSMNOP $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Distribute union over difference. (Contributed by NM, 17-May-1998.)
|
|
(Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
undif4 $p |- ( ( A i^i C ) = (/) ->
|
|
( A u. ( B \ C ) ) = ( ( A u. B ) \ C ) ) $=
|
|
( vx cv wcel wn wi wal cdif cun wb cin c0 wceq wo wa eldif elun 3bitr4g
|
|
pm2.621 olc impbid1 anbi2d orbi2i bitri anbi1i alimi disj1 dfcleq 3imtr4i
|
|
ordi ) DEZAFZUMCFGZHZDIUMABCJZKZFZUMABKZCJZFZLZDIACMNOURVAOUPVCDUPUNUMUQF
|
|
ZPZUMUTFZUOQZUSVBUPUNUMBFZPZUNUOPZQZVIUOQVEVGUPVJUOVIUPVJUOUNUOUAUOUNUBUC
|
|
UDVEUNVHUOQZPVKVDVLUNUMBCRUEUNVHUOULUFVFVIUOUMABSUGTUMAUQSUMUTCRTUHDACUID
|
|
URVAUJUK $.
|
|
|
|
$( Subset relation for disjoint classes. (Contributed by NM,
|
|
25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
disjssun $p |- ( ( A i^i B ) = (/) -> ( A C_ ( B u. C ) <-> A C_ C ) ) $=
|
|
( cin c0 wceq cun wss indi equncomi uneq2 un0 syl6eq syl5eq df-ss 3bitr4g
|
|
eqeq1d ) ABDZEFZABCGZDZAFACDZAFATHACHSUAUBASUAUBRGZUBUARUBABCIJSUCUBEGUBR
|
|
EUBKUBLMNQATOACOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass expressed in terms of difference. Exercise 7 of
|
|
[TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.) $)
|
|
ssdif0 $p |- ( A C_ B <-> ( A \ B ) = (/) ) $=
|
|
( vx cv wcel wi wal cdif wn wss c0 wceq wa eldif xchbinxr albii dfss2 eq0
|
|
iman 3bitr4i ) CDZAEZUABEZFZCGUAABHZEZIZCGABJUEKLUDUGCUDUBUCIMUFUBUCSUAAB
|
|
NOPCABQCUERT $.
|
|
$}
|
|
|
|
$( Universal class equality in terms of empty difference. (Contributed by
|
|
NM, 17-Sep-2003.) $)
|
|
vdif0 $p |- ( A = _V <-> ( _V \ A ) = (/) ) $=
|
|
( cvv wceq wss cdif c0 vss ssdif0 bitr3i ) ABCBADBAEFCAGBAHI $.
|
|
|
|
$( A proper subclass has a nonempty difference. (Contributed by NM,
|
|
3-May-1994.) $)
|
|
pssdifn0 $p |- ( ( A C_ B /\ A =/= B ) -> ( B \ A ) =/= (/) ) $=
|
|
( wss wne cdif c0 wceq ssdif0 eqss simplbi2 syl5bir necon3d imp ) ABCZABDBA
|
|
EZFDNOFABOFGBACZNABGZBAHQNPABIJKLM $.
|
|
|
|
$( A proper subclass has a nonempty difference. (Contributed by Mario
|
|
Carneiro, 27-Apr-2016.) $)
|
|
pssdif $p |- ( A C. B -> ( B \ A ) =/= (/) ) $=
|
|
( wpss wss wne wa cdif c0 df-pss pssdifn0 sylbi ) ABCABDABEFBAGHEABIABJK $.
|
|
|
|
$( A subclass missing a member is a proper subclass. (Contributed by NM,
|
|
12-Jan-2002.) $)
|
|
ssnelpss $p |- ( A C_ B -> ( ( C e. B /\ -. C e. A ) -> A C. B ) ) $=
|
|
( wcel wn wa wceq wss wpss nelneq2 eqcom sylnib dfpss2 baibr syl5ib ) CBDCA
|
|
DEFZABGZEZABHZABIZPBAGQCBAJBAKLTSRABMNO $.
|
|
|
|
${
|
|
ssnelpssd.1 $e |- ( ph -> A C_ B ) $.
|
|
ssnelpssd.2 $e |- ( ph -> C e. B ) $.
|
|
ssnelpssd.3 $e |- ( ph -> -. C e. A ) $.
|
|
$( Subclass inclusion with one element of the superclass missing is proper
|
|
subclass inclusion. Deduction form of ~ ssnelpss . (Contributed by
|
|
David Moews, 1-May-2017.) $)
|
|
ssnelpssd $p |- ( ph -> A C. B ) $=
|
|
( wcel wn wpss wss wa wi ssnelpss syl mp2and ) ADCHZDBHIZBCJZFGABCKQRLSME
|
|
BCDNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A proper subclass has a member in one argument that's not in both.
|
|
(Contributed by NM, 29-Feb-1996.) $)
|
|
pssnel $p |- ( A C. B -> E. x ( x e. B /\ -. x e. A ) ) $=
|
|
( wpss cv cdif wcel wex wn wa c0 wne pssdif n0 sylib eldif exbii ) BCDZAE
|
|
ZCBFZGZAHZSCGSBGIJZAHRTKLUBBCMATNOUAUCASCBPQO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Difference, intersection, and subclass relationship. (Contributed by
|
|
NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) $)
|
|
difin0ss $p |- ( ( ( A \ B ) i^i C ) = (/) -> ( C C_ A -> C C_ B ) ) $=
|
|
( vx cdif cin c0 wceq cv wcel wn wal wss wi eq0 wa iman elin eldif dfss2
|
|
anbi1i bitri ancom annim anbi2i 3bitr2i xchbinxr ax-2 sylbir al2imi sylbi
|
|
3imtr4g ) ABEZCFZGHDIZUNJZKZDLZCAMZCBMZNDUNOURUOCJZUOAJZNZDLVAUOBJZNZDLUS
|
|
UTUQVCVEDUQVAVBVDNZNZVCVENVGVAVFKZPZUPVAVFQUPVBVDKPZVAPZVAVJPVIUPUOUMJZVA
|
|
PVKUOUMCRVLVJVAUOABSUAUBVAVJUCVJVHVAVBVDUDUEUFUGVAVBVDUHUIUJDCATDCBTULUK
|
|
$.
|
|
|
|
$( Intersection, subclass, and difference relationship. (Contributed by
|
|
NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
|
(Proof shortened by Wolf Lammen, 30-Sep-2014.) $)
|
|
inssdif0 $p |- ( ( A i^i B ) C_ C <-> ( A i^i ( B \ C ) ) = (/) ) $=
|
|
( vx cv cin wcel wi wal cdif wn wss c0 wceq elin imbi1i iman bitri eldif
|
|
wa anbi2i anass 3bitr4ri xchbinx albii dfss2 eq0 3bitr4i ) DEZABFZGZUICGZ
|
|
HZDIUIABCJZFZGZKZDIUJCLUOMNUMUQDUMUIAGZUIBGZTZULKZTZUPUMUTULHVBKUKUTULUIA
|
|
BOPUTULQRURUIUNGZTURUSVATZTUPVBVCVDURUIBCSUAUIAUNOURUSVAUBUCUDUEDUJCUFDUO
|
|
UGUH $.
|
|
$}
|
|
|
|
$( The difference between a class and itself is the empty set. Proposition
|
|
5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28.
|
|
(Contributed by NM, 22-Apr-2004.) $)
|
|
difid $p |- ( A \ A ) = (/) $=
|
|
( wss cdif c0 wceq ssid ssdif0 mpbi ) AABAACDEAFAAGH $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The difference between a class and itself is the empty set. Proposition
|
|
5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28.
|
|
Alternate proof of ~ difid . (Contributed by David Abernethy,
|
|
17-Jun-2012.) (Proof modification is discouraged.)
|
|
(New usage is discouraged.) $)
|
|
difidALT $p |- ( A \ A ) = (/) $=
|
|
( vx cdif cv wcel wn crab c0 dfdif2 dfnul3 eqtr4i ) AACBDAEFBAGHBAAIBAJK
|
|
$.
|
|
$}
|
|
|
|
$( The difference between a class and the empty set. Part of Exercise 4.4 of
|
|
[Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) $)
|
|
dif0 $p |- ( A \ (/) ) = A $=
|
|
( cdif c0 difid difeq2i difdif eqtr3i ) AAABZBACBAHCAADEAAFG $.
|
|
|
|
$( The difference between the empty set and a class. Part of Exercise 4.4 of
|
|
[Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) $)
|
|
0dif $p |- ( (/) \ A ) = (/) $=
|
|
( c0 cdif wss wceq difss ss0 ax-mp ) BACZBDIBEBAFIGH $.
|
|
|
|
$( A class and its relative complement are disjoint. Theorem 38 of [Suppes]
|
|
p. 29. (Contributed by NM, 24-Mar-1998.) $)
|
|
disjdif $p |- ( A i^i ( B \ A ) ) = (/) $=
|
|
( cin wss cdif c0 wceq inss1 inssdif0 mpbi ) ABCADABAECFGABHABAIJ $.
|
|
|
|
$( The difference of a class from its intersection is empty. Theorem 37 of
|
|
[Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by
|
|
Andrew Salmon, 26-Jun-2011.) $)
|
|
difin0 $p |- ( ( A i^i B ) \ B ) = (/) $=
|
|
( cin wss cdif c0 wceq inss2 ssdif0 mpbi ) ABCZBDKBEFGABHKBIJ $.
|
|
|
|
$( The union of a class and its complement is the universe. Theorem 5.1(5)
|
|
of [Stoll] p. 17. (Contributed by NM, 17-Aug-2004.) $)
|
|
undifv $p |- ( A u. ( _V \ A ) ) = _V $=
|
|
( cvv cdif cun cin c0 dfun3 disjdif difeq2i dif0 3eqtri ) ABACZDBLBLCEZCBFC
|
|
BALGMFBLBHIBJK $.
|
|
|
|
$( Absorption of difference by union. This decomposes a union into two
|
|
disjoint classes (see ~ disjdif ). Theorem 35 of [Suppes] p. 29.
|
|
(Contributed by NM, 19-May-1998.) $)
|
|
undif1 $p |- ( ( A \ B ) u. B ) = ( A u. B ) $=
|
|
( cvv cdif cin undir invdif uneq1i uncom undifv eqtri ineq2i inv1 3eqtr3i
|
|
cun ) ACBDZEZBOABOZPBOZEZABDZBORAPBFQUABABGHTRCERSCRSBPOCPBIBJKLRMKN $.
|
|
|
|
$( Absorption of difference by union. This decomposes a union into two
|
|
disjoint classes (see ~ disjdif ). Part of proof of Corollary 6K of
|
|
[Enderton] p. 144. (Contributed by NM, 19-May-1998.) $)
|
|
undif2 $p |- ( A u. ( B \ A ) ) = ( A u. B ) $=
|
|
( cdif cun uncom undif1 3eqtri ) ABACZDHADBADABDAHEBAFBAEG $.
|
|
|
|
$( Absorption of difference by union. (Contributed by NM, 18-Aug-2013.) $)
|
|
undifabs $p |- ( A u. ( A \ B ) ) = A $=
|
|
( cdif cun undif3 unidm difeq1i difdif 3eqtri ) AABCDAADZBACZCAKCAAABEJAKAF
|
|
GABHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The intersection and class difference of a class with another class
|
|
unite to give the original class. (Contributed by Paul Chapman,
|
|
5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
inundif $p |- ( ( A i^i B ) u. ( A \ B ) ) = A $=
|
|
( vx cin cdif cv wcel wo wa wn elin eldif orbi12i pm4.42 bitr4i uneqri )
|
|
CABDZABEZACFZQGZSRGZHSAGZSBGZIZUBUCJIZHUBTUDUAUESABKSABLMUBUCNOP $.
|
|
$}
|
|
|
|
$( Absorption of union by difference. Theorem 36 of [Suppes] p. 29.
|
|
(Contributed by NM, 19-May-1998.) $)
|
|
difun2 $p |- ( ( A u. B ) \ B ) = ( A \ B ) $=
|
|
( cun cdif c0 difundir difid uneq2i un0 3eqtri ) ABCBDABDZBBDZCKECKABBFLEKB
|
|
GHKIJ $.
|
|
|
|
$( Union of complementary parts into whole. (Contributed by NM,
|
|
22-Mar-1998.) $)
|
|
undif $p |- ( A C_ B <-> ( A u. ( B \ A ) ) = B ) $=
|
|
( wss cun wceq cdif ssequn1 undif2 eqeq1i bitr4i ) ABCABDZBEABAFDZBEABGLKBA
|
|
BHIJ $.
|
|
|
|
$( A subset of a difference does not intersect the subtrahend. (Contributed
|
|
by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro,
|
|
24-Aug-2015.) $)
|
|
ssdifin0 $p |- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) ) $=
|
|
( cdif wss cin c0 wceq ssrin incom disjdif eqtri sseq0 sylancl ) ABCDZEACFZ
|
|
OCFZEQGHPGHAOCIQCOFGOCJCBKLPQMN $.
|
|
|
|
$( A class is a subclass of itself subtracted from another iff it is the
|
|
empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) $)
|
|
ssdifeq0 $p |- ( A C_ ( B \ A ) <-> A = (/) ) $=
|
|
( cdif wss wceq cin inidm ssdifin0 syl5eqr 0ss difeq2 sseq12d mpbiri impbii
|
|
c0 id ) ABACZDZAOEZRAAAFOAGABAHISROBOCZDTJSAOQTSPAOBKLMN $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( A condition equivalent to inclusion in the union of two classes.
|
|
(Contributed by NM, 26-Mar-2007.) $)
|
|
ssundif $p |- ( A C_ ( B u. C ) <-> ( A \ B ) C_ C ) $=
|
|
( vx cv wcel cun wi wal cdif wss wn wa wo pm5.6 eldif imbi1i imbi2i dfss2
|
|
elun 3bitr4ri albii 3bitr4i ) DEZAFZUDBCGZFZHZDIUDABJZFZUDCFZHZDIAUFKUICK
|
|
UHULDUEUDBFZLMZUKHUEUMUKNZHULUHUEUMUKOUJUNUKUDABPQUGUOUEUDBCTRUAUBDAUFSDU
|
|
ICSUC $.
|
|
$}
|
|
|
|
$( Swap the arguments of a class difference. (Contributed by NM,
|
|
29-Mar-2007.) $)
|
|
difcom $p |- ( ( A \ B ) C_ C <-> ( A \ C ) C_ B ) $=
|
|
( cun wss cdif uncom sseq2i ssundif 3bitr3i ) ABCDZEACBDZEABFCEACFBEKLABCGH
|
|
ABCIACBIJ $.
|
|
|
|
$( Two ways to express overlapping subsets. (Contributed by Stefan O'Rear,
|
|
31-Oct-2014.) $)
|
|
pssdifcom1 $p |- ( ( A C_ C /\ B C_ C ) ->
|
|
( ( C \ A ) C. B <-> ( C \ B ) C. A ) ) $=
|
|
( wss wa cdif wn wpss wb difcom ssconb ancoms notbid anbi12d dfpss3 3bitr4g
|
|
a1i ) ACDZBCDZEZCAFZBDZBUADZGZECBFZADZAUEDZGZEUABHUEAHTUBUFUDUHUBUFITCABJQT
|
|
UCUGSRUCUGIBACKLMNUABOUEAOP $.
|
|
|
|
$( Two ways to express non-covering pairs of subsets. (Contributed by Stefan
|
|
O'Rear, 31-Oct-2014.) $)
|
|
pssdifcom2 $p |- ( ( A C_ C /\ B C_ C ) ->
|
|
( B C. ( C \ A ) <-> A C. ( C \ B ) ) ) $=
|
|
( wss wa cdif wn wpss wb ssconb ancoms difcom notbid anbi12d dfpss3 3bitr4g
|
|
a1i ) ACDZBCDZEZBCAFZDZUABDZGZEACBFZDZUEADZGZEBUAHAUEHTUBUFUDUHSRUBUFIBACJK
|
|
TUCUGUCUGITCABLQMNBUAOAUEOP $.
|
|
|
|
$( Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16.
|
|
(Contributed by NM, 18-Aug-2004.) $)
|
|
difdifdir $p |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ ( B \ C ) ) $=
|
|
( cdif cvv cun cin dif32 invdif eqtr4i un0 indi disjdif incom eqtr3i uneq2i
|
|
c0 ddif indm difeq2i ineq2i 3eqtri ) ABDCDZACDZEBDZCFZGZUDEBCDZDZGUDUHDUCUD
|
|
UEGZQFZUGUCUJUKUCUDBDUJABCHUDBIJUJKJUGUJUDCGZFUKUDUECLQULUJCUDGQULCAMCUDNOP
|
|
JJUFUIUDUEEECDZDZFZUFUIUNCUECRPEBUMGZDUOUIBUMSUPUHEBCITOOUAUDUHIUB $.
|
|
|
|
$( Two ways to say that ` A ` and ` B ` partition ` C ` (when ` A ` and ` B `
|
|
don't overlap and ` A ` is a part of ` C ` ). (Contributed by FL,
|
|
17-Nov-2008.) $)
|
|
uneqdifeq $p |- ( ( A C_ C /\ ( A i^i B ) = (/) )
|
|
-> ( ( A u. B ) = C <-> ( C \ A ) = B ) ) $=
|
|
( wss cin c0 wceq wa cun cdif uncom eqcomd difeq1 difun2 incom expcom com12
|
|
wi eqtr adantl eqeq1i disj3 bitri eqcoms sylbi syl5com syl mpan difss sseq1
|
|
sylancl biimpi syl6bi mpi adantr imp eqimss ssundif sylibr adantlr eqssd ex
|
|
unss impbid ) ACDZABEZFGZHZABIZCGZCAJZBGZVGVJVLRVEVJVGVLBAIZVIGZVJVGVLRZBAK
|
|
VNVJHZCVMGZVOVPVMCVMVICSLVQVKVMAJZGZVRBAJZGZVOCVMAMBANVSWAHVKVTGZVGVLVKVRVT
|
|
SVGBVTGZWBVLRZVGBAEZFGWCVFWEFABOUABAUBUCWDVTBWBVTBGVLVKVTBSPUDUEUFUKUGUHQTV
|
|
HVLVJVHVLHVICVHVLVICDZVEVLWFRVGVLVEWFVLVKCDZVEWFRZCAUIVLWGBCDZWHVKBCUJVEWIW
|
|
FVEWIHWFABCVCULPUMUNQUOUPVEVLCVIDZVGVEVLHVKBDZWJVLWKVEVKBUQTCABURUSUTVAVBVD
|
|
$.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare
|
|
~ 19.2 ). The restricted version is valid only when the domain of
|
|
quantification is not empty. (Contributed by NM, 15-Nov-2003.) $)
|
|
r19.2z $p |- ( ( A =/= (/) /\ A. x e. A ph ) -> E. x e. A ph ) $=
|
|
( wral c0 wne wrex cv wex wa wi wal df-ral exintr sylbi n0 df-rex 3imtr4g
|
|
wcel impcom ) ABCDZCEFZABCGZUABHCSZBIZUDAJBIZUBUCUAUDAKBLUEUFKABCMUDABNOB
|
|
CPABCQRT $.
|
|
|
|
$( A response to the notion that the condition ` A =/= (/) ` can be removed
|
|
in ~ r19.2z . Interestingly enough, ` ph ` does not figure in the
|
|
left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.) $)
|
|
r19.2zb $p |- ( A =/= (/) <-> ( A. x e. A ph -> E. x e. A ph ) ) $=
|
|
( c0 wral wrex wi r19.2z ex wceq cv wcel noel pm2.21i rgen raleq necon3bi
|
|
wne mpbiri wex wa exsimpl df-rex n0 3imtr4i ja impbii ) CDRZABCEZABCFZGUH
|
|
UIUJABCHIUIUJUHUICDCDJUIABDEABDBKZDLAUKMNOABCDPSQUKCLZAUABTULBTUJUHULABUB
|
|
ABCUCBCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
r19.3rz.1 $e |- F/ x ph $.
|
|
$( Restricted quantification of wff not containing quantified variable.
|
|
(Contributed by FL, 3-Jan-2008.) $)
|
|
r19.3rz $p |- ( A =/= (/) -> ( ph <-> A. x e. A ph ) ) $=
|
|
( c0 wne cv wcel wex wi wral wb n0 biimt sylbi df-ral 19.23 bitri syl6bbr
|
|
wal ) CEFZABGCHZBIZAJZABCKZUAUCAUDLBCMUCANOUEUBAJBTUDABCPUBABDQRS $.
|
|
|
|
$( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is not empty.
|
|
(Contributed by NM, 26-Oct-2010.) $)
|
|
r19.28z $p |- ( A =/= (/) ->
|
|
( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) ) $=
|
|
( c0 wne wral wa r19.3rz anbi1d r19.26 syl6rbbr ) DFGZABCDHZIACDHZOIABICD
|
|
HNAPOACDEJKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ph $.
|
|
$( Restricted quantification of wff not containing quantified variable.
|
|
(Contributed by NM, 10-Mar-1997.) $)
|
|
r19.3rzv $p |- ( A =/= (/) -> ( ph <-> A. x e. A ph ) ) $=
|
|
( c0 wne cv wcel wex wi wb n0 biimt sylbi wal df-ral 19.23v bitri syl6bbr
|
|
wral ) CDEZABFCGZBHZAIZABCSZTUBAUCJBCKUBALMUDUAAIBNUCABCOUAABPQR $.
|
|
|
|
$( Restricted quantification of wff not containing quantified variable.
|
|
(Contributed by NM, 27-May-1998.) $)
|
|
r19.9rzv $p |- ( A =/= (/) -> ( ph <-> E. x e. A ph ) ) $=
|
|
( c0 wne wn wral wrex r19.3rzv bicomd con2bid dfrex2 syl6bbr ) CDEZAAFZBC
|
|
GZFABCHNPANOPOBCIJKABCLM $.
|
|
|
|
$( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is not empty.
|
|
(Contributed by NM, 19-Aug-2004.) $)
|
|
r19.28zv $p |- ( A =/= (/) ->
|
|
( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) ) $=
|
|
( c0 wne wral wa r19.3rzv anbi1d r19.26 syl6rbbr ) DEFZABCDGZHACDGZNHABHC
|
|
DGMAONACDIJABCDKL $.
|
|
|
|
$( Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is not empty.
|
|
(Contributed by Paul Chapman, 8-Oct-2007.) $)
|
|
r19.37zv $p |- ( A =/= (/) ->
|
|
( E. x e. A ( ph -> ps ) <-> ( ph -> E. x e. A ps ) ) ) $=
|
|
( c0 wne wrex wi wral r19.3rzv imbi1d r19.35 syl6rbbr ) DEFZABCDGZHACDIZO
|
|
HABHCDGNAPOACDJKABCDLM $.
|
|
|
|
$( Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed
|
|
by NM, 27-May-1998.) $)
|
|
r19.45zv $p |- ( A =/= (/) ->
|
|
( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) ) $=
|
|
( c0 wne wrex wo r19.9rzv orbi1d r19.43 syl6rbbr ) DEFZABCDGZHACDGZNHABHC
|
|
DGMAONACDIJABCDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
r19.27z.1 $e |- F/ x ps $.
|
|
$( Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is not empty.
|
|
(Contributed by NM, 26-Oct-2010.) $)
|
|
r19.27z $p |- ( A =/= (/) ->
|
|
( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) ) $=
|
|
( c0 wne wral wa r19.3rz anbi2d r19.26 syl6rbbr ) DFGZACDHZBIOBCDHZIABICD
|
|
HNBPOBCDEJKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
$( Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is not empty.
|
|
(Contributed by NM, 19-Aug-2004.) $)
|
|
r19.27zv $p |- ( A =/= (/) ->
|
|
( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) ) $=
|
|
( c0 wne wral wa r19.3rzv anbi2d r19.26 syl6rbbr ) DEFZACDGZBHNBCDGZHABHC
|
|
DGMBONBCDIJABCDKL $.
|
|
|
|
$( Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It
|
|
is valid only when the domain of quantification is not empty.
|
|
(Contributed by NM, 20-Sep-2003.) $)
|
|
r19.36zv $p |- ( A =/= (/) ->
|
|
( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> ps ) ) ) $=
|
|
( c0 wne wral wi wrex r19.9rzv imbi2d r19.35 syl6rbbr ) DEFZACDGZBHOBCDIZ
|
|
HABHCDINBPOBCDJKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Vacuous quantification is always true. (Contributed by NM,
|
|
11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
rzal $p |- ( A = (/) -> A. x e. A ph ) $=
|
|
( c0 wceq cv wcel ne0i necon2bi pm2.21d ralrimiv ) CDEZABCLBFZCGZANCDCMHI
|
|
JK $.
|
|
|
|
$( Restricted existential quantification implies its restriction is
|
|
nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) $)
|
|
rexn0 $p |- ( E. x e. A ph -> A =/= (/) ) $=
|
|
( c0 wne cv wcel ne0i a1d rexlimiv ) ACDEZBCBFZCGKACLHIJ $.
|
|
|
|
$( Idempotent law for restricted quantifier. (Contributed by NM,
|
|
28-Mar-1997.) $)
|
|
ralidm $p |- ( A. x e. A A. x e. A ph <-> A. x e. A ph ) $=
|
|
( c0 wceq wral wb rzal 2thd wn cv wcel wex neq0 wi biimt wal df-ral nfra1
|
|
19.23 bitri syl6rbbr sylbi pm2.61i ) CDEZABCFZBCFZUFGZUEUGUFUFBCHABCHIUEJ
|
|
BKCLZBMZUHBCNUJUFUJUFOZUGUJUFPUGUIUFOBQUKUFBCRUIUFBABCSTUAUBUCUD $.
|
|
$}
|
|
|
|
$( Vacuous universal quantification is always true. (Contributed by NM,
|
|
20-Oct-2005.) $)
|
|
ral0 $p |- A. x e. (/) ph $=
|
|
( c0 cv wcel noel pm2.21i rgen ) ABCBDZCEAIFGH $.
|
|
|
|
${
|
|
$d x A $.
|
|
rgenz.1 $e |- ( ( A =/= (/) /\ x e. A ) -> ph ) $.
|
|
$( Generalization rule that eliminates a non-zero class requirement.
|
|
(Contributed by NM, 8-Dec-2012.) $)
|
|
rgenz $p |- A. x e. A ph $=
|
|
( wral c0 rzal wne ralrimiva pm2.61ine ) ABCECFABCGCFHABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
ralf0.1 $e |- -. ph $.
|
|
$( The quantification of a falsehood is vacuous when true. (Contributed by
|
|
NM, 26-Nov-2005.) $)
|
|
ralf0 $p |- ( A. x e. A ph <-> A = (/) ) $=
|
|
( wral c0 wceq cv wcel wi wal wn con3 mpi alimi df-ral eq0 3imtr4i impbii
|
|
rzal ) ABCEZCFGZBHCIZAJZBKUCLZBKUAUBUDUEBUDALUEDUCAMNOABCPBCQRABCTS $.
|
|
$}
|
|
|
|
$( TODO - shorten r19.3zv, r19.27zv, r19.28zv, raaanv w/ non-v $)
|
|
${
|
|
$d x y A $.
|
|
raaan.1 $e |- F/ y ph $.
|
|
raaan.2 $e |- F/ x ps $.
|
|
$( Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) $)
|
|
raaan $p |- ( A. x e. A A. y e. A ( ph /\ ps ) <->
|
|
( A. x e. A ph /\ A. y e. A ps ) ) $=
|
|
( wa wral wb c0 wceq rzal pm5.1 syl12anc wne r19.28z ralbidv nfcv nfral
|
|
r19.27z bitrd pm2.61ine ) ABHDEIZCEIZACEIZBDEIZHZJZEKEKLUEUFUGUIUDCEMACEM
|
|
BDEMUEUHNOEKPZUEAUGHZCEIUHUJUDUKCEABDEFQRAUGCEBCDECESGTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y ph $. $d x ps $. $d x y A $.
|
|
$( Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) $)
|
|
raaanv $p |- ( A. x e. A A. y e. A ( ph /\ ps ) <->
|
|
( A. x e. A ph /\ A. y e. A ps ) ) $=
|
|
( wa wral wb wceq rzal pm5.1 syl12anc wne r19.28zv ralbidv r19.27zv bitrd
|
|
c0 pm2.61ine ) ABFDEGZCEGZACEGZBDEGZFZHZERERIUAUBUCUETCEJACEJBDEJUAUDKLER
|
|
MZUAAUCFZCEGUDUFTUGCEABDENOAUCCEPQS $.
|
|
$}
|
|
|
|
${
|
|
$d z y $. $d z x A $.
|
|
$( Set substitution into the first argument of a subset relation.
|
|
(Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario
|
|
Carneiro, 14-Nov-2016.) $)
|
|
sbss $p |- ( [ y / x ] x C_ A <-> y C_ A ) $=
|
|
( vz cv wss wsb vex sbequ sseq1 nfv sbie vtoclb ) AEZCFZADGDEZCFZOABGBEZC
|
|
FDRBHODBAIPRCJOQADQAKNPCJLM $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d C y $. $d D y $. $d x y $.
|
|
$( Distribute proper substitution through a subclass relation.
|
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander
|
|
van der Vekens, 23-Jul-2017.) $)
|
|
sbcss $p |- ( A e. B -> ( [. A / x ]. C C_ D <->
|
|
[_ A / x ]_ C C_ [_ A / x ]_ D ) ) $=
|
|
( vy wcel cv wi wal wsbc csb wss sbcalg sbcimg sbcel2g bitrd albidv dfss2
|
|
imbi12d sbcbii 3bitr4g ) BCGZFHZDGZUDEGZIZFJZABKZUDABDLZGZUDABELZGZIZFJZD
|
|
EMZABKUJULMUCUIUGABKZFJUOUGFABCNUCUQUNFUCUQUEABKZUFABKZIUNUEUFABCOUCURUKU
|
|
SUMABUDDCPABUDECPTQRQUPUHABFDESUAFUJULSUB $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Contraposition law for subset. (Contributed by SF, 11-Mar-2015.) $)
|
|
sscon34 $p |- ( A C_ B <-> ~ B C_ ~ A ) $=
|
|
( vx cv wcel wi wal ccompl wss wn con34b vex elcompl imbi12i bitr4i albii
|
|
dfss2 3bitr4i ) CDZAEZSBEZFZCGSBHZEZSAHZEZFZCGABIUCUEIUBUGCUBUAJZTJZFUGTU
|
|
AKUDUHUFUISBCLZMSAUJMNOPCABQCUCUEQR $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
"Weak deduction theorem" for set theory
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
In a Hilbert system of logic (which consists of a set of axioms, modus
|
|
ponens, and the generalization rule), converting a deduction to a proof using
|
|
the Deduction Theorem (taught in introductory logic books) involves an
|
|
exponential increase of the number of steps as hypotheses are successively
|
|
eliminated. Here is a trick that is not as general as the Deduction Theorem
|
|
but requires only a linear increase in the number of steps.
|
|
|
|
The general problem: We want to convert a deduction
|
|
P |- Q
|
|
into a proof of the theorem
|
|
|- P -> Q
|
|
i.e. we want to eliminate the hypothesis P. Normally this is done using the
|
|
Deduction (meta)Theorem, which looks at the microscopic steps of the
|
|
deduction and usually doubles or triples the number of these microscopic
|
|
steps for each hypothesis that is eliminated. We will look at a special case
|
|
of this problem, without appealing to the Deduction Theorem.
|
|
|
|
We assume ZF with class notation. A and B are arbitrary (possibly
|
|
proper) classes. P, Q, R, S and T are wffs.
|
|
|
|
We define the conditional operator, if(P,A,B), as follows:
|
|
if(P,A,B) =def= { x | (x \in A & P) v (x \in B & -. P) }
|
|
(where x does not occur in A, B, or P).
|
|
|
|
Lemma 1.
|
|
A = if(P,A,B) -> (P <-> R), B = if(P,A,B) -> (S <-> R), S |- R
|
|
Proof: Logic and Axiom of Extensionality.
|
|
|
|
Lemma 2.
|
|
A = if(P,A,B) -> (Q <-> T), T |- P -> Q
|
|
Proof: Logic and Axiom of Extensionality.
|
|
|
|
Here's a simple example that illustrates how it works. Suppose we have
|
|
a deduction
|
|
Ord A |- Tr A
|
|
which means, "Assume A is an ordinal class. Then A is a transitive class."
|
|
Note that A is a class variable that may be substituted with any class
|
|
expression, so this is really a deduction scheme.
|
|
|
|
We want to convert this to a proof of the theorem (scheme)
|
|
|- Ord A -> Tr A.
|
|
|
|
The catch is that we must be able to prove "Ord A" for at least one
|
|
object A (and this is what makes it weaker than the ordinary Deduction
|
|
Theorem). However, it is easy to prove |- Ord 0 (the empty set is
|
|
ordinal). (For a typical textbook "theorem," i.e. deduction, there is
|
|
usually at least one object satisfying each hypothesis, otherwise the
|
|
theorem would not be very useful. We can always go back to the standard
|
|
Deduction Theorem for those hypotheses where this is not the case.)
|
|
Continuing with the example:
|
|
|
|
Equality axioms (and Extensionality) yield
|
|
|- A = if(Ord A, A, 0) -> (Ord A <-> Ord if(Ord A, A, 0)) (1)
|
|
|- 0 = if(Ord A, A, 0) -> (Ord 0 <-> Ord if(Ord A, A, 0)) (2)
|
|
From (1), (2) and |- Ord 0, Lemma 1 yields
|
|
|- Ord if(Ord A, A, 0) (3)
|
|
From (3) and substituting if(Ord A, A, 0) for
|
|
A in the original deduction,
|
|
|- Tr if(Ord A, A, 0) (4)
|
|
Equality axioms (and Extensionality) yield
|
|
|- A = if(Ord A, A, 0) -> (Tr A <-> Tr if(Ord A, A, 0)) (5)
|
|
From (4) and (5), Lemma 2 yields
|
|
|- Ord A -> Tr A (Q.E.D.)
|
|
|
|
$)
|
|
|
|
$( These lemmas are used to convert hypotheses into antecedents,
|
|
when there is at least one class making the hypothesis true. $)
|
|
|
|
$( Declare new constant symbols. $)
|
|
$c if $. $( Conditional operator (was "ded" for "deduction class"). $)
|
|
|
|
$( Extend class notation to include the conditional operator. See ~ df-if
|
|
for a description. (In older databases this was denoted "ded".) $)
|
|
cif $a class if ( ph , A , B ) $.
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $.
|
|
$( Define the conditional operator. Read ` if ( ph , A , B ) ` as "if
|
|
` ph ` then ` A ` else ` B ` ." See ~ iftrue and ~ iffalse for its
|
|
values. In mathematical literature, this operator is rarely defined
|
|
formally but is implicit in informal definitions such as "let f(x)=0 if
|
|
x=0 and 1/x otherwise." (In older versions of this database, this
|
|
operator was denoted "ded" and called the "deduction class.")
|
|
|
|
An important use for us is in conjunction with the weak deduction
|
|
theorem, which converts a hypothesis into an antecedent. In that role,
|
|
` A ` is a class variable in the hypothesis and ` B ` is a class
|
|
(usually a constant) that makes the hypothesis true when it is
|
|
substituted for ` A ` . See ~ dedth for the main part of the weak
|
|
deduction theorem, ~ elimhyp to eliminate a hypothesis, and ~ keephyp to
|
|
keep a hypothesis. See the Deduction Theorem link on the Metamath Proof
|
|
Explorer Home Page for a description of the weak deduction theorem.
|
|
(Contributed by NM, 15-May-1999.) $)
|
|
df-if $a |- if ( ph , A , B ) =
|
|
{ x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $. $d x B $. $d x C $.
|
|
$( An alternate definition of the conditional operator ~ df-if with one
|
|
fewer connectives (but probably less intuitive to understand).
|
|
(Contributed by NM, 30-Jan-2006.) $)
|
|
dfif2 $p |- if ( ph , A , B ) =
|
|
{ x | ( ( x e. B -> ph ) -> ( x e. A /\ ph ) ) } $=
|
|
( cif cv wcel wa wn wo cab wi df-if df-or orcom iman imbi1i 3bitr4i abbii
|
|
eqtri ) ACDEBFZCGAHZUADGZAIHZJZBKUCALZUBLZBKABCDMUEUGBUDUBJUDIZUBLUEUGUDU
|
|
BNUBUDOUFUHUBUCAPQRST $.
|
|
|
|
$( An alternate definition of the conditional operator ~ df-if as a simple
|
|
class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) $)
|
|
dfif6 $p |- if ( ph , A , B ) =
|
|
( { x e. A | ph } u. { x e. B | -. ph } ) $=
|
|
( cv wcel wa cab wn cun wo crab cif unab df-rab uneq12i df-if 3eqtr4ri )
|
|
BEZCFAGZBHZSDFAIZGZBHZJTUCKBHABCLZUBBDLZJACDMTUCBNUEUAUFUDABCOUBBDOPABCDQ
|
|
R $.
|
|
|
|
$( Equality theorem for conditional operator. (Contributed by NM,
|
|
1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $)
|
|
ifeq1 $p |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) ) $=
|
|
( vx wceq crab wn cun cif rabeq uneq1d dfif6 3eqtr4g ) BCFZAEBGZAHEDGZIAE
|
|
CGZQIABDJACDJOPRQAEBCKLAEBDMAECDMN $.
|
|
|
|
$( Equality theorem for conditional operator. (Contributed by NM,
|
|
1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $)
|
|
ifeq2 $p |- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) ) $=
|
|
( vx wceq crab wn cun cif rabeq uneq2d dfif6 3eqtr4g ) BCFZAEDGZAHZEBGZIP
|
|
QECGZIADBJADCJORSPQEBCKLAEDBMAEDCMN $.
|
|
|
|
$( Value of the conditional operator when its first argument is true.
|
|
(Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
iftrue $p |- ( ph -> if ( ph , A , B ) = A ) $=
|
|
( vx cv wcel wi wa cab cif dedlem0a abbi2dv dfif2 syl6reqr ) ABDEZCFZAGOB
|
|
FZAHGZDIABCJARDBAQPKLADBCMN $.
|
|
|
|
$( Value of the conditional operator when its first argument is false.
|
|
(Contributed by NM, 14-Aug-1999.) $)
|
|
iffalse $p |- ( -. ph -> if ( ph , A , B ) = B ) $=
|
|
( vx wn cv wcel wa wo cab cif dedlemb abbi2dv df-if syl6reqr ) AEZCDFZBGZ
|
|
AHQCGZPHIZDJABCKPTDCARSLMADBCNO $.
|
|
$}
|
|
|
|
$( When values are unequal, but an "if" condition checks if they are equal,
|
|
then the "false" branch results. This is a simple utility to provide a
|
|
slight shortening and simplification of proofs vs. applying ~ iffalse
|
|
directly in this case. It happens, e.g., in oevn0 in set.mm.
|
|
(Contributed by David A. Wheeler, 15-May-2015.) $)
|
|
ifnefalse $p |- ( A =/= B -> if ( A = B , C , D ) = D ) $=
|
|
( wne wceq wn cif df-ne iffalse sylbi ) ABEABFZGLCDHDFABILCDJK $.
|
|
|
|
${
|
|
ifsb.1 $e |- ( if ( ph , A , B ) = A -> C = D ) $.
|
|
ifsb.2 $e |- ( if ( ph , A , B ) = B -> C = E ) $.
|
|
$( Distribute a function over an if-clause. (Contributed by Mario
|
|
Carneiro, 14-Aug-2013.) $)
|
|
ifsb $p |- C = if ( ph , D , E ) $=
|
|
( cif wceq iftrue syl eqtr4d wn iffalse pm2.61i ) ADAEFIZJADEQAABCIZBJDEJ
|
|
ABCKGLAEFKMANZDFQSRCJDFJABCOHLAEFOMP $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y ph $.
|
|
dfif3.1 $e |- C = { x | ph } $.
|
|
$( Alternate definition of the conditional operator ~ df-if . Note that
|
|
` ph ` is independent of ` x ` i.e. a constant true or false.
|
|
(Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro,
|
|
8-Sep-2013.) $)
|
|
dfif3 $p |- if ( ph , A , B )
|
|
= ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) ) $=
|
|
( vy cif crab wn cun cin cvv cdif dfif6 cab cv ineq2i dfrab3 eqtr4i biidd
|
|
wceq cbvabv eqtri notab difeq2i eqtr2i uneq12i ) ACDHAGCIZAJZGDIZKCELZDME
|
|
NZLZKAGCDOULUIUNUKULCAGPZLUIEUOCEABPUOFAABGBQGQUBAUAUCUDZRAGCSTUKDUJGPZLU
|
|
NUJGDSUQUMDUQMUONUMAGUEEUOMUPUFTRUGUHT $.
|
|
|
|
$( Alternate definition of the conditional operator ~ df-if . Note that
|
|
` ph ` is independent of ` x ` i.e. a constant true or false.
|
|
(Contributed by NM, 25-Aug-2013.) $)
|
|
dfif4 $p |- if ( ph , A , B )
|
|
= ( ( A u. B ) i^i ( ( A u. ( _V \ C ) ) i^i ( B u. C ) ) ) $=
|
|
( cif cin cvv cdif cun dfif3 undir undi uncom undifv ineq12i 3eqtri inass
|
|
inv1 eqtri ) ACDGCEHDIEJZHZKCUCKZEUCKZHZCDKZCUBKZDEKZHHZABCDEFLCEUCMUFUGU
|
|
HHZUIHUJUDUKUEUICDUBNUEEDKZEUBKZHUIIHUIEDUBNULUIUMIEDOEPQUITRQUGUHUISUAR
|
|
$.
|
|
|
|
$( Alternate definition of the conditional operator ~ df-if . Note that
|
|
` ph ` is independent of ` x ` i.e. a constant true or false (see also
|
|
~ abvor0 ). (Contributed by Gérard Lang, 18-Aug-2013.) $)
|
|
dfif5 $p |- if ( ph , A , B ) = ( ( A i^i B )
|
|
u. ( ( ( A \ B ) i^i C ) u. ( ( B \ A ) i^i ( _V \ C ) ) ) ) $=
|
|
( cun cdif cin undir unidm unass undi 3eqtr3ri ineq1i inabs eqtri 3eqtr4i
|
|
undifabs eqtr4i cvv cif inindi dfif4 uneq1i undif2 uneq12i unundi 3eqtrri
|
|
uncom uneq2i ineq2i ineq12i ) CDGZCUAEHZGZDEGZIIUNUPIZUNUQIZIZACDUBCDICDH
|
|
ZEIZDCHZUOIZGZGZUNUPUQUCABCDEFUDVFCVEGZDVEGZIUTCDVEJURVGUSVHURCVBGZCVDGZG
|
|
ZVGURCCDUOIZGZGZVKCCGZVLGVMVNURVOCVLCKUECCVLLCDUOMZNVICVJVMVICVAGZCEGZIZC
|
|
CVAEMVSCVRICVQCVRCDSOCEPQQCVCGZUPIURVJVMVTUNUPCDUFOCVCUOMVPRUGTCVBVDUHTCE
|
|
IZDGZDVBGZDVDGZGZUSVHWBDGWADDGZGWEWBWADDLWBWCDWDWBDVAGZUQIZWCDWAGDCGZUQIW
|
|
BWHDCEMWADUJWGWIUQDCUFORDVAEMTWDDVCGZDUOGZIDWKIDDVCUOMWJDWKDCSODUOPUIUGWF
|
|
DWADKUKNUSUNEDGZIWBUQWLUNDEUJULCEDJTDVBVDUHRUMTR $.
|
|
$}
|
|
|
|
$( Equality theorem for conditional operators. (Contributed by NM,
|
|
1-Sep-2004.) $)
|
|
ifeq12 $p |- ( ( A = B /\ C = D ) ->
|
|
if ( ph , A , C ) = if ( ph , B , D ) ) $=
|
|
( wceq cif ifeq1 ifeq2 sylan9eq ) BCFDEFABDGACDGACEGABCDHADECIJ $.
|
|
|
|
${
|
|
ifeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for conditional operator. (Contributed by NM,
|
|
16-Feb-2005.) $)
|
|
ifeq1d $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) ) $=
|
|
( wceq cif ifeq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
|
|
|
|
$( Equality deduction for conditional operator. (Contributed by NM,
|
|
16-Feb-2005.) $)
|
|
ifeq2d $p |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) ) $=
|
|
( wceq cif ifeq2 syl ) ACDGBECHBEDHGFBCDEIJ $.
|
|
|
|
ifeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for conditional operator. (Contributed by NM,
|
|
24-Mar-2015.) $)
|
|
ifeq12d $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) ) $=
|
|
( cif ifeq1d ifeq2d eqtrd ) ABCEIBDEIBDFIABCDEGJABEFDHKL $.
|
|
$}
|
|
|
|
$( Equivalence theorem for conditional operators. (Contributed by Raph
|
|
Levien, 15-Jan-2004.) $)
|
|
ifbi $p |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) ) $=
|
|
( wb wa wn wo cif wceq dfbi3 iftrue eqcomd sylan9eq iffalse jaoi sylbi ) AB
|
|
EABFZAGZBGZFZHACDIZBCDIZJZABKRUDUAABUBCUCACDLBUCCBCDLMNSTUBDUCACDOTUCDBCDOM
|
|
NPQ $.
|
|
|
|
${
|
|
ifbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalence deduction for conditional operators. (Contributed by NM,
|
|
18-Apr-2005.) $)
|
|
ifbid $p |- ( ph -> if ( ps , A , B ) = if ( ch , A , B ) ) $=
|
|
( wb cif wceq ifbi syl ) ABCGBDEHCDEHIFBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
ifbieq2i.1 $e |- ( ph <-> ps ) $.
|
|
ifbieq2i.2 $e |- A = B $.
|
|
$( Equivalence/equality inference for conditional operators. (Contributed
|
|
by Paul Chapman, 22-Jun-2011.) $)
|
|
ifbieq2i $p |- if ( ph , C , A ) = if ( ps , C , B ) $=
|
|
( cif wb wceq ifbi ax-mp ifeq2 eqtri ) AECHZBECHZBEDHZABIOPJFABECKLCDJPQJ
|
|
GBCDEMLN $.
|
|
$}
|
|
|
|
${
|
|
ifbieq2d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
ifbieq2d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equivalence/equality deduction for conditional operators. (Contributed
|
|
by Paul Chapman, 22-Jun-2011.) $)
|
|
ifbieq2d $p |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) ) $=
|
|
( cif ifbid ifeq2d eqtrd ) ABFDICFDICFEIABCFDGJACDEFHKL $.
|
|
$}
|
|
|
|
${
|
|
ifbieq12i.1 $e |- ( ph <-> ps ) $.
|
|
ifbieq12i.2 $e |- A = C $.
|
|
ifbieq12i.3 $e |- B = D $.
|
|
$( Equivalence deduction for conditional operators. (Contributed by NM,
|
|
18-Mar-2013.) $)
|
|
ifbieq12i $p |- if ( ph , A , B ) = if ( ps , C , D ) $=
|
|
( cif wceq ifeq1 ax-mp ifbieq2i eqtri ) ACDJZAEDJZBEFJCEKPQKHACEDLMABDFEG
|
|
INO $.
|
|
$}
|
|
|
|
${
|
|
ifbieq12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
ifbieq12d.2 $e |- ( ph -> A = C ) $.
|
|
ifbieq12d.3 $e |- ( ph -> B = D ) $.
|
|
$( Equivalence deduction for conditional operators. (Contributed by Jeff
|
|
Madsen, 2-Sep-2009.) $)
|
|
ifbieq12d $p |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) ) $=
|
|
( cif ifbid ifeq12d eqtrd ) ABDEKCDEKCFGKABCDEHLACDFEGIJMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y ph $. $d y ps $.
|
|
nfifd.2 $e |- ( ph -> F/ x ps ) $.
|
|
nfifd.3 $e |- ( ph -> F/_ x A ) $.
|
|
nfifd.4 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of ~ nfif . (Contributed by NM, 15-Feb-2013.)
|
|
(Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
nfifd $p |- ( ph -> F/_ x if ( ps , A , B ) ) $=
|
|
( vy cif cv wcel wi wa cab dfif2 nfv nfcrd nfimd nfand nfabd nfcxfrd ) AC
|
|
BDEJIKZELZBMZUCDLZBNZMZIOBIDEPAUHCIAIQAUEUGCAUDBCACIEHRFSAUFBCACIDGRFTSUA
|
|
UB $.
|
|
$}
|
|
|
|
${
|
|
nfif.1 $e |- F/ x ph $.
|
|
nfif.2 $e |- F/_ x A $.
|
|
nfif.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for a conditional operator.
|
|
(Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon,
|
|
26-Jun-2011.) $)
|
|
nfif $p |- F/_ x if ( ph , A , B ) $=
|
|
( cif wnfc wtru wnf a1i nfifd trud ) BACDHIJABCDABKJELBCIJFLBDIJGLMN $.
|
|
$}
|
|
|
|
${
|
|
ifeq1da.1 $e |- ( ( ph /\ ps ) -> A = B ) $.
|
|
$( Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
ifeq1da $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) ) $=
|
|
( cif wceq wa ifeq1d wn iffalse eqtr4d adantl pm2.61dan ) ABBCEGZBDEGZHZA
|
|
BIBCDEFJBKZRASPEQBCELBDELMNO $.
|
|
$}
|
|
|
|
${
|
|
ifeq2da.1 $e |- ( ( ph /\ -. ps ) -> A = B ) $.
|
|
$( Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
ifeq2da $p |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) ) $=
|
|
( cif wceq iftrue eqtr4d adantl wn wa ifeq2d pm2.61dan ) ABBECGZBEDGZHZBR
|
|
ABPEQBECIBEDIJKABLMBCDEFNO $.
|
|
$}
|
|
|
|
${
|
|
ifclda.1 $e |- ( ( ph /\ ps ) -> A e. C ) $.
|
|
ifclda.2 $e |- ( ( ph /\ -. ps ) -> B e. C ) $.
|
|
$( Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
ifclda $p |- ( ph -> if ( ps , A , B ) e. C ) $=
|
|
( cif wcel wa wceq iftrue adantl eqeltrd wn iffalse pm2.61dan ) ABBCDHZEI
|
|
ABJRCEBRCKABCDLMFNABOZJRDESRDKABCDPMGNQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d y ph $. $d x y $.
|
|
$( Distribute proper substitution through the conditional operator.
|
|
(Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro,
|
|
14-Nov-2016.) $)
|
|
csbifg $p |- ( A e. V -> [_ A / x ]_ if ( ph , B , C )
|
|
= if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) ) $=
|
|
( vy cv cif csb wsb wceq csbeq1 dfsbcq2 ifbieq12d eqeq12d nfcsb1v csbeq1a
|
|
wsbc vex nfs1v nfif weq sbequ12 csbief vtoclg ) BGHZADEIZJZABGKZBUGDJZBUG
|
|
EJZIZLBCUHJZABCSZBCDJZBCEJZIZLGCFUGCLZUIUNUMURBUGCUHMUSUJUOUKULUPUQABGCNB
|
|
UGCDMBUGCEMOPBUGUHUMGTUJBUKULABGUABUGDQBUGEQUBBGUCAUJDEUKULABGUDBUGDRBUGE
|
|
ROUEUF $.
|
|
$}
|
|
|
|
${
|
|
elimif.1 $e |- ( if ( ph , A , B ) = A -> ( ps <-> ch ) ) $.
|
|
elimif.2 $e |- ( if ( ph , A , B ) = B -> ( ps <-> th ) ) $.
|
|
$( Elimination of a conditional operator contained in a wff ` ps ` .
|
|
(Contributed by NM, 15-Feb-2005.) $)
|
|
elimif $p |- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) ) $=
|
|
( wn wo wa exmid biantrur andir cif wceq wb iftrue syl pm5.32i iffalse
|
|
orbi12i 3bitri ) BAAIZJZBKABKZUDBKZJACKZUDDKZJUEBALMAUDBNUFUHUGUIABCAAEFO
|
|
ZEPBCQAEFRGSTUDBDUDUJFPBDQAEFUAHSTUBUC $.
|
|
$}
|
|
|
|
${
|
|
ifboth.1 $e |- ( A = if ( ph , A , B ) -> ( ps <-> th ) ) $.
|
|
ifboth.2 $e |- ( B = if ( ph , A , B ) -> ( ch <-> th ) ) $.
|
|
${
|
|
ifbothda.3 $e |- ( ( et /\ ph ) -> ps ) $.
|
|
ifbothda.4 $e |- ( ( et /\ -. ph ) -> ch ) $.
|
|
$( A wff ` th ` containing a conditional operator is true when both of
|
|
its cases are true. (Contributed by NM, 15-Feb-2015.) $)
|
|
ifbothda $p |- ( et -> th ) $=
|
|
( wa wb cif wceq iftrue eqcomd syl adantl mpbid wn iffalse pm2.61dan )
|
|
EADEALBDJABDMZEAFAFGNZOUDAUEFAFGPQHRSTEAUAZLCDKUFCDMZEUFGUEOUGUFUEGAFGU
|
|
BQIRSTUC $.
|
|
$}
|
|
|
|
$( A wff ` th ` containing a conditional operator is true when both of its
|
|
cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario
|
|
Carneiro, 15-Feb-2015.) $)
|
|
ifboth $p |- ( ( ps /\ ch ) -> th ) $=
|
|
( wa simpll wn simplr ifbothda ) ABCDBCIEFGHBCAJBCAKLM $.
|
|
$}
|
|
|
|
$( Identical true and false arguments in the conditional operator.
|
|
(Contributed by NM, 18-Apr-2005.) $)
|
|
ifid $p |- if ( ph , A , A ) = A $=
|
|
( cif wceq iftrue iffalse pm2.61i ) AABBCBDABBEABBFG $.
|
|
|
|
$( Expansion of an equality with a conditional operator. (Contributed by NM,
|
|
14-Feb-2005.) $)
|
|
eqif $p |- ( A = if ( ph , B , C ) <->
|
|
( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) $=
|
|
( cif wceq eqeq2 elimif ) ABACDEZFBCFBDFCDICBGIDBGH $.
|
|
|
|
$( Membership in a conditional operator. (Contributed by NM,
|
|
14-Feb-2005.) $)
|
|
elif $p |- ( A e. if ( ph , B , C ) <->
|
|
( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) ) $=
|
|
( cif wcel eleq2 elimif ) ABACDEZFBCFBDFCDICBGIDBGH $.
|
|
|
|
$( Membership of a conditional operator. (Contributed by NM,
|
|
10-Sep-2005.) $)
|
|
ifel $p |- ( if ( ph , A , B ) e. C <->
|
|
( ( ph /\ A e. C ) \/ ( -. ph /\ B e. C ) ) ) $=
|
|
( cif wcel eleq1 elimif ) AABCEZDFBDFCDFBCIBDGICDGH $.
|
|
|
|
$( Membership (closure) of a conditional operator. (Contributed by NM,
|
|
4-Apr-2005.) $)
|
|
ifcl $p |- ( ( A e. C /\ B e. C ) -> if ( ph , A , B ) e. C ) $=
|
|
( wcel cif eleq1 ifboth ) ABDECDEABCFZDEBCBIDGCIDGH $.
|
|
|
|
$( The possible values of a conditional operator. (Contributed by NM,
|
|
17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
ifeqor $p |- ( if ( ph , A , B ) = A \/ if ( ph , A , B ) = B ) $=
|
|
( cif wceq wn iftrue con3i iffalse syl orri ) ABCDZBEZLCEZMFAFNAMABCGHABCIJ
|
|
K $.
|
|
|
|
$( Negating the first argument swaps the last two arguments of a conditional
|
|
operator. (Contributed by NM, 21-Jun-2007.) $)
|
|
ifnot $p |- if ( -. ph , A , B ) = if ( ph , B , A ) $=
|
|
( wn cif wceq notnot1 iffalse syl iftrue eqtr4d pm2.61i ) AADZBCEZACBEZFANC
|
|
OAMDNCFAGMBCHIACBJKMNBOMBCJACBHKL $.
|
|
|
|
$( Rewrite a conjunction in an if statement as two nested conditionals.
|
|
(Contributed by Mario Carneiro, 28-Jul-2014.) $)
|
|
ifan $p |- if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) $=
|
|
( wa cif wceq iftrue ifbid eqtr2d wn simpl con3i iffalse syl eqtr4d pm2.61i
|
|
ibar ) AABEZCDFZABCDFZDFZGAUBUATAUADHABSCDABRIJAKZTDUBUCSKTDGSAABLMSCDNOAUA
|
|
DNPQ $.
|
|
|
|
$( Rewrite a disjunction in an if statement as two nested conditionals.
|
|
(Contributed by Mario Carneiro, 28-Jul-2014.) $)
|
|
ifor $p |- if ( ( ph \/ ps ) , A , B ) = if ( ph , A , if ( ps , A , B ) ) $=
|
|
( wo cif wceq iftrue orcs eqtr4d wn iffalse biorf ifbid eqtr2d pm2.61i ) AA
|
|
BEZCDFZACBCDFZFZGARCTABRCGQCDHIACSHJAKZTSRACSLUABQCDABMNOP $.
|
|
|
|
${
|
|
dedth.1 $e |- ( A = if ( ph , A , B ) -> ( ps <-> ch ) ) $.
|
|
dedth.2 $e |- ch $.
|
|
$( Weak deduction theorem that eliminates a hypothesis ` ph ` , making it
|
|
become an antecedent. We assume that a proof exists for ` ph ` when the
|
|
class variable ` A ` is replaced with a specific class ` B ` . The
|
|
hypothesis ` ch ` should be assigned to the inference, and the
|
|
inference's hypothesis eliminated with ~ elimhyp . If the inference has
|
|
other hypotheses with class variable ` A ` , these can be kept by
|
|
assigning ~ keephyp to them. For more information, see the Deduction
|
|
Theorem ~ http://us.metamath.org/mpeuni/mmdeduction.html . (Contributed
|
|
by NM, 15-May-1999.) $)
|
|
dedth $p |- ( ph -> ps ) $=
|
|
( cif wceq wb iftrue eqcomd syl mpbiri ) ABCGADADEHZIBCJAODADEKLFMN $.
|
|
$}
|
|
|
|
${
|
|
dedth2h.1 $e |- ( A = if ( ph , A , C ) -> ( ch <-> th ) ) $.
|
|
dedth2h.2 $e |- ( B = if ( ps , B , D ) -> ( th <-> ta ) ) $.
|
|
dedth2h.3 $e |- ta $.
|
|
$( Weak deduction theorem eliminating two hypotheses. This theorem is
|
|
simpler to use than ~ dedth2v but requires that each hypothesis has
|
|
exactly one class variable. See also comments in ~ dedth .
|
|
(Contributed by NM, 15-May-1999.) $)
|
|
dedth2h $p |- ( ( ph /\ ps ) -> ch ) $=
|
|
( wi cif wceq imbi2d dedth imp ) ABCABCMBDMFHFAFHNOCDBJPBDEGIKLQQR $.
|
|
$}
|
|
|
|
${
|
|
dedth3h.1 $e |- ( A = if ( ph , A , D ) -> ( th <-> ta ) ) $.
|
|
dedth3h.2 $e |- ( B = if ( ps , B , R ) -> ( ta <-> et ) ) $.
|
|
dedth3h.3 $e |- ( C = if ( ch , C , S ) -> ( et <-> ze ) ) $.
|
|
dedth3h.4 $e |- ze $.
|
|
$( Weak deduction theorem eliminating three hypotheses. See comments in
|
|
~ dedth2h . (Contributed by NM, 15-May-1999.) $)
|
|
dedth3h $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
|
|
( wa wi cif wceq imbi2d dedth2h dedth 3impib ) ABCDABCRZDSUFESHKHAHKTUADE
|
|
UFNUBBCEFGIJLMOPQUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
dedth4h.1 $e |- ( A = if ( ph , A , R ) -> ( ta <-> et ) ) $.
|
|
dedth4h.2 $e |- ( B = if ( ps , B , S ) -> ( et <-> ze ) ) $.
|
|
dedth4h.3 $e |- ( C = if ( ch , C , F ) -> ( ze <-> si ) ) $.
|
|
dedth4h.4 $e |- ( D = if ( th , D , G ) -> ( si <-> rh ) ) $.
|
|
dedth4h.5 $e |- rh $.
|
|
$( Weak deduction theorem eliminating four hypotheses. See comments in
|
|
~ dedth2h . (Contributed by NM, 16-May-1999.) $)
|
|
dedth4h $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $=
|
|
( wa wi cif wceq imbi2d dedth2h imp ) ABUCCDUCZEABUJEUDUJFUDUJGUDJKNOJAJN
|
|
UEUFEFUJRUGKBKOUEUFFGUJSUGCDGHILMPQTUAUBUHUHUI $.
|
|
$}
|
|
|
|
${
|
|
dedth2v.1 $e |- ( A = if ( ph , A , C ) -> ( ps <-> ch ) ) $.
|
|
dedth2v.2 $e |- ( B = if ( ph , B , D ) -> ( ch <-> th ) ) $.
|
|
dedth2v.3 $e |- th $.
|
|
$( Weak deduction theorem for eliminating a hypothesis with 2 class
|
|
variables. Note: if the hypothesis can be separated into two
|
|
hypotheses, each with one class variable, then ~ dedth2h is simpler to
|
|
use. See also comments in ~ dedth . (Contributed by NM, 13-Aug-1999.)
|
|
(Proof shortened by Eric Schmidt, 28-Jul-2009.) $)
|
|
dedth2v $p |- ( ph -> ps ) $=
|
|
( dedth2h anidms ) ABAABCDEFGHIJKLM $.
|
|
$}
|
|
|
|
${
|
|
dedth3v.1 $e |- ( A = if ( ph , A , D ) -> ( ps <-> ch ) ) $.
|
|
dedth3v.2 $e |- ( B = if ( ph , B , R ) -> ( ch <-> th ) ) $.
|
|
dedth3v.3 $e |- ( C = if ( ph , C , S ) -> ( th <-> ta ) ) $.
|
|
dedth3v.4 $e |- ta $.
|
|
$( Weak deduction theorem for eliminating a hypothesis with 3 class
|
|
variables. See comments in ~ dedth2v . (Contributed by NM,
|
|
13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) $)
|
|
dedth3v $p |- ( ph -> ps ) $=
|
|
( dedth3h 3anidm12 anidms ) ABAABAAABCDEFGHIJKLMNOPQR $.
|
|
$}
|
|
|
|
${
|
|
dedth4v.1 $e |- ( A = if ( ph , A , R ) -> ( ps <-> ch ) ) $.
|
|
dedth4v.2 $e |- ( B = if ( ph , B , S ) -> ( ch <-> th ) ) $.
|
|
dedth4v.3 $e |- ( C = if ( ph , C , T ) -> ( th <-> ta ) ) $.
|
|
dedth4v.4 $e |- ( D = if ( ph , D , U ) -> ( ta <-> et ) ) $.
|
|
dedth4v.5 $e |- et $.
|
|
$( Weak deduction theorem for eliminating a hypothesis with 4 class
|
|
variables. See comments in ~ dedth2v . (Contributed by NM,
|
|
21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) $)
|
|
dedth4v $p |- ( ph -> ps ) $=
|
|
( anidms wa dedth4h ) ABAAUABAAAABCDEFGHIJKLMNOPQRSUBTT $.
|
|
$}
|
|
|
|
${
|
|
elimhyp.1 $e |- ( A = if ( ph , A , B ) -> ( ph <-> ps ) ) $.
|
|
elimhyp.2 $e |- ( B = if ( ph , A , B ) -> ( ch <-> ps ) ) $.
|
|
elimhyp.3 $e |- ch $.
|
|
$( Eliminate a hypothesis containing class variable ` A ` when it is known
|
|
for a specific class ` B ` . For more information, see comments in
|
|
~ dedth . (Contributed by NM, 15-May-1999.) $)
|
|
elimhyp $p |- ps $=
|
|
( cif wceq wb iftrue eqcomd syl ibi wn iffalse mpbii pm2.61i ) ABABADADEI
|
|
ZJABKATDADELMFNOAPZCBHUAETJCBKUATEADEQMGNRS $.
|
|
$}
|
|
|
|
${
|
|
elimhyp2v.1 $e |- ( A = if ( ph , A , C ) -> ( ph <-> ch ) ) $.
|
|
elimhyp2v.2 $e |- ( B = if ( ph , B , D ) -> ( ch <-> th ) ) $.
|
|
elimhyp2v.3 $e |- ( C = if ( ph , A , C ) -> ( ta <-> et ) ) $.
|
|
elimhyp2v.4 $e |- ( D = if ( ph , B , D ) -> ( et <-> th ) ) $.
|
|
elimhyp2v.5 $e |- ta $.
|
|
$( Eliminate a hypothesis containing 2 class variables. (Contributed by
|
|
NM, 14-Aug-1999.) $)
|
|
elimhyp2v $p |- th $=
|
|
( cif wceq wb iftrue eqcomd syl bitrd ibi wn iffalse mpbii pm2.61i ) ACAC
|
|
AABCAFAFHOZPABQAUGFAFHRSJTAGAGIOZPBCQAUHGAGIRSKTUAUBAUCZDCNUIDECUIHUGPDEQ
|
|
UIUGHAFHUDSLTUIIUHPECQUIUHIAGIUDSMTUAUEUF $.
|
|
$}
|
|
|
|
${
|
|
elimhyp3v.1 $e |- ( A = if ( ph , A , D ) -> ( ph <-> ch ) ) $.
|
|
elimhyp3v.2 $e |- ( B = if ( ph , B , R ) -> ( ch <-> th ) ) $.
|
|
elimhyp3v.3 $e |- ( C = if ( ph , C , S ) -> ( th <-> ta ) ) $.
|
|
elimhyp3v.4 $e |- ( D = if ( ph , A , D ) -> ( et <-> ze ) ) $.
|
|
elimhyp3v.5 $e |- ( R = if ( ph , B , R ) -> ( ze <-> si ) ) $.
|
|
elimhyp3v.6 $e |- ( S = if ( ph , C , S ) -> ( si <-> ta ) ) $.
|
|
elimhyp3v.7 $e |- et $.
|
|
$( Eliminate a hypothesis containing 3 class variables. (Contributed by
|
|
NM, 14-Aug-1999.) $)
|
|
elimhyp3v $p |- ta $=
|
|
( cif wceq wb iftrue eqcomd syl 3bitrd ibi wn iffalse mpbii pm2.61i ) ADA
|
|
DAABCDAHAHKUAZUBABUCAUMHAHKUDUENUFAIAILUAZUBBCUCAUNIAILUDUEOUFAJAJMUAZUBC
|
|
DUCAUOJAJMUDUEPUFUGUHAUIZEDTUPEFGDUPKUMUBEFUCUPUMKAHKUJUEQUFUPLUNUBFGUCUP
|
|
UNLAILUJUERUFUPMUOUBGDUCUPUOMAJMUJUESUFUGUKUL $.
|
|
$}
|
|
|
|
${
|
|
elimhyp4v.1 $e |- ( A = if ( ph , A , D ) -> ( ph <-> ch ) ) $.
|
|
elimhyp4v.2 $e |- ( B = if ( ph , B , R ) -> ( ch <-> th ) ) $.
|
|
elimhyp4v.3 $e |- ( C = if ( ph , C , S ) -> ( th <-> ta ) ) $.
|
|
elimhyp4v.4 $e |- ( F = if ( ph , F , G ) -> ( ta <-> ps ) ) $.
|
|
elimhyp4v.5 $e |- ( D = if ( ph , A , D ) -> ( et <-> ze ) ) $.
|
|
elimhyp4v.6 $e |- ( R = if ( ph , B , R ) -> ( ze <-> si ) ) $.
|
|
elimhyp4v.7 $e |- ( S = if ( ph , C , S ) -> ( si <-> rh ) ) $.
|
|
elimhyp4v.8 $e |- ( G = if ( ph , F , G ) -> ( rh <-> ps ) ) $.
|
|
elimhyp4v.9 $e |- et $.
|
|
$( Eliminate a hypothesis containing 4 class variables (for use with the
|
|
weak deduction theorem ~ dedth ). (Contributed by NM, 16-Apr-2005.) $)
|
|
elimhyp4v $p |- ps $=
|
|
( cif wceq wb iftrue eqcomd syl bitrd 3bitrd ibi wn iffalse mpbii pm2.61i
|
|
) ABABAADEBAACDAJAJMUGZUHACUIAUTJAJMUJUKRULAKAKNUGZUHCDUIAVAKAKNUJUKSULUM
|
|
ALALOUGZUHDEUIAVBLALOUJUKTULAPAPQUGZUHEBUIAVCPAPQUJUKUAULUNUOAUPZFBUFVDFH
|
|
IBVDFGHVDMUTUHFGUIVDUTMAJMUQUKUBULVDNVAUHGHUIVDVANAKNUQUKUCULUMVDOVBUHHIU
|
|
IVDVBOALOUQUKUDULVDQVCUHIBUIVDVCQAPQUQUKUEULUNURUS $.
|
|
$}
|
|
|
|
${
|
|
elimel.1 $e |- B e. C $.
|
|
$( Eliminate a membership hypothesis for weak deduction theorem, when
|
|
special case ` B e. C ` is provable. (Contributed by NM,
|
|
15-May-1999.) $)
|
|
elimel $p |- if ( A e. C , A , B ) e. C $=
|
|
( wcel cif eleq1 elimhyp ) ACEZIABFZCEBCEABAJCGBJCGDH $.
|
|
$}
|
|
|
|
${
|
|
elimdhyp.1 $e |- ( ph -> ps ) $.
|
|
elimdhyp.2 $e |- ( A = if ( ph , A , B ) -> ( ps <-> ch ) ) $.
|
|
elimdhyp.3 $e |- ( B = if ( ph , A , B ) -> ( th <-> ch ) ) $.
|
|
elimdhyp.4 $e |- th $.
|
|
$( Version of ~ elimhyp where the hypothesis is deduced from the final
|
|
antecedent. See ghomgrplem in set.mm for an example of its use.
|
|
(Contributed by Paul Chapman, 25-Mar-2008.) $)
|
|
elimdhyp $p |- ch $=
|
|
( cif wceq wb iftrue eqcomd syl mpbid wn iffalse mpbii pm2.61i ) ACABCGAE
|
|
AEFKZLBCMAUBEAEFNOHPQARZDCJUCFUBLDCMUCUBFAEFSOIPTUA $.
|
|
$}
|
|
|
|
${
|
|
keephyp.1 $e |- ( A = if ( ph , A , B ) -> ( ps <-> th ) ) $.
|
|
keephyp.2 $e |- ( B = if ( ph , A , B ) -> ( ch <-> th ) ) $.
|
|
keephyp.3 $e |- ps $.
|
|
keephyp.4 $e |- ch $.
|
|
$( Transform a hypothesis ` ps ` that we want to keep (but contains the
|
|
same class variable ` A ` used in the eliminated hypothesis) for use
|
|
with the weak deduction theorem. (Contributed by NM, 15-May-1999.) $)
|
|
keephyp $p |- th $=
|
|
( ifboth mp2an ) BCDIJABCDEFGHKL $.
|
|
$}
|
|
|
|
${
|
|
keephyp2v.1 $e |- ( A = if ( ph , A , C ) -> ( ps <-> ch ) ) $.
|
|
keephyp2v.2 $e |- ( B = if ( ph , B , D ) -> ( ch <-> th ) ) $.
|
|
keephyp2v.3 $e |- ( C = if ( ph , A , C ) -> ( ta <-> et ) ) $.
|
|
keephyp2v.4 $e |- ( D = if ( ph , B , D ) -> ( et <-> th ) ) $.
|
|
keephyp2v.5 $e |- ps $.
|
|
keephyp2v.6 $e |- ta $.
|
|
$( Keep a hypothesis containing 2 class variables (for use with the weak
|
|
deduction theorem ~ dedth ). (Contributed by NM, 16-Apr-2005.) $)
|
|
keephyp2v $p |- th $=
|
|
( wceq wb eqcomd syl cif iftrue bitrd mpbii wn iffalse pm2.61i ) ADABDOAB
|
|
CDAGAGIUAZQBCRAUHGAGIUBSKTAHAHJUAZQCDRAUIHAHJUBSLTUCUDAUEZEDPUJEFDUJIUHQE
|
|
FRUJUHIAGIUFSMTUJJUIQFDRUJUIJAHJUFSNTUCUDUG $.
|
|
$}
|
|
|
|
${
|
|
keephyp3v.1 $e |- ( A = if ( ph , A , D ) -> ( rh <-> ch ) ) $.
|
|
keephyp3v.2 $e |- ( B = if ( ph , B , R ) -> ( ch <-> th ) ) $.
|
|
keephyp3v.3 $e |- ( C = if ( ph , C , S ) -> ( th <-> ta ) ) $.
|
|
keephyp3v.4 $e |- ( D = if ( ph , A , D ) -> ( et <-> ze ) ) $.
|
|
keephyp3v.5 $e |- ( R = if ( ph , B , R ) -> ( ze <-> si ) ) $.
|
|
keephyp3v.6 $e |- ( S = if ( ph , C , S ) -> ( si <-> ta ) ) $.
|
|
keephyp3v.7 $e |- rh $.
|
|
keephyp3v.8 $e |- et $.
|
|
$( Keep a hypothesis containing 3 class variables. (Contributed by NM,
|
|
27-Sep-1999.) $)
|
|
keephyp3v $p |- ta $=
|
|
( cif wceq wb iftrue eqcomd syl 3bitrd mpbii wn iffalse pm2.61i ) ADAHDUA
|
|
AHBCDAIAILUCZUDHBUEAUNIAILUFUGOUHAJAJMUCZUDBCUEAUOJAJMUFUGPUHAKAKNUCZUDCD
|
|
UEAUPKAKNUFUGQUHUIUJAUKZEDUBUQEFGDUQLUNUDEFUEUQUNLAILULUGRUHUQMUOUDFGUEUQ
|
|
UOMAJMULUGSUHUQNUPUDGDUEUQUPNAKNULUGTUHUIUJUM $.
|
|
$}
|
|
|
|
${
|
|
keepel.1 $e |- A e. C $.
|
|
keepel.2 $e |- B e. C $.
|
|
$( Keep a membership hypothesis for weak deduction theorem, when special
|
|
case ` B e. C ` is provable. (Contributed by NM, 14-Aug-1999.) $)
|
|
keepel $p |- if ( ph , A , B ) e. C $=
|
|
( wcel cif eleq1 keephyp ) ABDGCDGABCHZDGBCBKDICKDIEFJ $.
|
|
$}
|
|
|
|
${
|
|
dedex.1 $e |- A e. _V $.
|
|
dedex.2 $e |- B e. _V $.
|
|
$( Conditional operator existence. (Contributed by NM, 2-Sep-2004.) $)
|
|
ifex $p |- if ( ph , A , B ) e. _V $=
|
|
( cvv keepel ) ABCFDEG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B y $. $d ph x y $.
|
|
$( Conditional operator existence. (Contributed by NM, 21-Mar-2011.) $)
|
|
ifexg $p |- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. _V ) $=
|
|
( vx vy cv cif cvv wcel wceq ifeq1 eleq1d ifeq2 vex ifex vtocl2g ) AFHZGH
|
|
ZIZJKABTIZJKABCIZJKFGBCDESBLUAUBJASBTMNTCLUBUCJATCBONASTFPGPQR $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Power classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the symbol for power class. $)
|
|
$c ~P $. $( Calligraphic P $)
|
|
|
|
$( Extend class notation to include power class. (The tilde in the Metamath
|
|
token is meant to suggest the calligraphic font of the P.) $)
|
|
cpw $a class ~P A $.
|
|
|
|
${
|
|
$d x A $. $d y A $. $d z x $. $d z y $. $d z A $.
|
|
$( Soundness justification theorem for ~ df-pw . (Contributed by Rodolfo
|
|
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
pwjust $p |- { x | x C_ A } = { y | y C_ A } $=
|
|
( vz cv wss cab sseq1 cbvabv eqtri ) AEZCFZAGDEZCFZDGBEZCFZBGLNADKMCHINPD
|
|
BMOCHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we
|
|
also let it apply to proper classes, i.e. those that are not members of
|
|
` _V ` . When applied to a set, this produces its power set. A power
|
|
set of S is the set of all subsets of S, including the empty set and S
|
|
itself. For example, if ` A = { ` 3 , 5 , 7 ` } ` , then
|
|
` ~P A = { (/) , { ` 3 ` } , { ` 5 ` } , { ` 7 ` } , { ` 3 , 5 ` } , `
|
|
` { ` 3 , 7 ` } , { ` 5 , 7 ` } , { ` 3 , 5 , 7 ` } } ` (ex-pw in
|
|
set.mm). We will later introduce the Axiom of Power Sets ax-pow in
|
|
set.mm, which can be expressed in class notation per ~ pwexg . Still
|
|
later we will prove, in hashpw in set.mm, that the size of the power set
|
|
of a finite set is 2 raised to the power of the size of the set.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
df-pw $a |- ~P A = { x | x C_ A } $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) $)
|
|
pweq $p |- ( A = B -> ~P A = ~P B ) $=
|
|
( vx wceq cv wss cab cpw sseq2 abbidv df-pw 3eqtr4g ) ABDZCEZAFZCGNBFZCGA
|
|
HBHMOPCABNIJCAKCBKL $.
|
|
$}
|
|
|
|
${
|
|
pweqi.1 $e |- A = B $.
|
|
$( Equality inference for power class. (Contributed by NM,
|
|
27-Nov-2013.) $)
|
|
pweqi $p |- ~P A = ~P B $=
|
|
( wceq cpw pweq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
pweqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for power class. (Contributed by NM,
|
|
27-Nov-2013.) $)
|
|
pweqd $p |- ( ph -> ~P A = ~P B ) $=
|
|
( wceq cpw pweq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
${
|
|
elpw.1 $e |- A e. _V $.
|
|
$( Membership in a power class. Theorem 86 of [Suppes] p. 47.
|
|
(Contributed by NM, 31-Dec-1993.) $)
|
|
elpw $p |- ( A e. ~P B <-> A C_ B ) $=
|
|
( vx cv wss cpw sseq1 df-pw elab2 ) DEZBFABFDABGCKABHDBIJ $.
|
|
$}
|
|
|
|
$( Membership in a power class. Theorem 86 of [Suppes] p. 47. See also
|
|
elpw2g in set.mm. (Contributed by NM, 6-Aug-2000.) $)
|
|
elpwg $p |- ( A e. V -> ( A e. ~P B <-> A C_ B ) ) $=
|
|
( vx cv cpw wcel wss eleq1 sseq1 vex elpw vtoclbg ) DEZBFZGNBHAOGABHDACNA
|
|
OINABJNBDKLM $.
|
|
$}
|
|
|
|
$( Subset relation implied by membership in a power class. (Contributed by
|
|
NM, 17-Feb-2007.) $)
|
|
elpwi $p |- ( A e. ~P B -> A C_ B ) $=
|
|
( cpw wcel wss elpwg ibi ) ABCZDABEABHFG $.
|
|
|
|
${
|
|
elpwid.1 $e |- ( ph -> A e. ~P B ) $.
|
|
$( An element of a power class is a subclass. Deduction form of ~ elpwi .
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
elpwid $p |- ( ph -> A C_ B ) $=
|
|
( cpw wcel wss elpwi syl ) ABCEFBCGDBCHI $.
|
|
$}
|
|
|
|
$( If ` A ` belongs to a part of ` C ` then ` A ` belongs to ` C ` .
|
|
(Contributed by FL, 3-Aug-2009.) $)
|
|
elelpwi $p |- ( ( A e. B /\ B e. ~P C ) -> A e. C ) $=
|
|
( cpw wcel elpwi sseld impcom ) BCDEZABEACEIBCABCFGH $.
|
|
|
|
${
|
|
$d y A $. $d x y $.
|
|
nfpw.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for power class. (Contributed by NM,
|
|
28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) $)
|
|
nfpw $p |- F/_ x ~P A $=
|
|
( vy cpw cv wss cab df-pw nfcv nfss nfab nfcxfr ) ABEDFZBGZDHDBIOADANBANJ
|
|
CKLM $.
|
|
$}
|
|
|
|
$( Membership of the original in a power set. (Contributed by Stefan O'Rear,
|
|
1-Feb-2015.) $)
|
|
pwidg $p |- ( A e. V -> A e. ~P A ) $=
|
|
( wcel cpw wss ssid elpwg mpbiri ) ABCAADCAAEAFAABGH $.
|
|
|
|
${
|
|
pwid.1 $e |- A e. _V $.
|
|
$( A set is a member of its power class. Theorem 87 of [Suppes] p. 47.
|
|
(Contributed by NM, 5-Aug-1993.) $)
|
|
pwid $p |- A e. ~P A $=
|
|
( cvv wcel cpw pwidg ax-mp ) ACDAAEDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass relationship for power class. (Contributed by NM,
|
|
21-Jun-2009.) $)
|
|
pwss $p |- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) ) $=
|
|
( cpw wss cv wcel wi wal dfss2 df-pw abeq2i imbi1i albii bitri ) BDZCEAFZ
|
|
PGZQCGZHZAIQBEZSHZAIAPCJTUBARUASUAAPABKLMNO $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Unordered and ordered pairs
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend class notation to include singleton. $)
|
|
csn $a class { A } $.
|
|
$( Extend class notation to include unordered pair. $)
|
|
cpr $a class { A , B } $.
|
|
$( Extend class notation to include unordered triplet. $)
|
|
ctp $a class { A , B , C } $.
|
|
|
|
${
|
|
$d x A $. $d y A $. $d z x $. $d z y $. $d z A $.
|
|
$( Soundness justification theorem for ~ df-sn . (Contributed by Rodolfo
|
|
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
snjust $p |- { x | x = A } = { y | y = A } $=
|
|
( vz cv wceq cab eqeq1 cbvabv eqtri ) AEZCFZAGDEZCFZDGBEZCFZBGLNADKMCHINP
|
|
DBMOCHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For
|
|
convenience, it is well-defined for proper classes, i.e., those that are
|
|
not elements of ` _V ` , although it is not very meaningful in this
|
|
case. For an alternate definition see ~ dfsn2 . (Contributed by NM,
|
|
5-Aug-1993.) $)
|
|
df-sn $a |- { A } = { x | x = A } $.
|
|
$}
|
|
|
|
$( Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For
|
|
example, ` A e. { ` 1 , -u 1 ` } -> ( A ` ^ 2 ` ) = ` 1 (ex-pr in
|
|
set.mm). They are unordered, so ` { A , B } = { B , A } ` as proven by
|
|
~ prcom . For a more traditional definition, but requiring a dummy
|
|
variable, see ~ dfpr2 . (Contributed by NM, 5-Aug-1993.) $)
|
|
df-pr $a |- { A , B } = ( { A } u. { B } ) $.
|
|
|
|
$( Define unordered triple of classes. Definition of [Enderton] p. 19.
|
|
(Contributed by NM, 9-Apr-1994.) $)
|
|
df-tp $a |- { A , B , C } = ( { A , B } u. { C } ) $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring]
|
|
p. 15. (Contributed by NM, 5-Aug-1993.) $)
|
|
sneq $p |- ( A = B -> { A } = { B } ) $=
|
|
( vx wceq cv cab csn eqeq2 abbidv df-sn 3eqtr4g ) ABDZCEZADZCFMBDZCFAGBGL
|
|
NOCABMHICAJCBJK $.
|
|
$}
|
|
|
|
${
|
|
sneqi.1 $e |- A = B $.
|
|
$( Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) $)
|
|
sneqi $p |- { A } = { B } $=
|
|
( wceq csn sneq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
sneqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) $)
|
|
sneqd $p |- ( ph -> { A } = { B } ) $=
|
|
( wceq csn sneq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( Alternate definition of singleton. Definition 5.1 of [TakeutiZaring]
|
|
p. 15. (Contributed by NM, 24-Apr-1994.) $)
|
|
dfsn2 $p |- { A } = { A , A } $=
|
|
( cpr csn cun df-pr unidm eqtr2i ) AABACZHDHAAEHFG $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
|
|
p. 15. (Contributed by NM, 5-Aug-1993.) $)
|
|
elsn $p |- ( x e. { A } <-> x = A ) $=
|
|
( cv wceq csn df-sn abeq2i ) ACBDABEABFG $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Alternate definition of unordered pair. Definition 5.1 of
|
|
[TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) $)
|
|
dfpr2 $p |- { A , B } = { x | ( x = A \/ x = B ) } $=
|
|
( cpr csn cun cv wceq cab df-pr wcel elun elsn orbi12i bitri abbi2i eqtri
|
|
wo ) BCDBEZCEZFZAGZBHZUBCHZRZAIBCJUEAUAUBUAKUBSKZUBTKZRUEUBSTLUFUCUGUDABM
|
|
ACMNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( A member of an unordered pair of classes is one or the other of them.
|
|
Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM,
|
|
13-Sep-1995.) $)
|
|
elprg $p |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) ) $=
|
|
( vx cv wceq wo cpr eqeq1 orbi12d dfpr2 elab2g ) EFZBGZNCGZHABGZACGZHEABC
|
|
IDNAGOQPRNABJNACJKEBCLM $.
|
|
$}
|
|
|
|
${
|
|
elpr.1 $e |- A e. _V $.
|
|
$( A member of an unordered pair of classes is one or the other of them.
|
|
Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM,
|
|
13-Sep-1995.) $)
|
|
elpr $p |- ( A e. { B , C } <-> ( A = B \/ A = C ) ) $=
|
|
( cvv wcel cpr wceq wo wb elprg ax-mp ) AEFABCGFABHACHIJDABCEKL $.
|
|
$}
|
|
|
|
${
|
|
elpr2.1 $e |- B e. _V $.
|
|
elpr2.2 $e |- C e. _V $.
|
|
$( A member of an unordered pair of classes is one or the other of them.
|
|
Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM,
|
|
14-Oct-2005.) $)
|
|
elpr2 $p |- ( A e. { B , C } <-> ( A = B \/ A = C ) ) $=
|
|
( cpr wcel wceq wo elprg ibi cvv wb eleq1 mpbiri jaoi syl ibir impbii ) A
|
|
BCFZGZABHZACHZIZUAUDABCTJKUDUAUDALGZUAUDMUBUEUCUBUEBLGDABLNOUCUECLGEACLNO
|
|
PABCLJQRS $.
|
|
$}
|
|
|
|
$( If a class is an element of a pair, then it is one of the two paired
|
|
elements. (Contributed by Scott Fenton, 1-Apr-2011.) $)
|
|
elpri $p |- ( A e. { B , C } -> ( A = B \/ A = C ) ) $=
|
|
( cpr wcel wceq wo elprg ibi ) ABCDZEABFACFGABCJHI $.
|
|
|
|
${
|
|
nelpri.1 $e |- A =/= B $.
|
|
nelpri.2 $e |- A =/= C $.
|
|
$( If an element doesn't match the items in an unordered pair, it is not in
|
|
the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
nelpri $p |- -. A e. { B , C } $=
|
|
( wne cpr wcel wn wa wceq wo neanior elpri con3i sylbi mp2an ) ABFZACFZAB
|
|
CGHZIZDERSJABKACKLZIUAABACMTUBABCNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
|
|
p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof
|
|
shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
elsncg $p |- ( A e. V -> ( A e. { B } <-> A = B ) ) $=
|
|
( vx cv wceq csn eqeq1 df-sn elab2g ) DEZBFABFDABGCKABHDBIJ $.
|
|
$}
|
|
|
|
${
|
|
elsnc.1 $e |- A e. _V $.
|
|
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
|
|
p. 15. (Contributed by NM, 13-Sep-1995.) $)
|
|
elsnc $p |- ( A e. { B } <-> A = B ) $=
|
|
( cvv wcel csn wceq wb elsncg ax-mp ) ADEABFEABGHCABDIJ $.
|
|
$}
|
|
|
|
$( There is only one element in a singleton. (Contributed by NM,
|
|
5-Jun-1994.) $)
|
|
elsni $p |- ( A e. { B } -> A = B ) $=
|
|
( csn wcel wceq elsncg ibi ) ABCZDABEABHFG $.
|
|
|
|
$( A set is a member of its singleton. Part of Theorem 7.6 of [Quine]
|
|
p. 49. (Contributed by NM, 28-Oct-2003.) $)
|
|
snidg $p |- ( A e. V -> A e. { A } ) $=
|
|
( wcel csn wceq eqid elsncg mpbiri ) ABCAADCAAEAFAABGH $.
|
|
|
|
$( A class is a set iff it is a member of its singleton. (Contributed by NM,
|
|
5-Apr-2004.) $)
|
|
snidb $p |- ( A e. _V <-> A e. { A } ) $=
|
|
( cvv wcel csn snidg elex impbii ) ABCAADZCABEAHFG $.
|
|
|
|
${
|
|
snid.1 $e |- A e. _V $.
|
|
$( A set is a member of its singleton. Part of Theorem 7.6 of [Quine]
|
|
p. 49. (Contributed by NM, 31-Dec-1993.) $)
|
|
snid $p |- A e. { A } $=
|
|
( cvv wcel csn snidb mpbi ) ACDAAEDBAFG $.
|
|
$}
|
|
|
|
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
|
|
p. 15. This variation requires only that ` B ` , rather than ` A ` , be a
|
|
set. (Contributed by NM, 28-Oct-2003.) $)
|
|
elsnc2g $p |- ( B e. V -> ( A e. { B } <-> A = B ) ) $=
|
|
( wcel csn wceq elsni snidg eleq1 syl5ibrcom impbid2 ) BCDZABEZDZABFZABGLNO
|
|
BMDBCHABMIJK $.
|
|
|
|
${
|
|
elsnc2.1 $e |- B e. _V $.
|
|
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
|
|
p. 15. This variation requires only that ` B ` , rather than ` A ` , be
|
|
a set. (Contributed by NM, 12-Jun-1994.) $)
|
|
elsnc2 $p |- ( A e. { B } <-> A = B ) $=
|
|
( cvv wcel csn wceq wb elsnc2g ax-mp ) BDEABFEABGHCABDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d ps x $.
|
|
$( Substitution expressed in terms of quantification over a singleton.
|
|
(Contributed by Mario Carneiro, 23-Apr-2015.) $)
|
|
ralsns $p |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) ) $=
|
|
( wcel wsbc cv wceq wal csn wral sbc6g df-ral imbi1i albii bitri syl6rbbr
|
|
wi elsn ) CDEABCFBGZCHZARZBIZABCJZKZABCDLUETUDEZARZBIUCABUDMUGUBBUFUAABCS
|
|
NOPQ $.
|
|
|
|
$( Restricted existential quantification over a singleton. (Contributed by
|
|
Mario Carneiro, 23-Apr-2015.) $)
|
|
rexsns $p |- ( A e. V -> ( E. x e. { A } ph <-> [. A / x ]. ph ) ) $=
|
|
( wcel wsbc cv wceq wa wex csn wrex wb sbc5 a1i df-rex anbi1i exbii bitri
|
|
elsn syl6rbbr ) CDEZABCFZBGZCHZAIZBJZABCKZLZUCUGMUBABCNOUIUDUHEZAIZBJUGAB
|
|
UHPUKUFBUJUEABCTQRSUA $.
|
|
|
|
ralsng.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Substitution expressed in terms of quantification over a singleton.
|
|
(Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro,
|
|
23-Apr-2015.) $)
|
|
ralsng $p |- ( A e. V -> ( A. x e. { A } ph <-> ps ) ) $=
|
|
( wcel csn wral wsbc ralsns sbcieg bitrd ) DEGACDHIACDJBACDEKABCDEFLM $.
|
|
|
|
$( Restricted existential quantification over a singleton. (Contributed by
|
|
NM, 29-Jan-2012.) $)
|
|
rexsng $p |- ( A e. V -> ( E. x e. { A } ph <-> ps ) ) $=
|
|
( wcel csn wrex wsbc rexsns sbcieg bitrd ) DEGACDHIACDJBACDEKABCDEFLM $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d ps x $.
|
|
ralsn.1 $e |- A e. _V $.
|
|
ralsn.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Convert a quantification over a singleton to a substitution.
|
|
(Contributed by NM, 27-Apr-2009.) $)
|
|
ralsn $p |- ( A. x e. { A } ph <-> ps ) $=
|
|
( cvv wcel csn wral wb ralsng ax-mp ) DGHACDIJBKEABCDGFLM $.
|
|
|
|
$( Restricted existential quantification over a singleton. (Contributed by
|
|
Jeff Madsen, 5-Jan-2011.) $)
|
|
rexsn $p |- ( E. x e. { A } ph <-> ps ) $=
|
|
( cvv wcel csn wrex wb rexsng ax-mp ) DGHACDIJBKEABCDGFLM $.
|
|
$}
|
|
|
|
$( Members of an unordered triple of classes. (Contributed by FL,
|
|
2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) $)
|
|
eltpg $p |- ( A e. V -> ( A e. { B , C , D } <->
|
|
( A = B \/ A = C \/ A = D ) ) ) $=
|
|
( wcel cpr csn wo wceq ctp w3o elprg elsncg orbi12d df-tp eleq2i elun bitri
|
|
cun df-3or 3bitr4g ) AEFZABCGZFZADHZFZIZABJZACJZIZADJZIABCDKZFZUIUJULLUCUEU
|
|
KUGULABCEMADENOUNAUDUFTZFUHUMUOABCDPQAUDUFRSUIUJULUAUB $.
|
|
|
|
$( A member of an unordered triple of classes is one of them. (Contributed
|
|
by Mario Carneiro, 11-Feb-2015.) $)
|
|
eltpi $p |- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) ) $=
|
|
( ctp wcel wceq w3o eltpg ibi ) ABCDEZFABGACGADGHABCDKIJ $.
|
|
|
|
${
|
|
eltp.1 $e |- A e. _V $.
|
|
$( A member of an unordered triple of classes is one of them. Special case
|
|
of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM,
|
|
8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.) $)
|
|
eltp $p |- ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) $=
|
|
( cvv wcel ctp wceq w3o wb eltpg ax-mp ) AFGABCDHGABIACIADIJKEABCDFLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Alternate definition of unordered triple of classes. Special case of
|
|
Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM,
|
|
8-Apr-1994.) $)
|
|
dftp2 $p |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) } $=
|
|
( cv wceq w3o ctp vex eltp abbi2i ) AEZBFLCFLDFGABCDHLBCDAIJK $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
nfpr.1 $e |- F/_ x A $.
|
|
nfpr.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for unordered pairs. (Contributed by
|
|
NM, 14-Nov-1995.) $)
|
|
nfpr $p |- F/_ x { A , B } $=
|
|
( vy cpr cv wceq wo cab dfpr2 nfeq2 nfor nfab nfcxfr ) ABCGFHZBIZQCIZJZFK
|
|
FBCLTAFRSAAQBDMAQCEMNOP $.
|
|
$}
|
|
|
|
$( Membership of a conditional operator in an unordered pair. (Contributed
|
|
by NM, 17-Jun-2007.) $)
|
|
ifpr $p |- ( ( A e. C /\ B e. D ) -> if ( ph , A , B ) e. { A , B } ) $=
|
|
( wcel cvv cif cpr elex wa ifcl wceq wo ifeqor elprg mpbiri syl syl2an ) BD
|
|
FBGFZCGFZABCHZBCIFZCEFBDJCEJTUAKUBGFZUCABCGLUDUCUBBMUBCMNABCOUBBCGPQRS $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ps $. $d x ch $. $d x th $.
|
|
ralprg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ralprg.2 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
$( Convert a quantification over a pair to a conjunction. (Contributed by
|
|
NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
ralprg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( A. x e. { A , B } ph <-> ( ps /\ ch ) ) ) $=
|
|
( cpr wral csn wa wcel cun df-pr raleqi ralunb ralsng bi2anan9 syl5bb
|
|
bitri ) ADEFKZLZADEMZLZADFMZLZNZEGOZFHOZNBCNUEADUFUHPZLUJADUDUMEFQRADUFUH
|
|
SUCUKUGBULUICABDEGITACDFHJTUAUB $.
|
|
|
|
$( Convert a quantification over a pair to a disjunction. (Contributed by
|
|
NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
rexprg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( E. x e. { A , B } ph <-> ( ps \/ ch ) ) ) $=
|
|
( cpr wrex csn wo wcel wa cun df-pr rexeqi rexsng orbi1d orbi2d sylan9bb
|
|
rexun bitri syl5bb ) ADEFKZLZADEMZLZADFMZLZNZEGOZFHOZPBCNZUHADUIUKQZLUMAD
|
|
UGUQEFRSADUIUKUDUEUNUMBULNUOUPUNUJBULABDEGITUAUOULCBACDFHJTUBUCUF $.
|
|
|
|
raltpg.3 $e |- ( x = C -> ( ph <-> th ) ) $.
|
|
$( Convert a quantification over a triple to a conjunction. (Contributed
|
|
by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
raltpg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) ) $=
|
|
( wcel w3a cpr wral csn wa ctp wb ralprg ralsng bi2anan9 3impa cun raleqi
|
|
df-tp ralunb bitri df-3an 3bitr4g ) FIOZGJOZHKOZPAEFGQZRZAEHSZRZTZBCTZDTZ
|
|
AEFGHUAZRZBCDPUNUOUPVAVCUBUNUOTURVBUPUTDABCEFGIJLMUCADEHKNUDUEUFVEAEUQUSU
|
|
GZRVAAEVDVFFGHUIUHAEUQUSUJUKBCDULUM $.
|
|
|
|
$( Convert a quantification over a triple to a disjunction. (Contributed
|
|
by Mario Carneiro, 23-Apr-2015.) $)
|
|
rextpg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) ) $=
|
|
( wcel w3a cpr wrex csn wo ctp wb wa rexprg orbi1d rexsng orbi2d sylan9bb
|
|
w3o 3impa cun df-tp rexeqi rexun bitri df-3or 3bitr4g ) FIOZGJOZHKOZPAEFG
|
|
QZRZAEHSZRZTZBCTZDTZAEFGHUAZRZBCDUIURUSUTVEVGUBURUSUCZVEVFVDTUTVGVJVBVFVD
|
|
ABCEFGIJLMUDUEUTVDDVFADEHKNUFUGUHUJVIAEVAVCUKZRVEAEVHVKFGHULUMAEVAVCUNUOB
|
|
CDUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $. $d x ch $.
|
|
ralpr.1 $e |- A e. _V $.
|
|
ralpr.2 $e |- B e. _V $.
|
|
ralpr.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ralpr.4 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
$( Convert a quantification over a pair to a conjunction. (Contributed by
|
|
NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
ralpr $p |- ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) $=
|
|
( cvv wcel cpr wral wa wb ralprg mp2an ) EKLFKLADEFMNBCOPGHABCDEFKKIJQR
|
|
$.
|
|
|
|
$( Convert an existential quantification over a pair to a disjunction.
|
|
(Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro,
|
|
23-Apr-2015.) $)
|
|
rexpr $p |- ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) $=
|
|
( cvv wcel cpr wrex wo wb rexprg mp2an ) EKLFKLADEFMNBCOPGHABCDEFKKIJQR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x ps $. $d x ch $. $d x th $.
|
|
raltp.1 $e |- A e. _V $.
|
|
raltp.2 $e |- B e. _V $.
|
|
raltp.3 $e |- C e. _V $.
|
|
raltp.4 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
raltp.5 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
raltp.6 $e |- ( x = C -> ( ph <-> th ) ) $.
|
|
$( Convert a quantification over a triple to a conjunction. (Contributed
|
|
by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
raltp $p |- ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) $=
|
|
( cvv wcel ctp wral w3a wb raltpg mp3an ) FOPGOPHOPAEFGHQRBCDSTIJKABCDEFG
|
|
HOOOLMNUAUB $.
|
|
|
|
$( Convert a quantification over a triple to a disjunction. (Contributed
|
|
by Mario Carneiro, 23-Apr-2015.) $)
|
|
rextp $p |- ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) $=
|
|
( cvv wcel ctp wrex w3o wb rextpg mp3an ) FOPGOPHOPAEFGHQRBCDSTIJKABCDEFG
|
|
HOOOLMNUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( TODO - make obsolete; use ralsnsSBC instead - also,
|
|
shorten posn w/ ralsn or ralsng $)
|
|
$( Substitution expressed in terms of quantification over a singleton.
|
|
(Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro,
|
|
23-Apr-2015.) $)
|
|
sbcsng $p |- ( A e. V -> ( [. A / x ]. ph <-> A. x e. { A } ph ) ) $=
|
|
( wcel csn wral wsbc ralsns bicomd ) CDEABCFGABCHABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
nfsn.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for singletons. (Contributed by NM,
|
|
14-Nov-1995.) $)
|
|
nfsn $p |- F/_ x { A } $=
|
|
( csn cpr dfsn2 nfpr nfcxfr ) ABDBBEBFABBCCGH $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d V y $. $d x y $.
|
|
$( Distribute proper substitution through the singleton of a class.
|
|
~ csbsng is derived from the virtual deduction proof csbsngVD in
|
|
set.mm. (Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbsng $p |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } ) $=
|
|
( vy wcel wceq cab csb csn wsbc csbabg sbceq2g abbidv eqtrd df-sn csbeq2i
|
|
cv 3eqtr4g ) BDFZABERZCGZEHZIZUAABCIZGZEHZABCJZIUEJTUDUBABKZEHUGUBAEBDLTU
|
|
IUFEABUACDMNOABUHUCECPQEUEPS $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Intersection with the singleton of a non-member is disjoint.
|
|
(Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) $)
|
|
disjsn $p |- ( ( A i^i { B } ) = (/) <-> -. B e. A ) $=
|
|
( vx csn cin c0 wceq cv wcel wn wi wal wa disj1 con2b imbi1i imnan 3bitri
|
|
elsn albii wex alnex df-clel xchbinxr ) ABDZEFGCHZAIZUFUEIZJKZCLUFBGZUGMZ
|
|
JZCLZBAIZJCAUENUIULCUIUHUGJZKUJUOKULUGUHOUHUJUOCBSPUJUGQRTUMUKCUAUNUKCUBC
|
|
BAUCUDR $.
|
|
$}
|
|
|
|
$( Intersection of distinct singletons is disjoint. (Contributed by NM,
|
|
25-May-1998.) $)
|
|
disjsn2 $p |- ( A =/= B -> ( { A } i^i { B } ) = (/) ) $=
|
|
( wne csn wcel wn cin c0 wceq elsni eqcomd necon3ai disjsn sylibr ) ABCBADZ
|
|
EZFOBDGHIPABPBABAJKLOBMN $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( The singleton of a proper class (one that doesn't exist) is the empty
|
|
set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.) $)
|
|
snprc $p |- ( -. A e. _V <-> { A } = (/) ) $=
|
|
( vx csn c0 wceq cvv wcel cv wex wn elsn exbii neq0 isset 3bitr4i con1bii
|
|
) ACZDEZAFGZBHZQGZBITAEZBIRJSUAUBBBAKLBQMBANOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x B $.
|
|
r19.12sn.1 $e |- A e. _V $.
|
|
$( Special case of ~ r19.12 where its converse holds. (Contributed by NM,
|
|
19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
r19.12sn $p |- ( E. x e. { A } A. y e. B ph
|
|
<-> A. y e. B E. x e. { A } ph ) $=
|
|
( cvv wcel wral csn wrex wb wsbc sbcralg rexsns ralbidv 3bitr4d ax-mp ) D
|
|
GHZACEIZBDJZKZABUAKZCEIZLFSTBDMABDMZCEIUBUDABCDEGNTBDGOSUCUECEABDGOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Condition where a restricted class abstraction is a singleton.
|
|
(Contributed by NM, 28-May-2006.) $)
|
|
rabsn $p |- ( B e. A -> { x e. A | x = B } = { B } ) $=
|
|
( wcel cv wceq wa cab crab csn eleq1 pm5.32ri abbidv df-rab df-sn 3eqtr4g
|
|
baib ) CBDZAEZBDZSCFZGZAHUAAHUAABICJRUBUAAUBRUAUATRSCBKLQMUAABNACOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $. $d y A $.
|
|
$( Another way to express existential uniqueness of a wff: its class
|
|
abstraction is a singleton. (Contributed by Mario Carneiro,
|
|
14-Nov-2016.) $)
|
|
euabsn2 $p |- ( E! x ph <-> E. y { x | ph } = { y } ) $=
|
|
( weu cv wceq wb wal wex cab csn df-eu wcel abeq1 elsn bibi2i albii bitri
|
|
exbii bitr4i ) ABDABEZCEZFZGZBHZCIABJUBKZFZCIABCLUGUECUGAUAUFMZGZBHUEABUF
|
|
NUIUDBUHUCABUBOPQRST $.
|
|
|
|
$( Another way to express existential uniqueness of a wff: its class
|
|
abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) $)
|
|
euabsn $p |- ( E! x ph <-> E. x { x | ph } = { x } ) $=
|
|
( vy weu cab cv csn wceq wex euabsn2 nfab1 nfeq1 sneq eqeq2d cbvex bitr4i
|
|
nfv ) ABDABEZCFZGZHZCIRBFZGZHZBIABCJUDUABCUDCQBRTABKLUBSHUCTRUBSMNOP $.
|
|
|
|
$( A way to express restricted existential uniqueness of a wff: its
|
|
restricted class abstraction is a singleton. (Contributed by NM,
|
|
30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) $)
|
|
reusn $p |- ( E! x e. A ph <-> E. y { x e. A | ph } = { y } ) $=
|
|
( cv wcel wa weu cab csn wceq wex wreu euabsn2 df-reu df-rab eqeq1i exbii
|
|
crab 3bitr4i ) BEDFAGZBHUABIZCEJZKZCLABDMABDSZUCKZCLUABCNABDOUFUDCUEUBUCA
|
|
BDPQRT $.
|
|
|
|
$( Restricted existential uniqueness determined by a singleton.
|
|
(Contributed by NM, 29-May-2006.) $)
|
|
absneu $p |- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph ) $=
|
|
( vy wcel cab csn wceq wa wex weu sneq eqeq2d spcegv imp euabsn2 sylibr
|
|
cv ) CDFZABGZCHZIZJUAESZHZIZEKZABLTUCUGUFUCECDUDCIUEUBUAUDCMNOPABEQR $.
|
|
|
|
$( Restricted existential uniqueness determined by a singleton.
|
|
(Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro,
|
|
23-Dec-2016.) $)
|
|
rabsneu $p |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x e. B ph ) $=
|
|
( wcel crab csn wceq wa cv weu df-rab eqeq1i absneu sylan2b df-reu sylibr
|
|
wreu cab ) CEFZABDGZCHZIZJBKDFAJZBLZABDSUDUAUEBTZUCIUFUBUGUCABDMNUEBCEOPA
|
|
BDQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Two ways to express " ` A ` is a singleton." (Contributed by NM,
|
|
30-Oct-2010.) $)
|
|
eusn $p |- ( E! x x e. A <-> E. x A = { x } ) $=
|
|
( cv wcel weu cab csn wceq wex euabsn abid2 eqeq1i exbii bitri ) ACZBDZAE
|
|
PAFZOGZHZAIBRHZAIPAJSTAQBRABKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
rabsnt.1 $e |- B e. _V $.
|
|
rabsnt.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( Truth implied by equality of a restricted class abstraction and a
|
|
singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario
|
|
Carneiro, 23-Dec-2016.) $)
|
|
rabsnt $p |- ( { x e. A | ph } = { B } -> ps ) $=
|
|
( crab csn wceq wcel snid id syl5eleqr elrab simprbi syl ) ACDHZEIZJZERKZ
|
|
BTESREFLTMNUAEDKBABCEDGOPQ $.
|
|
$}
|
|
|
|
$( Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) $)
|
|
prcom $p |- { A , B } = { B , A } $=
|
|
( csn cun cpr uncom df-pr 3eqtr4i ) ACZBCZDJIDABEBAEIJFABGBAGH $.
|
|
|
|
$( Equality theorem for unordered pairs. (Contributed by NM,
|
|
29-Mar-1998.) $)
|
|
preq1 $p |- ( A = B -> { A , C } = { B , C } ) $=
|
|
( wceq csn cun cpr sneq uneq1d df-pr 3eqtr4g ) ABDZAEZCEZFBEZNFACGBCGLMONAB
|
|
HIACJBCJK $.
|
|
|
|
$( Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) $)
|
|
preq2 $p |- ( A = B -> { C , A } = { C , B } ) $=
|
|
( wceq cpr preq1 prcom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.
|
|
|
|
$( Equality theorem for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq12 $p |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } ) $=
|
|
( wceq cpr preq1 preq2 sylan9eq ) ACEBDEABFCBFCDFACBGBDCHI $.
|
|
|
|
${
|
|
preq1i.1 $e |- A = B $.
|
|
$( Equality inference for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq1i $p |- { A , C } = { B , C } $=
|
|
( wceq cpr preq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq2i $p |- { C , A } = { C , B } $=
|
|
( wceq cpr preq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
preq12i.2 $e |- C = D $.
|
|
$( Equality inference for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq12i $p |- { A , C } = { B , D } $=
|
|
( wceq cpr preq12 mp2an ) ABGCDGACHBDHGEFACBDIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
preq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq1d $p |- ( ph -> { A , C } = { B , C } ) $=
|
|
( wceq cpr preq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq2d $p |- ( ph -> { C , A } = { C , B } ) $=
|
|
( wceq cpr preq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
preq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for unordered pairs. (Contributed by NM,
|
|
19-Oct-2012.) $)
|
|
preq12d $p |- ( ph -> { A , C } = { B , D } ) $=
|
|
( wceq cpr preq12 syl2anc ) ABCHDEHBDICEIHFGBDCEJK $.
|
|
$}
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
13-Sep-2011.) $)
|
|
tpeq1 $p |- ( A = B -> { A , C , D } = { B , C , D } ) $=
|
|
( wceq cpr csn cun ctp preq1 uneq1d df-tp 3eqtr4g ) ABEZACFZDGZHBCFZPHACDIB
|
|
CDINOQPABCJKACDLBCDLM $.
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
13-Sep-2011.) $)
|
|
tpeq2 $p |- ( A = B -> { C , A , D } = { C , B , D } ) $=
|
|
( wceq cpr csn cun ctp preq2 uneq1d df-tp 3eqtr4g ) ABEZCAFZDGZHCBFZPHCADIC
|
|
BDINOQPABCJKCADLCBDLM $.
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
13-Sep-2011.) $)
|
|
tpeq3 $p |- ( A = B -> { C , D , A } = { C , D , B } ) $=
|
|
( wceq cpr csn cun ctp sneq uneq2d df-tp 3eqtr4g ) ABEZCDFZAGZHOBGZHCDAICDB
|
|
INPQOABJKCDALCDBLM $.
|
|
|
|
${
|
|
tpeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
22-Jun-2014.) $)
|
|
tpeq1d $p |- ( ph -> { A , C , D } = { B , C , D } ) $=
|
|
( wceq ctp tpeq1 syl ) ABCGBDEHCDEHGFBCDEIJ $.
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
22-Jun-2014.) $)
|
|
tpeq2d $p |- ( ph -> { C , A , D } = { C , B , D } ) $=
|
|
( wceq ctp tpeq2 syl ) ABCGDBEHDCEHGFBCDEIJ $.
|
|
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
22-Jun-2014.) $)
|
|
tpeq3d $p |- ( ph -> { C , D , A } = { C , D , B } ) $=
|
|
( wceq ctp tpeq3 syl ) ABCGDEBHDECHGFBCDEIJ $.
|
|
|
|
tpeq123d.2 $e |- ( ph -> C = D ) $.
|
|
tpeq123d.3 $e |- ( ph -> E = F ) $.
|
|
$( Equality theorem for unordered triples. (Contributed by NM,
|
|
22-Jun-2014.) $)
|
|
tpeq123d $p |- ( ph -> { A , C , E } = { B , D , F } ) $=
|
|
( ctp tpeq1d tpeq2d tpeq3d 3eqtrd ) ABDFKCDFKCEFKCEGKABCDFHLADECFIMAFGCEJ
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Rotation of the elements of an unordered triple. (Contributed by Alan
|
|
Sare, 24-Oct-2011.) $)
|
|
tprot $p |- { A , B , C } = { B , C , A } $=
|
|
( vx cv wceq w3o cab ctp 3orrot abbii dftp2 3eqtr4i ) DEZAFZNBFZNCFZGZDHP
|
|
QOGZDHABCIBCAIRSDOPQJKDABCLDBCALM $.
|
|
$}
|
|
|
|
$( Swap 1st and 2nd members of an undordered triple. (Contributed by NM,
|
|
22-May-2015.) $)
|
|
tpcoma $p |- { A , B , C } = { B , A , C } $=
|
|
( cpr csn cun ctp prcom uneq1i df-tp 3eqtr4i ) ABDZCEZFBADZMFABCGBACGLNMABH
|
|
IABCJBACJK $.
|
|
|
|
$( Swap 2nd and 3rd members of an undordered triple. (Contributed by NM,
|
|
22-May-2015.) $)
|
|
tpcomb $p |- { A , B , C } = { A , C , B } $=
|
|
( ctp tpcoma tprot 3eqtr4i ) BCADCBADABCDACBDBCAEABCFACBFG $.
|
|
|
|
$( Split off the first element of an unordered triple. (Contributed by Mario
|
|
Carneiro, 5-Jan-2016.) $)
|
|
tpass $p |- { A , B , C } = ( { A } u. { B , C } ) $=
|
|
( ctp cpr csn cun df-tp tprot uncom 3eqtr4i ) BCADBCEZAFZGABCDMLGBCAHABCIML
|
|
JK $.
|
|
|
|
$( Two ways to write an unordered quadruple. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
qdass $p |- ( { A , B } u. { C , D } ) = ( { A , B , C } u. { D } ) $=
|
|
( cpr csn cun ctp unass df-tp uneq1i df-pr uneq2i 3eqtr4ri ) ABEZCFZGZDFZGO
|
|
PRGZGABCHZRGOCDEZGOPRITQRABCJKUASOCDLMN $.
|
|
|
|
$( Two ways to write an unordered quadruple. (Contributed by Mario Carneiro,
|
|
5-Jan-2016.) $)
|
|
qdassr $p |- ( { A , B } u. { C , D } ) = ( { A } u. { B , C , D } ) $=
|
|
( csn cun cpr ctp unass df-pr uneq1i tpass uneq2i 3eqtr4i ) AEZBEZFZCDGZFOP
|
|
RFZFABGZRFOBCDHZFOPRITQRABJKUASOBCDLMN $.
|
|
|
|
$( Unordered triple ` { A , A , B } ` is just an overlong way to write
|
|
` { A , B } ` . (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
tpidm12 $p |- { A , A , B } = { A , B } $=
|
|
( csn cun cpr ctp dfsn2 uneq1i df-pr df-tp 3eqtr4ri ) ACZBCZDAAEZMDABEAABFL
|
|
NMAGHABIAABJK $.
|
|
|
|
$( Unordered triple ` { A , B , A } ` is just an overlong way to write
|
|
` { A , B } ` . (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
tpidm13 $p |- { A , B , A } = { A , B } $=
|
|
( ctp cpr tprot tpidm12 eqtr3i ) AABCABACABDAABEABFG $.
|
|
|
|
$( Unordered triple ` { A , B , B } ` is just an overlong way to write
|
|
` { A , B } ` . (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
tpidm23 $p |- { A , B , B } = { A , B } $=
|
|
( ctp cpr tprot tpidm12 prcom 3eqtri ) ABBCBBACBADABDABBEBAFBAGH $.
|
|
|
|
$( Unordered triple ` { A , A , A } ` is just an overlong way to write
|
|
` { A } ` . (Contributed by David A. Wheeler, 10-May-2015.) $)
|
|
tpidm $p |- { A , A , A } = { A } $=
|
|
( ctp cpr csn tpidm12 dfsn2 eqtr4i ) AAABAACADAAEAFG $.
|
|
|
|
$( An unordered pair contains its first member. Part of Theorem 7.6 of
|
|
[Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) $)
|
|
prid1g $p |- ( A e. V -> A e. { A , B } ) $=
|
|
( wcel cpr wceq wo eqid orci elprg mpbiri ) ACDAABEDAAFZABFZGLMAHIAABCJK $.
|
|
|
|
$( An unordered pair contains its second member. Part of Theorem 7.6 of
|
|
[Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) $)
|
|
prid2g $p |- ( B e. V -> B e. { A , B } ) $=
|
|
( wcel cpr prid1g prcom syl6eleq ) BCDBBAEABEBACFBAGH $.
|
|
|
|
${
|
|
prid1.1 $e |- A e. _V $.
|
|
$( An unordered pair contains its first member. Part of Theorem 7.6 of
|
|
[Quine] p. 49. (Contributed by NM, 5-Aug-1993.) $)
|
|
prid1 $p |- A e. { A , B } $=
|
|
( cvv wcel cpr prid1g ax-mp ) ADEAABFECABDGH $.
|
|
$}
|
|
|
|
${
|
|
prid2.1 $e |- B e. _V $.
|
|
$( An unordered pair contains its second member. Part of Theorem 7.6 of
|
|
[Quine] p. 49. (Contributed by NM, 5-Aug-1993.) $)
|
|
prid2 $p |- B e. { A , B } $=
|
|
( cpr prid1 prcom eleqtri ) BBADABDBACEBAFG $.
|
|
$}
|
|
|
|
${
|
|
tpid1.1 $e |- A e. _V $.
|
|
$( One of the three elements of an unordered triple. (Contributed by NM,
|
|
7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
tpid1 $p |- A e. { A , B , C } $=
|
|
( ctp wcel wceq w3o eqid 3mix1i eltp mpbir ) AABCEFAAGZABGZACGZHMNOAIJAAB
|
|
CDKL $.
|
|
$}
|
|
|
|
${
|
|
tpid2.1 $e |- B e. _V $.
|
|
$( One of the three elements of an unordered triple. (Contributed by NM,
|
|
7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
tpid2 $p |- B e. { A , B , C } $=
|
|
( ctp wcel wceq w3o eqid 3mix2i eltp mpbir ) BABCEFBAGZBBGZBCGZHNMOBIJBAB
|
|
CDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x D $.
|
|
$( Closed theorem form of ~ tpid3 . This proof was automatically generated
|
|
from the virtual deduction proof tpid3gVD in set.mm using a translation
|
|
program. (Contributed by Alan Sare, 24-Oct-2011.) $)
|
|
tpid3g $p |- ( A e. B -> A e. { C , D , A } ) $=
|
|
( vx wcel cv wceq wex ctp elisset w3o cab wi 3mix3 a1i abid syl6ibr dftp2
|
|
eleq2i eleq1 mpbidi exlimdv mpd ) ABFZEGZAHZEIACDAJZFZEABKUEUGUIEUGUFUHFZ
|
|
UIUEUEUGUFUFCHZUFDHZUGLZEMZFZUJUEUGUMUOUGUMNUEUGUKULOPUMEQRUHUNUFECDASTRU
|
|
FAUHUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
tpid3.1 $e |- C e. _V $.
|
|
$( One of the three elements of an unordered triple. (Contributed by NM,
|
|
7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
tpid3 $p |- C e. { A , B , C } $=
|
|
( ctp wcel wceq w3o eqid 3mix3i eltp mpbir ) CABCEFCAGZCBGZCCGZHOMNCIJCAB
|
|
CDKL $.
|
|
$}
|
|
|
|
$( The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.) $)
|
|
snnzg $p |- ( A e. V -> { A } =/= (/) ) $=
|
|
( wcel csn c0 wne snidg ne0i syl ) ABCAADZCJEFABGJAHI $.
|
|
|
|
${
|
|
snnz.1 $e |- A e. _V $.
|
|
$( The singleton of a set is not empty. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
snnz $p |- { A } =/= (/) $=
|
|
( cvv wcel csn c0 wne snnzg ax-mp ) ACDAEFGBACHI $.
|
|
$}
|
|
|
|
${
|
|
prnz.1 $e |- A e. _V $.
|
|
$( A pair containing a set is not empty. (Contributed by NM,
|
|
9-Apr-1994.) $)
|
|
prnz $p |- { A , B } =/= (/) $=
|
|
( cpr wcel c0 wne prid1 ne0i ax-mp ) AABDZEKFGABCHKAIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A pair containing a set is not empty. (Contributed by FL,
|
|
19-Sep-2011.) $)
|
|
prnzg $p |- ( A e. V -> { A , B } =/= (/) ) $=
|
|
( vx cv cpr c0 wne wceq preq1 neeq1d vex prnz vtoclg ) DEZBFZGHABFZGHDACO
|
|
AIPQGOABJKOBDLMN $.
|
|
$}
|
|
|
|
${
|
|
tpnz.1 $e |- A e. _V $.
|
|
$( A triplet containing a set is not empty. (Contributed by NM,
|
|
10-Apr-1994.) $)
|
|
tpnz $p |- { A , B , C } =/= (/) $=
|
|
( ctp wcel c0 wne tpid1 ne0i ax-mp ) AABCEZFLGHABCDILAJK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
snss.1 $e |- A e. _V $.
|
|
$( The singleton of an element of a class is a subset of the class.
|
|
Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.) $)
|
|
snss $p |- ( A e. B <-> { A } C_ B ) $=
|
|
( vx cv csn wcel wi wal wceq wss elsn imbi1i albii dfss2 clel2 3bitr4ri )
|
|
DEZAFZGZRBGZHZDIRAJZUAHZDISBKABGUBUDDTUCUADALMNDSBODABCPQ $.
|
|
$}
|
|
|
|
$( Membership in a set with an element removed. (Contributed by NM,
|
|
10-Oct-2007.) $)
|
|
eldifsn $p |- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) ) $=
|
|
( csn cdif wcel wn wa wne eldif elsncg necon3bbid pm5.32i bitri ) ABCDZEFAB
|
|
FZAOFZGZHPACIZHABOJPRSPQACACBKLMN $.
|
|
|
|
$( Membership in a set with an element removed. (Contributed by NM,
|
|
10-Mar-2015.) $)
|
|
eldifsni $p |- ( A e. ( B \ { C } ) -> A =/= C ) $=
|
|
( csn cdif wcel wne eldifsn simprbi ) ABCDEFABFACGABCHI $.
|
|
|
|
$( ` A ` is not in ` ( B \ { A } ) ` . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
neldifsn $p |- -. A e. ( B \ { A } ) $=
|
|
( csn cdif wcel wne neirr eldifsni mto ) ABACDEAAFAGABAHI $.
|
|
|
|
$( ` A ` is not in ` ( B \ { A } ) ` . Deduction form. (Contributed by
|
|
David Moews, 1-May-2017.) $)
|
|
neldifsnd $p |- ( ph -> -. A e. ( B \ { A } ) ) $=
|
|
( csn cdif wcel wn neldifsn a1i ) BCBDEFGABCHI $.
|
|
|
|
$( Restricted existential quantification over a set with an element removed.
|
|
(Contributed by NM, 4-Feb-2015.) $)
|
|
rexdifsn $p |- ( E. x e. ( A \ { B } ) ph
|
|
<-> E. x e. A ( x =/= B /\ ph ) ) $=
|
|
( cv wne wa csn cdif wcel eldifsn anbi1i anass bitri rexbii2 ) ABEZDFZAGZBC
|
|
DHIZCPSJZAGPCJZQGZAGUARGTUBAPCDKLUAQAMNO $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( The singleton of an element of a class is a subset of the class.
|
|
Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) $)
|
|
snssg $p |- ( A e. V -> ( A e. B <-> { A } C_ B ) ) $=
|
|
( vx cv wcel csn wss eleq1 wceq sneq sseq1d vex snss vtoclbg ) DEZBFPGZBH
|
|
ABFAGZBHDACPABIPAJQRBPAKLPBDMNO $.
|
|
|
|
$( An element not in a set can be removed without affecting the set.
|
|
(Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
difsn $p |- ( -. A e. B -> ( B \ { A } ) = B ) $=
|
|
( vx wcel wn csn cdif cv wne wa eldifsn simpl wceq eleq1 biimpcd necon3bd
|
|
com12 ancld impbid2 syl5bb eqrdv ) ABDZEZCBAFGZBCHZUDDUEBDZUEAIZJZUCUFUEB
|
|
AKUCUHUFUFUGLUCUFUGUFUCUGUFUBUEAUEAMUFUBUEABNOPQRSTUA $.
|
|
|
|
$( Removal of a singleton from an unordered pair. (Contributed by NM,
|
|
16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
difprsnss $p |- ( { A , B } \ { A } ) C_ { B } $=
|
|
( vx cpr csn cdif cv wcel wn wa wceq wo elpr elsn notbii biimparc syl2anb
|
|
vex biorf eldif 3imtr4i ssriv ) CABDZAEZFZBEZCGZUCHZUGUDHZIZJUGBKZUGUEHUG
|
|
UFHUHUGAKZUKLZULIZUKUJUGABCRMUIULCANOUNUKUMULUKSPQUGUCUDTCBNUAUB $.
|
|
$}
|
|
|
|
$( Removal of a singleton from an unordered pair. (Contributed by Thierry
|
|
Arnoux, 4-Feb-2017.) $)
|
|
difprsn1 $p |- ( A =/= B -> ( { A , B } \ { A } ) = { B } ) $=
|
|
( wne cpr csn cdif wceq necom cin c0 disjsn2 disj3 sylib cun df-pr equncomi
|
|
difeq1i difun2 eqtri syl6reqr sylbir ) ABCBACZABDZAEZFZBEZGBAHUBUFUFUDFZUEU
|
|
BUFUDIJGUFUGGBAKUFUDLMUEUFUDNZUDFUGUCUHUDUCUDUFABOPQUFUDRSTUA $.
|
|
|
|
$( Removal of a singleton from an unordered pair. (Contributed by Alexander
|
|
van der Vekens, 5-Oct-2017.) $)
|
|
difprsn2 $p |- ( A =/= B -> ( { A , B } \ { B } ) = { A } ) $=
|
|
( wne cpr csn cdif prcom difeq1i wceq necom difprsn1 sylbi syl5eq ) ABCZABD
|
|
ZBEZFBADZPFZAEZOQPABGHNBACRSIABJBAKLM $.
|
|
|
|
$( Removal of a singleton from an unordered triple. (Contributed by
|
|
Alexander van der Vekens, 5-Oct-2017.) $)
|
|
diftpsn3 $p |- ( ( A =/= C /\ B =/= C )
|
|
-> ( { A , B , C } \ { C } ) = { A , B } ) $=
|
|
( wne wa ctp csn cdif cpr cun a1i c0 cin necom disjsn2 sylbi uneq12d syl6eq
|
|
wceq 3eqtrd df-tp difeq1d df-pr ineq1d incom indi eqtri adantr adantl unidm
|
|
difundir disj3 sylib eqcomd difid un0 ) ACDZBCDZEZABCFZCGZHABIZVAJZVAHZVBVA
|
|
HZVAVAHZJZVBUSUTVCVAUTVCSUSABCUAKUBVDVGSUSVBVAVAUKKUSVGVBLJVBUSVEVBVFLUSVBV
|
|
EUSVBVAMZLSVBVESUSVHAGZBGZJZVAMZVAVIMZVAVJMZJZLUSVBVKVAVBVKSUSABUCKUDVLVOSU
|
|
SVLVAVKMVOVKVAUEVAVIVJUFUGKUSVOLLJLUSVMLVNLUQVMLSZURUQCADVPACNCAOPUHURVNLSZ
|
|
UQURCBDVQBCNCBOPUIQLUJRTVBVAULUMUNVFLSUSVAUOKQVBUPRT $.
|
|
|
|
$( ` ( B \ { A } ) ` equals ` B ` if and only if ` A ` is not a member of
|
|
` B ` . Generalization of ~ difsn . (Contributed by David Moews,
|
|
1-May-2017.) $)
|
|
difsnb $p |- ( -. A e. B <-> ( B \ { A } ) = B ) $=
|
|
( wcel wn csn cdif wceq difsn neldifsnd nelne1 mpdan necomd necon2bi impbii
|
|
wne ) ABCZDBAEFZBGABHPQBPBQPAQCDBQOPABIABQJKLMN $.
|
|
|
|
$( ` ( B \ { A } ) ` is a proper subclass of ` B ` if and only if ` A ` is a
|
|
member of ` B ` . (Contributed by David Moews, 1-May-2017.) $)
|
|
difsnpss $p |- ( A e. B <-> ( B \ { A } ) C. B ) $=
|
|
( wcel wn csn cdif wpss notnot wne wss wa biantrur difsnb necon3bbii df-pss
|
|
difss 3bitr4i bitri ) ABCZSDZDZBAEZFZBGZSHUCBIZUCBJZUEKUAUDUFUEBUBPLTUCBABM
|
|
NUCBOQR $.
|
|
|
|
$( The singleton of an element of a class is a subset of the class.
|
|
(Contributed by NM, 6-Jun-1994.) $)
|
|
snssi $p |- ( A e. B -> { A } C_ B ) $=
|
|
( wcel csn wss snssg ibi ) ABCADBEABBFG $.
|
|
|
|
${
|
|
snssd.1 $e |- ( ph -> A e. B ) $.
|
|
$( The singleton of an element of a class is a subset of the class
|
|
(deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
snssd $p |- ( ph -> { A } C_ B ) $=
|
|
( wcel csn wss wb snssg syl mpbid ) ABCEZBFCGZDALLMHDBCCIJK $.
|
|
$}
|
|
|
|
$( If we remove a single element from a class then put it back in, we end up
|
|
with the original class. (Contributed by NM, 2-Oct-2006.) $)
|
|
difsnid $p |- ( B e. A -> ( ( A \ { B } ) u. { B } ) = A ) $=
|
|
( wcel csn cdif cun uncom wss wceq snssi undif sylib syl5eq ) BACZABDZEZOFO
|
|
PFZAPOGNOAHQAIBAJOAKLM $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Compute the power set of the power set of the empty set. (See ~ pw0 for
|
|
the power set of the empty set.) Theorem 90 of [Suppes] p. 48.
|
|
Although this theorem is a special case of ~ pwsn , we have chosen to
|
|
show a direct elementary proof. (Contributed by NM, 7-Aug-1994.) $)
|
|
pwpw0 $p |- ~P { (/) } = { (/) , { (/) } } $=
|
|
( vx vy cv c0 csn wss cab wceq wo cpw cpr wn wa wcel wal dfss2 elsn sylbi
|
|
wi wex imbi2i albii bitri neq0 exintr syl5bi exancom df-clel bitr4i snssi
|
|
syl6 anc2li eqss syl6ibr orrd sseq1 mpbiri eqimss jaoi impbii abbii df-pw
|
|
0ss dfpr2 3eqtr4i ) ACZDEZFZAGVFDHZVFVGHZIZAGVGJDVGKVHVKAVHVKVHVIVJVHVILZ
|
|
VHVGVFFZMVJVHVLVMVHBCZVFNZVNDHZSZBOZVLVMSVHVOVNVGNZSZBOVRBVFVGPVTVQBVSVPV
|
|
OBDQUAUBUCVRVLVOVPMBTZVMVLVOBTVRWABVFUDVOVPBUEUFWADVFNZVMWAVPVOMBTWBVOVPB
|
|
UGBDVFUHUIDVFUJRUKRULVFVGUMUNUOVIVHVJVIVHDVGFVGVCVFDVGUPUQVFVGURUSUTVAAVG
|
|
VBADVGVDVE $.
|
|
$}
|
|
|
|
$( A singleton is a subset of an unordered pair containing its member.
|
|
(Contributed by NM, 27-Aug-2004.) $)
|
|
snsspr1 $p |- { A } C_ { A , B } $=
|
|
( csn cun cpr ssun1 df-pr sseqtr4i ) ACZIBCZDABEIJFABGH $.
|
|
|
|
$( A singleton is a subset of an unordered pair containing its member.
|
|
(Contributed by NM, 2-May-2009.) $)
|
|
snsspr2 $p |- { B } C_ { A , B } $=
|
|
( csn cun cpr ssun2 df-pr sseqtr4i ) BCZACZIDABEIJFABGH $.
|
|
|
|
$( A singleton is a subset of an unordered triple containing its member.
|
|
(Contributed by NM, 9-Oct-2013.) $)
|
|
snsstp1 $p |- { A } C_ { A , B , C } $=
|
|
( csn cpr cun ctp snsspr1 ssun1 sstri df-tp sseqtr4i ) ADZABEZCDZFZABCGMNPA
|
|
BHNOIJABCKL $.
|
|
|
|
$( A singleton is a subset of an unordered triple containing its member.
|
|
(Contributed by NM, 9-Oct-2013.) $)
|
|
snsstp2 $p |- { B } C_ { A , B , C } $=
|
|
( csn cpr cun ctp snsspr2 ssun1 sstri df-tp sseqtr4i ) BDZABEZCDZFZABCGMNPA
|
|
BHNOIJABCKL $.
|
|
|
|
$( A singleton is a subset of an unordered triple containing its member.
|
|
(Contributed by NM, 9-Oct-2013.) $)
|
|
snsstp3 $p |- { C } C_ { A , B , C } $=
|
|
( csn cpr cun ctp ssun2 df-tp sseqtr4i ) CDZABEZKFABCGKLHABCIJ $.
|
|
|
|
${
|
|
prss.1 $e |- A e. _V $.
|
|
prss.2 $e |- B e. _V $.
|
|
$( A pair of elements of a class is a subset of the class. Theorem 7.5 of
|
|
[Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by
|
|
Andrew Salmon, 29-Jun-2011.) $)
|
|
prss $p |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) $=
|
|
( csn wss wa cun wcel cpr unss snss anbi12i df-pr sseq1i 3bitr4i ) AFZCGZ
|
|
BFZCGZHRTIZCGACJZBCJZHABKZCGRTCLUCSUDUAACDMBCEMNUEUBCABOPQ $.
|
|
$}
|
|
|
|
$( A pair of elements of a class is a subset of the class. Theorem 7.5 of
|
|
[Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by
|
|
Andrew Salmon, 29-Jun-2011.) $)
|
|
prssg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) ) $=
|
|
( wcel wa csn wss cpr snssg bi2anan9 cun unss df-pr sseq1i bitr4i syl6bb )
|
|
ADFZBEFZGACFZBCFZGAHZCIZBHZCIZGZABJZCIZSUAUDTUBUFACDKBCEKLUGUCUEMZCIUIUCUEC
|
|
NUHUJCABOPQR $.
|
|
|
|
$( A pair of elements of a class is a subset of the class. (Contributed by
|
|
NM, 16-Jan-2015.) $)
|
|
prssi $p |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C ) $=
|
|
( wcel wa cpr wss prssg ibi ) ACDBCDEABFCGABCCCHI $.
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The subsets of a singleton. (Contributed by NM, 24-Apr-2004.) $)
|
|
sssn $p |- ( A C_ { B } <-> ( A = (/) \/ A = { B } ) ) $=
|
|
( vx csn wss c0 wceq wo wn wa wcel cv wex neq0 ssel elsni syl6 eleq1 ibd
|
|
wb exlimdv syl5bi snssi anc2li eqss syl6ibr orrd sseq1 mpbiri eqimss jaoi
|
|
0ss impbii ) ABDZEZAFGZAUNGZHUOUPUQUOUPIZUOUNAEZJUQUOURUSUOURBAKZUSURCLZA
|
|
KZCMUOUTCANUOVBUTCUOVBUTUOVBVABGZVBUTTUOVBVAUNKVCAUNVAOVABPQVABARQSUAUBBA
|
|
UCQUDAUNUEUFUGUPUOUQUPUOFUNEUNULAFUNUHUIAUNUJUKUM $.
|
|
|
|
$( The property of being sandwiched between two sets naturally splits under
|
|
union with a singleton. This is the induction hypothesis for the
|
|
determination of large powersets such as ~ pwtp . (Contributed by Mario
|
|
Carneiro, 2-Jul-2016.) $)
|
|
ssunsn2 $p |- ( ( B C_ A /\ A C_ ( C u. { D } ) ) <->
|
|
( ( B C_ A /\ A C_ C ) \/
|
|
( ( B u. { D } ) C_ A /\ A C_ ( C u. { D } ) ) ) ) $=
|
|
( wcel wss csn cun wa wo wb syl wi anim12d pm4.72 sylib bitrd wceq bitr3i
|
|
a1i snssi unss bicomi rbaibr anbi1d biimpi expcom ssun3 wn cdif c0 disjsn
|
|
cin disj3 sseq1 sylbi uncom sseq2i ssundif syl6rbbr anbi2d simplbi biimpd
|
|
orcom syl6bb pm2.61i ) DAEZBAFZACDGZHZFZIZVHACFZIZBVIHAFZVKIZJZKVGVLVPVQV
|
|
GVHVOVKVGVIAFZVHVOKDAUAZVOVHVRVHVRIZVOBVIAUBZUCZUDLUEVGVNVPMVPVQKVGVHVOVM
|
|
VKVGVRVHVOMVSVHVRVOVTVOWAUFUGLVMVKMVGACVIUHTNVNVPOPQVGUIZVLVNVQWCVKVMVHWC
|
|
VMAVIUJZCFZVKWCAWDRZVMWEKWCAVIUMUKRWFADULAVIUNSAWDCUOUPVKAVICHZFWEWGVJAVI
|
|
CUQURAVICUSSUTZVAWCVNVPVNJZVQWCVPVNMVNWIKWCVOVHVKVMVOVHMWCVOVHVRWBVBTWCVK
|
|
VMWHVCNVPVNOPVPVNVDVEQVF $.
|
|
|
|
$( Possible values for a set sandwiched between another set and it plus a
|
|
singleton. (Contributed by Mario Carneiro, 2-Jul-2016.) $)
|
|
ssunsn $p |- ( ( B C_ A /\ A C_ ( B u. { C } ) ) <->
|
|
( A = B \/ A = ( B u. { C } ) ) ) $=
|
|
( wss csn cun wa wo wceq ssunsn2 ancom eqss bitr4i orbi12i bitri ) BADZAB
|
|
CEFZDZGPABDZGZQADZRGZHABIZAQIZHABBCJTUCUBUDTSPGUCPSKABLMUBRUAGUDUARKAQLMN
|
|
O $.
|
|
|
|
$( Two ways to express that a nonempty set equals a singleton.
|
|
(Contributed by NM, 15-Dec-2007.) $)
|
|
eqsn $p |- ( A =/= (/) -> ( A = { B } <-> A. x e. A x = B ) ) $=
|
|
( c0 wne csn wceq wss cv wral eqimss wn df-ne wo sssn biimpi syl5bi com12
|
|
ord impbid2 wcel dfss3 elsn ralbii bitri syl6bb ) BDEZBCFZGZBUHHZAIZCGZAB
|
|
JZUGUIUJBUHKUJUGUIUGBDGZLUJUIBDMUJUNUIUJUNUINBCOPSQRTUJUKUHUAZABJUMABUHUB
|
|
UOULABACUCUDUEUF $.
|
|
$}
|
|
|
|
$( Possible values for a set sandwiched between another set and it plus a
|
|
singleton. (Contributed by Mario Carneiro, 2-Jul-2016.) $)
|
|
ssunpr $p |- ( ( B C_ A /\ A C_ ( B u. { C , D } ) ) <->
|
|
( ( A = B \/ A = ( B u. { C } ) ) \/
|
|
( A = ( B u. { D } ) \/ A = ( B u. { C , D } ) ) ) ) $=
|
|
( wss cpr cun wa csn wo wceq df-pr uneq2i unass eqtr4i sseq2i anbi2i ssunsn
|
|
ssunsn2 3bitri un23 eqtr2i eqeq2i orbi2i orbi12i ) BAEZABCDFZGZEZHUFABCIZGZ
|
|
DIZGZEZHUFAUKEHZBULGZAEZUNHZJABKAUKKJZAUPKZAUHKZJZJUIUNUFUHUMAUHBUJULGZGUMU
|
|
GVCBCDLMBUJULNOZPQABUKDSUOUSURVBABCRURUQAUPUJGZEZHUTAVEKZJVBUNVFUQUMVEABUJU
|
|
LUAZPQAUPCRVGVAUTVEUHAUHUMVEVDVHUBUCUDTUET $.
|
|
|
|
$( The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof
|
|
shortened by Mario Carneiro, 2-Jul-2016.) $)
|
|
sspr $p |- ( A C_ { B , C } <->
|
|
( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) ) $=
|
|
( cpr wss c0 cun wa wceq csn wo uncom un0 sseq2i 0ss biantrur bitr3i eqeq2i
|
|
eqtri orbi12i ssunpr orbi2i 3bitri ) ABCDZEZFAEZAFUDGZEZHZAFIZAFBJZGZIZKZAF
|
|
CJZGZIZAUGIZKZKUJAUKIZKZAUOIZAUDIZKZKUEUHUIUGUDAUGUDFGUDFUDLUDMSZNUFUHAOPQA
|
|
FBCUAUNVAUSVDUMUTUJULUKAULUKFGUKFUKLUKMSRUBUQVBURVCUPUOAUPUOFGUOFUOLUOMSRUG
|
|
UDAVERTTUC $.
|
|
|
|
$( The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.) $)
|
|
sstp $p |- ( A C_ { B , C , D } <->
|
|
( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/
|
|
( ( A = { D } \/ A = { B , D } ) \/
|
|
( A = { C , D } \/ A = { B , C , D } ) ) ) ) $=
|
|
( wss cpr csn cun c0 wa wceq wo sseq2i biantrur uncom eqtr4i eqeq2i orbi12i
|
|
df-pr 3bitri ctp df-tp 0ss ssunsn2 sspr bitr3i sseq1i anbi12i ssunpr orbi2i
|
|
un0 eqtri eqtr2i bitri ) ABCDUAZEABCFZDGZHZEZIAEZUSJZAIKABGZKLACGZKAUPKLLZA
|
|
UQKZABDFZKZLZACDFZKZAUOKZLZLZLZUOURABCDUBZMUTUSAUCZNVAUTAUPEZJZIUQHZAEZUSJZ
|
|
LVNAIUPDUDVRVDWAVMVRVQVDUTVQVPNABCUEUFWAUQAEZAUQUPHZEZJVEAUQVBHZKZLZAUQVCHZ
|
|
KZAWCKZLZLVMVTWBUSWDVSUQAVSUQIHUQIUQOUQUKULUGURWCAUPUQOZMUHAUQBCUIWGVHWKVLW
|
|
FVGVEWEVFAWEVBUQHVFUQVBOBDSPQUJWIVJWJVKWHVIAWHVCUQHVIUQVCOCDSPQWCUOAUOURWCV
|
|
OWLUMQRRTRUNT $.
|
|
|
|
${
|
|
tpss.1 $e |- A e. _V $.
|
|
tpss.2 $e |- B e. _V $.
|
|
tpss.3 $e |- C e. _V $.
|
|
$( A triplet of elements of a class is a subset of the class. (Contributed
|
|
by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
tpss $p |- ( ( A e. D /\ B e. D /\ C e. D ) <-> { A , B , C } C_ D ) $=
|
|
( cpr wss csn wa cun wcel w3a ctp unss df-3an prss snss anbi12i 3bitr4i
|
|
bitri df-tp sseq1i ) ABHZDIZCJZDIZKZUEUGLZDIADMZBDMZCDMZNZABCOZDIUEUGDPUN
|
|
UKULKZUMKUIUKULUMQUPUFUMUHABDEFRCDGSTUBUOUJDABCUCUDUA $.
|
|
$}
|
|
|
|
${
|
|
sneqr.1 $e |- A e. _V $.
|
|
$( If the singletons of two sets are equal, the two sets are equal. Part
|
|
of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM,
|
|
27-Aug-1993.) $)
|
|
sneqr $p |- ( { A } = { B } -> A = B ) $=
|
|
( csn wceq wcel snid eleq2 mpbii elsnc sylib ) ADZBDZEZAMFZABENALFOACGLMA
|
|
HIABCJK $.
|
|
|
|
$( If a singleton is a subset of another, their members are equal.
|
|
(Contributed by NM, 28-May-2006.) $)
|
|
snsssn $p |- ( { A } C_ { B } -> A = B ) $=
|
|
( csn wss c0 wceq wo sssn wne wn snnz df-ne mpbi pm2.21i sneqr jaoi sylbi
|
|
) ADZBDZESFGZSTGZHABGZSBIUAUCUBUAUCSFJUAKACLSFMNOABCPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Closed form of ~ sneqr . (Contributed by Scott Fenton, 1-Apr-2011.) $)
|
|
sneqrg $p |- ( A e. V -> ( { A } = { B } -> A = B ) ) $=
|
|
( vx cv csn wceq wi sneq eqeq1d eqeq1 imbi12d vex sneqr vtoclg ) DEZFZBFZ
|
|
GZPBGZHAFZRGZABGZHDACPAGZSUBTUCUDQUARPAIJPABKLPBDMNO $.
|
|
|
|
$}
|
|
|
|
$( Two singletons of sets are equal iff their elements are equal.
|
|
(Contributed by Scott Fenton, 16-Apr-2012.) $)
|
|
sneqbg $p |- ( A e. V -> ( { A } = { B } <-> A = B ) ) $=
|
|
( wcel csn wceq sneqrg sneq impbid1 ) ACDAEBEFABFABCGABHI $.
|
|
|
|
${
|
|
sneqb.1 $e |- A e. _V $.
|
|
$( Biconditional equality for singletons. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
sneqb $p |- ( { A } = { B } <-> A = B ) $=
|
|
( cvv wcel csn wceq wb sneqbg ax-mp ) ADEAFBFGABGHCABDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The singleton of a class is a subset of its power class. (Contributed
|
|
by NM, 5-Aug-1993.) $)
|
|
snsspw $p |- { A } C_ ~P A $=
|
|
( vx csn cpw cv wceq wss wcel eqimss elsn df-pw abeq2i 3imtr4i ssriv ) BA
|
|
CZADZBEZAFQAGZQOHQPHQAIBAJRBPBAKLMN $.
|
|
$}
|
|
|
|
${
|
|
prsspw.1 $e |- A e. _V $.
|
|
prsspw.2 $e |- B e. _V $.
|
|
$( An unordered pair belongs to the power class of a class iff each member
|
|
belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof
|
|
shortened by Andrew Salmon, 26-Jun-2011.) $)
|
|
prsspw $p |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) $=
|
|
( cpr cpw wss wcel wa prss elpw anbi12i bitr3i ) ABFCGZHAOIZBOIZJACHZBCHZ
|
|
JABODEKPRQSACDLBCELMN $.
|
|
$}
|
|
|
|
${
|
|
$d B x $. $d ps x $.
|
|
ralunsn.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( Restricted quantification over the union of a set and a singleton, using
|
|
implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
|
|
(Revised by Mario Carneiro, 23-Apr-2015.) $)
|
|
ralunsn $p |- ( B e. C -> ( A. x e. ( A u. { B } ) ph <->
|
|
( A. x e. A ph /\ ps ) ) ) $=
|
|
( csn cun wral wa wcel ralunb ralsng anbi2d syl5bb ) ACDEHZIJACDJZACQJZKE
|
|
FLZRBKACDQMTSBRABCEFGNOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x y $. $d C x $. $d ch x $. $d ps y $. $d th x $.
|
|
2ralunsn.1 $e |- ( x = B -> ( ph <-> ch ) ) $.
|
|
2ralunsn.2 $e |- ( y = B -> ( ph <-> ps ) ) $.
|
|
2ralunsn.3 $e |- ( x = B -> ( ps <-> th ) ) $.
|
|
$( Double restricted quantification over the union of a set and a
|
|
singleton, using implicit substitution. (Contributed by Paul Chapman,
|
|
17-Nov-2012.) $)
|
|
2ralunsn $p |- ( B e. C ->
|
|
( A. x e. ( A u. { B } ) A. y e. ( A u. { B } ) ph <->
|
|
( ( A. x e. A A. y e. A ph /\ A. x e. A ps ) /\
|
|
( A. y e. A ch /\ th ) ) ) ) $=
|
|
( wcel csn cun wral wa ralunsn ralbidv cv wceq r19.26 anbi1i syl6bb bitrd
|
|
anbi12d ) HIMZAFGHNOZPZEUHPAFGPZBQZEUHPZUJEGPBEGPQZCFGPZDQZQZUGUIUKEUHABF
|
|
GHIKRSUGULUKEGPZUOQUPUKUOEGHIETHUAZUJUNBDURACFGJSLUFRUQUMUOUJBEGUBUCUDUE
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( The power set of a singleton. (Contributed by NM, 5-Jun-2006.) $)
|
|
pwsn $p |- ~P { A } = { (/) , { A } } $=
|
|
( vx cv csn wss cab c0 wceq wo cpw cpr sssn abbii df-pw dfpr2 3eqtr4i ) B
|
|
CZADZEZBFQGHQRHIZBFRJGRKSTBQALMBRNBGROP $.
|
|
|
|
$d x y $. $d y A $.
|
|
$( The power set of a singleton (direct proof). TO DO - should we keep
|
|
this? (Contributed by NM, 5-Jun-2006.)
|
|
(Proof modification is discouraged.) (New usage is discouraged.) $)
|
|
pwsnALT $p |- ~P { A } = { (/) , { A } } $=
|
|
( vx vy cv csn wss cab c0 wceq wo cpw cpr wn wa wcel wal dfss2 wex sylbi
|
|
wi elsn imbi2i albii bitri neq0 exintr syl5bi df-clel exancom bitr2i syl6
|
|
snssi anc2li eqss syl6ibr orrd 0ss sseq1 mpbiri eqimss impbii abbii df-pw
|
|
jaoi dfpr2 3eqtr4i ) BDZAEZFZBGVGHIZVGVHIZJZBGVHKHVHLVIVLBVIVLVIVJVKVIVJM
|
|
ZVIVHVGFZNVKVIVMVNVICDZVGOZVOAIZTZCPZVMVNTVIVPVOVHOZTZCPVSCVGVHQWAVRCVTVQ
|
|
VPCAUAUBUCUDVSVMVPVQNCRZVNVMVPCRVSWBCVGUEVPVQCUFUGWBAVGOZVNWCVQVPNCRWBCAV
|
|
GUHVQVPCUIUJAVGULSUKSUMVGVHUNUOUPVJVIVKVJVIHVHFVHUQVGHVHURUSVGVHUTVDVAVBB
|
|
VHVCBHVHVEVF $.
|
|
|
|
$( The power set of an unordered pair. (Contributed by NM, 1-May-2009.) $)
|
|
pwpr $p |- ~P { A , B } = ( { (/) , { A } } u. { { B } , { A , B } } ) $=
|
|
( vx cpr cpw c0 csn cun cv wss wcel wo wceq sspr elpr orbi12i bitr4i elpw
|
|
vex elun 3bitr4i eqriv ) CABDZEZFAGZDZBGZUCDZHZCIZUCJZUJUFKZUJUHKZLZUJUDK
|
|
UJUIKUKUJFMUJUEMLZUJUGMUJUCMLZLUNUJABNULUOUMUPUJFUECSZOUJUGUCUQOPQUJUCUQR
|
|
UJUFUHTUAUB $.
|
|
|
|
$( The power set of an unordered triple. (Contributed by Mario Carneiro,
|
|
2-Jul-2016.) $)
|
|
pwtp $p |- ~P { A , B , C } =
|
|
( ( { (/) , { A } } u. { { B } , { A , B } } ) u.
|
|
( { { C } , { A , C } } u. { { B , C } , { A , B , C } } ) ) $=
|
|
( vx ctp cpw c0 csn cpr cun cv wcel wss vex wceq elun elpr orbi12i bitri
|
|
wo elpw sstp 3bitr4ri eqriv ) DABCEZFZGAHZIZBHZABIZIZJZCHZACIZIZBCIZUEIZJ
|
|
ZJZDKZUFLUTUEMZUTUSLZUTUEDNZUAUTULLZUTURLZTUTGOUTUGOTZUTUIOUTUJOTZTZUTUMO
|
|
UTUNOTZUTUPOUTUEOTZTZTVBVAVDVHVEVKVDUTUHLZUTUKLZTVHUTUHUKPVLVFVMVGUTGUGVC
|
|
QUTUIUJVCQRSVEUTUOLZUTUQLZTVKUTUOUQPVNVIVOVJUTUMUNVCQUTUPUEVCQRSRUTULURPU
|
|
TABCUBUCSUD $.
|
|
$}
|
|
|
|
$( Compute the power set of the power set of the power set of the empty set.
|
|
(See also ~ pw0 and ~ pwpw0 .) (Contributed by NM, 2-May-2009.) $)
|
|
pwpwpw0 $p |- ~P { (/) , { (/) } } =
|
|
( { (/) , { (/) } } u. { { { (/) } } , { (/) , { (/) } } } ) $=
|
|
( c0 csn pwpr ) AABC $.
|
|
|
|
$( The power class of the universe is the universe. Exercise 4.12(d) of
|
|
[Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) $)
|
|
pwv $p |- ~P _V = _V $=
|
|
( vx cvv cpw cv wcel wss ssv vex elpw mpbir 2th eqriv ) ABCZBADZMEZNBEONBFN
|
|
GNBAHZIJPKL $.
|
|
|
|
${
|
|
unsneqsn.1 $e |- B e. _V $.
|
|
$( If union with a singleton yields a singleton, then the first argument is
|
|
either also the singleton or is the empty set. (Contributed by SF,
|
|
15-Jan-2015.) $)
|
|
unsneqsn $p |- ( ( A u. { B } ) = { C } -> ( A = (/) \/ A = { B } ) ) $=
|
|
( csn cun wceq wss c0 wcel ssun2 snid sselii eleq2 mpbii elsni syl eqeq2d
|
|
wo sneq biimprd mpcom ssequn1 sylibr sssn sylib ) ABEZFZCEZGZAUGHZAIGAUGG
|
|
SUJUHUGGZUKBCGZUJULUJBUIJZUMUJBUHJUNUGUHBUGAKBDLMUHUIBNOBCPQUMULUJUMUGUIU
|
|
HBCTRUAUBAUGUCUDABUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Alternate definition of proper subset. Theorem IX.4.21 of [Rosser]
|
|
p. 236. (Contributed by SF, 19-Jan-2015.) $)
|
|
dfpss4 $p |- ( A C. B <-> ( A C_ B /\ E. x e. B -. x e. A ) ) $=
|
|
( wpss wss wn wa wcel wrex dfpss3 wral dfss3 dfral2 bitr2i con1bii anbi2i
|
|
cv bitri ) BCDBCEZCBEZFZGSAQBHZFACIZGBCJUAUCSUCTTUBACKUCFACBLUBACMNOPR $.
|
|
$}
|
|
|
|
$( Adjoining a new element is one-to-one. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
adj11 $p |- ( ( -. C e. A /\ -. C e. B ) ->
|
|
( ( A u. { C } ) = ( B u. { C } ) <-> A = B ) ) $=
|
|
( wcel wn wa csn cun wceq cdif difeq1 difun2 3eqtr3g difsn eqeqan12d syl5ib
|
|
uneq1 impbid1 ) CADEZCBDEZFZACGZHZBUBHZIZABIZUEAUBJZBUBJZIUAUFUEUCUBJUDUBJU
|
|
GUHUCUDUBKAUBLBUBLMSTUGAUHBCANCBNOPABUBQR $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Two ways of saying that two classes are disjoint. (Contributed by SF,
|
|
5-Feb-2015.) $)
|
|
disj5 $p |- ( ( A i^i B ) = (/) <-> A C_ ~ B ) $=
|
|
( vx cv wcel wn wral ccompl wi wal cin wceq wss vex elcompl ralbii df-ral
|
|
c0 bitr3i disj dfss2 3bitr4i ) CDZBEFZCAGZUCAEUCBHZEZICJZABKRLAUFMUEUGCAG
|
|
UHUGUDCAUCBCNOPUGCAQSCABTCAUFUAUB $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The union of a class
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare class union symbol. $)
|
|
$c U. $. $( Big cup $)
|
|
|
|
$( Extend class notation to include the union of a class (read: 'union
|
|
` A ` ') $)
|
|
cuni $a class U. A $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Define the union of a class i.e. the collection of all members of the
|
|
members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For
|
|
example, ` U. { { ` 1 , 3 ` } , { ` 1 , 8 ` } } = { ` 1 , 3 , 8 ` } `
|
|
(ex-uni in set.mm). This is similar to the union of two classes
|
|
~ df-un . (Contributed by NM, 23-Aug-1993.) $)
|
|
df-uni $a |- U. A = { x | E. y ( x e. y /\ y e. A ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of class union. (Contributed by NM,
|
|
28-Jun-1998.) $)
|
|
dfuni2 $p |- U. A = { x | E. y e. A x e. y } $=
|
|
( cuni cv wcel wa wex cab wrex df-uni exancom df-rex bitr4i abbii eqtri )
|
|
CDAEBEZFZQCFZGBHZAIRBCJZAIABCKTUAATSRGBHUARSBLRBCMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d x B y $.
|
|
$( Membership in class union. (Contributed by NM, 22-May-1994.) $)
|
|
eluni $p |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) $=
|
|
( vy cuni wcel cvv cv wa wex elex adantr exlimiv wceq eleq1 anbi1d exbidv
|
|
df-uni elab2g pm5.21nii ) BCEZFBGFZBAHZFZUCCFZIZAJZBUAKUFUBAUDUBUEBUCKLMD
|
|
HZUCFZUEIZAJUGDBUAGUHBNZUJUFAUKUIUDUEUHBUCOPQDACRST $.
|
|
|
|
$( Membership in class union. Restricted quantifier version. (Contributed
|
|
by NM, 31-Aug-1999.) $)
|
|
eluni2 $p |- ( A e. U. B <-> E. x e. B A e. x ) $=
|
|
( cv wcel wa wex cuni wrex exancom eluni df-rex 3bitr4i ) BADZEZNCEZFAGPO
|
|
FAGBCHEOACIOPAJABCKOACLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Membership in class union. (Contributed by NM, 24-Mar-1995.) $)
|
|
elunii $p |- ( ( A e. B /\ B e. C ) -> A e. U. C ) $=
|
|
( vx wcel wa cv cuni wceq eleq2 eleq1 anbi12d spcegv anabsi7 eluni sylibr
|
|
wex ) ABEZBCEZFZADGZEZUACEZFZDQZACHERSUEUDTDBCUABIUBRUCSUABAJUABCKLMNDACO
|
|
P $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $.
|
|
nfuni.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for union. (Contributed by NM,
|
|
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
nfuni $p |- F/_ x U. A $=
|
|
( vy vz cuni cv wcel wrex cab dfuni2 nfv nfrex nfab nfcxfr ) ABFDGEGHZEBI
|
|
ZDJDEBKQADPAEBCPALMNO $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $. $d y z ph $.
|
|
nfunid.3 $e |- ( ph -> F/_ x A ) $.
|
|
$( Deduction version of ~ nfuni . (Contributed by NM, 18-Feb-2013.) $)
|
|
nfunid $p |- ( ph -> F/_ x U. A ) $=
|
|
( vy vz cuni cv wcel wrex cab dfuni2 nfv nfvd nfrexd nfabd nfcxfrd ) ABCG
|
|
EHFHIZFCJZEKEFCLASBEAEMARBFCAFMDARBNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d B y z $. $d V y z $. $d x y z $.
|
|
$( Distribute proper substitution through the union of a class.
|
|
(Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbunig $p |- ( A e. V -> [_ A / x ]_ U. B = U. [_ A / x ]_ B ) $=
|
|
( vz vy wcel cv wa wex cab csb cuni wsbc csbabg sbcexg sbcang sbcg df-uni
|
|
bitrd sbcel2g anbi12d exbidv abbidv eqtrd csbeq2i 3eqtr4g ) BDGZABEHFHZGZ
|
|
UICGZIZFJZEKZLZUJUIABCLZGZIZFJZEKZABCMZLUPMUHUOUMABNZEKUTUMAEBDOUHVBUSEUH
|
|
VBULABNZFJUSULFABDPUHVCURFUHVCUJABNZUKABNZIURUJUKABDQUHVDUJVEUQUJABDRABUI
|
|
CDUAUBTUCTUDUEABVAUNEFCSUFEFUPSUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Equality theorem for class union. Exercise 15 of [TakeutiZaring]
|
|
p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew
|
|
Salmon, 29-Jun-2011.) $)
|
|
unieq $p |- ( A = B -> U. A = U. B ) $=
|
|
( vy vx wceq cv wcel wrex cab cuni rexeq abbidv dfuni2 3eqtr4g ) ABEZCFDF
|
|
GZDAHZCIPDBHZCIAJBJOQRCPDABKLCDAMCDBMN $.
|
|
$}
|
|
|
|
${
|
|
unieqi.1 $e |- A = B $.
|
|
$( Inference of equality of two class unions. (Contributed by NM,
|
|
30-Aug-1993.) $)
|
|
unieqi $p |- U. A = U. B $=
|
|
( wceq cuni unieq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
unieqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Deduction of equality of two class unions. (Contributed by NM,
|
|
21-Apr-1995.) $)
|
|
unieqd $p |- ( ph -> U. A = U. B ) $=
|
|
( wceq cuni unieq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d ph y $.
|
|
$( Membership in union of a class abstraction. (Contributed by NM,
|
|
11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.) $)
|
|
eluniab $p |- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) ) $=
|
|
( vy cab cuni wcel cv wa wex eluni nfv nfsab1 nfan wceq eleq2 abid syl6bb
|
|
eleq1 anbi12d cbvex bitri ) CABEZFGCDHZGZUDUCGZIZDJCBHZGZAIZBJDCUCKUGUJDB
|
|
UEUFBUEBLABDMNUJDLUDUHOZUEUIUFAUDUHCPUKUFUHUCGAUDUHUCSABQRTUAUB $.
|
|
|
|
$( Membership in union of a class abstraction. (Contributed by NM,
|
|
4-Oct-2006.) $)
|
|
elunirab $p |- ( A e. U. { x e. B | ph } <->
|
|
E. x e. B ( A e. x /\ ph ) ) $=
|
|
( cv wcel cab cuni wex crab wrex eluniab df-rab unieqi eleq2i df-rex an12
|
|
wa exbii bitri 3bitr4i ) CBEZDFZARZBGZHZFCUBFZUDRZBIZCABDJZHZFUGARZBDKZUD
|
|
BCLUKUFCUJUEABDMNOUMUCULRZBIUIULBDPUNUHBUCUGAQSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
unipr.1 $e |- A e. _V $.
|
|
unipr.2 $e |- B e. _V $.
|
|
$( The union of a pair is the union of its members. Proposition 5.7 of
|
|
[TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) $)
|
|
unipr $p |- U. { A , B } = ( A u. B ) $=
|
|
( vx vy cpr cuni cun wel cv wcel wa wex wceq wo vex bitri clel3 exancom
|
|
elpr anbi2i andi exbii 19.43 eluni elun orbi12i 3bitr4i eqriv ) EABGZHZAB
|
|
IZEFJZFKZUKLZMZFNZUNUOAOZMZFNZUNUOBOZMZFNZPZEKZULLVFUMLZURUTVCPZFNVEUQVHF
|
|
UQUNUSVBPZMVHUPVIUNUOABFQUAUBUNUSVBUCRUDUTVCFUERFVFUKUFVGVFALZVFBLZPVEVFA
|
|
BUGVJVAVKVDVJUSUNMFNVAFVFACSUSUNFTRVKVBUNMFNVDFVFBDSVBUNFTRUHRUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( The union of a pair is the union of its members. Proposition 5.7 of
|
|
[TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) $)
|
|
uniprg $p |- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) ) $=
|
|
( vx vy cv cpr cuni cun wceq preq1 unieqd uneq1 eqeq12d preq2 uneq2 unipr
|
|
vex vtocl2g ) EGZFGZHZIZUAUBJZKAUBHZIZAUBJZKABHZIZABJZKEFABCDUAAKZUDUGUEU
|
|
HULUCUFUAAUBLMUAAUBNOUBBKZUGUJUHUKUMUFUIUBBAPMUBBAQOUAUBESFSRT $.
|
|
$}
|
|
|
|
${
|
|
unisn.1 $e |- A e. _V $.
|
|
$( A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|
|
(Contributed by NM, 30-Aug-1993.) $)
|
|
unisn $p |- U. { A } = A $=
|
|
( csn cuni cpr cun dfsn2 unieqi unipr unidm 3eqtri ) ACZDAAEZDAAFALMAGHAA
|
|
BBIAJK $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|
|
(Contributed by NM, 13-Aug-2002.) $)
|
|
unisng $p |- ( A e. V -> U. { A } = A ) $=
|
|
( vx cv csn cuni wceq sneq unieqd id eqeq12d vex unisn vtoclg ) CDZEZFZOG
|
|
AEZFZAGCABOAGZQSOATPROAHITJKOCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( An alternative statement of the effective freeness of a class ` A ` ,
|
|
when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) $)
|
|
dfnfc2 $p |- ( A. x A e. V -> ( F/_ x A <-> A. y F/ x y = A ) ) $=
|
|
( wcel wal wnfc cv wceq wnf nfcvd id nfeqd alrimiv wa csn cuni simpr elsn
|
|
df-nfc nfbii albii bitri sylibr nfunid nfa1 nfnf1 nfal nfan unisng adantr
|
|
sps nfceqdf mpbid ex impbid2 ) CDEZAFZACGZBHZCIZAJZBFZUSVBBUSAUTCUSAUTKUS
|
|
LMNURVCUSURVCOZACPZQZGUSVDAVEVDVCAVEGZURVCRVGUTVEEZAJZBFVCABVETVIVBBVHVAA
|
|
BCSUAUBUCUDUEVDAVFCURVCAUQAUFVBABVAAUGUHUIURVFCIZVCUQVJACDUJULUKUMUNUOUP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The class union of the union of two classes. Theorem 8.3 of [Quine]
|
|
p. 53. (Contributed by NM, 20-Aug-1993.) $)
|
|
uniun $p |- U. ( A u. B ) = ( U. A u. U. B ) $=
|
|
( vx vy cun cuni cv wcel wa wo 19.43 elun anbi2i andi bitri exbii orbi12i
|
|
wex eluni 3bitr4i eqriv ) CABEZFZAFZBFZEZCGZDGZHZUHUBHZIZDRZUGUDHZUGUEHZJ
|
|
ZUGUCHUGUFHUIUHAHZIZUIUHBHZIZJZDRUQDRZUSDRZJULUOUQUSDKUKUTDUKUIUPURJZIUTU
|
|
JVCUIUHABLMUIUPURNOPUMVAUNVBDUGASDUGBSQTDUGUBSUGUDUELTUA $.
|
|
|
|
$( The class union of the intersection of two classes. Exercise 4.12(n) of
|
|
[Mendelson] p. 235. See uniinqs in set.mm for a condition where
|
|
equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by
|
|
Andrew Salmon, 29-Jun-2011.) $)
|
|
uniin $p |- U. ( A i^i B ) C_ ( U. A i^i U. B ) $=
|
|
( vx vy cin cuni cv wcel wex 19.40 elin anbi2i anandi bitri exbii anbi12i
|
|
wa eluni 3imtr4i ssriv ) CABEZFZAFZBFZEZCGZDGZHZUGUAHZQZDIZUFUCHZUFUDHZQZ
|
|
UFUBHUFUEHUHUGAHZQZUHUGBHZQZQZDIUPDIZURDIZQUKUNUPURDJUJUSDUJUHUOUQQZQUSUI
|
|
VBUHUGABKLUHUOUQMNOULUTUMVADUFARDUFBRPSDUFUARUFUCUDKST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
|
|
(Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon,
|
|
29-Jun-2011.) $)
|
|
uniss $p |- ( A C_ B -> U. A C_ U. B ) $=
|
|
( vx vy wss cuni cv wcel wa wex ssel anim2d eximdv eluni 3imtr4g ssrdv )
|
|
ABEZCAFZBFZQCGZDGZHZUAAHZIZDJUBUABHZIZDJTRHTSHQUDUFDQUCUEUBABUAKLMDTANDTB
|
|
NOP $.
|
|
|
|
$( Subclass relationship for class union. (Contributed by NM,
|
|
24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) $)
|
|
ssuni $p |- ( ( A C_ B /\ B e. C ) -> A C_ U. C ) $=
|
|
( vy vx wcel wss cuni cv wi wal eleq2 imbi1d elunii expcom vtoclga imim2d
|
|
wceq alimdv dfss2 3imtr4g impcom ) BCFZABGZACHZGZUCDIZAFZUGBFZJZDKUHUGUEF
|
|
ZJZDKUDUFUCUJULDUCUIUKUHUGEIZFZUKJUIUKJEBCUMBRUNUIUKUMBUGLMUNUMCFUKUGUMCN
|
|
OPQSDABTDAUETUAUB $.
|
|
$}
|
|
|
|
${
|
|
unissi.1 $e |- A C_ B $.
|
|
$( Subclass relationship for subclass union. Inference form of ~ uniss .
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
unissi $p |- U. A C_ U. B $=
|
|
( wss cuni uniss ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
unissd.1 $e |- ( ph -> A C_ B ) $.
|
|
$( Subclass relationship for subclass union. Deduction form of ~ uniss .
|
|
(Contributed by David Moews, 1-May-2017.) $)
|
|
unissd $p |- ( ph -> U. A C_ U. B ) $=
|
|
( wss cuni uniss syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The union of a set is empty iff the set is included in the singleton of
|
|
the empty set. (Contributed by NM, 12-Sep-2004.) $)
|
|
uni0b $p |- ( U. A = (/) <-> A C_ { (/) } ) $=
|
|
( vx vy cv c0 csn wcel wral wceq wss cuni elsn ralbii dfss3 wex wrex neq0
|
|
wn rexcom4 3bitr4ri rexbii eluni2 exbii rexnal 3bitri con4bii ) BDZEFZGZB
|
|
AHUGEIZBAHZAUHJAKZEIZUIUJBABELMBAUHNUMUKUMRCDZULGZCOZUJRZBAPZUKRCULQUNUGG
|
|
ZCOZBAPUSBAPZCOURUPUSBCASUQUTBACUGQUAUOVACBUNAUBUCTUJBAUDUEUFT $.
|
|
|
|
$( The union of a set is empty iff all of its members are empty.
|
|
(Contributed by NM, 16-Aug-2006.) $)
|
|
uni0c $p |- ( U. A = (/) <-> A. x e. A x = (/) ) $=
|
|
( cuni c0 wceq csn wss cv wcel wral uni0b dfss3 elsn ralbii 3bitri ) BCDE
|
|
BDFZGAHZPIZABJQDEZABJBKABPLRSABADMNO $.
|
|
$}
|
|
|
|
$( The union of the empty set is the empty set. Theorem 8.7 of [Quine]
|
|
p. 54. (Reproved without relying on ax-nul in set.mm by Eric Schmidt.)
|
|
(Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt,
|
|
4-Apr-2007.) $)
|
|
uni0 $p |- U. (/) = (/) $=
|
|
( c0 cuni wceq csn wss 0ss uni0b mpbir ) ABACAADZEIFAGH $.
|
|
|
|
$( An element of a class is a subclass of its union. Theorem 8.6 of [Quine]
|
|
p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40.
|
|
(Contributed by NM, 6-Jun-1994.) $)
|
|
elssuni $p |- ( A e. B -> A C_ U. B ) $=
|
|
( wss wcel cuni ssid ssuni mpan ) AACABDABECAFAABGH $.
|
|
|
|
$( Condition turning a subclass relationship for union into an equality.
|
|
(Contributed by NM, 18-Jul-2006.) $)
|
|
unissel $p |- ( ( U. A C_ B /\ B e. A ) -> U. A = B ) $=
|
|
( cuni wss wcel wa simpl elssuni adantl eqssd ) ACZBDZBAEZFKBLMGMBKDLBAHIJ
|
|
$.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Relationship involving membership, subset, and union. Exercise 5 of
|
|
[Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) $)
|
|
unissb $p |- ( U. A C_ B <-> A. x e. A x C_ B ) $=
|
|
( vy cv cuni wcel wi wal wss wa wex eluni imbi1i 19.23v albii bitri dfss2
|
|
wral 3bitr4i bitr4i alcom 19.21v impexp bi2.04 imbi2i df-ral ) DEZBFZGZUH
|
|
CGZHZDIZAEZBGZUNCJZHZAIZUICJUPABSUMUHUNGZUOKZUKHZAIZDIZURULVBDULUTALZUKHV
|
|
BUJVDUKAUHBMNUTUKAOUAPVCVADIZAIURVADAUBVEUQAUOUSUKHZHZDIUOVFDIZHVEUQUOVFD
|
|
UCVAVGDVAUSUOUKHHVGUSUOUKUDUSUOUKUEQPUPVHUODUNCRUFTPQQDUICRUPABUGT $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $.
|
|
$( A subclass condition on the members of two classes that implies a
|
|
subclass relation on their unions. Proposition 8.6 of [TakeutiZaring]
|
|
p. 59. See ~ iunss2 for a generalization to indexed unions.
|
|
(Contributed by NM, 22-Mar-2004.) $)
|
|
uniss2 $p |- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B ) $=
|
|
( cv wss wrex wral cuni wcel ssuni expcom rexlimiv ralimi unissb sylibr )
|
|
AEZBEZFZBDGZACHQDIZFZACHCIUAFTUBACSUBBDSRDJUBQRDKLMNACUAOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( If the difference ` A \ B ` contains the largest members of ` A ` , then
|
|
the union of the difference is the union of ` A ` . (Contributed by NM,
|
|
22-Mar-2004.) $)
|
|
unidif $p |- ( A. x e. A E. y e. ( A \ B ) x C_ y ->
|
|
U. ( A \ B ) = U. A ) $=
|
|
( cv wss cdif wrex wral cuni wa wceq uniss2 difss uniss ax-mp eqss sylibr
|
|
jctil ) AEBEFBCDGZHACIZTJZCJZFZUCUBFZKUBUCLUAUEUDABCTMTCFUDCDNTCOPSUBUCQR
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Relationship implying union. (Contributed by NM, 10-Nov-1999.) $)
|
|
ssunieq $p |- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B ) $=
|
|
( wcel cv wss wral cuni wceq elssuni unissb biimpri anim12i eqss sylibr
|
|
wa ) BCDZAEBFACGZPBCHZFZSBFZPBSIQTRUABCJUARACBKLMBSNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Any member of a class is the largest of those members that it includes.
|
|
(Contributed by NM, 13-Aug-2002.) $)
|
|
unimax $p |- ( A e. B -> U. { x e. B | x C_ A } = A ) $=
|
|
( vy wcel cv wss crab wral cuni wceq ssid sseq1 elrab3 elrab simprbi rgen
|
|
mpbiri wa ssunieq eqcomd sylancl ) BCEZBAFZBGZACHZEZDFZBGZDUFIZUFJZBKUCUG
|
|
BBGZBLUEULABCUDBBMNRUIDUFUHUFEUHCEUIUEUIAUHCUDUHBMOPQUGUJSBUKDBUFTUAUB $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The intersection of a class
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare class intersection symbol. $)
|
|
$c |^| $. $( Big cap $)
|
|
|
|
$( Extend class notation to include the intersection of a class (read:
|
|
'intersect ` A ` '). $)
|
|
cint $a class |^| A $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Define the intersection of a class. Definition 7.35 of [TakeutiZaring]
|
|
p. 44. For example, ` |^| { { ` 1 , 3 ` } , { ` 1 , 8 ` } } = { ` 1
|
|
` } ` . Compare this with the intersection of two classes, ~ df-in .
|
|
(Contributed by NM, 18-Aug-1993.) $)
|
|
df-int $a |- |^| A = { x | A. y ( y e. A -> x e. y ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of class intersection. (Contributed by NM,
|
|
28-Jun-1998.) $)
|
|
dfint2 $p |- |^| A = { x | A. y e. A x e. y } $=
|
|
( cint cv wcel wi wal cab wral df-int df-ral abbii eqtr4i ) CDBEZCFAEOFZG
|
|
BHZAIPBCJZAIABCKRQAPBCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Equality law for intersection. (Contributed by NM, 13-Sep-1999.) $)
|
|
inteq $p |- ( A = B -> |^| A = |^| B ) $=
|
|
( vx vy wceq cv wcel wral cab cint raleq abbidv dfint2 3eqtr4g ) ABEZCFDF
|
|
GZDAHZCIPDBHZCIAJBJOQRCPDABKLCDAMCDBMN $.
|
|
$}
|
|
|
|
${
|
|
inteqi.1 $e |- A = B $.
|
|
$( Equality inference for class intersection. (Contributed by NM,
|
|
2-Sep-2003.) $)
|
|
inteqi $p |- |^| A = |^| B $=
|
|
( wceq cint inteq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
inteqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for class intersection. (Contributed by NM,
|
|
2-Sep-2003.) $)
|
|
inteqd $p |- ( ph -> |^| A = |^| B ) $=
|
|
( wceq cint inteq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x A y $. $d x B y $.
|
|
elint.1 $e |- A e. _V $.
|
|
$( Membership in class intersection. (Contributed by NM, 21-May-1994.) $)
|
|
elint $p |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) ) $=
|
|
( vy cv wcel wi wal cint wceq eleq1 imbi2d albidv df-int elab2 ) AFZCGZEF
|
|
ZQGZHZAIRBQGZHZAIEBCJDSBKZUAUCAUDTUBRSBQLMNEACOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
elint2.1 $e |- A e. _V $.
|
|
$( Membership in class intersection. (Contributed by NM, 14-Oct-1999.) $)
|
|
elint2 $p |- ( A e. |^| B <-> A. x e. B A e. x ) $=
|
|
( cint wcel cv wi wal wral elint df-ral bitr4i ) BCEFAGZCFBNFZHAIOACJABCD
|
|
KOACLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Membership in class intersection, with the sethood requirement expressed
|
|
as an antecedent. (Contributed by NM, 20-Nov-2003.) $)
|
|
elintg $p |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) ) $=
|
|
( vy cv cint wcel wral eleq1 wceq ralbidv vex elint2 vtoclbg ) EFZCGZHPAF
|
|
ZHZACIBQHBRHZACIEBDPBQJPBKSTACPBRJLAPCEMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|
|
(Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
elinti $p |- ( A e. |^| B -> ( C e. B -> A e. C ) ) $=
|
|
( vx cint wcel wi cv wral elintg eleq2 rspccv syl6bi pm2.43i ) ABEZFZCBFA
|
|
CFZGZPPADHZFZDBIRDABOJTQDCBSCAKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $.
|
|
nfint.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for intersection. (Contributed by NM,
|
|
2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) $)
|
|
nfint $p |- F/_ x |^| A $=
|
|
( vy vz cint cv wcel wral cab dfint2 nfv nfral nfab nfcxfr ) ABFDGEGHZEBI
|
|
ZDJDEBKQADPAEBCPALMNO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d ph y $.
|
|
inteqab.1 $e |- A e. _V $.
|
|
$( Membership in the intersection of a class abstraction. (Contributed by
|
|
NM, 30-Aug-1993.) $)
|
|
elintab $p |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) $=
|
|
( vy cab cint wcel cv wi wal elint nfsab1 nfv nfim wceq eleq1 abid syl6bb
|
|
eleq2 imbi12d cbval bitri ) CABFZGHEIZUDHZCUEHZJZEKACBIZHZJZBKECUDDLUHUKE
|
|
BUFUGBABEMUGBNOUKENUEUIPZUFAUGUJULUFUIUDHAUEUIUDQABRSUEUICTUAUBUC $.
|
|
|
|
$( Membership in the intersection of a class abstraction. (Contributed by
|
|
NM, 17-Oct-1999.) $)
|
|
elintrab $p |- ( A e. |^| { x e. B | ph } <->
|
|
A. x e. B ( ph -> A e. x ) ) $=
|
|
( cv wcel wa cab cint wi wal crab wral elintab impexp albii df-rab inteqi
|
|
bitri eleq2i df-ral 3bitr4i ) CBFZDGZAHZBIZJZGZUEACUDGZKZKZBLZCABDMZJZGUK
|
|
BDNUIUFUJKZBLUMUFBCEOUPULBUEAUJPQTUOUHCUNUGABDRSUAUKBDUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d y ph $.
|
|
$( Membership in the intersection of a class abstraction. (Contributed by
|
|
NM, 17-Feb-2007.) $)
|
|
elintrabg $p |- ( A e. V -> ( A e. |^| { x e. B | ph } <->
|
|
A. x e. B ( ph -> A e. x ) ) ) $=
|
|
( vy cv crab cint wcel wi wral eleq1 wceq imbi2d ralbidv elintrab vtoclbg
|
|
vex ) FGZABDHIZJATBGZJZKZBDLCUAJACUBJZKZBDLFCETCUAMTCNZUDUFBDUGUCUEATCUBM
|
|
OPABTDFSQR $.
|
|
|
|
$( The intersection of the empty set is the universal class. Exercise 2 of
|
|
[TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) $)
|
|
int0 $p |- |^| (/) = _V $=
|
|
( vy vx cv c0 wcel wi wal cab wceq cint cvv noel pm2.21i ax-gen 2th abbii
|
|
eqid df-int df-v 3eqtr4i ) ACZDEZBCZUAEZFZAGZBHUCUCIZBHDJKUFUGBUFUGUEAUBU
|
|
DUALMNUCQOPBADRBST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y ph $.
|
|
$( An element of a class includes the intersection of the class. Exercise
|
|
4 of [TakeutiZaring] p. 44 (with correction), generalized to classes.
|
|
(Contributed by NM, 18-Nov-1995.) $)
|
|
intss1 $p |- ( A e. B -> |^| B C_ A ) $=
|
|
( vx vy wcel cint cv wal vex elint wceq eleq1 eleq2 imbi12d spcgv pm2.43a
|
|
wi syl5bi ssrdv ) ABEZCBFZACGZUAEDGZBEZUBUCEZQZDHZTUBAEZDUBBCIJUGTUHUFTUH
|
|
QDABUCAKUDTUEUHUCABLUCAUBMNOPRS $.
|
|
|
|
$( Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
|
|
and its converse. (Contributed by NM, 14-Oct-1999.) $)
|
|
ssint $p |- ( A C_ |^| B <-> A. x e. B A C_ x ) $=
|
|
( vy cint wss cv wcel wral dfss3 vex elint2 ralbii ralcom bitr4i 3bitri )
|
|
BCEZFDGZQHZDBIRAGZHZACIZDBIZBTFZACIZDBQJSUBDBARCDKLMUCUADBIZACIUEUADABCNU
|
|
DUFACDBTJMOP $.
|
|
|
|
$( Subclass of the intersection of a class abstraction. (Contributed by
|
|
NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
ssintab $p |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) ) $=
|
|
( vy cab cint wss cv wral wi wal ssint sseq2 ralab2 bitri ) CABEZFGCDHZGZ
|
|
DPIACBHZGZJBKDCPLARTDBQSCMNO $.
|
|
|
|
$( Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.) $)
|
|
ssintub $p |- A C_ |^| { x e. B | A C_ x } $=
|
|
( vy cv wss crab cint ssint wcel sseq2 elrab simprbi mprgbir ) BBAEZFZACG
|
|
ZHFBDEZFZDQDBQIRQJRCJSPSARCORBKLMN $.
|
|
|
|
$( Subclass of the minimum value of class of supersets. (Contributed by
|
|
NM, 10-Aug-2006.) $)
|
|
ssmin $p |- A C_ |^| { x | ( A C_ x /\ ph ) } $=
|
|
( cv wss wa cab cint wi ssintab simpl mpgbir ) CCBDEZAFZBGHENMIBNBCJMAKL
|
|
$.
|
|
|
|
$( Any member of a class is the smallest of those members that include it.
|
|
(Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon,
|
|
9-Jul-2011.) $)
|
|
intmin $p |- ( A e. B -> |^| { x e. B | A C_ x } = A ) $=
|
|
( vy wcel cv wss crab cint wi wral elintrab ssid wceq sseq2 eleq2 imbi12d
|
|
vex rspcv mpii syl5bi ssrdv ssintub a1i eqssd ) BCEZBAFZGZACHIZBUFDUIBDFZ
|
|
UIEUHUJUGEZJZACKZUFUJBEZUHAUJCDRLUFUMBBGZUNBMULUOUNJABCUGBNUHUOUKUNUGBBOU
|
|
GBUJPQSTUAUBBUIGUFABCUCUDUE $.
|
|
|
|
$( Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) $)
|
|
intss $p |- ( A C_ B -> |^| B C_ |^| A ) $=
|
|
( vy vx cv wcel wal cint wss imim1 al2imi vex elint 3imtr4g alrimiv dfss2
|
|
wi 3imtr4i ) CEZAFZSBFZQZCGZDEZBHZFZUDAHZFZQZDGABIUEUGIUCUIDUCUAUDSFZQZCG
|
|
TUJQZCGUFUHUBUKULCTUAUJJKCUDBDLZMCUDAUMMNOCABPDUEUGPR $.
|
|
|
|
$( The intersection of a nonempty set is a subclass of its union.
|
|
(Contributed by NM, 29-Jul-2006.) $)
|
|
intssuni $p |- ( A =/= (/) -> |^| A C_ U. A ) $=
|
|
( vx vy c0 wne cint cuni cv wcel wral r19.2z ex vex elint2 eluni2 3imtr4g
|
|
wrex ssrdv ) ADEZBAFZAGZSBHZCHIZCAJZUCCAQZUBTIUBUAISUDUEUCCAKLCUBABMNCUBA
|
|
OPR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( Subclass of the intersection of a restricted class builder.
|
|
(Contributed by NM, 30-Jan-2015.) $)
|
|
ssintrab $p |- ( A C_ |^| { x e. B | ph }
|
|
<-> A. x e. B ( ph -> A C_ x ) ) $=
|
|
( crab cint wss cv wcel wa cab wral df-rab inteqi sseq2i wal impexp albii
|
|
wi ssintab df-ral 3bitr4i bitri ) CABDEZFZGCBHZDIZAJZBKZFZGZACUFGZSZBDLZU
|
|
EUJCUDUIABDMNOUHULSZBPUGUMSZBPUKUNUOUPBUGAULQRUHBCTUMBDUAUBUC $.
|
|
$}
|
|
|
|
$( If the union of a class is included in its intersection, the class is
|
|
either the empty set or a singleton ( ~ uniintsn ). (Contributed by NM,
|
|
30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
unissint $p |- ( U. A C_ |^| A <-> ( A = (/) \/ U. A = |^| A ) ) $=
|
|
( cuni cint wss c0 wo wn wa simpl wne df-ne intssuni sylbir adantl eqssd ex
|
|
wceq orrd cvv ssv int0 sseqtr4i inteq syl5sseqr eqimss jaoi impbii ) ABZACZ
|
|
DZAEQZUHUIQZFUJUKULUJUKGZULUJUMHUHUIUJUMIUMUIUHDZUJUMAEJUNAEKALMNOPRUKUJULU
|
|
KECZUHUIUHSUOUHTUAUBAEUCUDUHUIUEUFUG $.
|
|
|
|
$( Subclass relationship for intersection and union. (Contributed by NM,
|
|
29-Jul-2006.) $)
|
|
intssuni2 $p |- ( ( A C_ B /\ A =/= (/) ) -> |^| A C_ U. B ) $=
|
|
( c0 wne wss cint cuni intssuni uniss sylan9ssr ) ACDABEAFAGBGAHABIJ $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
intminss.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( Under subset ordering, the intersection of a restricted class
|
|
abstraction is less than or equal to any of its members. (Contributed
|
|
by NM, 7-Sep-2013.) $)
|
|
intminss $p |- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A ) $=
|
|
( wcel wa crab cint wss elrab intss1 sylbir ) DEGBHDACEIZGOJDKABCDEFLDOMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
intmin2.1 $e |- A e. _V $.
|
|
$( Any set is the smallest of all sets that include it. (Contributed by
|
|
NM, 20-Sep-2003.) $)
|
|
intmin2 $p |- |^| { x | A C_ x } = A $=
|
|
( cv wss cvv crab cint cab rabab inteqi wcel wceq intmin ax-mp eqtr3i ) B
|
|
ADEZAFGZHZQAIZHBRTQAJKBFLSBMCABFNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x ps $.
|
|
intmin3.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
intmin3.3 $e |- ps $.
|
|
$( Under subset ordering, the intersection of a class abstraction is less
|
|
than or equal to any of its members. (Contributed by NM,
|
|
3-Jul-2005.) $)
|
|
intmin3 $p |- ( A e. V -> |^| { x | ph } C_ A ) $=
|
|
( wcel cab cint wss elabg mpbiri intss1 syl ) DEHZDACIZHZQJDKPRBGABCDEFLM
|
|
DQNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( Elimination of a conjunct in a class intersection. (Contributed by NM,
|
|
31-Jul-2006.) $)
|
|
intmin4 $p |- ( A C_ |^| { x | ph } ->
|
|
|^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } ) $=
|
|
( vy cab cint wss cv wa wcel wi wal wb ssintab simpr impbid2 imbi1d alimi
|
|
ancr elintab albi syl sylbi vex 3bitr4g eqrdv ) CABEFZGZDCBHZGZAIZBEFZUGU
|
|
HUKDHZUIJZKZBLZAUNKZBLZUMULJUMUGJUHAUJKZBLZUPURMZABCNUTUOUQMZBLVAUSVBBUSU
|
|
KAUNUSUKAUJAOAUJSPQRUOUQBUAUBUCUKBUMDUDZTABUMVCTUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d x z ph $. $d x y z $.
|
|
intab.1 $e |- A e. _V $.
|
|
intab.2 $e |- { x | E. y ( ph /\ x = A ) } e. _V $.
|
|
$( The intersection of a special case of a class abstraction. ` y ` may be
|
|
free in ` ph ` and ` A ` , which can be thought of a ` ph ( y ) ` and
|
|
` A ( y ) ` . Typically, abrexex2 in set.mm or abexssex in set.mm can
|
|
be used to satisfy the second hypothesis. (Contributed by NM,
|
|
28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) $)
|
|
intab $p |- |^| { x | A. y ( ph -> A e. x ) } =
|
|
{ x | E. y ( ph /\ x = A ) } $=
|
|
( vz cv wcel wi wal cab cint wceq wa wex wss ex alrimiv sylibr cvv anbi2d
|
|
eqeq1 exbidv cbvabv eqeltri nfe1 nfab nfeq2 eleq2 imbi2d albid elab 19.8a
|
|
wsbc sbc6 df-sbc sylib mpgbir intss1 19.29r simplr pm3.35 adantlr eqeltrd
|
|
ax-mp exlimiv syl vex elintab abssi eqssi eqtri ) ADBHZIZJZCKZBLZMZAGHZDN
|
|
ZOZCPZGLZAVNDNZOZCPZBLZVSWDWDVRIZVSWDQWIADWDIZJZCVQWKCKBWDWDWHUAWCWGGBVTV
|
|
NNZWBWFCWLWAWEAVTVNDUCUBUDUEZFUFVNWDNZVPWKCCVNWDWCCGWBCUGUHUIWNVOWJAVNWDD
|
|
UJUKULUMAWCGDUOZWJAWAWCJZGKWOAWPGAWAWCWBCUNRSWCGDEUPTWCGDUQURUSWDVRUTVFWC
|
|
GVSWCVQVTVNIZJZBKVTVSIWCWRBWCVQWQWCVQOWBVPOZCPWQWBVPCVAWSWQCWSVTDVNAWAVPV
|
|
BAVPVOWAAVOVCVDVEVGVHRSVQBVTGVIVJTVKVLWMVM $.
|
|
$}
|
|
|
|
$( The intersection of a class containing the empty set is empty.
|
|
(Contributed by NM, 24-Apr-2004.) $)
|
|
int0el $p |- ( (/) e. A -> |^| A = (/) ) $=
|
|
( c0 wcel cint intss1 wss 0ss a1i eqssd ) BACZADZBBAEBKFJKGHI $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The class intersection of the union of two classes. Theorem 78 of
|
|
[Suppes] p. 42. (Contributed by NM, 22-Sep-2002.) $)
|
|
intun $p |- |^| ( A u. B ) = ( |^| A i^i |^| B ) $=
|
|
( vx vy cun cint cin cv wcel wi wal wa 19.26 elun imbi1i jaob bitri elint
|
|
wo 3bitr4i albii vex anbi12i elin eqriv ) CABEZFZAFZBFZGZDHZUFIZCHZUKIZJZ
|
|
DKZUMUHIZUMUIIZLZUMUGIUMUJIUKAIZUNJZUKBIZUNJZLZDKVADKZVCDKZLUPUSVAVCDMUOV
|
|
DDUOUTVBSZUNJVDULVGUNUKABNOUTUNVBPQUAUQVEURVFDUMACUBZRDUMBVHRUCTDUMUFVHRU
|
|
MUHUIUDTUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
intpr.1 $e |- A e. _V $.
|
|
intpr.2 $e |- B e. _V $.
|
|
$( The intersection of a pair is the intersection of its members. Theorem
|
|
71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.) $)
|
|
intpr $p |- |^| { A , B } = ( A i^i B ) $=
|
|
( vx vy cpr cint cin cv wcel wi wal wa wceq 19.26 wo vex clel4 3bitr4i
|
|
elpr imbi1i jaob bitri albii anbi12i elint elin eqriv ) EABGZHZABIZFJZUJK
|
|
ZEJZUMKZLZFMZUOAKZUOBKZNZUOUKKUOULKUMAOZUPLZUMBOZUPLZNZFMVCFMZVEFMZNURVAV
|
|
CVEFPUQVFFUQVBVDQZUPLVFUNVIUPUMABFRUAUBVBUPVDUCUDUEUSVGUTVHFUOACSFUOBDSUF
|
|
TFUOUJERUGUOABUHTUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( The intersection of a pair is the intersection of its members. Closed
|
|
form of ~ intpr . Theorem 71 of [Suppes] p. 42. (Contributed by FL,
|
|
27-Apr-2008.) $)
|
|
intprg $p |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) ) $=
|
|
( vx vy cv cpr cint cin wceq preq1 inteqd ineq1 eqeq12d preq2 ineq2 intpr
|
|
vex vtocl2g ) EGZFGZHZIZUAUBJZKAUBHZIZAUBJZKABHZIZABJZKEFABCDUAAKZUDUGUEU
|
|
HULUCUFUAAUBLMUAAUBNOUBBKZUGUJUHUKUMUFUIUBBAPMUBBAQOUAUBESFSRT $.
|
|
$}
|
|
|
|
$( Intersection of a singleton. (Contributed by Stefan O'Rear,
|
|
22-Feb-2015.) $)
|
|
intsng $p |- ( A e. V -> |^| { A } = A ) $=
|
|
( wcel csn cint cpr dfsn2 inteqi cin wceq intprg anidms inidm syl6eq syl5eq
|
|
) ABCZADZEAAFZEZAQRAGHPSAAIZAPSTJAABBKLAMNO $.
|
|
|
|
${
|
|
intsn.1 $e |- A e. _V $.
|
|
$( The intersection of a singleton is its member. Theorem 70 of [Suppes]
|
|
p. 41. (Contributed by NM, 29-Sep-2002.) $)
|
|
intsn $p |- |^| { A } = A $=
|
|
( cvv wcel csn cint wceq intsng ax-mp ) ACDAEFAGBACHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y ph $.
|
|
$( Two ways to express " ` A ` is a singleton." See also en1 in set.mm,
|
|
en1b in set.mm, card1 in set.mm, and ~ eusn . (Contributed by NM,
|
|
2-Aug-2010.) $)
|
|
uniintsn $p |- ( U. A = |^| A <-> E. x A = { x } ) $=
|
|
( vy cuni cint wceq cv wex wcel wa wal c0 inteq syl6eq adantl unieq sylib
|
|
wne cvv wss csn wi vn0 int0 eqeq1 syl5ib imp eqtr3d ex necon3d mpi n0 cpr
|
|
uni0 vex prss cun cin uniss simpl sseqtrd intss sstrd unipr intpr 3sstr3g
|
|
inss1 ssun1 sstri jctir eqss uneqin bitr3i syl5bi alrimivv jca weu euabsn
|
|
cab eleq1 abid2 eqeq1i exbii 3bitr3i unisn intsn 3eqtr4a exlimiv impbii
|
|
eu4 ) BDZBEZFZBAGZUAZFZAHZWMWNBIZAHZWRCGZBIZJZWNWTFZUBZCKAKZJZWQWMWSXEWMB
|
|
LRZWSWMSLRXGUCWMBLSLWMBLFZSLFWMXHJWLSLXHWLSFWMXHWLLESBLMUDNOWMXHWLLFZXHWK
|
|
LFWMXIXHWKLDLBLPUNNWKWLLUEUFUGUHUIUJUKABULQWMXDACXBWNWTUMZBTZWMXCWNWTBAUO
|
|
ZCUOZUPWMXKXCWMXKJZWNWTUQZWNWTURZTZXPXOTZJZXCXNXQXRXNXJDZXJEZXOXPXNXTWLYA
|
|
XNXTWKWLXKXTWKTWMXJBUSOWMXKUTVAXKWLYATWMXJBVBOVCWNWTXLXMVDWNWTXLXMVEVFXPW
|
|
NXOWNWTVGWNWTVHVIVJXSXOXPFXCXOXPVKWNWTVLVMQUIVNVOVPWRAVQWRAVSZWOFZAHXFWQW
|
|
RAVRWRXAACWNWTBVTWJYCWPAYBBWOABWAWBWCWDQWPWMAWPWODWNWKWLWNXLWEBWOPWPWLWOE
|
|
WNBWOMWNXLWFNWGWHWI $.
|
|
|
|
$( The union and the intersection of a class abstraction are equal exactly
|
|
when there is a unique satisfying value of ` ph ( x ) ` . (Contributed
|
|
by Mario Carneiro, 24-Dec-2016.) $)
|
|
uniintab $p |- ( E! x ph <-> U. { x | ph } = |^| { x | ph } ) $=
|
|
( vy weu cab cv csn wceq wex cuni cint euabsn2 uniintsn bitr4i ) ABDABEZC
|
|
FGHCIOJOKHABCLCOMN $.
|
|
$}
|
|
|
|
${
|
|
intunsn.1 $e |- B e. _V $.
|
|
$( Theorem joining a singleton to an intersection. (Contributed by NM,
|
|
29-Sep-2002.) $)
|
|
intunsn $p |- |^| ( A u. { B } ) = ( |^| A i^i B ) $=
|
|
( csn cun cint cin intun intsn ineq2i eqtri ) ABDZEFAFZLFZGMBGALHNBMBCIJK
|
|
$.
|
|
$}
|
|
|
|
$( Relative intersection of an empty set. (Contributed by Stefan O'Rear,
|
|
3-Apr-2015.) $)
|
|
rint0 $p |- ( X = (/) -> ( A i^i |^| X ) = A ) $=
|
|
( c0 wceq cint cin inteq ineq2d cvv int0 ineq2i inv1 eqtri syl6eq ) BCDZABE
|
|
ZFACEZFZAOPQABCGHRAIFAQIAJKALMN $.
|
|
|
|
${
|
|
$d B y $. $d X y $.
|
|
$( Membership in a restricted intersection. (Contributed by Stefan O'Rear,
|
|
3-Apr-2015.) $)
|
|
elrint $p |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) ) $=
|
|
( cint cin wcel wa cv wral elin elintg pm5.32i bitri ) DBCEZFGDBGZDOGZHPD
|
|
AIGACJZHDBOKPQRADCBLMN $.
|
|
|
|
$( Membership in a restricted intersection. (Contributed by Stefan O'Rear,
|
|
3-Apr-2015.) $)
|
|
elrint2 $p |- ( X e. A -> ( X e. ( A i^i |^| B ) <->
|
|
A. y e. B X e. y ) ) $=
|
|
( cint cin wcel cv wral elrint baib ) DBCEFGDBGDAHGACIABCDJK $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Indexed union and intersection
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c U_ $. $( Underlined big cup. $)
|
|
$c |^|_ $. $( Underlined big cap. $)
|
|
|
|
$( Extend class notation to include indexed union. Note: Historically
|
|
(prior to 21-Oct-2005), set.mm used the notation ` U. x e. A B ` , with
|
|
the same union symbol as ~ cuni . While that syntax was unambiguous, it
|
|
did not allow for LALR parsing of the syntax constructions in set.mm. The
|
|
new syntax uses as distinguished symbol ` U_ ` instead of ` U. ` and does
|
|
allow LALR parsing. Thanks to Peter Backes for suggesting this change. $)
|
|
ciun $a class U_ x e. A B $.
|
|
|
|
$( Extend class notation to include indexed intersection. Note:
|
|
Historically (prior to 21-Oct-2005), set.mm used the notation
|
|
` |^| x e. A B ` , with the same intersection symbol as ~ cint . Although
|
|
that syntax was unambiguous, it did not allow for LALR parsing of the
|
|
syntax constructions in set.mm. The new syntax uses a distinguished
|
|
symbol ` |^|_ ` instead of ` |^| ` and does allow LALR parsing. Thanks to
|
|
Peter Backes for suggesting this change. $)
|
|
ciin $a class |^|_ x e. A B $.
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( Define indexed union. Definition indexed union in [Stoll] p. 45. In
|
|
most applications, ` A ` is independent of ` x ` (although this is not
|
|
required by the definition), and ` B ` depends on ` x ` i.e. can be read
|
|
informally as ` B ( x ) ` . We call ` x ` the index, ` A ` the index
|
|
set, and ` B ` the indexed set. In most books, ` x e. A ` is written as
|
|
a subscript or underneath a union symbol ` U. ` . We use a special
|
|
union symbol ` U_ ` to make it easier to distinguish from plain class
|
|
union. In many theorems, you will see that ` x ` and ` A ` are in the
|
|
same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and
|
|
that ` B ` and ` x ` do not share a distinct variable group (meaning
|
|
that can be thought of as ` B ( x ) ` i.e. can be substituted with a
|
|
class expression containing ` x ` ). An alternate definition tying
|
|
indexed union to ordinary union is ~ dfiun2 . Theorem ~ uniiun provides
|
|
a definition of ordinary union in terms of indexed union. Theorems
|
|
~ fniunfv and ~ funiunfv are useful when ` B ` is a function.
|
|
(Contributed by NM, 27-Jun-1998.) $)
|
|
df-iun $a |- U_ x e. A B = { y | E. x e. A y e. B } $.
|
|
|
|
$( Define indexed intersection. Definition of [Stoll] p. 45. See the
|
|
remarks for its sibling operation of indexed union ~ df-iun . An
|
|
alternate definition tying indexed intersection to ordinary intersection
|
|
is ~ dfiin2 . Theorem ~ intiin provides a definition of ordinary
|
|
intersection in terms of indexed intersection. (Contributed by NM,
|
|
27-Jun-1998.) $)
|
|
df-iin $a |- |^|_ x e. A B = { y | A. x e. A y e. B } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d y C $.
|
|
$( Membership in indexed union. (Contributed by NM, 3-Sep-2003.) $)
|
|
eliun $p |- ( A e. U_ x e. B C <-> E. x e. B A e. C ) $=
|
|
( vy ciun wcel wrex elex rexlimivw cv wceq eleq1 rexbidv df-iun pm5.21nii
|
|
cvv elab2g ) BACDFZGBQGZBDGZACHZBSIUATACBDIJEKZDGZACHUBEBSQUCBLUDUAACUCBD
|
|
MNAECDORP $.
|
|
|
|
$( Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.) $)
|
|
eliin $p |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) ) $=
|
|
( vy cv wcel wral ciin wceq eleq1 ralbidv df-iin elab2g ) FGZDHZACIBDHZAC
|
|
IFBACDJEPBKQRACPBDLMAFCDNO $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x z B $. $d z C $. $d x y $.
|
|
$( Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.) $)
|
|
iuncom $p |- U_ x e. A U_ y e. B C = U_ y e. B U_ x e. A C $=
|
|
( vz ciun cv wcel wrex rexcom eliun rexbii 3bitr4i eqriv ) FACBDEGZGZBDAC
|
|
EGZGZFHZPIZACJZTRIZBDJZTQITSITEIZBDJZACJUEACJZBDJUBUDUEABCDKUAUFACBTDELMU
|
|
CUGBDATCELMNATCPLBTDRLNO $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d x y z $.
|
|
$( Commutation of union with indexed union. (Contributed by Mario
|
|
Carneiro, 18-Jan-2014.) $)
|
|
iuncom4 $p |- U_ x e. A U. B = U. U_ x e. A B $=
|
|
( vy vz cuni ciun cv wcel wa wex df-rex rexbii rexcom4 bitri exbii eluni2
|
|
wrex eliun 3bitr4i r19.41v anbi1i eqriv ) DABCFZGZABCGZFZDHZUDIZABRZUHEHZ
|
|
IZEUFRZUHUEIUHUGIULECRZABRZUKCIZABRZULJZEKZUJUMUOUPULJZABRZEKZUSUOUTEKZAB
|
|
RVBUNVCABULECLMUTAEBNOVAUREUPULABUAPOUIUNABEUHCQMUMUKUFIZULJZEKUSULEUFLVE
|
|
UREVDUQULAUKBCSUBPOTAUHBUDSEUHUFQTUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Indexed union of a constant class, i.e. where ` B ` does not depend on
|
|
` x ` . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew
|
|
Salmon, 25-Jul-2011.) $)
|
|
iunconst $p |- ( A =/= (/) -> U_ x e. A B = B ) $=
|
|
( vy c0 wne ciun cv wcel wrex r19.9rzv eliun syl6rbbr eqrdv ) BEFZDABCGZC
|
|
ODHZCIZRABJQPIRABKAQBCLMN $.
|
|
|
|
$( Indexed intersection of a constant class, i.e. where ` B ` does not
|
|
depend on ` x ` . (Contributed by Mario Carneiro, 6-Feb-2015.) $)
|
|
iinconst $p |- ( A =/= (/) -> |^|_ x e. A B = B ) $=
|
|
( vy c0 wne ciin cv wcel wral r19.3rzv cvv vex eliin ax-mp syl6rbbr eqrdv
|
|
wb ) BEFZDABCGZCSDHZCIZUBABJZUATIZUBABKUALIUDUCRDMAUABCLNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z A $. $d x z B $. $d z C $.
|
|
$( Law combining indexed union with indexed intersection. Eq. 14 in
|
|
[KuratowskiMostowski] p. 109. This theorem also appears as the last
|
|
example at ~ http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29 .
|
|
(Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon,
|
|
25-Jul-2011.) $)
|
|
iuniin $p |- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C $=
|
|
( vz ciin ciun cv wcel wrex wral r19.12 cvv vex eliin ax-mp eliun 3imtr4i
|
|
wb rexbii ralbii ssriv ) FACBDEGZHZBDACEHZGZFIZUDJZACKZUHUFJZBDLZUHUEJUHU
|
|
GJZUHEJZBDLZACKUNACKZBDLUJULUNABCDMUIUOACUHNJZUIUOTFOZBUHDENPQUAUKUPBDAUH
|
|
CERUBSAUHCUDRUQUMULTURBUHDUFNPQSUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $.
|
|
$( Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iunss1 $p |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C ) $=
|
|
( vy wss ciun cv wcel wrex ssrexv eliun 3imtr4g ssrdv ) BCFZEABDGZACDGZOE
|
|
HZDIZABJSACJRPIRQISABCKARBDLARCDLMN $.
|
|
|
|
$( Subclass theorem for indexed union. (Contributed by NM,
|
|
24-Jan-2012.) $)
|
|
iinss1 $p |- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C ) $=
|
|
( vy wss ciin cv wcel wral ssralv cvv wb vex eliin ax-mp 3imtr4g ssrdv )
|
|
BCFZEACDGZABDGZSEHZDIZACJZUCABJZUBTIZUBUAIZUCABCKUBLIZUFUDMENZAUBCDLOPUHU
|
|
GUEMUIAUBBDLOPQR $.
|
|
|
|
$( Equality theorem for indexed union. (Contributed by NM,
|
|
27-Jun-1998.) $)
|
|
iuneq1 $p |- ( A = B -> U_ x e. A C = U_ x e. B C ) $=
|
|
( wss wa ciun wceq iunss1 anim12i eqss 3imtr4i ) BCEZCBEZFABDGZACDGZEZPOE
|
|
ZFBCHOPHMQNRABCDIACBDIJBCKOPKL $.
|
|
|
|
$( Equality theorem for restricted existential quantifier. (Contributed by
|
|
NM, 27-Jun-1998.) $)
|
|
iineq1 $p |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C ) $=
|
|
( vy wceq cv wcel wral cab ciin raleq abbidv df-iin 3eqtr4g ) BCFZEGDHZAB
|
|
IZEJQACIZEJABDKACDKPRSEQABCLMAEBDNAECDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $.
|
|
$( Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
ss2iun $p |- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C ) $=
|
|
( vy wss wral ciun cv wcel wrex ssel ralimi rexim syl eliun 3imtr4g ssrdv
|
|
wi ) CDFZABGZEABCHZABDHZUAEIZCJZABKZUDDJZABKZUDUBJUDUCJUAUEUGSZABGUFUHSTU
|
|
IABCDUDLMUEUGABNOAUDBCPAUDBDPQR $.
|
|
|
|
$( Equality theorem for indexed union. (Contributed by NM,
|
|
22-Oct-2003.) $)
|
|
iuneq2 $p |- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C ) $=
|
|
( wss wral wa ciun wceq ss2iun anim12i eqss ralbii r19.26 bitri 3imtr4i )
|
|
CDEZABFZDCEZABFZGZABCHZABDHZEZUCUBEZGCDIZABFZUBUCIRUDTUEABCDJABDCJKUGQSGZ
|
|
ABFUAUFUHABCDLMQSABNOUBUCLP $.
|
|
|
|
$( Equality theorem for indexed intersection. (Contributed by NM,
|
|
22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iineq2 $p |- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C ) $=
|
|
( vy wceq wral cv wcel cab ciin wb eleq2 ralimi syl abbidv df-iin 3eqtr4g
|
|
ralbi ) CDFZABGZEHZCIZABGZEJUBDIZABGZEJABCKABDKUAUDUFEUAUCUELZABGUDUFLTUG
|
|
ABCDUBMNUCUEABSOPAEBCQAEBDQR $.
|
|
$}
|
|
|
|
${
|
|
iuneq2i.1 $e |- ( x e. A -> B = C ) $.
|
|
$( Equality inference for indexed union. (Contributed by NM,
|
|
22-Oct-2003.) $)
|
|
iuneq2i $p |- U_ x e. A B = U_ x e. A C $=
|
|
( wceq ciun iuneq2 mprg ) CDFABCGABDGFABABCDHEI $.
|
|
|
|
$( Equality inference for indexed intersection. (Contributed by NM,
|
|
22-Oct-2003.) $)
|
|
iineq2i $p |- |^|_ x e. A B = |^|_ x e. A C $=
|
|
( wceq ciin iineq2 mprg ) CDFABCGABDGFABABCDHEI $.
|
|
$}
|
|
|
|
${
|
|
iineq2d.1 $e |- F/ x ph $.
|
|
iineq2d.2 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Equality deduction for indexed intersection. (Contributed by NM,
|
|
7-Dec-2011.) $)
|
|
iineq2d $p |- ( ph -> |^|_ x e. A B = |^|_ x e. A C ) $=
|
|
( wceq wral ciin cv wcel ex ralrimi iineq2 syl ) ADEHZBCIBCDJBCEJHAQBCFAB
|
|
KCLQGMNBCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
iuneq2dv.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Equality deduction for indexed union. (Contributed by NM,
|
|
3-Aug-2004.) $)
|
|
iuneq2dv $p |- ( ph -> U_ x e. A B = U_ x e. A C ) $=
|
|
( wceq wral ciun ralrimiva iuneq2 syl ) ADEGZBCHBCDIBCEIGAMBCFJBCDEKL $.
|
|
|
|
$( Equality deduction for indexed intersection. (Contributed by NM,
|
|
3-Aug-2004.) $)
|
|
iineq2dv $p |- ( ph -> |^|_ x e. A B = |^|_ x e. A C ) $=
|
|
( nfv iineq2d ) ABCDEABGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
iuneq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for indexed union, deduction version. (Contributed by
|
|
Drahflow, 22-Oct-2015.) $)
|
|
iuneq1d $p |- ( ph -> U_ x e. A C = U_ x e. B C ) $=
|
|
( wceq ciun iuneq1 syl ) ACDGBCEHBDEHGFBCDEIJ $.
|
|
|
|
${
|
|
$d x ph $.
|
|
iuneq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for indexed union, deduction version. (Contributed
|
|
by Drahflow, 22-Oct-2015.) $)
|
|
iuneq12d $p |- ( ph -> U_ x e. A C = U_ x e. B D ) $=
|
|
( ciun iuneq1d wceq cv wcel adantr iuneq2dv eqtrd ) ABCEIBDEIBDFIABCDEG
|
|
JABDEFAEFKBLDMHNOP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d x A $.
|
|
iuneq2d.2 $e |- ( ph -> B = C ) $.
|
|
$( Equality deduction for indexed union. (Contributed by Drahflow,
|
|
22-Oct-2015.) $)
|
|
iuneq2d $p |- ( ph -> U_ x e. A B = U_ x e. A C ) $=
|
|
( wceq cv wcel adantr iuneq2dv ) ABCDEADEGBHCIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d x z $. $d y z $.
|
|
nfiun.1 $e |- F/_ y A $.
|
|
nfiun.2 $e |- F/_ y B $.
|
|
$( Bound-variable hypothesis builder for indexed union. (Contributed by
|
|
Mario Carneiro, 25-Jan-2014.) $)
|
|
nfiun $p |- F/_ y U_ x e. A B $=
|
|
( vz ciun cv wcel wrex cab df-iun nfcri nfrex nfab nfcxfr ) BACDHGIDJZACK
|
|
ZGLAGCDMSBGRBACEBGDFNOPQ $.
|
|
|
|
$( Bound-variable hypothesis builder for indexed intersection.
|
|
(Contributed by Mario Carneiro, 25-Jan-2014.) $)
|
|
nfiin $p |- F/_ y |^|_ x e. A B $=
|
|
( vz ciin cv wcel wral cab df-iin nfcri nfral nfab nfcxfr ) BACDHGIDJZACK
|
|
ZGLAGCDMSBGRBACEBGDFNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( Bound-variable hypothesis builder for indexed union. (Contributed by
|
|
NM, 12-Oct-2003.) $)
|
|
nfiu1 $p |- F/_ x U_ x e. A B $=
|
|
( vy ciun cv wcel wrex cab df-iun nfre1 nfab nfcxfr ) AABCEDFCGZABHZDIADB
|
|
CJOADNABKLM $.
|
|
|
|
$( Bound-variable hypothesis builder for indexed intersection.
|
|
(Contributed by NM, 15-Oct-2003.) $)
|
|
nfii1 $p |- F/_ x |^|_ x e. A B $=
|
|
( vy ciin cv wcel wral cab df-iin nfra1 nfab nfcxfr ) AABCEDFCGZABHZDIADB
|
|
CJOADNABKLM $.
|
|
$}
|
|
|
|
${
|
|
$d y z w A $. $d y z w B $. $d w C z $. $d w x y z $.
|
|
$( Alternate definition of indexed union when ` B ` is a set. Definition
|
|
15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof
|
|
shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
dfiun2g $p |- ( A. x e. A B e. C ->
|
|
U_ x e. A B = U. { y | E. x e. A y = B } ) $=
|
|
( vz wcel wral ciun cv wceq wrex cab cuni wa wex nfra1 wb rsp syl6bb syl6
|
|
clel3g rexbida rexcom4 r19.41v exbii exancom bitri eliun eluniab 3bitr4g
|
|
imp eqrdv ) DEGZACHZFACDIZBJZDKZACLZBMNZUOFJZDGZACLZVAUQGZUSOBPZVAUPGVAUT
|
|
GUOVCURVDOZACLZBPZVEUOVCVFBPZACLVHUOVBVIACUNACQUOAJCGZVBVIRZUOVJUNVKUNACS
|
|
BVADEUBUAULUCVFABCUDTVHUSVDOZBPVEVGVLBURVDACUEUFUSVDBUGUHTAVACDUIUSBVAUJU
|
|
KUM $.
|
|
|
|
$( Alternate definition of indexed intersection when ` B ` is a set.
|
|
(Contributed by Jeff Hankins, 27-Aug-2009.) $)
|
|
dfiin2g $p |- ( A. x e. A B e. C
|
|
-> |^|_ x e. A B = |^| { y | E. x e. A y = B } ) $=
|
|
( vw vz wcel wral cv cab wceq wrex wi wal df-ral wb eqeq1 albii bitr4i
|
|
ciin cint eleq2 biimprcd alrimiv eqid imbi12d mpii impbid2 imim2i pm5.74d
|
|
spcgv alimi albi syl sylbi alcom r19.23v vex rexbidv elab imbi1i 3bitr3ri
|
|
19.21v syl6bb syl5bb abbidv df-iin df-int 3eqtr4g ) DEHZACIZFJZDHZACIZFKG
|
|
JZBJZDLZACMZBKZHZVMVPHZNZGOZFKACDUAVTUBVLVOWDFVOAJCHZVNNZAOZVLWDVNACPVLWG
|
|
WEVPDLZWBNZGOZNZAOZWDVLWEVKNZAOZWGWLQZVKACPWNWFWKQZAOWOWMWPAWMWEVNWJVKVNW
|
|
JQWEVKVNWJVNWIGWHWBVNVPDVMUCZUDUEVKWJDDLZVNDUFWIWRVNNGDEWHWHWRWBVNVPDDRWQ
|
|
UGULUHUIUJUKUMWFWKAUNUOUPWIACIZGOZWEWINZGOZAOZWDWLWTXAAOZGOXCWSXDGWIACPSX
|
|
AAGUQTWSWCGWSWHACMZWBNWCWHWBACURWAXEWBVSXEBVPGUSVQVPLVRWHACVQVPDRUTVAVBTS
|
|
XBWKAWEWIGVDSVCVEVFVGAFCDVHFGVTVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
dfiun2.1 $e |- B e. _V $.
|
|
$( Alternate definition of indexed union when ` B ` is a set. Definition
|
|
15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by
|
|
David Abernethy, 19-Jun-2012.) $)
|
|
dfiun2 $p |- U_ x e. A B = U. { y | E. x e. A y = B } $=
|
|
( cvv wcel ciun cv wceq wrex cab cuni dfiun2g a1i mprg ) DFGZACDHBIDJACKB
|
|
LMJACABCDFNQAICGEOP $.
|
|
|
|
$( Alternate definition of indexed intersection when ` B ` is a set.
|
|
Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
dfiin2 $p |- |^|_ x e. A B = |^| { y | E. x e. A y = B } $=
|
|
( cvv wcel ciin cv wceq wrex cab cint dfiin2g a1i mprg ) DFGZACDHBIDJACKB
|
|
LMJACABCDFNQAICGEOP $.
|
|
$}
|
|
|
|
${
|
|
$d z y A $. $d z x A $. $d z B $. $d z C $.
|
|
cbviun.1 $e |- F/_ y B $.
|
|
cbviun.2 $e |- F/_ x C $.
|
|
cbviun.3 $e |- ( x = y -> B = C ) $.
|
|
$( Rule used to change the bound variables in an indexed union, with the
|
|
substitution specified implicitly by the hypothesis. (Contributed by
|
|
NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) $)
|
|
cbviun $p |- U_ x e. A B = U_ y e. A C $=
|
|
( vz cv wcel wrex cab ciun nfcri wceq eleq2d cbvrex abbii df-iun 3eqtr4i
|
|
) IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOAJBJPDEUBHQRSAICDT
|
|
BICETUA $.
|
|
|
|
$( Change bound variables in an indexed intersection. (Contributed by Jeff
|
|
Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
cbviin $p |- |^|_ x e. A B = |^|_ y e. A C $=
|
|
( vz cv wcel wral cab ciin nfcri wceq eleq2d cbvral abbii df-iin 3eqtr4i
|
|
) IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOAJBJPDEUBHQRSAICDT
|
|
BICETUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d y A $. $d y B $. $d x C $.
|
|
cbviunv.1 $e |- ( x = y -> B = C ) $.
|
|
$( Rule used to change the bound variables in an indexed union, with the
|
|
substitution specified implicitly by the hypothesis. (Contributed by
|
|
NM, 15-Sep-2003.) $)
|
|
cbviunv $p |- U_ x e. A B = U_ y e. A C $=
|
|
( nfcv cbviun ) ABCDEBDGAEGFH $.
|
|
|
|
$( Change bound variables in an indexed intersection. (Contributed by Jeff
|
|
Hankins, 26-Aug-2009.) $)
|
|
cbviinv $p |- |^|_ x e. A B = |^|_ y e. A C $=
|
|
( nfcv cbviin ) ABCDEBDGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d y A $. $d y B $.
|
|
$( Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iunss $p |- ( U_ x e. A B C_ C <-> A. x e. A B C_ C ) $=
|
|
( vy ciun wss cv wcel wrex cab wal wral df-iun sseq1i abss ralbii ralcom4
|
|
wi dfss2 r19.23v albii 3bitrri 3bitri ) ABCFZDGEHZCIZABJZEKZDGUHUFDIZSZEL
|
|
ZCDGZABMZUEUIDAEBCNOUHEDPUNUGUJSZELZABMUOABMZELULUMUPABECDTQUOAEBRUQUKEUG
|
|
UJABUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d y A $. $d y B $.
|
|
$( Subset implication for an indexed union. (Contributed by NM,
|
|
3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
ssiun $p |- ( E. x e. A C C_ B -> C C_ U_ x e. A B ) $=
|
|
( vy wss wrex ciun cv wcel ssel reximi r19.37av syl eliun syl6ibr ssrdv
|
|
wi ) DCFZABGZEDABCHZTEIZDJZUBCJZABGZUBUAJTUCUDRZABGUCUERSUFABDCUBKLUCUDAB
|
|
MNAUBBCOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y $.
|
|
$( Identity law for subset of an indexed union. (Contributed by NM,
|
|
12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
ssiun2 $p |- ( x e. A -> B C_ U_ x e. A B ) $=
|
|
( vy cv wcel ciun wrex rspe ex eliun syl6ibr ssrdv ) AEBFZDCABCGZNDEZCFZQ
|
|
ABHZPOFNQRQABIJAPBCKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $.
|
|
ssiun2s.1 $e |- ( x = C -> B = D ) $.
|
|
$( Subset relationship for an indexed union. (Contributed by NM,
|
|
26-Oct-2003.) $)
|
|
ssiun2s $p |- ( C e. A -> D C_ U_ x e. A B ) $=
|
|
( ciun wss nfcv nfiu1 nfss cv wceq sseq1d ssiun2 vtoclgaf ) CABCGZHEQHADB
|
|
ADIAEQAEIABCJKALDMCEQFNABCOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x B $. $d y C $. $d x D $.
|
|
$( A subclass condition on the members of two indexed classes ` C ( x ) `
|
|
and ` D ( y ) ` that implies a subclass relation on their indexed
|
|
unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59.
|
|
Compare ~ uniss2 . (Contributed by NM, 9-Dec-2004.) $)
|
|
iunss2 $p |- ( A. x e. A E. y e. B C C_ D ->
|
|
U_ x e. A C C_ U_ y e. B D ) $=
|
|
( wss wrex wral ciun ssiun ralimi iunss sylibr ) EFGBDHZACIEBDFJZGZACIACE
|
|
JPGOQACBDFEKLACEPMN $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y $. $d x B $.
|
|
$( The indexed union of a class abstraction. (Contributed by NM,
|
|
27-Dec-2004.) $)
|
|
iunab $p |- U_ x e. A { y | ph } = { y | E. x e. A ph } $=
|
|
( cab ciun wrex wceq cv wcel wb nfcv nfab1 nfiun cleqf abid eliun 3bitr4i
|
|
rexbii mpgbir ) BDACEZFZABDGZCEZHCIZUBJZUEUDJZKCCUBUDBCDUACDLACMNUCCMOUEU
|
|
AJZBDGUCUFUGUHABDACPSBUEDUAQUCCPRT $.
|
|
|
|
$( The indexed union of a restricted class abstraction. (Contributed by
|
|
NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) $)
|
|
iunrab $p |- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph } $=
|
|
( cv wcel cab ciun wrex crab iunab wceq df-rab a1i iuneq2i r19.42v eqtr4i
|
|
wa abbii 3eqtr4i ) BDCFEGZASZCHZIUCBDJZCHZBDACEKZIABDJZCEKZUCBCDLBDUGUDUG
|
|
UDMBFDGACENOPUIUBUHSZCHUFUHCENUEUJCUBABDQTRUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $.
|
|
iunxdif2.1 $e |- ( x = y -> C = D ) $.
|
|
$( Indexed union with a class difference as its index. (Contributed by NM,
|
|
10-Dec-2004.) $)
|
|
iunxdif2 $p |- ( A. x e. A E. y e. ( A \ B ) C C_ D ->
|
|
U_ y e. ( A \ B ) D = U_ x e. A C ) $=
|
|
( wss cdif wrex wral ciun wceq iunss2 difss iunss1 ax-mp cbviunv sseqtr4i
|
|
wa jctil eqss sylibr ) EFHBCDIZJACKZBUDFLZACELZHZUGUFHZTUFUGMUEUIUHABCUDE
|
|
FNUFBCFLZUGUDCHUFUJHCDOBUDCFPQABCEFGRSUAUFUGUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d x y $.
|
|
ssiinf.1 $e |- F/_ x C $.
|
|
$( Subset theorem for an indexed intersection. (Contributed by FL,
|
|
15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.) $)
|
|
ssiinf $p |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B ) $=
|
|
( vy cv ciin wcel wral wss cvv vex eliin ax-mp ralbii nfcv ralcomf dfss3
|
|
wb bitri 3bitr4i ) FGZABCHZIZFDJZUCCIZFDJZABJZDUDKDCKZABJUFUGABJZFDJUIUEU
|
|
KFDUCLIUEUKTFMAUCBCLNOPUGFADBEFBQRUAFDUDSUJUHABFDCSPUB $.
|
|
$}
|
|
|
|
${
|
|
$d x C $.
|
|
$( Subset theorem for an indexed intersection. (Contributed by NM,
|
|
15-Oct-2003.) $)
|
|
ssiin $p |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B ) $=
|
|
( nfcv ssiinf ) ABCDADEF $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d y A $. $d y B $.
|
|
$( Subset implication for an indexed intersection. (Contributed by NM,
|
|
15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iinss $p |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C ) $=
|
|
( vy wss wrex ciin cv wcel wral cvv wb vex eliin ax-mp wi reximi r19.36av
|
|
ssel syl syl5bi ssrdv ) CDFZABGZEABCHZDEIZUFJZUGCJZABKZUEUGDJZUGLJUHUJMEN
|
|
AUGBCLOPUEUIUKQZABGUJUKQUDULABCDUGTRUIUKABSUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d x y $.
|
|
$( An indexed intersection is included in any of its members. (Contributed
|
|
by FL, 15-Oct-2012.) $)
|
|
iinss2 $p |- ( x e. A -> |^|_ x e. A B C_ B ) $=
|
|
( vy cv wcel ciin wral wi cvv wb vex eliin ax-mp rsp sylbi com12 ssrdv )
|
|
AEBFZDABCGZCDEZTFZSUACFZUBUCABHZSUCIUAJFUBUDKDLAUABCJMNUCABOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Class union in terms of indexed union. Definition in [Stoll] p. 43.
|
|
(Contributed by NM, 28-Jun-1998.) $)
|
|
uniiun $p |- U. A = U_ x e. A x $=
|
|
( vy cuni cv wcel wrex cab ciun dfuni2 df-iun eqtr4i ) BDCEAEZFABGCHABMIC
|
|
ABJACBMKL $.
|
|
|
|
$( Class intersection in terms of indexed intersection. Definition in
|
|
[Stoll] p. 44. (Contributed by NM, 28-Jun-1998.) $)
|
|
intiin $p |- |^| A = |^|_ x e. A x $=
|
|
( vy cint cv wcel wral cab ciin dfint2 df-iin eqtr4i ) BDCEAEZFABGCHABMIC
|
|
ABJACBMKL $.
|
|
|
|
$( An indexed union of singletons recovers the index set. (Contributed by
|
|
NM, 6-Sep-2005.) $)
|
|
iunid $p |- U_ x e. A { x } = A $=
|
|
( vy cv csn ciun wceq cab wcel df-sn equcom abbii eqtri a1i iuneq2i iunab
|
|
wrex risset abid2 3eqtr2i ) ABADZEZFABUACDZGZCHZFZBABUBUEUBUEGUABIUBUCUAG
|
|
ZCHUECUAJUGUDCCAKLMNOUFUDABQZCHUCBIZCHBUDACBPUIUHCAUCBRLCBSTM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( An indexed union of the empty set is empty. (Contributed by NM,
|
|
26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iun0 $p |- U_ x e. A (/) = (/) $=
|
|
( vy c0 ciun cv wcel wrex wn noel a1i nrex eliun mtbir 2false eqriv ) CAB
|
|
DEZDCFZQGZRDGZSTABHTABTIAFBGRJZKLARBDMNUAOP $.
|
|
|
|
$( An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
0iun $p |- U_ x e. (/) A = (/) $=
|
|
( vy c0 ciun cv wcel wrex rex0 eliun mtbir noel 2false eqriv ) CADBEZDCFZ
|
|
OGZPDGQPBGZADHRAIAPDBJKPLMN $.
|
|
|
|
$( An empty indexed intersection is the universal class. (Contributed by
|
|
NM, 20-Oct-2005.) $)
|
|
0iin $p |- |^|_ x e. (/) A = _V $=
|
|
( vy c0 ciin cv wcel wral cab cvv df-iin vex ral0 2th abbi2i eqtr4i ) ADB
|
|
ECFZBGZADHZCIJACDBKSCJQJGSCLRAMNOP $.
|
|
|
|
$( Indexed intersection with a universal index class. When ` A ` doesn't
|
|
depend on ` x ` , this evaluates to ` A ` by ~ 19.3 and ~ abid2 . When
|
|
` A = x ` , this evaluates to ` (/) ` by ~ intiin and intv in set.mm.
|
|
(Contributed by NM, 11-Sep-2008.) $)
|
|
viin $p |- |^|_ x e. _V A = { y | A. x y e. A } $=
|
|
( cvv ciin cv wcel wral cab wal df-iin ralv abbii eqtri ) ADCEBFCGZADHZBI
|
|
OAJZBIABDCKPQBOALMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( There is a non-empty class in an indexed collection ` B ( x ) ` iff the
|
|
indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.)
|
|
(Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
|
|
iunn0 $p |- ( E. x e. A B =/= (/) <-> U_ x e. A B =/= (/) ) $=
|
|
( vy cv wcel wex wrex c0 wne rexcom4 eliun exbii bitr4i n0 rexbii 3bitr4i
|
|
ciun ) DEZCFZDGZABHZSABCRZFZDGZCIJZABHUCIJUBTABHZDGUETADBKUDUGDASBCLMNUFU
|
|
AABDCOPDUCOQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y $.
|
|
$( Indexed intersection of a class builder. (Contributed by NM,
|
|
6-Dec-2011.) $)
|
|
iinab $p |- |^|_ x e. A { y | ph } = { y | A. x e. A ph } $=
|
|
( cab ciin wral wceq cv wcel nfcv nfab1 nfiin cleqf abid ralbii cvv eliin
|
|
wb vex ax-mp 3bitr4i mpgbir ) BDACEZFZABDGZCEZHCIZUEJZUHUGJZSCCUEUGBCDUDC
|
|
DKACLMUFCLNUHUDJZBDGZUFUIUJUKABDACOPUHQJUIULSCTBUHDUDQRUAUFCOUBUC $.
|
|
|
|
$d x A $. $d x B $.
|
|
$( Indexed intersection of a restricted class builder. (Contributed by NM,
|
|
6-Dec-2011.) $)
|
|
iinrab $p |- ( A =/= (/)
|
|
-> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } ) $=
|
|
( c0 wne cv wcel wa wral cab crab ciin r19.28zv abbidv df-rab a1i iineq2i
|
|
wceq iinab eqtri 3eqtr4g ) DFGZCHEIZAJZBDKZCLZUEABDKZJZCLBDACEMZNZUICEMUD
|
|
UGUJCUEABDOPULBDUFCLZNUHBDUKUMUKUMTBHDIACEQRSUFBCDUAUBUICEQUC $.
|
|
|
|
$d y B $.
|
|
$( Indexed intersection of a restricted class builder. (Contributed by NM,
|
|
6-Dec-2011.) $)
|
|
iinrab2 $p |- ( |^|_ x e. A { y e. B | ph } i^i B )
|
|
= { y e. B | A. x e. A ph } $=
|
|
( crab ciin cin wral wceq c0 cvv iineq1 0iin syl6eq incom inv1 eqtri rzal
|
|
ineq1d rabid2 ralcom bitr2i sylib wne iinrab wss ssrab2 dfss mpbi syl6eqr
|
|
eqtrd pm2.61ine ) BDACEFZGZEHZABDIZCEFZJDKDKJZUPEURUSUPLEHZEUSUOLEUSUOBKU
|
|
NGLBDKUNMBUNNOTUTELHELEPEQROUSACEIZBDIZEURJZVABDSVCUQCEIVBUQCEUAACBEDUBUC
|
|
UDULDKUEZUPUREHZURVDUOUREABCDEUFTUREUGURVEJUQCEUHUREUIUJUKUM $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y B $. $d y C $.
|
|
$( Indexed union of intersection. Generalization of half of theorem
|
|
"Distributive laws" in [Enderton] p. 30. Use ~ uniiun to recover
|
|
Enderton's theorem. (Contributed by NM, 26-Mar-2004.) $)
|
|
iunin2 $p |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C ) $=
|
|
( vy cin ciun cv wcel wrex r19.42v elin rexbii eliun anbi2i 3bitr4i eqriv
|
|
wa ) EABCDFZGZCABDGZFZEHZSIZABJZUCCIZUCUAIZRZUCTIUCUBIUFUCDIZRZABJUFUIABJ
|
|
ZRUEUHUFUIABKUDUJABUCCDLMUGUKUFAUCBDNOPAUCBSNUCCUALPQ $.
|
|
|
|
$( Indexed union of intersection. Generalization of half of theorem
|
|
"Distributive laws" in [Enderton] p. 30. Use ~ uniiun to recover
|
|
Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.) $)
|
|
iunin1 $p |- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B ) $=
|
|
( cin ciun iunin2 wceq cv wcel incom a1i iuneq2i 3eqtr4i ) ABCDEZFCABDFZE
|
|
ABDCEZFPCEABCDGABQOQOHAIBJDCKLMPCKN $.
|
|
|
|
$( Indexed intersection of union. Generalization of half of theorem
|
|
"Distributive laws" in [Enderton] p. 30. Use ~ intiin to recover
|
|
Enderton's theorem. (Contributed by NM, 19-Aug-2004.) $)
|
|
iinun2 $p |- |^|_ x e. A ( B u. C ) = ( B u. |^|_ x e. A C ) $=
|
|
( vy cun ciin cv wcel wral wo r19.32v elun ralbii cvv eliin ax-mp 3bitr4i
|
|
wb vex orbi2i eqriv ) EABCDFZGZCABDGZFZEHZUCIZABJZUGCIZUGUEIZKZUGUDIZUGUF
|
|
IUJUGDIZKZABJUJUNABJZKUIULUJUNABLUHUOABUGCDMNUKUPUJUGOIZUKUPSETZAUGBDOPQU
|
|
ARUQUMUISURAUGBUCOPQUGCUEMRUB $.
|
|
|
|
$( Indexed union of class difference. Generalization of half of theorem
|
|
"De Morgan's laws" in [Enderton] p. 31. Use ~ intiin to recover
|
|
Enderton's theorem. (Contributed by NM, 19-Aug-2004.) $)
|
|
iundif2 $p |- U_ x e. A ( B \ C ) = ( B \ |^|_ x e. A C ) $=
|
|
( vy cdif ciun ciin cv wcel wrex wn wa eldif rexbii r19.42v rexnal cvv wb
|
|
wral vex eliin ax-mp xchbinxr anbi2i 3bitri eliun 3bitr4i eqriv ) EABCDFZ
|
|
GZCABDHZFZEIZUJJZABKZUNCJZUNULJZLZMZUNUKJUNUMJUPUQUNDJZLZMZABKUQVBABKZMUT
|
|
UOVCABUNCDNOUQVBABPVDUSUQVDVAABTZURVAABQUNRJURVESEUAAUNBDRUBUCUDUEUFAUNBU
|
|
JUGUNCULNUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d y C $. $d x D $. $d x y $.
|
|
$( Rearrange indexed unions over intersection. (Contributed by NM,
|
|
18-Dec-2008.) $)
|
|
2iunin $p |- U_ x e. A U_ y e. B ( C i^i D )
|
|
= ( U_ x e. A C i^i U_ y e. B D ) $=
|
|
( cin ciun wceq cv wcel iunin2 a1i iuneq2i iunin1 eqtri ) ACBDEFGHZHACEBD
|
|
FHZGZHACEHRGACQSQSIAJCKBDEFLMNACREOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $.
|
|
$( Indexed intersection of class difference. Generalization of half of
|
|
theorem "De Morgan's laws" in [Enderton] p. 31. Use ~ uniiun to recover
|
|
Enderton's theorem. (Contributed by NM, 5-Oct-2006.) $)
|
|
iindif2 $p |- ( A =/= (/) ->
|
|
|^|_ x e. A ( B \ C ) = ( B \ U_ x e. A C ) ) $=
|
|
( vy c0 wne cdif ciin ciun cv wcel wn wa r19.28zv eldif bicomi ralbii cvv
|
|
wral wrex ralnex eliun xchbinxr anbi2i 3bitr3g wb vex eliin ax-mp 3bitr4g
|
|
eqrdv ) BFGZEABCDHZIZCABDJZHZUMEKZUNLZABTZURCLZURUPLZMZNZURUOLZURUQLUMVAU
|
|
RDLZMZNZABTVAVGABTZNUTVDVAVGABOVHUSABUSVHURCDPQRVIVCVAVIVFABUAVBVFABUBAUR
|
|
BDUCUDUEUFURSLVEUTUGEUHAURBUNSUIUJURCUPPUKUL $.
|
|
|
|
$( Indexed intersection of intersection. Generalization of half of theorem
|
|
"Distributive laws" in [Enderton] p. 30. Use ~ intiin to recover
|
|
Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.) $)
|
|
iinin2 $p |- ( A =/= (/) ->
|
|
|^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) ) $=
|
|
( vy c0 wne cin ciin cv wcel wral wa r19.28zv elin wb eliin ax-mp 3bitr4g
|
|
cvv ralbii vex anbi2i eqrdv ) BFGZEABCDHZIZCABDIZHZUEEJZUFKZABLZUJCKZUJUH
|
|
KZMZUJUGKZUJUIKUEUMUJDKZMZABLUMUQABLZMULUOUMUQABNUKURABUJCDOUAUNUSUMUJTKZ
|
|
UNUSPEUBZAUJBDTQRUCSUTUPULPVAAUJBUFTQRUJCUHOSUD $.
|
|
|
|
$( Indexed intersection of intersection. Generalization of half of theorem
|
|
"Distributive laws" in [Enderton] p. 30. Use ~ intiin to recover
|
|
Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.) $)
|
|
iinin1 $p |- ( A =/= (/) ->
|
|
|^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) ) $=
|
|
( c0 wne cin ciin iinin2 wceq cv wcel incom a1i iineq2i 3eqtr4g ) BEFABCD
|
|
GZHCABDHZGABDCGZHRCGABCDIABSQSQJAKBLDCMNORCMP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d X x y $. $d B x $.
|
|
$( Elementhood in a relative intersection. (Contributed by Mario Carneiro,
|
|
30-Dec-2016.) $)
|
|
elriin $p |- ( B e. ( A i^i |^|_ x e. X S ) <->
|
|
( B e. A /\ A. x e. X B e. S ) ) $=
|
|
( ciin cin wcel wa wral elin eliin pm5.32i bitri ) CBAEDFZGHCBHZCOHZIPCDH
|
|
AEJZICBOKPQRACEDBLMN $.
|
|
|
|
$( Relative intersection of an empty family. (Contributed by Stefan
|
|
O'Rear, 3-Apr-2015.) $)
|
|
riin0 $p |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A ) $=
|
|
( c0 wceq ciin cin iineq1 ineq2d cvv 0iin ineq2i inv1 eqtri syl6eq ) DEFZ
|
|
BADCGZHBAECGZHZBQRSBADECIJTBKHBSKBACLMBNOP $.
|
|
|
|
$( Relative intersection of a nonempty family. (Contributed by Stefan
|
|
O'Rear, 3-Apr-2015.) $)
|
|
riinn0 $p |- ( ( A. x e. X S C_ A /\ X =/= (/) ) ->
|
|
( A i^i |^|_ x e. X S ) = |^|_ x e. X S ) $=
|
|
( wss wral c0 wne wa ciin incom wceq wrex r19.2z ancoms iinss df-ss sylib
|
|
cin syl syl5eq ) CBEZADFZDGHZIZBADCJZSUFBSZUFBUFKUEUFBEZUGUFLUEUBADMZUHUD
|
|
UCUIUBADNOADCBPTUFBQRUA $.
|
|
|
|
$( Relative intersection of a relative abstraction. (Contributed by Stefan
|
|
O'Rear, 3-Apr-2015.) $)
|
|
riinrab $p |- ( A i^i |^|_ x e. X { y e. A | ph } ) =
|
|
{ y e. A | A. x e. X ph } $=
|
|
( crab ciin cin wral wceq c0 riin0 rzal ralrimivw rabid2 sylibr eqtrd wne
|
|
wss ssrab2 rgenw riinn0 mpan iinrab pm2.61ine ) DBEACDFZGZHZABEIZCDFZJEKE
|
|
KJZUHDUJBDUFELUKUICDIDUJJUKUICDABEMNUICDOPQEKRZUHUGUJUFDSZBEIULUHUGJUMBEA
|
|
CDTUABDUFEUBUCABCEDUDQUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d y V $.
|
|
iinxsng.1 $e |- ( x = A -> B = C ) $.
|
|
$( A singleton index picks out an instance of an indexed intersection's
|
|
argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario
|
|
Carneiro, 17-Nov-2016.) $)
|
|
iinxsng $p |- ( A e. V -> |^|_ x e. { A } B = C ) $=
|
|
( vy wcel csn ciin cv wral cab df-iin wceq eleq2d ralsng abbi1dv syl5eq )
|
|
BEHZABIZCJGKZCHZAUALZGMDAGUACNTUDGDUCUBDHABEAKBOCDUBFPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x y D $. $d x y E $. $d y V $.
|
|
$d y W $.
|
|
iinxprg.1 $e |- ( x = A -> C = D ) $.
|
|
iinxprg.2 $e |- ( x = B -> C = E ) $.
|
|
$( Indexed intersection with an unordered pair index. (Contributed by NM,
|
|
25-Jan-2012.) $)
|
|
iinxprg $p |- ( ( A e. V /\ B e. W )
|
|
-> |^|_ x e. { A , B } C = ( D i^i E ) ) $=
|
|
( vy wcel wa cpr ciin cin cv wceq eleq2d cvv wral ralprg eliin ax-mp elin
|
|
wb vex 3bitr4g eqrdv ) BGLCHLMZKABCNZDOZEFPZUJKQZDLZAUKUAZUNELZUNFLZMUNUL
|
|
LZUNUMLUOUQURABCGHAQZBRDEUNISUTCRDFUNJSUBUNTLUSUPUFKUGAUNUKDTUCUDUNEFUEUH
|
|
UI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d y V $.
|
|
iunxsng.1 $e |- ( x = A -> B = C ) $.
|
|
$( A singleton index picks out an instance of an indexed union's argument.
|
|
(Contributed by Mario Carneiro, 25-Jun-2016.) $)
|
|
iunxsng $p |- ( A e. V -> U_ x e. { A } B = C ) $=
|
|
( vy wcel csn ciun cv wrex eliun wceq eleq2d rexsng syl5bb eqrdv ) BEHZGA
|
|
BIZCJZDGKZUAHUBCHZATLSUBDHZAUBTCMUCUDABEAKBNCDUBFOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $.
|
|
iunxsn.1 $e |- A e. _V $.
|
|
iunxsn.2 $e |- ( x = A -> B = C ) $.
|
|
$( A singleton index picks out an instance of an indexed union's argument.
|
|
(Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro,
|
|
25-Jun-2016.) $)
|
|
iunxsn $p |- U_ x e. { A } B = C $=
|
|
( cvv wcel csn ciun wceq iunxsng ax-mp ) BGHABICJDKEABCDGFLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $. $d y C $.
|
|
$( Separate a union in an indexed union. (Contributed by NM,
|
|
27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
|
|
iunun $p |- U_ x e. A ( B u. C ) = ( U_ x e. A B u. U_ x e. A C ) $=
|
|
( vy cun ciun cv wcel wrex r19.43 elun rexbii eliun orbi12i 3bitr4i eqriv
|
|
wo ) EABCDFZGZABCGZABDGZFZEHZSIZABJZUDUAIZUDUBIZRZUDTIUDUCIUDCIZUDDIZRZAB
|
|
JUJABJZUKABJZRUFUIUJUKABKUEULABUDCDLMUGUMUHUNAUDBCNAUDBDNOPAUDBSNUDUAUBLP
|
|
Q $.
|
|
|
|
$( Separate a union in the index of an indexed union. (Contributed by NM,
|
|
26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
|
|
iunxun $p |- U_ x e. ( A u. B ) C = ( U_ x e. A C u. U_ x e. B C ) $=
|
|
( vy cun ciun cv wcel wrex rexun eliun orbi12i bitr4i elun 3bitr4i eqriv
|
|
wo ) EABCFZDGZABDGZACDGZFZEHZDIZASJZUDUAIZUDUBIZRZUDTIUDUCIUFUEABJZUEACJZ
|
|
RUIUEABCKUGUJUHUKAUDBDLAUDCDLMNAUDSDLUDUAUBOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d x z A $. $d z B $. $d y z C $.
|
|
$( Separate an indexed union in the index of an indexed union.
|
|
(Contributed by Mario Carneiro, 5-Dec-2016.) $)
|
|
iunxiun $p |- U_ x e. U_ y e. A B C = U_ y e. A U_ x e. B C $=
|
|
( vz ciun cv wcel wa wex eliun anbi1i r19.41v bitr4i exbii rexcom4 df-rex
|
|
wrex 3bitr4i bitri rexbii eqriv ) FABCDGZEGZBCADEGZGZFHZEIZAUDSZUHUFIZBCS
|
|
ZUHUEIUHUGIAHZUDIZUIJZAKZUMDIZUIJZAKZBCSZUJULUPURBCSZAKUTUOVAAUOUQBCSZUIJ
|
|
VAUNVBUIBUMCDLMUQUIBCNOPURBACQOUIAUDRUKUSBCUKUIADSUSAUHDELUIADRUAUBTAUHUD
|
|
ELBUHCUFLTUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( A relationship involving union and indexed intersection. Exercise 23 of
|
|
[Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened
|
|
by Mario Carneiro, 17-Nov-2016.) $)
|
|
iinuni $p |- ( A u. |^| B ) = |^|_ x e. B ( A u. x ) $=
|
|
( vy cint cun cv ciin wcel wral wel r19.32v elun ralbii vex elint2 orbi2i
|
|
wo 3bitr4ri cvv wb eliin ax-mp 3bitr4i eqriv ) DBCEZFZACBAGZFZHZDGZBIZUKU
|
|
FIZRZUKUIIZACJZUKUGIUKUJIZULDAKZRZACJULURACJZRUPUNULURACLUOUSACUKBUHMNUMU
|
|
TULAUKCDOZPQSUKBUFMUKTIUQUPUAVAAUKCUITUBUCUDUE $.
|
|
|
|
$( A relationship involving union and indexed union. Exercise 25 of
|
|
[Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened
|
|
by Mario Carneiro, 17-Nov-2016.) $)
|
|
iununi $p |- ( ( B = (/) -> A = (/) ) <->
|
|
( A u. U. B ) = U_ x e. B ( A u. x ) ) $=
|
|
( c0 wceq wi cuni cun cv ciun wn wne iunconst sylbir iun0 iuneq2d 3eqtr4a
|
|
df-ne id syl6eq ja eqcomd uneq1d uniiun uneq2i iunun 3eqtr4g unieq uneq2d
|
|
uni0 un0 iuneq1 0iun eqeq12d biimpcd impbii ) CDEZBDEZFZBCGZHZACBAIZHZJZE
|
|
ZUSBACVBJZHACBJZVFHVAVDUSBVGVFUSVGBUQURVGBEZUQKCDLVHCDRACBMNURACDJDVGBACO
|
|
URACBDURSZPVIQUAUBUCUTVFBACUDUEACBVBUFUGUQVEURUQVABVDDUQVABDHBUQUTDBUQUTD
|
|
GDCDUHUJTUIBUKTUQVDADVCJDACDVCULAVCUMTUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Subclass relationship for power class and union. (Contributed by NM,
|
|
18-Jul-2006.) $)
|
|
sspwuni $p |- ( A C_ ~P B <-> U. A C_ B ) $=
|
|
( vx cv cpw wcel wral wss cuni vex elpw ralbii dfss3 unissb 3bitr4i ) CDZ
|
|
BEZFZCAGPBHZCAGAQHAIBHRSCAPBCJKLCAQMCABNO $.
|
|
|
|
$( Two ways to express a collection of subclasses. (Contributed by NM,
|
|
19-Jul-2006.) $)
|
|
pwssb $p |- ( A C_ ~P B <-> A. x e. A x C_ B ) $=
|
|
( cpw wss cuni cv wral sspwuni unissb bitri ) BCDEBFCEAGCEABHBCIABCJK $.
|
|
$}
|
|
|
|
$( Relationship for power class and union. (Contributed by NM,
|
|
18-Jul-2006.) $)
|
|
elpwuni $p |- ( B e. A -> ( A C_ ~P B <-> U. A = B ) ) $=
|
|
( cpw wss cuni wcel wceq sspwuni unissel expcom eqimss impbid1 syl5bb ) ABC
|
|
DAEZBDZBAFZNBGZABHPOQOPQABIJNBKLM $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The power class of an intersection in terms of indexed intersection.
|
|
Exercise 24(a) of [Enderton] p. 33. (Contributed by NM,
|
|
29-Nov-2003.) $)
|
|
iinpw $p |- ~P |^| A = |^|_ x e. A ~P x $=
|
|
( vy cint cpw cv ciin wss wcel wral ssint vex elpw ralbii bitr4i wb eliin
|
|
cvv ax-mp 3bitr4i eqriv ) CBDZEZABAFZEZGZCFZUBHZUGUEIZABJZUGUCIUGUFIZUHUG
|
|
UDHZABJUJAUGBKUIULABUGUDCLZMNOUGUBUMMUGRIUKUJPUMAUGBUERQSTUA $.
|
|
|
|
$( Inclusion of an indexed union of a power class in the power class of the
|
|
union of its index. Part of Exercise 24(b) of [Enderton] p. 33.
|
|
(Contributed by NM, 25-Nov-2003.) $)
|
|
iunpwss $p |- U_ x e. A ~P x C_ ~P U. A $=
|
|
( vy cpw ciun cuni wss wrex wcel ssiun eliun vex elpw rexbii bitri uniiun
|
|
cv sseq2i 3imtr4i ssriv ) CABAQZDZEZBFZDZCQZUAGZABHZUFABUAEZGZUFUCIZUFUEI
|
|
ZABUAUFJUKUFUBIZABHUHAUFBUBKUMUGABUFUACLZMNOULUFUDGUJUFUDUNMUDUIUFABPROST
|
|
$.
|
|
$}
|
|
|
|
$( Relative intersection of a nonempty set. (Contributed by Stefan O'Rear,
|
|
3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.) $)
|
|
rintn0 $p |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X ) $=
|
|
( cpw wss c0 wne wa cint incom wceq cuni intssuni2 ssid sspwuni mpbi syl6ss
|
|
cin df-ss sylib syl5eq ) BACZDBEFGZABHZQUCAQZUCAUCIUBUCADUDUCJUBUCUAKZABUAL
|
|
UAUADUEADUAMUAANOPUCARST $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The Kuratowski ordered pair
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new symbols needed. $)
|
|
$c << $. $( Double bracket $)
|
|
$c >> $. $( Double bracket $)
|
|
|
|
$( Extend class notation to include Kuratowski ordered pair. $)
|
|
copk $a class << A , B >> $.
|
|
|
|
$( Define the Kuratowski ordered pair. This ordered pair definition is
|
|
standard for ZFC set theory, but we do not use it beyond establishing
|
|
~ df-op , since it is not type-level. We state this definition since it
|
|
is a simple definition that can be used by the set construction axioms
|
|
that follow this section. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-opk $a |- << A , B >> = { { A } , { A , B } } $.
|
|
|
|
$( Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) $)
|
|
opkeq1 $p |- ( A = B -> << A , C >> = << B , C >> ) $=
|
|
( wceq csn cpr copk sneq preq1 preq12d df-opk 3eqtr4g ) ABDZAEZACFZFBEZBCFZ
|
|
FACGBCGMNPOQABHABCIJACKBCKL $.
|
|
|
|
$( Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) $)
|
|
opkeq2 $p |- ( A = B -> << C , A >> = << C , B >> ) $=
|
|
( wceq csn cpr copk preq2 preq2d df-opk 3eqtr4g ) ABDZCEZCAFZFMCBFZFCAGCBGL
|
|
NOMABCHICAJCBJK $.
|
|
|
|
$( Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.) $)
|
|
opkeq12 $p |- ( ( A = C /\ B = D ) -> << A , B >> = << C , D >> ) $=
|
|
( wceq copk opkeq1 opkeq2 sylan9eq ) ACEBDEABFCBFCDFACBGBDCHI $.
|
|
|
|
${
|
|
opkeq1i.1 $e |- A = B $.
|
|
$( Equality inference for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
opkeq1i $p |- << A , C >> = << B , C >> $=
|
|
( wceq copk opkeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
opkeq2i $p |- << C , A >> = << C , B >> $=
|
|
( wceq copk opkeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
opkeq12i.2 $e |- C = D $.
|
|
$( Equality inference for ordered pairs. (The proof was shortened by
|
|
Eric Schmidt, 4-Apr-2007.) (Contributed by NM, 16-Dec-2006.) $)
|
|
opkeq12i $p |- << A , C >> = << B , D >> $=
|
|
( wceq copk opkeq12 mp2an ) ABGCDGACHBDHGEFACBDIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
opkeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
opkeq1d $p |- ( ph -> << A , C >> = << B , C >> ) $=
|
|
( wceq copk opkeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for ordered pairs. (Contributed by NM,
|
|
16-Dec-2006.) $)
|
|
opkeq2d $p |- ( ph -> << C , A >> = << C , B >> ) $=
|
|
( wceq copk opkeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
${
|
|
opkeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for ordered pairs. (The proof was shortened by
|
|
Andrew Salmon, 29-Jun-2011.) (Contributed by NM, 16-Dec-2006.) $)
|
|
opkeq12d $p |- ( ph -> << A , C >> = << B , D >> ) $=
|
|
( wceq copk opkeq12 syl2anc ) ABCHDEHBDICEIHFGBDCEJK $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
nfopk.1 $e |- F/_ x A $.
|
|
nfopk.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for ordered pairs. (Contributed by
|
|
NM, 14-Nov-1995.) $)
|
|
nfopk $p |- F/_ x << A , B >> $=
|
|
( copk csn cpr df-opk nfsn nfpr nfcxfr ) ABCFBGZBCHZHBCIAMNABDJABCDEKKL
|
|
$.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
More Boolean set operations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Complement in terms of difference. (Contributed by SF, 2-Jan-2018.) $)
|
|
compldif $p |- ~ A = ( _V \ A ) $=
|
|
( cvv cdif ccompl cin df-dif incom inv1 3eqtrri ) BACBADZEJBEJBAFBJGJHI $.
|
|
|
|
$( The complement of the universe is the empty set. (Contributed by SF,
|
|
2-Jan-2018.) $)
|
|
complV $p |- ~ _V = (/) $=
|
|
( cvv ccompl cdif c0 compldif df-nul eqtr4i ) ABAACDAEFG $.
|
|
|
|
$( The complement of the empty set is the universe. (Contributed by SF,
|
|
2-Jan-2018.) $)
|
|
compl0 $p |- ~ (/) = _V $=
|
|
( cvv ccompl c0 complV compleqi dblcompl eqtr3i ) ABZBCBAHCDEAFG $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Anti-intersection with complement. (Contributed by SF, 2-Jan-2018.) $)
|
|
nincompl $p |- ( A -i^i ~ A ) = _V $=
|
|
( vx ccompl cnin cvv wceq cv wcel eqv wn wa pm3.24 wnan vex elnin elcompl
|
|
nanbi2i df-nan 3bitri mpbir mpgbir ) AACZDZEFBGZUCHZBBUCIUEUDAHZUFJZKJZUF
|
|
LUEUFUDUBHZMUFUGMUHUDAUBBNZOUIUGUFUDAUJPQUFUGRSTUA $.
|
|
$}
|
|
|
|
$( Intersection with complement. (Contributed by SF, 2-Jan-2018.) $)
|
|
incompl $p |- ( A i^i ~ A ) = (/) $=
|
|
( ccompl cin cnin cvv c0 df-in nincompl compleqi complV 3eqtri ) AABZCALDZB
|
|
EBFALGMEAHIJK $.
|
|
|
|
$( Union with complement. (Contributed by SF, 2-Jan-2018.) $)
|
|
uncompl $p |- ( A u. ~ A ) = _V $=
|
|
( ccompl cun cnin cvv df-un nincompl eqtri ) AABZCIIBDEAIFIGH $.
|
|
|
|
$( The intersection of an intersection and a difference is empty.
|
|
(Contributed by set.mm contributors, 10-Mar-2015.) $)
|
|
inindif $p |- ( ( A i^i B ) i^i ( A \ B ) ) = (/) $=
|
|
( cin cdif ccompl c0 df-dif ineq2i inindi incompl in0 eqtri 3eqtr2i ) ABCZA
|
|
BDZCNABEZCZCABPCZCZFOQNABGHABPISAFCFRFABJHAKLM $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( Condition for subset when ` A ` is already known to be a subset.
|
|
(Contributed by SF, 13-Jan-2015.) $)
|
|
ssofss $p |- ( A C_ C -> ( A C_ B <-> A. x e. C ( x e. A -> x e. B ) ) ) $=
|
|
( wss cv wcel wi wral ccompl wa vex elcompl ssel con3d syl5bi imp pm2.21d
|
|
wn cvv ralrimiva biantrud cun wal ralv raleqi dfss2 3bitr4ri ralunb bitri
|
|
uncompl syl6rbbr ) BDEZAFZBGZUNCGZHZADIZURUQADJZIZKZBCEZUMUTURUMUQAUSUMUN
|
|
USGZKUOUPUMVCUOSZVCUNDGZSUMVDUNDALMUMUOVEBDUNNOPQRUAUBVBUQADUSUCZIZVAUQAT
|
|
IUQAUDVGVBUQAUEUQAVFTDUKUFABCUGUHUQADUSUIUJUL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
$( When ` A ` and ` B ` are subsets of ` C ` , equality depends only on the
|
|
elements of ` C ` . (Contributed by SF, 13-Jan-2015.) $)
|
|
ssofeq $p |- ( ( A C_ C /\ B C_ C ) ->
|
|
( A = B <-> A. x e. C ( x e. A <-> x e. B ) ) ) $=
|
|
( wss wa cv wcel wi wral wceq wb ssofss bi2anan9 eqss ralbiim 3bitr4g ) B
|
|
DEZCDEZFBCEZCBEZFAGZBHZUBCHZIADJZUDUCIADJZFBCKUCUDLADJRTUESUAUFABCDMACBDM
|
|
NBCOUCUDADPQ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
NF Set Theory - add the Set Construction Axioms
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Introduce the set construction axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( State the axiom of anti-intersection. Axiom P1 of [Hailperin] p. 6.
|
|
This axiom sets up boolean operations on sets.
|
|
|
|
Note on this and the following axioms: this axiom, ~ ax-xp , ~ ax-cnv ,
|
|
~ ax-1c , ~ ax-sset , ~ ax-si , ~ ax-ins2 , ~ ax-ins3 , and
|
|
~ ax-typlower are from Hailperin and are designed to implement the
|
|
Stratification Axiom of Quine.
|
|
|
|
A well-formed formula using only propositional symbols, predicate
|
|
symbols, and ` e. ` is "stratified" iff you can make a (metalogical)
|
|
mapping from the variables to the natural numbers such that any formulas
|
|
of the form ` x = y ` have the same number, and any formulas of the form
|
|
` x e. y ` have ` x ` as one less than ` y ` . Quine's stratification
|
|
axiom states that there is a set corresponding to any stratified
|
|
formula.
|
|
|
|
Since we cannot state stratification from within the logic, we use
|
|
Hailperin's axioms and prove existence of stratified sets using
|
|
Hailperin's algorithm. (Contributed by SF, 12-Jan-2015.) $)
|
|
ax-nin $a |- E. z A. w ( w e. z <-> ( w e. x -/\ w e. y ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w t $.
|
|
$( State the axiom of cross product. This axiom guarantees the existence
|
|
of the (Kuratowski) cross product of ` _V ` with ` x ` . Axiom P5 of
|
|
[Hailperin] p. 10. (Contributed by SF, 12-Jan-2015.) $)
|
|
ax-xp $a |- E. y A. z ( z e. y <->
|
|
E. w E. t ( z = << w , t >> /\ t e. x ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( State the axiom of converse. This axiom guarantees the existence of the
|
|
Kuratowski converse of ` x ` . Axiom P7 of [Hailperin] p. 10.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
ax-cnv $a |- E. y A. z A. w ( << z , w >> e. y <-> << w , z >> e. x ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( State the axiom of cardinal one. This axiom guarantees the existence of
|
|
the set of all singletons, which will become cardinal one later in our
|
|
development. Axiom P8 of [Hailperin] p. 10. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
ax-1c $a |- E. x A. y ( y e. x <-> E. z A. w ( w e. y <-> w = z ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( State the axiom of the subset relationship. This axiom guarantees the
|
|
existence of the Kuratowski relationship representing subset. Slight
|
|
generalization of axiom P9 of [Hailperin] p. 10. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
ax-sset $a |- E. x A. y A. z ( << y , z >> e. x <->
|
|
A. w ( w e. y -> w e. z ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( State the axiom of the singleton image. This axiom guarantees that
|
|
guarantees the existence of a set that raises the "type" of another set
|
|
when considered as a relationship. Axiom P2 of [Hailperin] p. 10.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
ax-si $a |- E. y A. z A. w ( << { z } , { w } >> e. y <->
|
|
<< z , w >> e. x ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w t $.
|
|
$( State the insertion two axiom. This axiom sets up a set that inserts an
|
|
extra variable at the second place of the relationship described by
|
|
` x ` . Axiom P3 of [Hailperin] p. 10. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
ax-ins2 $a |- E. y A. z A. w A. t (
|
|
<< { { z } } , << w , t >> >> e. y <-> << z , t >> e. x ) $.
|
|
|
|
$( State the insertion three axiom. This axiom sets up a set that inserts
|
|
an extra variable at the third place of the relationship described by
|
|
` x ` . Axiom P4 of [Hailperin] p. 10. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
ax-ins3 $a |- E. y A. z A. w A. t (
|
|
<< { { z } } , << w , t >> >> e. y <-> << z , w >> e. x ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( The type lowering axiom. This axiom eventually sets up both the
|
|
existence of the sum set and the existence of the range of a
|
|
relationship. Axiom P6 of [Hailperin] p. 10. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
ax-typlower $a |- E. y A. z ( z e. y <-> A. w << w , { z } >> e. x ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The singleton axiom. This axiom sets up the existence of a singleton
|
|
set. This appears to have been an oversight on Hailperin's part, as it
|
|
is needed to prove the properties of Kuratowski ordered pairs.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
ax-sn $a |- E. y A. z ( z e. y <-> z = x ) $.
|
|
$}
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Primitive forms for some axioms
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d a c $. $d B c $.
|
|
$( Lemma for the primitive axioms. Primitive form of equality to a
|
|
singleton. (Contributed by SF, 25-Mar-2015.) $)
|
|
axprimlem1 $p |- ( a = { B } <-> A. c ( c e. a <-> c = B ) ) $=
|
|
( cv csn wceq wel wcel wb wal dfcleq elsn bibi2i albii bitri ) BDZAEZFCBG
|
|
ZCDZQHZIZCJRSAFZIZCJCPQKUAUCCTUBRCALMNO $.
|
|
$}
|
|
|
|
${
|
|
$d a d $. $d B d $. $d B e $. $d B f $. $d C d $. $d C f $. $d d e $.
|
|
$d d f $.
|
|
$( Lemma for the primitive axioms. Primitive form of equality to a
|
|
Kuratowski ordered pair. (Contributed by SF, 25-Mar-2015.) $)
|
|
axprimlem2 $p |- ( a = << B , C >> <-> A. d ( d e. a <-> ( A. e ( e e. d
|
|
<-> e = B ) \/ A. f ( f e. d <-> ( f = B \/ f = C ) ) ) ) ) $=
|
|
( cv wceq cpr wel wb wal wo wcel dfcleq vex elpr bibi2i albii bitri copk
|
|
csn df-opk eqeq2i axprimlem1 orbi12i ) EGZABUAZHUGAUBZABIZIZHZFEJZCFJCGAH
|
|
KCLZDFJZDGZAHUPBHMZKZDLZMZKZFLZUHUKUGABUCUDULUMFGZUKNZKZFLVBFUGUKOVEVAFVD
|
|
UTUMVDVCUIHZVCUJHZMUTVCUIUJFPQVFUNVGUSAFCUEVGUOUPUJNZKZDLUSDVCUJOVIURDVHU
|
|
QUOUPABDPQRSTUFTRSTT $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a c $. $d a t $. $d a w $. $d a z $. $d b w $. $d c t $.
|
|
$d c w $. $d t w $. $d t x $. $d t y $. $d t z $. $d w x $. $d w y $.
|
|
$d w z $. $d x y $. $d x z $. $d y z $.
|
|
$( ~ ax-xp presented without any set theory definitions. (Contributed by
|
|
SF, 25-Mar-2015.) $)
|
|
axxpprim $p |- E. y A. z ( z e. y <-> E. w E. t ( A. a ( a e. z <-> ( A. b
|
|
( b e. a <-> b = w ) \/ A. c ( c e. a <-> ( c = w \/ c = t ) ) ) ) /\ t
|
|
e. x ) ) $=
|
|
( wel cv copk wceq wa wex wb wal weq wo ax-xp axprimlem2 albii exbii mpbi
|
|
anbi1i 2exbii bibi2i ) CBIZCJDJZEJZKLZEAIZMZENDNZOZCPZBNUGFCIGFIGDQOGPHFI
|
|
HDQHEQROHPROFPZUKMZENDNZOZCPZBNABCDESUOUTBUNUSCUMURUGULUQDEUJUPUKUHUIGHCF
|
|
TUDUEUFUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a w $. $d a y $. $d a z $. $d b c $. $d b d $. $d b w $.
|
|
$d b z $. $d c z $. $d d w $. $d d z $. $d e f $. $d e w $. $d e x $.
|
|
$d e z $. $d f g $. $d f h $. $d f w $. $d f z $. $d g w $. $d h w $.
|
|
$d h z $. $d w x $. $d w y $. $d w z $. $d x y $. $d x z $. $d y z $.
|
|
$( ~ ax-cnv presented without any set theory definitions. (Contributed by
|
|
SF, 25-Mar-2015.) $)
|
|
axcnvprim $p |- E. y A. z A. w ( E. a ( A. b ( b e. a <-> ( A. c ( c e. b
|
|
<-> c = z ) \/ A. d ( d e. b <-> ( d = z \/ d = w ) ) ) ) /\ a e. y )
|
|
<-> E. e ( A. f ( f e. e <-> ( A. g ( g e. f <-> g = w ) \/ A. h ( h e.
|
|
f <-> ( h = w \/ h = z ) ) ) ) /\ e e. x ) ) $=
|
|
( cv wb wal wex wel weq wo wa copk ax-cnv df-clel axprimlem2 anbi1i exbii
|
|
wcel wceq bitri bibi12i 2albii mpbi ) CMZDMZUAZBMZUGZUNUMUAZAMZUGZNZDOCOZ
|
|
BPJIQKJQKCRNKOLJQLCRLDRSNLOSNJOZIBQZTZIPZFEQGFQGDRNGOHFQHDRHCRSNHOSNFOZEA
|
|
QZTZEPZNZDOCOZBPABCDUBVBVLBVAVKCDUQVFUTVJUQIMUOUHZVDTZIPVFIUOUPUCVNVEIVMV
|
|
CVDUMUNKLIJUDUEUFUIUTEMURUHZVHTZEPVJEURUSUCVPVIEVOVGVHUNUMGHEFUDUEUFUIUJU
|
|
KUFUL $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a x $. $d a y $. $d a z $. $d b c $. $d b d $. $d b y $.
|
|
$d b z $. $d c y $. $d d y $. $d d z $. $d e x $. $d e y $. $d e z $.
|
|
$d x y $. $d x z $. $d y z $.
|
|
$( ~ ax-sset presented without any set theory definitions. (Contributed by
|
|
SF, 25-Mar-2015.) $)
|
|
axssetprim $p |- E. x A. y A. z ( E. a ( A. b ( b e. a <-> ( A. c ( c e. b
|
|
<-> c = y ) \/ A. d ( d e. b <-> ( d = y \/ d = z ) ) ) ) /\ a e. x )
|
|
<-> A. e ( e e. y -> e e. z ) ) $=
|
|
( cv copk wcel wel wi wal wb wex weq wo wa exbii ax-sset axprimlem2 bitri
|
|
wceq df-clel anbi1i bibi1i 2albii mpbi ) BIZCIZJZAIZKZDBLDCLMDNZOZCNBNZAP
|
|
FELGFLGBQOGNHFLHBQHCQROHNROFNZEALZSZEPZUOOZCNBNZAPABCDUAUQVCAUPVBBCUNVAUO
|
|
UNEIULUDZUSSZEPVAEULUMUEVEUTEVDURUSUJUKGHEFUBUFTUCUGUHTUI $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a w $. $d a y $. $d a z $. $d b c $. $d b e $. $d b w $.
|
|
$d b z $. $d c d $. $d c z $. $d d z $. $d e f $. $d e g $. $d e w $.
|
|
$d e z $. $d f z $. $d g w $. $d h i $. $d h w $. $d h x $. $d h z $.
|
|
$d i j $. $d i k $. $d i w $. $d i z $. $d j z $. $d k w $. $d k z $.
|
|
$d w x $. $d w y $. $d w z $. $d x y $. $d x z $. $d y z $.
|
|
$( ~ ax-si presented without any set theory definitions. (Contributed by
|
|
SF, 25-Mar-2015.) $)
|
|
axsiprim $p |- E. y A. z A. w ( E. a ( A. b ( b e. a <-> ( A. c ( c e. b
|
|
<-> A. d ( d e. c <-> d = z ) ) \/ A. e ( e e. b <-> ( A. f ( f e. e
|
|
<-> f = z ) \/ A. g ( g e. e <-> g = w ) ) ) ) ) /\ a e. y ) <-> E. h (
|
|
A. i ( i e. h <-> ( A. j ( j e. i <-> j = z ) \/ A. k ( k e. i <-> ( k
|
|
= z \/ k = w ) ) ) ) /\ h e. x ) ) $=
|
|
( cv wb wal wex wel csn copk wcel weq wo wa ax-si wceq df-clel axprimlem2
|
|
axprimlem1 bibi2i albii orbi12i bitri anbi1i exbii bibi12i 2albii mpbi )
|
|
CPZUAZDPZUAZUBZBPZUCZVAVCUBZAPZUCZQZDRCRZBSMLTZNMTZONTOCUDQORZQZNRZEMTZFE
|
|
TFCUDQFRZGETGDUDQGRZUEZQZERZUEZQZMRZLBTZUFZLSZIHTJITJCUDQJRKITKCUDKDUDUEQ
|
|
KRUEQIRZHATZUFZHSZQZDRCRZBSABCDUGVLWOBVKWNCDVGWIVJWMVGLPVEUHZWGUFZLSWILVE
|
|
VFUIWQWHLWPWFWGWPVMVNNPVBUHZQZNRZVREPZVBUHZXAVDUHZUEZQZERZUEZQZMRWFVBVDNE
|
|
LMUJXHWEMXGWDVMWTVQXFWCWSVPNWRVOVNVANOUKULUMXEWBEXDWAVRXBVSXCVTVAEFUKVCEG
|
|
UKUNULUMUNULUMUOUPUQUOVJHPVHUHZWKUFZHSWMHVHVIUIXJWLHXIWJWKVAVCJKHIUJUPUQU
|
|
OURUSUQUT $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a w $. $d a x $. $d a z $. $d b c $. $d b d $. $d b w $.
|
|
$d b z $. $d c w $. $d d e $. $d d w $. $d d z $. $d e z $. $d w x $.
|
|
$d w y $. $d w z $. $d x y $. $d x z $. $d y z $.
|
|
$( ~ ax-typlower presented without any set theory definitions.
|
|
(Contributed by SF, 25-Mar-2015.) $)
|
|
axtyplowerprim $p |- E. y A. z ( z e. y <-> A. w E. a ( A. b ( b e. a <-> (
|
|
A. c ( c e. b <-> c = w ) \/ A. d ( d e. b <-> ( d = w \/ A. e ( e e. d
|
|
<-> e = z ) ) ) ) ) /\ a e. x ) ) $=
|
|
( wel cv wal wb wex weq wo wa wceq bibi2i albii csn copk wcel ax-typlower
|
|
df-clel axprimlem2 axprimlem1 orbi2i bitri anbi1i exbii mpbi ) CBJZDKZCKZ
|
|
UAZUBZAKZUCZDLZMZCLZBNUMGFJZHGJHDOMHLZIGJZIDOZEIJECOMELZPZMZILZPZMZGLZFAJ
|
|
ZQZFNZDLZMZCLZBNABCDUDVBVSBVAVRCUTVQUMUSVPDUSFKUQRZVNQZFNVPFUQURUEWAVOFVT
|
|
VMVNVTVCVDVEVFIKUPRZPZMZILZPZMZGLVMUNUPHIFGUFWGVLGWFVKVCWEVJVDWDVIIWCVHVE
|
|
WBVGVFUOIEUGUHSTUHSTUIUJUKUITSTUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a t $. $d a w $. $d a y $. $d a z $. $d b c $. $d b f $.
|
|
$d b t $. $d b w $. $d b z $. $d c d $. $d c z $. $d d e $. $d d z $.
|
|
$d e g $. $d e z $. $d f g $. $d f h $. $d f t $. $d f w $. $d f z $.
|
|
$d g z $. $d h i $. $d h j $. $d h t $. $d h w $. $d i w $. $d j t $.
|
|
$d j w $. $d k l $. $d k t $. $d k x $. $d k z $. $d l m $. $d l n $.
|
|
$d l t $. $d l z $. $d m z $. $d n t $. $d n z $. $d t w $. $d t x $.
|
|
$d t y $. $d t z $. $d w x $. $d w y $. $d w z $. $d x y $. $d x z $.
|
|
$d y z $.
|
|
$( ~ ax-ins2 presented without any set theory definitions. (Contributed by
|
|
SF, 25-Mar-2015.) $)
|
|
axins2prim $p |- E. y A. z A. w A. t ( E. a ( A. b ( b e. a <-> ( A. c ( c
|
|
e. b <-> A. d ( d e. c <-> A. e ( e e. d <-> e = z ) ) ) \/ A. f ( f e.
|
|
b <-> ( A. g ( g e. f <-> A. e ( e e. g <-> e = z ) ) \/ A. h ( h e. f
|
|
<-> ( A. i ( i e. h <-> i = w ) \/ A. j ( j e. h <-> ( j = w \/ j = t )
|
|
) ) ) ) ) ) ) /\ a e. y ) <-> E. k ( A. l ( l e. k <-> ( A. m ( m e. l
|
|
<-> m = z ) \/ A. n ( n e. l <-> ( n = z \/ n = t ) ) ) ) /\ k e. x ) )
|
|
$=
|
|
( wal cv csn copk wcel wb wex wel weq wo wa ax-ins2 axprimlem2 axprimlem1
|
|
wceq df-clel bibi2i albii bitri orbi12i anbi1i exbii bibi12i 2albii mpbi
|
|
) CUAZUBZUBZDUAZEUAZUCZUCZBUAZUDZVEVIUCZAUAZUDZUEZETZDTCTZBUFPOUGZQPUGZRQ
|
|
UGZFRUGFCUHZUEFTZUEZRTZUEZQTZGPUGZHGUGZFHUGWCUEFTZUEZHTZIGUGJIUGJDUHUEJTK
|
|
IUGKDUHKEUHUIUEKTUIUEITZUIZUEZGTZUIZUEZPTZOBUGZUJZOUFZSLUGMSUGMCUHUEMTNSU
|
|
GNCUHNEUHUIUENTUIUESTZLAUGZUJZLUFZUEZETZDTCTZBUFABCDEUKVSXJBVRXICDVQXHEVM
|
|
XCVPXGVMOUAVKUNZXAUJZOUFXCOVKVLUOXLXBOXKWTXAXKVTWAQUAVGUNZUEZQTZWIGUAZVGU
|
|
NZXPVJUNZUIZUEZGTZUIZUEZPTWTVGVJQGOPULYCWSPYBWRVTXOWHYAWQXNWGQXMWFWAXMWBR
|
|
UAVFUNZUEZRTWFVFQRUMYEWERYDWDWBVERFUMUPUQURUPUQXTWPGXSWOWIXQWMXRWNXQWJHUA
|
|
VFUNZUEZHTWMVFGHUMYGWLHYFWKWJVEHFUMUPUQURVHVIJKGIULUSUPUQUSUPUQURUTVAURVP
|
|
LUAVNUNZXEUJZLUFXGLVNVOUOYIXFLYHXDXEVEVIMNLSULUTVAURVBUQVCVAVD $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a t $. $d a w $. $d a y $. $d a z $. $d b c $. $d b f $.
|
|
$d b t $. $d b w $. $d b z $. $d c d $. $d c z $. $d d e $. $d d z $.
|
|
$d e g $. $d e z $. $d f g $. $d f h $. $d f t $. $d f w $. $d f z $.
|
|
$d g z $. $d h i $. $d h j $. $d h t $. $d h w $. $d i w $. $d j t $.
|
|
$d j w $. $d k l $. $d k w $. $d k x $. $d k z $. $d l m $. $d l n $.
|
|
$d l w $. $d l z $. $d m z $. $d n w $. $d n z $. $d t w $. $d t x $.
|
|
$d t y $. $d t z $. $d w x $. $d w y $. $d w z $. $d x y $. $d x z $.
|
|
$d y z $.
|
|
$( ~ ax-ins3 presented without any set theory definitions. (Contributed by
|
|
SF, 25-Mar-2015.) $)
|
|
axins3prim $p |- E. y A. z A. w A. t ( E. a ( A. b ( b e. a <-> ( A. c ( c
|
|
e. b <-> A. d ( d e. c <-> A. e ( e e. d <-> e = z ) ) ) \/ A. f ( f e.
|
|
b <-> ( A. g ( g e. f <-> A. e ( e e. g <-> e = z ) ) \/ A. h ( h e. f
|
|
<-> ( A. i ( i e. h <-> i = w ) \/ A. j ( j e. h <-> ( j = w \/ j = t )
|
|
) ) ) ) ) ) ) /\ a e. y ) <-> E. k ( A. l ( l e. k <-> ( A. m ( m e. l
|
|
<-> m = z ) \/ A. n ( n e. l <-> ( n = z \/ n = w ) ) ) ) /\ k e. x ) )
|
|
$=
|
|
( wal cv csn copk wcel wb wex wel weq wo wa ax-ins3 axprimlem2 axprimlem1
|
|
wceq df-clel bibi2i albii bitri orbi12i anbi1i exbii bibi12i 2albii mpbi
|
|
) CUAZUBZUBZDUAZEUAZUCZUCZBUAZUDZVEVHUCZAUAZUDZUEZETZDTCTZBUFPOUGZQPUGZRQ
|
|
UGZFRUGFCUHZUEFTZUEZRTZUEZQTZGPUGZHGUGZFHUGWCUEFTZUEZHTZIGUGJIUGJDUHUEJTK
|
|
IUGKDUHKEUHUIUEKTUIUEITZUIZUEZGTZUIZUEZPTZOBUGZUJZOUFZSLUGMSUGMCUHUEMTNSU
|
|
GNCUHNDUHUIUENTUIUESTZLAUGZUJZLUFZUEZETZDTCTZBUFABCDEUKVSXJBVRXICDVQXHEVM
|
|
XCVPXGVMOUAVKUNZXAUJZOUFXCOVKVLUOXLXBOXKWTXAXKVTWAQUAVGUNZUEZQTZWIGUAZVGU
|
|
NZXPVJUNZUIZUEZGTZUIZUEZPTWTVGVJQGOPULYCWSPYBWRVTXOWHYAWQXNWGQXMWFWAXMWBR
|
|
UAVFUNZUEZRTWFVFQRUMYEWERYDWDWBVERFUMUPUQURUPUQXTWPGXSWOWIXQWMXRWNXQWJHUA
|
|
VFUNZUEZHTWMVFGHUMYGWLHYFWKWJVEHFUMUPUQURVHVIJKGIULUSUPUQUSUPUQURUTVAURVP
|
|
LUAVNUNZXEUJZLUFXGLVNVOUOYIXFLYHXDXEVEVHMNLSULUTVAURVBUQVCVAVD $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Initial existence theorems
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A x y z w $. $d B x y z w $.
|
|
$( The anti-intersection of two sets is a set. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
ninexg $p |- ( ( A e. V /\ B e. W ) -> ( A -i^i B ) e. _V ) $=
|
|
( vx vy vw vz cv cnin cvv wcel wceq nineq1 eleq1d wel wb wal wex bitri
|
|
nineq2 wnan ax-nin isset dfcleq elnin bibi2i albii exbii mpbir vtocl2g
|
|
vex ) EIZFIZJZKLZAUNJZKLABJZKLEFABCDUMAMUOUQKUMAUNNOUNBMUQURKUNBAUAOUPGHP
|
|
ZGEPGFPUBZQZGRZHSZEFHGUCUPHIZUOMZHSVCHUOUDVEVBHVEUSGIZUOLZQZGRVBGVDUOUEVH
|
|
VAGVGUTUSVFUMUNGULUFUGUHTUITUJUK $.
|
|
$}
|
|
|
|
${
|
|
ninex.1 $e |- A e. _V $.
|
|
ninex.2 $e |- B e. _V $.
|
|
$( The anti-intersection of two sets is a set. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
ninex $p |- ( A -i^i B ) e. _V $=
|
|
( cvv wcel cnin ninexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
$( The complement of a set is a set. (Contributed by SF, 12-Jan-2015.) $)
|
|
complexg $p |- ( A e. V -> ~ A e. _V ) $=
|
|
( wcel ccompl cnin cvv df-compl ninexg anidms syl5eqel ) ABCZADAAEZFAGKLFCA
|
|
ABBHIJ $.
|
|
|
|
$( The intersection of two sets is a set. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
inexg $p |- ( ( A e. V /\ B e. W ) -> ( A i^i B ) e. _V ) $=
|
|
( wcel wa cin cnin ccompl cvv df-in ninexg complexg syl syl5eqel ) ACEBDEFZ
|
|
ABGABHZIZJABKPQJERJEABCDLQJMNO $.
|
|
|
|
$( The union of two sets is a set. (Contributed by SF, 12-Jan-2015.) $)
|
|
unexg $p |- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V ) $=
|
|
( wcel wa cun ccompl cnin cvv df-un complexg ninexg syl2an syl5eqel ) ACEZB
|
|
DEZFABGAHZBHZIZJABKPRJESJETJEQACLBDLRSJJMNO $.
|
|
|
|
$( The difference of two sets is a set. (Contributed by SF, 12-Jan-2015.) $)
|
|
difexg $p |- ( ( A e. V /\ B e. W ) -> ( A \ B ) e. _V ) $=
|
|
( wcel wa cdif ccompl cin cvv df-dif complexg inexg sylan2 syl5eqel ) ACEZB
|
|
DEZFABGABHZIZJABKQPRJESJEBDLARCJMNO $.
|
|
|
|
$( The symmetric difference of two sets is a set. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
symdifexg $p |- ( ( A e. V /\ B e. W ) -> ( A (+) B ) e. _V ) $=
|
|
( wcel csymdif cdif cun cvv df-symdif difexg ancoms unexg syl2anc syl5eqel
|
|
wa ) ACEZBDEZPZABFABGZBAGZHZIABJSTIEUAIEZUBIEABCDKRQUCBADCKLTUAIIMNO $.
|
|
|
|
${
|
|
boolex.1 $e |- A e. _V $.
|
|
$( The complement of a set is a set. (Contributed by SF, 12-Jan-2015.) $)
|
|
complex $p |- ~ A e. _V $=
|
|
( cvv wcel ccompl complexg ax-mp ) ACDAECDBACFG $.
|
|
|
|
boolex.2 $e |- B e. _V $.
|
|
$( The intersection of two sets is a set. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
inex $p |- ( A i^i B ) e. _V $=
|
|
( cvv wcel cin inexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
|
|
$( The union of two sets is a set. (Contributed by SF, 12-Jan-2015.) $)
|
|
unex $p |- ( A u. B ) e. _V $=
|
|
( cvv wcel cun unexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
|
|
$( The difference of two sets is a set. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
difex $p |- ( A \ B ) e. _V $=
|
|
( cvv wcel cdif difexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
|
|
$( The symmetric difference of two sets is a set. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
symdifex $p |- ( A (+) B ) e. _V $=
|
|
( cvv wcel csymdif symdifexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
$( The universal class exists. This marks a major departure from ZFC set
|
|
theory, where ` _V ` is a proper class. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
vvex $p |- _V e. _V $=
|
|
( vx cv ccompl cun cvv uncompl vex complex unex eqeltrri ) ABZKCZDEEKFKLAGZ
|
|
KMHIJ $.
|
|
|
|
$( The empty class exists. (Contributed by SF, 12-Jan-2015.) $)
|
|
0ex $p |- (/) e. _V $=
|
|
( cvv ccompl c0 complV vvex complex eqeltrri ) ABCADAEFG $.
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( A singleton always exists. (Contributed by SF, 12-Jan-2015.) $)
|
|
snex $p |- { A } e. _V $=
|
|
( vx vz vy cvv wcel csn cv wceq sneq eleq1d wel weq wb wal wex axprimlem1
|
|
ax-sn isset c0 exbii bitri mpbir vtoclg snprc biimpi 0ex syl6eqel pm2.61i
|
|
wn ) AEFZAGZEFZBHZGZEFZUMBAEUNAIUOULEUNAJKUPCDLCBMNCOZDPZBDCRUPDHUOIZDPUR
|
|
DUOSUSUQDUNDCQUAUBUCUDUKUJZULTEUTULTIAUEUFUGUHUI $.
|
|
$}
|
|
|
|
$( An unordered pair exists. (Contributed by SF, 12-Jan-2015.) $)
|
|
prex $p |- { A , B } e. _V $=
|
|
( cpr csn cun cvv df-pr snex unex eqeltri ) ABCADZBDZEFABGKLAHBHIJ $.
|
|
|
|
$( A Kuratowski ordered pair exists. (Contributed by SF, 12-Jan-2015.) $)
|
|
opkex $p |- << A , B >> e. _V $=
|
|
( copk csn cpr cvv df-opk prex eqeltri ) ABCADZABEZEFABGJKHI $.
|
|
|
|
$( A singleton of a set belongs to a power class of a set containing it.
|
|
(Contributed by SF, 1-Feb-2015.) $)
|
|
snelpwg $p |- ( A e. V -> ( { A } e. ~P B <-> A e. B ) ) $=
|
|
( wcel csn wss cpw snssg snex elpw syl6rbbr ) ACDABDAEZBFLBGDABCHLBAIJK $.
|
|
|
|
${
|
|
snelpw.1 $e |- A e. _V $.
|
|
$( A singleton of a set belongs to a power class of a set containing it.
|
|
(Contributed by SF, 1-Feb-2015.) $)
|
|
snelpw $p |- ( { A } e. ~P B <-> A e. B ) $=
|
|
( cvv wcel csn cpw wb snelpwg ax-mp ) ADEAFBGEABEHCABDIJ $.
|
|
$}
|
|
|
|
$( A singleton of a set belongs to the power class of a class containing the
|
|
set. (Contributed by Alan Sare, 25-Aug-2011.) $)
|
|
snelpwi $p |- ( A e. B -> { A } e. ~P B ) $=
|
|
( wcel csn wss cpw snssi snex elpw sylibr ) ABCADZBEKBFCABGKBAHIJ $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( A class equals the union of its power class. Exercise 6(a) of
|
|
[Enderton] p. 38. (The proof was shortened by Alan Sare, 28-Dec-2008.)
|
|
(Contributed by SF, 14-Oct-1996.) (Revised by SF, 29-Dec-2008.) $)
|
|
unipw $p |- U. ~P A = A $=
|
|
( vx vy cpw cuni cv wcel wa wex eluni wss wi vex elpw ssel impcom exlimiv
|
|
sylbi csn snid snelpwi elunii sylancr impbii eqriv ) BADZEZABFZUGGZUHAGZU
|
|
IUHCFZGZUKUFGZHZCIUJCUHUFJUNUJCUMULUJUMUKAKULUJLUKACMNUKAUHORPQRUJUHUHSZG
|
|
UOUFGUIUHBMTUHAUAUHUOUFUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Classes are subclasses if and only if their power classes are
|
|
subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by SF,
|
|
13-Oct-1996.) $)
|
|
sspwb $p |- ( A C_ B <-> ~P A C_ ~P B ) $=
|
|
( vx wss cpw wcel sstr2 com12 vex elpw 3imtr4g ssrdv csn ssel snex bitr4i
|
|
cv snss 3imtr3g impbii ) ABDZAEZBEZDZUACUBUCUACQZADZUEBDZUEUBFUEUCFUFUAUG
|
|
UEABGHUEACIZJUEBUHJKLUDCABUDUEMZUBFZUIUCFZUEAFZUEBFZUBUCUINUJUIADULUIAUEO
|
|
ZJUEAUHRPUKUIBDUMUIBUNJUEBUHRPSLT $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d A x $. $d a z $. $d A z $. $d b z $.
|
|
$d X a $. $d X b $. $d X x $. $d x z $. $d X z $.
|
|
|
|
$( Compute the power class of an adjoinment. (Contributed by SF,
|
|
30-Jan-2015.) $)
|
|
pwadjoin $p |- ~P ( A u. { X } ) =
|
|
( ~P A u. { a | E. b e. ~P A a = ( b u. { X } ) } ) $=
|
|
( vz vx cun cpw cv wceq wrex wss wo wcel wn wa bitri vex elpw com12 uncom
|
|
csn cab cdif sseq2i ssundif biimpi adantr snex difex sylibr eqcomd adantl
|
|
difsnid uneq1 eqeq2d rspcev syl2anc ex con3d wel wi ssel elun elsn orbi2i
|
|
ax-1 eleq1 anbi1d pm2.21 impcom syl6bi jaoi sylbi exp3a syld imp3a orcomd
|
|
ssrdv orrd ssun3 unss1 sseq1 syl5ibrcom rexlimiv impbii weq eqeq1 rexbidv
|
|
elab orbi12i 3bitr4i eqriv ) EABUBZGZHZAHZCIZDIZWNGZJZDWQKZCUCZGZEIZWOLZX
|
|
EALZXEWTJZDWQKZMZXEWPNXEXDNZXFXJXFXIXGXFXIXGXFXIOBXENZOZXGXFXLXIXFXLXIXFX
|
|
LPZXEWNUDZWQNZXEXOWNGZJZXIXNXOALZXPXFXSXLXFXSXFXEWNAGZLXSWOXTXEAWNUAUEXEW
|
|
NAUFQUGUHXOAXEWNERZBUIUJSUKXLXRXFXLXQXEXEBUNULUMXHXRDXOWQWSXOJWTXQXEWSXOW
|
|
NUOUPUQURUSUTXFXMXGXFXMPZFXEAFEVAZYBFIZANZYCXFXMYEYCXFYDWONZXMYEVBZXFYCYF
|
|
XEWOYDVCTYFYCYGYFYCXMYEYFYEYDBJZMZYCXMPZYEVBZYFYEYDWNNZMYIYDAWNVDYLYHYEFB
|
|
VEVFQYEYKYHYEYJVGYHYJXLXMPYEYHYCXLXMYDBXEVHVIXMXLYEXLYEVJVKVLVMVNVOTVPVQT
|
|
VSUSVPVTVRXGXFXIXEAWNWAXHXFDWQWSWQNZXFXHWTWOLZYMWSALYNWSADRSWSAWNWBVNXEWT
|
|
WOWCWDWEVMWFXEWOYASXKXEWQNZXEXCNZMXJXEWQXCVDYOXGYPXIXEAYASXBXICXEYACEWGXA
|
|
XHDWQWRXEWTWHWIWJWKQWLWM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Singletons and pairs (continued)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( A singleton is a subset of an unordered pair. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
snprss1 $p |- { A } C_ { A , B } $=
|
|
( csn cun cpr ssun1 df-pr sseqtr4i ) ACZIBCZDABEIJFABGH $.
|
|
|
|
$( A singleton is a subset of an unordered pair. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
snprss2 $p |- { A } C_ { B , A } $=
|
|
( csn cpr snprss1 prcom sseqtri ) ACABDBADABEABFG $.
|
|
|
|
$( An unordered pair of a proper class. (Contributed by SF, 12-Jan-2015.) $)
|
|
prprc2 $p |- ( -. A e. _V -> { B , A } = { B } ) $=
|
|
( cvv wcel wn cpr csn cun df-pr wceq snprc biimpi uneq2d un0 syl6eq syl5eq
|
|
c0 ) ACDEZBAFBGZAGZHZSBAIRUASQHSRTQSRTQJAKLMSNOP $.
|
|
|
|
$( An unordered pair of a proper class. (Contributed by SF, 12-Jan-2015.) $)
|
|
prprc1 $p |- ( -. A e. _V -> { A , B } = { B } ) $=
|
|
( cvv wcel wn cpr csn prcom prprc2 syl5eq ) ACDEABFBAFBGABHABIJ $.
|
|
|
|
${
|
|
preqr1.1 $e |- A e. _V $.
|
|
preqr1.2 $e |- B e. _V $.
|
|
$( Reverse equality lemma for unordered pairs. If two unordered pairs have
|
|
the same second element, the first elements are equal. (Contributed by
|
|
NM, 18-Oct-1995.) $)
|
|
preqr1 $p |- ( { A , C } = { B , C } -> A = B ) $=
|
|
( cpr wceq wcel wo prid1 eleq2 mpbii elpr sylib mpbiri eqcom eqeq2 oplem1
|
|
) ACFZBCFZGZABGZACGZBAGZBCGZUAATHZUBUCIUAASHUFACDJSTAKLABCDMNUABSHZUDUEIU
|
|
AUGBTHBCEJSTBKOBACEMNABPACBQR $.
|
|
$}
|
|
|
|
${
|
|
preqr2.1 $e |- A e. _V $.
|
|
preqr2.2 $e |- B e. _V $.
|
|
$( Reverse equality lemma for unordered pairs. If two unordered pairs have
|
|
the same first element, the second elements are equal. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
preqr2 $p |- ( { C , A } = { C , B } -> A = B ) $=
|
|
( cpr wceq prcom eqeq12i preqr1 sylbi ) CAFZCBFZGACFZBCFZGABGLNMOCAHCBHIA
|
|
BCDEJK $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $.
|
|
$( Reverse equality lemma for unordered pairs. If two unordered pairs have
|
|
the same first element, the second elements are equal. (Contributed by
|
|
SF, 12-Jan-2015.) $)
|
|
preqr2g $p |- ( ( A e. V /\ B e. W ) ->
|
|
( { C , A } = { C , B } -> A = B ) ) $=
|
|
( vx vy cv cpr wceq weq wi preq2 eqeq1d eqeq1 imbi12d eqeq2d eqeq2 preqr2
|
|
vex vtocl2g ) CFHZIZCGHZIZJZFGKZLCAIZUEJZAUDJZLUHCBIZJZABJZLFGABDEUBAJZUF
|
|
UIUGUJUNUCUHUEUBACMNUBAUDOPUDBJZUIULUJUMUOUEUKUHUDBCMQUDBARPUBUDCFTGTSUA
|
|
$.
|
|
$}
|
|
|
|
${
|
|
preq12b.1 $e |- A e. _V $.
|
|
preq12b.2 $e |- B e. _V $.
|
|
preq12b.3 $e |- C e. _V $.
|
|
preq12b.4 $e |- D e. _V $.
|
|
$( Equality relationship for two unordered pairs. (Contributed by NM,
|
|
17-Oct-1996.) $)
|
|
preq12b $p |- ( { A , B } = { C , D } <->
|
|
( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) $=
|
|
( cpr wceq wa wo wcel preq1 eqeq1d preqr2 syl6bi com12 ancld prcom eqeq2i
|
|
prid1 eleq2 mpbii elpr sylib wi sylbi orim12d preq12 syl6eq sylan9eq jaoi
|
|
mpd impbii ) ABIZCDIZJZACJZBDJZKZADJZBCJZKZLZURUSVBLZVEURAUQMZVFURAUPMVGA
|
|
BEUBUPUQAUCUDACDEUEUFURUSVAVBVDURUSUTUSURUTUSURCBIZUQJUTUSUPVHUQACBNOBDCF
|
|
HPQRSURVBVCURUPDCIZJZVBVCUGUQVIUPCDTUAVBVJVCVBVJDBIZVIJVCVBUPVKVIADBNZOBC
|
|
DFGPQRUHSUIUNVAURVDABCDUJVBVCUPBDIZUQVBUPVKVMVLDBTUKBCDNULUMUO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $. $d B x y z w $. $d C x y z w $. $d D x y z w $.
|
|
$d V x y z w $. $d W x y z w $. $d X x y z w $. $d Y x y z w $.
|
|
$( Closed form of ~ preq12b . (Contributed by Scott Fenton,
|
|
28-Mar-2014.) $)
|
|
preq12bg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) ->
|
|
( { A , B } = { C , D } <->
|
|
( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) ) $=
|
|
( vx vy vz vw wcel wa cpr wceq wo wb wi cv weq preq1 eqeq1d eqeq1 orbi12d
|
|
anbi1d bibi12d imbi2d preq2 anbi2d eqeq2 w3a vex preq12b vtoclbg vtocl3ga
|
|
eqeq2d a1i 3expa impr ) AEMZBFMZNCGMZDHMZABOZCDOZPZACPZBDPZNZADPZBCPZNZQZ
|
|
RZVAVBVCVDVOSZVDITZJTZOZKTZDOZPZIKUAZVRDPZNZVQDPZJKUAZNZQZRZSZVDAVROZWAPZ
|
|
AVTPZWDNZVKWGNZQZRZSVDVEWAPZWNVINZVKBVTPZNZQZRZSVPIJKABCEFGVQAPZWJWRVDXEW
|
|
BWMWIWQXEVSWLWAVQAVRUBUCXEWEWOWHWPXEWCWNWDVQAVTUDUFXEWFVKWGVQADUDUFUEUGUH
|
|
VRBPZWRXDVDXFWMWSWQXCXFWLVEWAVRBAUIUCXFWOWTWPXBXFWDVIWNVRBDUDUJXFWGXAVKVR
|
|
BVTUDUJUEUGUHVTCPZXDVOVDXGWSVGXCVNXGWAVFVEVTCDUBUQXGWTVJXBVMXGWNVHVIVTCAU
|
|
KUFXGXAVLVKVTCBUKUJUEUGUHWKVQEMVRFMVTGMULVSVTLTZOZPWCJLUAZNZILUAZWGNZQWBW
|
|
ILDHXHDPZXIWAVSXHDVTUIUQXNXKWEXMWHXNXJWDWCXHDVRUKUJXNXLWFWGXHDVQUKUFUEVQV
|
|
RVTXHIUMJUMKUMLUMUNUOURUPUSUT $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Kuratowski ordered pairs (continued)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Membership in a Kuratowski ordered pair. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
elopk $p |- ( A e. << B , C >> <-> ( A = { B } \/ A = { B , C } ) ) $=
|
|
( copk wcel csn cpr wceq wo df-opk eleq2i snex prex elpr2 bitri ) ABCDZEABF
|
|
ZBCGZGZEAQHARHIPSABCJKAQRBLBCMNO $.
|
|
|
|
$( Equality of the first member of a Kuratowski ordered pair, which holds
|
|
regardless of the sethood of the second members. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
opkth1g $p |- ( ( A e. V /\ << A , B >> = << C , D >> ) -> A = C ) $=
|
|
( wcel copk wceq wa csn cpr wo eqid orci elopk mpbir eleq2 snidg syl5ibrcom
|
|
biimprd prid1g jaod syl5bi sylan9r mpi wb elsncg adantr mpbid ) AEFZABGZCDG
|
|
ZHZIZACJZFZACHZUNUOULFZUPURUOUOHZUOCDKHZLUSUTUOMNUOCDOPUMURUOUKFZUJUPUMVAUR
|
|
UKULUOQTVAUOAJZHZUOABKZHZLUJUPUOABOUJVCUPVEUJUPVCAVBFAERUOVBAQSUJUPVEAVDFAB
|
|
EUAUOVDAQSUBUCUDUEUJUPUQUFUMACEUGUHUI $.
|
|
|
|
$( Two Kuratowski ordered pairs are equal iff their components are equal.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
opkthg $p |- ( ( A e. V /\ B e. W /\ D e. T ) ->
|
|
( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) ) ) $=
|
|
( wcel w3a copk wceq wa simp1 opkth1g sylan wi cpr df-opk prex syl5 jca csn
|
|
simp2 simp3 opkeq1 eqeq1d biimpd impcom eqeq12i preqr2 sylbi exp3a imp jcai
|
|
preqr2g ex opkeq12 impbid1 ) AFHZBGHZDEHZIZABJZCDJZKZACKZBDKZLZVBVEVHVBVELV
|
|
FVGVBUSVEVFUSUTVAMABCDFNOVBUTVALZVEVFVGPZVBUTVAUSUTVAUCUSUTVAUDUAVIVEVJVIVE
|
|
VFVGVEVFLCBJZVDKZVIVGVFVEVLVFVEVLVFVCVKVDACBUEUFUGUHVLCBQZCDQZKZVIVGVLCUBZV
|
|
MQZVPVNQZKVOVKVQVDVRCBRCDRUIVMVNVPCBSCDSUJUKBDCGEUOTTULUMOUNUPABCDUQUR $.
|
|
|
|
${
|
|
opkth.1 $e |- A e. _V $.
|
|
opkth.2 $e |- B e. _V $.
|
|
opkth.3 $e |- D e. _V $.
|
|
$( Two Kuratowski ordered pairs are equal iff their components are equal.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
opkth $p |- ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) ) $=
|
|
( cvv wcel copk wceq wa wb opkthg mp3an ) AHIBHIDHIABJCDJKACKBDKLMEFGABCD
|
|
HHHNO $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Cardinal one, unit unions, and unit power classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c 1c $. $( Cardinal one $)
|
|
$c ~P1 $. $( Unit power class: calligraphic P with subscript 1. $)
|
|
$c U.1 $. $( Unitary union. $)
|
|
|
|
$( Extend class notation to include the unit union of a class (read: 'unit
|
|
union ` A ` ') $)
|
|
cuni1 $a class U.1 A $.
|
|
|
|
$( Extend the definition of a class to include cardinal one. $)
|
|
c1c $a class 1c $.
|
|
|
|
$( Extend class notation to include unit power class. $)
|
|
cpw1 $a class ~P1 A $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define cardinal one. This is the set of all singletons, or the set of
|
|
all sets of size one. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-1c $a |- 1c = { x | E. y x = { y } } $.
|
|
$}
|
|
|
|
$( Define unit power class. Definition from [Rosser] p. 252. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
df-pw1 $a |- ~P1 A = ( ~P A i^i 1c ) $.
|
|
|
|
$( Define the unit union of a class. This operation is used implicitly in
|
|
both Holmes and Hailperin to complete their stratification algorithms,
|
|
although neither provide explicit notation for it. See ~ eluni1 for
|
|
membership condition. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-uni1 $a |- U.1 A = U. ( A i^i 1c ) $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Membership in cardinal one. (Contributed by SF, 12-Jan-2015.) $)
|
|
el1c $p |- ( A e. 1c <-> E. x A = { x } ) $=
|
|
( vy c1c wcel cvv cv csn wceq elex snex eleq1 mpbiri exlimiv eqeq1 exbidv
|
|
wex df-1c elab2g pm5.21nii ) BDEBFEZBAGZHZIZAQZBDJUDUAAUDUAUCFEUBKBUCFLMN
|
|
CGZUCIZAQUECBDFUFBIUGUDAUFBUCOPCARST $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
snel1c.1 $e |- A e. _V $.
|
|
$( A singleton is an element of cardinal one. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
snel1c $p |- { A } e. 1c $=
|
|
( vx csn c1c wcel cv wceq wex eqid sneq eqeq2d spcev ax-mp el1c mpbir ) A
|
|
DZEFQCGZDZHZCIZQQHZUAQJTUBCABRAHSQQRAKLMNCQOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( A singleton is an element of cardinal one. (Contributed by SF,
|
|
30-Jan-2015.) $)
|
|
snel1cg $p |- ( A e. V -> { A } e. 1c ) $=
|
|
( vx cv csn c1c wcel wceq sneq eleq1d vex snel1c vtoclg ) CDZEZFGAEZFGCAB
|
|
NAHOPFNAIJNCKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( Cardinal one is a set. (Contributed by SF, 12-Jan-2015.) $)
|
|
1cex $p |- 1c e. _V $=
|
|
( vy vx vw vz c1c cvv wcel wel weq wb wal wex ax-1c cv isset bitri bibi2i
|
|
wceq albii exbii csn cab df-1c eqeq2i abeq2 dfcleq df-sn abeq2i mpbir ) E
|
|
FGZABHZCAHZCDIZJZCKZDLZJZAKZBLZBADCMUJBNZERZBLUSBEOVAURBVAUKANZDNZUAZRZDL
|
|
ZJZAKZURVAUTVFAUBZRVHEVIUTADUCUDVFAUTUEPVGUQAVFUPUKVEUODVEULCNVDGZJZCKUOC
|
|
VBVDUFVKUNCVJUMULUMCVDCVCUGUHQSPTQSPTPUI $.
|
|
$}
|
|
|
|
$( Equality theorem for unit power class. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
pw1eq $p |- ( A = B -> ~P1 A = ~P1 B ) $=
|
|
( wceq cpw c1c cin cpw1 pweq ineq1d df-pw1 3eqtr4g ) ABCZADZEFBDZEFAGBGLMNE
|
|
ABHIAJBJK $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Membership in a unit power class. (Contributed by SF, 13-Jan-2015.) $)
|
|
elpw1 $p |- ( A e. ~P1 B <-> E. x e. B A = { x } ) $=
|
|
( cpw1 wcel cpw c1c wa cv csn wceq wrex cin df-pw1 eleq2i elin bitri el1c
|
|
wex bitr4i anbi2i 19.42v eleq1 wss snex elpw vex snss syl6bb exbii df-rex
|
|
pm5.32ri ) BCDZEZBCFZEZBGEZHZBAIZJZKZACLZUNBUOGMZEURUMVCBCNOBUOGPQURUPVAH
|
|
ZASZVBURUPVAASZHVEUQVFUPABRUAUPVAAUBTVEUSCEZVAHZASVBVDVHAVAUPVGVAUPUTUOEZ
|
|
VGBUTUOUCVIUTCUDVGUTCUSUEUFUSCAUGUHTUIULUJVAACUKTQQ $.
|
|
|
|
$( Membership in a unit power class applied twice. (Contributed by SF,
|
|
15-Jan-2015.) $)
|
|
elpw12 $p |- ( A e. ~P1 ~P1 B <-> E. x e. B A = { { x } } ) $=
|
|
( vy cpw1 wcel cv csn wceq wrex wa wex elpw1 anbi1i r19.41v bitr4i df-rex
|
|
exbii rexcom4 3bitr4i snex sneq eqeq2d ceqsexv rexbii 3bitri ) BCEZEFBDGZ
|
|
HZIZDUGJZUHAGZHZIZUJKZDLZACJZBUMHZIZACJDBUGMUHUGFZUJKZDLUOACJZDLUKUQVAVBD
|
|
VAUNACJZUJKVBUTVCUJAUHCMNUNUJACOPRUJDUGQUOADCSTUPUSACUJUSDUMULUAUNUIURBUH
|
|
UMUBUCUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Membership of a singleton in a unit power class. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
snelpw1 $p |- ( { A } e. ~P1 B <-> A e. B ) $=
|
|
( vx csn cv wceq wrex cpw1 wcel eqcom vex sneqb bitri rexbii elpw1 risset
|
|
3bitr4i ) ADZCEZDZFZCBGSAFZCBGRBHIABIUAUBCBUATRFUBRTJSACKLMNCRBOCABPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Membership in ` ~P1 1c ` (Contributed by SF, 13-Jan-2015.) $)
|
|
elpw11c $p |- ( A e. ~P1 1c <-> E. x A = { { x } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wrex wex elpw1 df-rex anbi1i 19.41v bitr4i
|
|
wa el1c exbii bitri excom snex sneq eqeq2d ceqsexv 3bitri ) BDEFBCGZHZIZC
|
|
DJZUGAGZHZIZUIQZAKZCKZBULHZIZAKZCBDLUJUGDFZUIQZCKUPUICDMVAUOCVAUMAKZUIQUO
|
|
UTVBUIAUGRNUMUIAOPSTUPUNCKZAKUSUNCAUAVCURAUIURCULUKUBUMUHUQBUGULUCUDUESTU
|
|
F $.
|
|
|
|
$( Membership in ` ~P1 ~P1 1c ` (Contributed by SF, 13-Jan-2015.) $)
|
|
elpw121c $p |- ( A e. ~P1 ~P1 1c <-> E. x A = { { { x } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wrex elpw1 wa df-rex elpw11c anbi1i 19.41v
|
|
wex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEZEFBCGZHZIZCUF
|
|
JZBAGHZHZHZIZAQZCBUFKUJUGULIZUILZAQZCQZUOUJUGUFFZUILZCQUSUICUFMVAURCVAUPA
|
|
QZUILURUTVBUIAUGNOUPUIAPRSTUSUQCQZAQUOUQCAUAVCUNAUIUNCULUKUBUPUHUMBUGULUC
|
|
UDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 1c ` (Contributed by SF, 14-Jan-2015.) $)
|
|
elpw131c $p |- ( A e. ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { x } } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wex elpw1 wa df-rex elpw121c anbi1i 19.41v
|
|
wrex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEZEFBCGZHZIZC
|
|
UFQZBAGHHZHZHZIZAJZCBUFKUJUGULIZUILZAJZCJZUOUJUGUFFZUILZCJUSUICUFMVAURCVA
|
|
UPAJZUILURUTVBUIAUGNOUPUIAPRSTUSUQCJZAJUOUQCAUAVCUNAUIUNCULUKUBUPUHUMBUGU
|
|
LUCUDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 ~P1 1c ` (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
elpw141c $p |- ( A e. ~P1 ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { { x } } } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wex elpw1 wa df-rex elpw131c anbi1i 19.41v
|
|
wrex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEEZEFBCGZHZIZ
|
|
CUFQZBAGHHHZHZHZIZAJZCBUFKUJUGULIZUILZAJZCJZUOUJUGUFFZUILZCJUSUICUFMVAURC
|
|
VAUPAJZUILURUTVBUIAUGNOUPUIAPRSTUSUQCJZAJUOUQCAUAVCUNAUIUNCULUKUBUPUHUMBU
|
|
GULUCUDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 ~P1 ~P1 1c ` (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
elpw151c $p |- ( A e. ~P1 ~P1 ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { { { x } } } } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wex elpw1 wa df-rex elpw141c anbi1i 19.41v
|
|
wrex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEEEZEFBCGZHZI
|
|
ZCUFQZBAGHHHHZHZHZIZAJZCBUFKUJUGULIZUILZAJZCJZUOUJUGUFFZUILZCJUSUICUFMVAU
|
|
RCVAUPAJZUILURUTVBUIAUGNOUPUIAPRSTUSUQCJZAJUOUQCAUAVCUNAUIUNCULUKUBUPUHUM
|
|
BUGULUCUDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c ` (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
elpw161c $p |- ( A e. ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { { { { x } } } } } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wex elpw1 wa df-rex elpw151c anbi1i 19.41v
|
|
wrex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEEEEZEFBCGZHZ
|
|
IZCUFQZBAGHHHHHZHZHZIZAJZCBUFKUJUGULIZUILZAJZCJZUOUJUGUFFZUILZCJUSUICUFMV
|
|
AURCVAUPAJZUILURUTVBUIAUGNOUPUIAPRSTUSUQCJZAJUOUQCAUAVCUNAUIUNCULUKUBUPUH
|
|
UMBUGULUCUDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c ` (Contributed by SF,
|
|
15-Jan-2015.) $)
|
|
elpw171c $p |- ( A e. ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { { { { { x } } } } } } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wex elpw1 wa df-rex elpw161c anbi1i 19.41v
|
|
wrex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEEEEEZEFBCGZH
|
|
ZIZCUFQZBAGHHHHHHZHZHZIZAJZCBUFKUJUGULIZUILZAJZCJZUOUJUGUFFZUILZCJUSUICUF
|
|
MVAURCVAUPAJZUILURUTVBUIAUGNOUPUIAPRSTUSUQCJZAJUOUQCAUAVCUNAUIUNCULUKUBUP
|
|
UHUMBUGULUCUDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c ` (Contributed by SF,
|
|
15-Jan-2015.) $)
|
|
elpw181c $p |- (
|
|
A e. ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { { { { { { x } } } } } } } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wex elpw1 wa df-rex elpw171c anbi1i 19.41v
|
|
wrex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEEEEEEZEFBCGZ
|
|
HZIZCUFQZBAGHHHHHHHZHZHZIZAJZCBUFKUJUGULIZUILZAJZCJZUOUJUGUFFZUILZCJUSUIC
|
|
UFMVAURCVAUPAJZUILURUTVBUIAUGNOUPUIAPRSTUSUQCJZAJUOUQCAUAVCUNAUIUNCULUKUB
|
|
UPUHUMBUGULUCUDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c ` (Contributed by
|
|
SF, 24-Jan-2015.) $)
|
|
elpw191c $p |- (
|
|
A e. ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { { { { { { { x } } } } } } } } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wex elpw1 wa df-rex elpw181c anbi1i 19.41v
|
|
wrex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEEEEEEEZEFBCG
|
|
ZHZIZCUFQZBAGHHHHHHHHZHZHZIZAJZCBUFKUJUGULIZUILZAJZCJZUOUJUGUFFZUILZCJUSU
|
|
ICUFMVAURCVAUPAJZUILURUTVBUIAUGNOUPUIAPRSTUSUQCJZAJUOUQCAUAVCUNAUIUNCULUK
|
|
UBUPUHUMBUGULUCUDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c `
|
|
(Contributed by SF, 24-Jan-2015.) $)
|
|
elpw1101c $p |- (
|
|
A e. ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { { { { { { { { x } } } } } } } } } } } ) $=
|
|
( vy c1c cpw1 wcel cv csn wceq wex elpw1 wa df-rex elpw191c anbi1i 19.41v
|
|
wrex bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEEEEEEEEZEFBC
|
|
GZHZIZCUFQZBAGHHHHHHHHHZHZHZIZAJZCBUFKUJUGULIZUILZAJZCJZUOUJUGUFFZUILZCJU
|
|
SUICUFMVAURCVAUPAJZUILURUTVBUIAUGNOUPUIAPRSTUSUQCJZAJUOUQCAUAVCUNAUIUNCUL
|
|
UKUBUPUHUMBUGULUCUDUESTTT $.
|
|
|
|
$( Membership in ` ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c `
|
|
(Contributed by SF, 24-Jan-2015.) $)
|
|
elpw1111c $p |- (
|
|
A e. ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c <->
|
|
E. x A = { { { { { { { { { { { { x } } } } } } } } } } } } ) $=
|
|
( vy c1c cpw1 wcel csn wceq wrex wex elpw1 df-rex elpw1101c anbi1i 19.41v
|
|
cv wa bitr4i exbii bitri excom snex sneq eqeq2d ceqsexv ) BDEEEEEEEEEEZEF
|
|
BCPZGZHZCUFIZBAPGGGGGGGGGGZGZGZHZAJZCBUFKUJUGULHZUIQZAJZCJZUOUJUGUFFZUIQZ
|
|
CJUSUICUFLVAURCVAUPAJZUIQURUTVBUIAUGMNUPUIAORSTUSUQCJZAJUOUQCAUAVCUNAUIUN
|
|
CULUKUBUPUHUMBUGULUCUDUESTTT $.
|
|
$}
|
|
|
|
$( A unit power class is a subset of ` 1c ` . (Contributed by SF,
|
|
22-Jan-2015.) $)
|
|
pw1ss1c $p |- ~P1 A C_ 1c $=
|
|
( cpw1 cpw c1c cin df-pw1 inss2 eqsstri ) ABACZDEDAFIDGH $.
|
|
|
|
$( The empty class is not a singleton. (Contributed by SF, 22-Jan-2015.) $)
|
|
0nel1c $p |- -. (/) e. 1c $=
|
|
( vx c0 c1c wcel cv csn wceq wex wn cvv vex snprc eqcom bitri con1bii mpbir
|
|
nex el1c mtbir ) BCDBAEZFZGZAHUBAUBITJDZAKUCUBUCIUABGUBTLUABMNOPQABRS $.
|
|
|
|
$( Note that ` x ` is a dummy variable in the proof below. $)
|
|
$( Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47.
|
|
(The proof was shortened by Andrew Salmon, 29-Jun-2011.) (Contributed by
|
|
SF, 5-Aug-1993.) (Revised by SF, 29-Jun-2011.) $)
|
|
pw0 $p |- ~P (/) = { (/) } $=
|
|
( vx cv c0 wss cab wceq cpw csn ss0b abbii df-pw df-sn 3eqtr4i ) ABZCDZAENC
|
|
FZAECGCHOPANIJACKACLM $.
|
|
|
|
$( Compute the unit power class of ` (/) ` (Contributed by SF,
|
|
22-Jan-2015.) $)
|
|
pw10 $p |- ~P1 (/) = (/) $=
|
|
( vx c0 cpw1 cpw c1c cin csn df-pw1 ineq1i wceq cv wcel wn disj 0nel1c elsn
|
|
pw0 wb eleq1 sylbi mtbiri mprgbir 3eqtri ) BCBDZEFBGZEFZBBHUDUEEQIUFBJAKZEL
|
|
ZMAUEAUEENUGUELZUHBELZOUIUGBJUHUJRABPUGBESTUAUBUC $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( A condition for equality to unit power class. (Contributed by SF,
|
|
21-Jan-2015.) $)
|
|
eqpw1 $p |- ( A = ~P1 B <-> ( A C_ 1c /\ A. x ( { x } e. A <-> x e. B )
|
|
) ) $=
|
|
( vy cpw1 wceq c1c wss cv csn wcel wb pw1ss1c sseq1 wi bitr4i albii bitri
|
|
wal eleq1 mpbiri wral ssofeq mpan2 df-ral wex imbi1i 19.23v alcom bibi12d
|
|
el1c snex ceqsalv snelpw1 bibi2i syl6bb biadan2 ) BCEZFZBGHZAIZJZBKZVACKZ
|
|
LZASZUSUTURGHZCMZBURGNUAUTUSDIZBKZVIURKZLZDGUBZVFUTVGUSVMLVHDBURGUCUDVMVI
|
|
VBFZVLOZDSZASZVFVMVIGKZVLOZDSZVQVLDGUEVTVOASZDSVQVSWADVSVNAUFZVLOWAVRWBVL
|
|
AVIUKUGVNVLAUHPQVOADUIPRVPVEAVPVCVBURKZLZVEVLWDDVBVAULVNVJVCVKWCVIVBBTVIV
|
|
BURTUJUMWCVDVCVACUNUORQRUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Unit power class distributes over union. (Contributed by SF,
|
|
22-Jan-2015.) $)
|
|
pw1un $p |- ~P1 ( A u. B ) = ( ~P1 A u. ~P1 B ) $=
|
|
( vx vy cun cpw1 cv wceq wrex wcel rexun elpw1 elun orbi12i bitri 3bitr4i
|
|
csn wo eqriv ) CABEZFZAFZBFZEZCGZDGQHZDTIUFDAIZUFDBIZRZUEUAJUEUDJZUFDABKD
|
|
UETLUJUEUBJZUEUCJZRUIUEUBUCMUKUGULUHDUEALDUEBLNOPS $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Unit power class distributes over intersection. (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
pw1in $p |- ~P1 ( A i^i B ) = ( ~P1 A i^i ~P1 B ) $=
|
|
( vx vy cin cpw1 cv csn wceq wrex wcel wa ancom eleq1 snelpw1 syl6bb elin
|
|
anbi2d elpw1 3bitr4i syl6bbr pm5.32ri an12 3bitr3i rexbii2 anbi1i r19.41v
|
|
eqriv ) CABEZFZAFZBFZEZCGZDGZHZIZDUIJUQUNULKZLZDAJZUNUJKUNUMKZUQUSDUIAUOA
|
|
KZURLZUQLUQVCLUOUIKZUQLVBUSLVCUQMUQVCVDUQVCVBUOBKZLVDUQURVEVBUQURUPULKVEU
|
|
NUPULNUOBOPRUOABQUAUBUQVBURUCUDUEDUNUISUNUKKZURLUQDAJZURLVAUTVFVGURDUNASU
|
|
FUNUKULQUQURDAUGTTUH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
pw1sn.1 $e |- A e. _V $.
|
|
$( Compute the unit power class of a singleton. (Contributed by SF,
|
|
22-Jan-2015.) $)
|
|
pw1sn $p |- ~P1 { A } = { { A } } $=
|
|
( vx vy cpw1 cv wceq wrex wcel sneq eqeq2d rexsn elpw1 elsn 3bitr4i eqriv
|
|
csn ) CAQZEZRQZCFZDFZQZGZDRHUARGZUASIUATIUDUEDABUBAGUCRUAUBAJKLDUARMCRNOP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( The unit power class of a class is empty iff the class itself is empty.
|
|
(Contributed by SF, 22-Jan-2015.) $)
|
|
pw10b $p |- ( ~P1 A = (/) <-> A = (/) ) $=
|
|
( vx cpw1 c0 wceq wne cv wcel wex csn snelpw1 ne0i sylbir exlimiv necon4i
|
|
n0 sylbi pw1eq pw10 syl6eq impbii ) ACZDEADEZADUBDADFBGZAHZBIUBDFZBAPUEUF
|
|
BUEUDJZUBHUFUDAKUBUGLMNQOUCUBDCDADRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Two unit power classes are disjoint iff the classes themselves are
|
|
disjoint. (Contributed by SF, 26-Jan-2015.) $)
|
|
pw1disj $p |- ( ( ~P1 A i^i ~P1 B ) = (/) <-> ( A i^i B ) = (/) ) $=
|
|
( vx vy cpw1 cin c0 wceq cv wcel wn wral csn wi disj eleq1 notbid snelpw1
|
|
sylbi ralrimiv rspccv notbii 3imtr3g sylibr wrex elpw1 rsp imp syl5ibrcom
|
|
wa syl6bb rexlimdva syl5bi impbii ) AEZBEZFGHZABFGHZUQCIZBJZKZCALURUQVACA
|
|
UQUSMZUOJZVBUPJZKZUSAJVAUQDIZUPJZKZDUOLVCVENDUOUPOVHVEDVBUOVFVBHVGVDVFVBU
|
|
PPQUASUSARVDUTUSBRUBUCTCABOUDURUSUPJZKZCUOLUQURVJCUOUSUOJUSVFMZHZDAUEURVJ
|
|
DUSAUFURVLVJDAURVFAJZUJVJVLVFBJZKZURVMVOURVODALVMVONDABOVODAUGSUHVLVIVNVL
|
|
VIVKUPJVNUSVKUPPVFBRUKQUIULUMTCUOUPOUDUN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Cardinal one is the unit power class of the universe. (Contributed by
|
|
SF, 29-Jan-2015.) $)
|
|
df1c2 $p |- 1c = ~P1 _V $=
|
|
( vx vy c1c cvv cpw1 cv csn wceq wrex wcel rexv elpw1 el1c 3bitr4ri eqriv
|
|
wex ) ACDEZAFZBFGHZBDISBPRQJRCJSBKBRDLBRMNO $.
|
|
$}
|
|
|
|
$( Unit power set preserves subset. (Contributed by SF, 3-Feb-2015.) $)
|
|
pw1ss $p |- ( A C_ B -> ~P1 A C_ ~P1 B ) $=
|
|
( wss cpw c1c cin cpw1 sspwb ssrin sylbi df-pw1 3sstr4g ) ABCZADZEFZBDZEFZA
|
|
GBGMNPCOQCABHNPEIJAKBKL $.
|
|
|
|
${
|
|
$d A t $. $d A x $. $d B t $. $d B x $. $d t x $.
|
|
$( The unit power class operation is one-to-one. (Contributed by SF,
|
|
26-Feb-2015.) $)
|
|
pw111 $p |- ( ~P1 A = ~P1 B <-> A = B ) $=
|
|
( vt vx cv csn wceq cpw1 wcel wb wi wal eleq1 snelpw1 bitri albii c1c wss
|
|
snex pw1ss1c bibi12d ceqsalv bibi12i wral ssofeq mp2an df-ral el1c imbi1i
|
|
wex 19.23v bitr4i alcom dfcleq 3bitr4i ) CEZDEZFZGZUPAHZIZUPBHZIZJZKZCLZD
|
|
LZUQAIZUQBIZJZDLUTVBGZABGVFVJDVFURUTIZURVBIZJZVJVDVNCURUQSUSVAVLVCVMUPURU
|
|
TMUPURVBMUAUBVLVHVMVIUQANUQBNUCOPVKVDCQUDZVGUTQRVBQRVKVOJATBTCUTVBQUEUFVO
|
|
UPQIZVDKZCLZVGVDCQUGVRVEDLZCLVGVQVSCVQUSDUJZVDKVSVPVTVDDUPUHUIUSVDDUKULPV
|
|
ECDUMOOODABUNUO $.
|
|
$}
|
|
|
|
$( A unit power class is a subset of a power class. (Contributed by SF,
|
|
10-Mar-2015.) $)
|
|
pw1sspw $p |- ~P1 A C_ ~P A $=
|
|
( cpw1 cpw c1c cin df-pw1 inss1 eqsstri ) ABACZDEIAFIDGH $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Membership in a unit union. (Contributed by SF, 15-Mar-2015.) $)
|
|
eluni1g $p |- ( A e. V -> ( A e. U.1 B <-> { A } e. B ) ) $=
|
|
( vy vx cuni1 wcel wceq csn wex c1c cin cuni df-uni1 anbi1i bitr4i 3bitri
|
|
cv wa exbii eleq2i eluni elin ancom el1c 19.41v anbi2i 19.42v excom eleq2
|
|
an12 vex elsnc2 syl6bb eleq1 anbi12d ceqsexv eqcom eleq1d ceqsexgv syl5bb
|
|
snex sneq ) ABFZGZDRZAHZVFIZBGZSZDJZACGAIZBGZVEABKLZMZGAERZGZVPVNGZSZEJZV
|
|
KVDVOABNUAEAVNUBVTVQVPVHHZVPBGZSZSZDJZEJWDEJZDJVKVSWEEVSVQWCDJZSWEVRWGVQV
|
|
RWBVPKGZSWHWBSZWGVPBKUCWBWHUDWIWADJZWBSWGWHWJWBDVPUEOWAWBDUFPQUGVQWCDUHPT
|
|
WDEDUIWFVJDWFWAVQWBSZSZEJAVFHZVISZVJWDWLEVQWAWBUKTWKWNEVHVFVBWAVQWMWBVIWA
|
|
VQAVHGWMVPVHAUJAVFDULUMUNVPVHBUOUPUQWMVGVIAVFUROQTQQVIVMDACVGVHVLBVFAVCUS
|
|
UTVA $.
|
|
$}
|
|
|
|
${
|
|
eluni1.1 $e |- A e. _V $.
|
|
$( Membership in a unit union. (Contributed by SF, 15-Mar-2015.) $)
|
|
eluni1 $p |- ( A e. U.1 B <-> { A } e. B ) $=
|
|
( cvv wcel cuni1 csn wb eluni1g ax-mp ) ADEABFEAGBEHCABDIJ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Kuratowski relationships
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c X._k $. $( Times symbol (cross product symbol) (read: 'cross') $)
|
|
$c `'_k $. $( Small elevated smiley (converse operation) $)
|
|
$c "_k $. $( Left quote (image symbol) $)
|
|
$c o._k $. $( Small circle (composition symbol) $)
|
|
$c Ins2_k $. $( Insert two operator. $)
|
|
$c Ins3_k $. $( Insert three operator. $)
|
|
$c P6 $. $( P6 operator $)
|
|
$c SI_k $. $( Singleton image functor $)
|
|
$c _S_k $. $( Subset relationship $)
|
|
$c Image_k $. $( Image functor $)
|
|
$c _I_k $. $( Identity relationship $)
|
|
|
|
$( Extend the definition of a class to include the Kuratowski cross
|
|
product. $)
|
|
cxpk $a class ( A X._k B ) $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski converse of a
|
|
class. $)
|
|
ccnvk $a class `'_k A $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski second
|
|
insertion operator. $)
|
|
cins2k $a class Ins2_k A $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski third insertion
|
|
operator. $)
|
|
cins3k $a class Ins3_k A $.
|
|
|
|
$( Extend the definition of a class to include the P6 operator (the set
|
|
guaranteed by ~ ax-typlower ). $)
|
|
cp6 $a class P6 A $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski image of a
|
|
class. (Read: The image of ` B ` under ` A ` .) $)
|
|
cimak $a class ( A "_k B ) $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski composition of
|
|
two classes. (Read: The composition of ` A ` and ` B ` .) $)
|
|
ccomk $a class ( A o._k B ) $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski singleton
|
|
image. $)
|
|
csik $a class SI_k A $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski image
|
|
functor. $)
|
|
cimagek $a class Image_k A $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski subset
|
|
relationship. $)
|
|
cssetk $a class _S_k $.
|
|
|
|
$( Extend the definition of a class to include the Kuratowski identity
|
|
relationship. $)
|
|
cidk $a class _I_k $.
|
|
|
|
${
|
|
$d A x y z t u v $. $d B x y z t u v $.
|
|
$( Define the Kuratowski cross product. This definition through ~ df-idk
|
|
set up the Kuratowski relationships. These are used mainly to prove the
|
|
properties of ~ df-op , and are not used thereafter. (Contributed by
|
|
SF, 12-Jan-2015.) $)
|
|
df-xpk $a |- ( A X._k B ) =
|
|
{ x | E. y E. z ( x = << y , z >> /\ ( y e. A /\ z e. B ) ) } $.
|
|
|
|
$( Define the Kuratowski converse. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-cnvk $a |- `'_k A =
|
|
{ x | E. y E. z ( x = << y , z >> /\ << z , y >> e. A ) } $.
|
|
|
|
$( Define the Kuratowski second insertion operator. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
df-ins2k $a |- Ins2_k A =
|
|
{ x | E. y E. z ( x = << y , z >> /\
|
|
E. t E. u E. v ( y = { { t } } /\ z = << u , v >> /\
|
|
<< t , v >> e. A ) ) } $.
|
|
|
|
$( Define the Kuratowski third insertion operator. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
df-ins3k $a |- Ins3_k A =
|
|
{ x | E. y E. z ( x = << y , z >> /\
|
|
E. t E. u E. v ( y = { { t } } /\ z = << u , v >> /\
|
|
<< t , u >> e. A ) ) } $.
|
|
|
|
$( Define the Kuratowski image operator. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
df-imak $a |- ( A "_k B ) = { x | E. y e. B << y , x >> e. A } $.
|
|
|
|
$( Define the Kuratowski composition operator. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
df-cok $a |- ( A o._k B ) =
|
|
( ( Ins2_k A i^i Ins3_k `'_k B ) "_k _V ) $.
|
|
|
|
$( Define the P6 operator. This is the set guaranteed by ~ ax-typlower .
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
df-p6 $a |- P6 A = { x | ( _V X._k { { x } } ) C_ A } $.
|
|
|
|
$( Define the Kuratowski singleton image operation. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
df-sik $a |- SI_k A = { x |
|
|
E. y E. z ( x = << y , z >> /\
|
|
E. t E. u ( y = { t } /\ z = { u } /\ << t , u >> e. A ) ) } $.
|
|
|
|
$( Define the Kuratowski subset relationship. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
df-ssetk $a |- _S_k = { x | E. y E. z ( x = << y , z >> /\ y C_ z ) } $.
|
|
|
|
$( Define the Kuratowski image function. See ~ opkelimagek for
|
|
membership. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-imagek $a |- Image_k A =
|
|
( ( _V X._k _V ) \
|
|
( ( Ins2_k _S_k (+) Ins3_k ( _S_k o._k `'_k SI_k A ) ) "_k
|
|
~P1 ~P1 1c ) ) $.
|
|
|
|
$( Define the Kuratowski identity relationship. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
df-idk $a |- _I_k = { x | E. y E. z ( x = << y , z >> /\ y = z ) } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $. $d B x y z w $. $d C x y z w $.
|
|
|
|
$( Membership in a Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
elxpk $p |- ( A e. ( B X._k C ) <-> E. x E. y ( A = << x , y >> /\
|
|
( x e. B /\ y e. C ) ) ) $=
|
|
( vw cxpk wcel cvv cv copk wceq wa wex opkex eleq1 mpbiri adantr exlimivv
|
|
elex eqeq1 anbi1d 2exbidv df-xpk elab2g pm5.21nii ) CDEGZHCIHZCAJZBJZKZLZ
|
|
UIDHUJEHMZMZBNANZCUGTUNUHABULUHUMULUHUKIHUIUJOCUKIPQRSFJZUKLZUMMZBNANUOFC
|
|
UGIUPCLZURUNABUSUQULUMUPCUKUAUBUCFABDEUDUEUF $.
|
|
|
|
$( Membership in a cross product. (Contributed by SF, 12-Jan-2015.) $)
|
|
elxpk2 $p |- ( A e. ( B X._k C ) <->
|
|
E. x e. B E. y e. C A = << x , y >> ) $=
|
|
( cv wcel wa copk wceq wex wrex cxpk ancom 2exbii r2ex elxpk 3bitr4ri ) A
|
|
FZDGBFZEGHZCSTIJZHZBKAKUBUAHZBKAKUBBELADLCDEMGUCUDABUAUBNOUBABDEPABCDEQR
|
|
$.
|
|
|
|
$d x y z A $. $d x y z B $. $d x y z C $.
|
|
$( Equality theorem for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq1 $p |- ( A = B -> ( A X._k C ) = ( B X._k C ) ) $=
|
|
( vx vy vz wceq cxpk cv copk wrex wcel rexeq elxpk2 3bitr4g eqrdv ) ABGZD
|
|
ACHZBCHZQDIZEIFIJGFCKZEAKUAEBKTRLTSLUAEABMEFTACNEFTBCNOP $.
|
|
|
|
$( Equality theorem for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq2 $p |- ( A = B -> ( C X._k A ) = ( C X._k B ) ) $=
|
|
( vx vy vz wceq cxpk cv copk wrex wcel rexeq rexbidv elxpk2 3bitr4g eqrdv
|
|
) ABGZDCAHZCBHZRDIZEIFIJGZFAKZECKUBFBKZECKUASLUATLRUCUDECUBFABMNEFUACAOEF
|
|
UACBOPQ $.
|
|
$}
|
|
|
|
$( Equality theorem for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq12 $p |- ( ( A = B /\ C = D ) -> ( A X._k C ) = ( B X._k D ) ) $=
|
|
( wceq cxpk xpkeq1 xpkeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
xpkeq1i.1 $e |- A = B $.
|
|
$( Equality inference for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq1i $p |- ( A X._k C ) = ( B X._k C ) $=
|
|
( wceq cxpk xpkeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq2i $p |- ( C X._k A ) = ( C X._k B ) $=
|
|
( wceq cxpk xpkeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
xpkeq12i.1 $e |- A = B $.
|
|
xpkeq12i.2 $e |- C = D $.
|
|
$( Equality inference for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq12i $p |- ( A X._k C ) = ( B X._k D ) $=
|
|
( wceq cxpk xpkeq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
xpkeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq1d $p |- ( ph -> ( A X._k C ) = ( B X._k C ) ) $=
|
|
( wceq cxpk xpkeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq2d $p |- ( ph -> ( C X._k A ) = ( C X._k B ) ) $=
|
|
( wceq cxpk xpkeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
xpkeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for Kuratowski cross product. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
xpkeq12d $p |- ( ph -> ( A X._k C ) = ( B X._k D ) ) $=
|
|
( wceq cxpk xpkeq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Membership in ` ( _V X._k _V ) ` (Contributed by SF, 13-Jan-2015.) $)
|
|
elvvk $p |- ( A e. ( _V X._k _V ) <-> E. x E. y A = << x , y >> ) $=
|
|
( cvv cxpk wcel cv copk wceq wex elxpk vex pm3.2i biantru 2exbii bitr4i
|
|
wa ) CDDEFCAGZBGZHIZRDFZSDFZQZQZBJAJTBJAJABCDDKTUDABUCTUAUBALBLMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d t w $. $d t y $. $d t z $. $d w y $. $d w z $. $d x y $. $d x z $.
|
|
$( Any Kuratowski ordered pair abstraction is a subset of
|
|
` ( _V X._k _V ) ` . (Contributed by SF, 13-Jan-2015.) $)
|
|
opkabssvvk $p |- { x | E. y E. z ( x = << y , z >> /\ ph ) } C_
|
|
( _V X._k _V ) $=
|
|
( vw vt cv copk wceq wex cvv cxpk wcel eqid vex weq opkeq12 eqeq2d spc2ev
|
|
wa ax-mp elvvk mpbir eleq1 mpbiri adantr exlimivv abssi ) BGZCGZDGZHZIZAT
|
|
ZDJCJBKKLZUNUIUOMZCDUMUPAUMUPULUOMZUQULEGZFGZHZIZFJEJZULULIZVBULNVAVCEFUJ
|
|
UKCODOECPFDPTUTULULURUSUJUKQRSUAEFULUBUCUIULUOUDUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d x z $.
|
|
opkabssvvki.1 $e |- A = { x | E. y E. z ( x = << y , z >> /\ ph ) } $.
|
|
$( Any Kuratowski ordered pair abstraction is a subset of
|
|
` ( _V X._k _V ) ` . (Contributed by SF, 13-Jan-2015.) $)
|
|
opkabssvvki $p |- A C_ ( _V X._k _V ) $=
|
|
( cv copk wceq wa wex cab cvv cxpk opkabssvvk eqsstri ) EBGCGDGHIAJDKCKBL
|
|
MMNFABCDOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Any Kuratowski cross product is a subset of ` ( _V X._k _V ) ` .
|
|
(Contributed by SF, 13-Jan-2015.) $)
|
|
xpkssvvk $p |- ( A X._k B ) C_ ( _V X._k _V ) $=
|
|
( vy vz vx cv wcel wa cxpk df-xpk opkabssvvki ) CFAGDFBGHECDABIECDABJK $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Subset for Kuratowski relationships. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
ssrelk $p |- ( A C_ ( _V X._k _V ) ->
|
|
( A C_ B <-> A. x A. y ( << x , y >> e. A -> << x , y >> e. B ) ) ) $=
|
|
( vz cvv cxpk wss cv wcel wi wral copk wal ssofss wceq df-ral bitri eleq1
|
|
wex elvvk imbi1i 19.23vv bitr4i albii alrot3 opkex imbi12d ceqsalv 2albii
|
|
syl6bb ) CFFGZHCDHEIZCJZUMDJZKZEULLZAIZBIZMZCJZUTDJZKZBNANZECDULOUQUMUTPZ
|
|
UPKZENZBNANZVDUQUMULJZUPKZENZVHUPEULQVKVFBNANZENVHVJVLEVJVEBTATZUPKVLVIVM
|
|
UPABUMUAUBVEUPABUCUDUEVFEABUFRRVGVCABUPVCEUTURUSUGVEUNVAUOVBUMUTCSUMUTDSU
|
|
HUIUJRUK $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Equality for two Kuratowski relationships. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
eqrelk $p |- ( ( A C_ ( _V X._k _V ) /\ B C_ ( _V X._k _V ) ) ->
|
|
( A = B <-> A. x A. y ( << x , y >> e. A <-> << x , y >> e. B ) ) ) $=
|
|
( vz cvv cxpk wss wa wceq cv wcel wb wral copk wal ssofeq wi wex eleq1
|
|
df-ral elvvk imbi1i 19.23vv bitr4i albii alrot3 bitri opkex 2albii 3bitri
|
|
bibi12d ceqsalv syl6bb ) CFFGZHDUOHICDJEKZCLZUPDLZMZEUONZAKZBKZOZCLZVCDLZ
|
|
MZBPAPZECDUOQUTUPUOLZUSRZEPZUPVCJZUSRZEPZBPAPZVGUSEUOUAVJVLBPAPZEPVNVIVOE
|
|
VIVKBSASZUSRVOVHVPUSABUPUBUCVKUSABUDUEUFVLEABUGUHVMVFABUSVFEVCVAVBUIVKUQV
|
|
DURVEUPVCCTUPVCDTULUMUJUKUN $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
eqrelkriiv.1 $e |- A C_ ( _V X._k _V ) $.
|
|
eqrelkriiv.2 $e |- B C_ ( _V X._k _V ) $.
|
|
${
|
|
eqrelkriiv.3 $e |- ( << x , y >> e. A <-> << x , y >> e. B ) $.
|
|
$( Equality for two Kuratowski relationships. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
eqrelkriiv $p |- A = B $=
|
|
( wceq cv copk wcel wb wal gen2 cvv cxpk wss eqrelk mp2an mpbir ) CDHZA
|
|
IBIJZCKUBDKLZBMAMZUCABGNCOOPZQDUEQUAUDLEFABCDRST $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y $.
|
|
eqrelkrdv.3 $e |- ( ph -> ( << x , y >> e. A <-> << x , y >> e. B ) ) $.
|
|
$( Equality for two Kuratowski relationships. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
eqrelkrdv $p |- ( ph -> A = B ) $=
|
|
( cv copk wcel wb wal wceq alrimivv cvv cxpk wss eqrelk mp2an sylibr )
|
|
ABICIJZDKUBEKLZCMBMZDENZAUCBCHODPPQZREUFRUEUDLFGBCDESTUA $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Equality theorem for Kuratowski converse. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
cnvkeq $p |- ( A = B -> `'_k A = `'_k B ) $=
|
|
( vx vy vz wceq cv copk wcel wa wex cab ccnvk eleq2 anbi2d 2exbidv abbidv
|
|
df-cnvk 3eqtr4g ) ABFZCGDGZEGZHFZUBUAHZAIZJZEKDKZCLUCUDBIZJZEKDKZCLAMBMTU
|
|
GUJCTUFUIDETUEUHUCABUDNOPQCDEARCDEBRS $.
|
|
$}
|
|
|
|
${
|
|
cnvkeqi.1 $e |- A = B $.
|
|
$( Equality inference for Kuratowski converse. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
cnvkeqi $p |- `'_k A = `'_k B $=
|
|
( wceq ccnvk cnvkeq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
cnvkeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for Kuratowski converse. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
cnvkeqd $p |- ( ph -> `'_k A = `'_k B ) $=
|
|
( wceq ccnvk cnvkeq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w t u $. $d B x y z w t u $.
|
|
$( Equality theorem for the Kuratowski insert two operator. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
ins2keq $p |- ( A = B -> Ins2_k A = Ins2_k B ) $=
|
|
( vx vy vz vw vt vu wceq cv copk csn w3a wex wa cab cins2k eleq2 df-ins2k
|
|
wcel 3anbi3d 3exbidv anbi2d 2exbidv abbidv 3eqtr4g ) ABIZCJDJZEJZKIZUHFJZ
|
|
LLIZUIGJHJZKIZUKUMKZATZMZHNGNFNZOZENDNZCPUJULUNUOBTZMZHNGNFNZOZENDNZCPAQB
|
|
QUGUTVECUGUSVDDEUGURVCUJUGUQVBFGHUGUPVAULUNABUORUAUBUCUDUECDEHGFASCDEHGFB
|
|
SUF $.
|
|
|
|
$( Equality theorem for the Kuratowski insert three operator. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
ins3keq $p |- ( A = B -> Ins3_k A = Ins3_k B ) $=
|
|
( vx vy vz vw vt vu wceq cv copk csn w3a wex wa cab cins3k eleq2 df-ins3k
|
|
wcel 3anbi3d 3exbidv anbi2d 2exbidv abbidv 3eqtr4g ) ABIZCJDJZEJZKIZUHFJZ
|
|
LLIZUIGJZHJKIZUKUMKZATZMZHNGNFNZOZENDNZCPUJULUNUOBTZMZHNGNFNZOZENDNZCPAQB
|
|
QUGUTVECUGUSVDDEUGURVCUJUGUQVBFGHUGUPVAULUNABUORUAUBUCUDUECDEHGFASCDEHGFB
|
|
SUF $.
|
|
$}
|
|
|
|
${
|
|
inskeqi.1 $e |- A = B $.
|
|
$( Equality inference for Kuratowski insert two operator. (Contributed by
|
|
SF, 12-Jan-2015.) $)
|
|
ins2keqi $p |- Ins2_k A = Ins2_k B $=
|
|
( wceq cins2k ins2keq ax-mp ) ABDAEBEDCABFG $.
|
|
|
|
$( Equality inference for Kuratowski insert three operator. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
ins3keqi $p |- Ins3_k A = Ins3_k B $=
|
|
( wceq cins3k ins3keq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
inskeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for Kuratowski insert two operator. (Contributed by
|
|
SF, 12-Jan-2015.) $)
|
|
ins2keqd $p |- ( ph -> Ins2_k A = Ins2_k B ) $=
|
|
( wceq cins2k ins2keq syl ) ABCEBFCFEDBCGH $.
|
|
|
|
$( Equality deduction for Kuratowski insert three operator. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
ins3keqd $p |- ( ph -> Ins3_k A = Ins3_k B ) $=
|
|
( wceq cins3k ins3keq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $.
|
|
$( Equality theorem for Kuratowski image. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
imakeq1 $p |- ( A = B -> ( A "_k C ) = ( B "_k C ) ) $=
|
|
( vy vx wceq cv copk wcel wrex cimak eleq2 rexbidv abbidv df-imak 3eqtr4g
|
|
cab ) ABFZDGEGHZAIZDCJZEQSBIZDCJZEQACKBCKRUAUCERTUBDCABSLMNEDACOEDBCOP $.
|
|
|
|
$( Equality theorem for Kuratowski image. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
imakeq2 $p |- ( A = B -> ( C "_k A ) = ( C "_k B ) ) $=
|
|
( vy vx wceq cv copk wcel wrex cab cimak rexeq abbidv df-imak 3eqtr4g ) A
|
|
BFZDGEGHCIZDAJZEKRDBJZEKCALCBLQSTERDABMNEDCAOEDCBOP $.
|
|
$}
|
|
|
|
${
|
|
imakeq1i.1 $e |- A = B $.
|
|
$( Equality theorem for image. (Contributed by SF, 12-Jan-2015.) $)
|
|
imakeq1i $p |- ( A "_k C ) = ( B "_k C ) $=
|
|
( wceq cimak imakeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality theorem for Kuratowski image. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
imakeq2i $p |- ( C "_k A ) = ( C "_k B ) $=
|
|
( wceq cimak imakeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
imakeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for Kuratowski image. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
imakeq1d $p |- ( ph -> ( A "_k C ) = ( B "_k C ) ) $=
|
|
( wceq cimak imakeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality theorem for Kuratowski image. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
imakeq2d $p |- ( ph -> ( C "_k A ) = ( C "_k B ) ) $=
|
|
( wceq cimak imakeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
$}
|
|
|
|
$( Equality theorem for Kuratowski composition of two classes. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
cokeq1 $p |- ( A = B -> ( A o._k C ) = ( B o._k C ) ) $=
|
|
( wceq cins2k ccnvk cins3k cin cimak ins2keq ineq1d imakeq1d df-cok 3eqtr4g
|
|
cvv ccomk ) ABDZAEZCFGZHZOIBEZSHZOIACPBCPQTUBOQRUASABJKLACMBCMN $.
|
|
|
|
$( Equality theorem for Kuratowski composition of two classes. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
cokeq2 $p |- ( A = B -> ( C o._k A ) = ( C o._k B ) ) $=
|
|
( wceq cins2k ccnvk cins3k cin cimak cnvkeq ins3keqd ineq2d imakeq1d df-cok
|
|
cvv ccomk 3eqtr4g ) ABDZCEZAFZGZHZOISBFZGZHZOICAPCBPRUBUEORUAUDSRTUCABJKLMC
|
|
ANCBNQ $.
|
|
|
|
${
|
|
cokeq1i.1 $e |- A = B $.
|
|
$( Equality inference for Kuratowski composition of two classes.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
cokeq1i $p |- ( A o._k C ) = ( B o._k C ) $=
|
|
( wceq ccomk cokeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for Kuratowski composition of two classes.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
cokeq2i $p |- ( C o._k A ) = ( C o._k B ) $=
|
|
( wceq ccomk cokeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
cokeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for Kuratowski composition of two classes.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
cokeq1d $p |- ( ph -> ( A o._k C ) = ( B o._k C ) ) $=
|
|
( wceq ccomk cokeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for Kuratowski composition of two classes.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
cokeq2d $p |- ( ph -> ( C o._k A ) = ( C o._k B ) ) $=
|
|
( wceq ccomk cokeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
cokeq12i.1 $e |- A = B $.
|
|
cokeq12i.2 $e |- C = D $.
|
|
$( Equality inference for Kuratowski composition of two classes.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
cokeq12i $p |- ( A o._k C ) = ( B o._k D ) $=
|
|
( ccomk cokeq1i cokeq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
|
|
${
|
|
cokeq12d.1 $e |- ( ph -> A = B ) $.
|
|
cokeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for Kuratowski composition of two classes.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
cokeq12d $p |- ( ph -> ( A o._k C ) = ( B o._k D ) ) $=
|
|
( ccomk cokeq1d cokeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Equality theorem for P6 operation. (Contributed by SF, 12-Jan-2015.) $)
|
|
p6eq $p |- ( A = B -> P6 A = P6 B ) $=
|
|
( vx wceq cvv cv csn cxpk wss cab cp6 sseq2 abbidv df-p6 3eqtr4g ) ABDZEC
|
|
FGGHZAIZCJQBIZCJAKBKPRSCABQLMCANCBNO $.
|
|
$}
|
|
|
|
${
|
|
p6eqi.1 $e |- A = B $.
|
|
$( Equality inference for the P6 operation. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
p6eqi $p |- P6 A = P6 B $=
|
|
( wceq cp6 p6eq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
p6eqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for the P6 operation. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
p6eqd $p |- ( ph -> P6 A = P6 B ) $=
|
|
( wceq cp6 p6eq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w t $. $d B x y z w t $.
|
|
$( Equality theorem for Kuratowski singleton image. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
sikeq $p |- ( A = B -> SI_k A = SI_k B ) $=
|
|
( vx vy vz vw vt wceq cv copk csn wcel w3a wex wa cab csik 2exbidv df-sik
|
|
eleq2 3anbi3d anbi2d abbidv 3eqtr4g ) ABHZCIDIZEIZJHZUFFIZKHZUGGIZKHZUIUK
|
|
JZALZMZGNFNZOZENDNZCPUHUJULUMBLZMZGNFNZOZENDNZCPAQBQUEURVCCUEUQVBDEUEUPVA
|
|
UHUEUOUTFGUEUNUSUJULABUMTUARUBRUCCDEGFASCDEGFBSUD $.
|
|
$}
|
|
|
|
${
|
|
sikeqi.1 $e |- A = B $.
|
|
$( Equality inference for Kuratowski singleton image. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
sikeqi $p |- SI_k A = SI_k B $=
|
|
( wceq csik sikeq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
sikeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for Kuratowski singleton image. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
sikeqd $p |- ( ph -> SI_k A = SI_k B ) $=
|
|
( wceq csik sikeq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( Equality theorem for image operation. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
imagekeq $p |- ( A = B -> Image_k A = Image_k B ) $=
|
|
( wceq cvv cxpk cssetk cins2k csik ccnvk ccomk cins3k csymdif c1c cpw1 cdif
|
|
cimak cimagek sikeq cnvkeqd df-imagek cokeq2d ins3keqd symdifeq2d imakeq1d
|
|
difeq2d 3eqtr4g ) ABCZDDEZFGZFAHZIZJZKZLZMNNZPZOUHUIFBHZIZJZKZLZUOPZOAQBQUG
|
|
UPVBUHUGUNVAUOUGUMUTUIUGULUSUGUKURFUGUJUQABRSUAUBUCUDUEATBTUF $.
|
|
|
|
${
|
|
$d A y z $. $d B x $. $d B y $. $d B z $. $d C x $. $d C y $.
|
|
$d C z $. $d ch z $. $d ph x $. $d ps y $. $d x y z $.
|
|
opkelopkabg.1 $e |- A = { x | E. y E. z ( x = << y , z >> /\ ph ) } $.
|
|
opkelopkabg.2 $e |- ( y = B -> ( ph <-> ps ) ) $.
|
|
opkelopkabg.3 $e |- ( z = C -> ( ps <-> ch ) ) $.
|
|
$( Kuratowski ordered pair membership in an abstraction of Kuratowski
|
|
ordered pairs. (Contributed by SF, 12-Jan-2015.) $)
|
|
opkelopkabg $p |- ( ( B e. V /\ C e. W ) ->
|
|
( << B , C >> e. A <-> ch ) ) $=
|
|
( wcel wceq wa wex cvv wb copk cv opkex eqeq1 eqcom syl6bb anbi1d 2exbidv
|
|
elab2 elex vex opkthg mp3an12 adantl exbidv 19.42v anbi2d ceqsexgv adantr
|
|
anass 3bitrd syl2an syl5bb ) HIUAZGOEUBZFUBZUAZVDPZAQZFRZERZHJOZIKOZQCDUB
|
|
ZVGPZAQZFRERVKDVDGHIUCVNVDPZVPVIEFVQVOVHAVQVOVDVGPVHVNVDVGUDVDVGUEUFUGUHL
|
|
UIVLHSOZISOZVKCTVMHJUJIKUJVRVSQZVKVEHPZVFIPZAQZFRZQZERZWBBQZFRZCVTVJWEEVT
|
|
VJWAWCQZFRWEVTVIWIFVTVIWAWBQZAQWIVTVHWJAVSVHWJTZVRVESOVFSOVSWKEUKFUKVEVFH
|
|
ISSSULUMUNUGWAWBAUTUFUOWAWCFUPUFUOVRWFWHTVSWDWHEHSWAWCWGFWAABWBMUQUOURUSV
|
|
SWHCTVRBCFISNURUNVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d A z $. $d B x $. $d B y $. $d B z $. $d C x $. $d C y $.
|
|
$d C z $. $d ch z $. $d ph x $. $d ps y $. $d x y $. $d x z $.
|
|
$d y z $.
|
|
opkelopkab.1 $e |- A = { x | E. y E. z ( x = << y , z >> /\ ph ) } $.
|
|
opkelopkab.2 $e |- ( y = B -> ( ph <-> ps ) ) $.
|
|
opkelopkab.3 $e |- ( z = C -> ( ps <-> ch ) ) $.
|
|
opkelopkab.4 $e |- B e. _V $.
|
|
opkelopkab.5 $e |- C e. _V $.
|
|
$( Kuratowski ordered pair membership in an abstraction of Kuratowski
|
|
ordered pairs. (Contributed by SF, 12-Jan-2015.) $)
|
|
opkelopkab $p |- ( << B , C >> e. A <-> ch ) $=
|
|
( cvv wcel copk wb opkelopkabg mp2an ) HOPIOPHIQGPCRMNABCDEFGHIOOJKLST $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $.
|
|
$( Kuratowski ordered pair membership in a Kuratowski cross product.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
opkelxpkg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. ( C X._k D ) <-> ( A e. C /\ B e. D ) ) ) $=
|
|
( vx vy vz cv wcel wa cxpk df-xpk wceq eleq1 anbi1d anbi2d opkelopkabg )
|
|
GJZCKZHJZDKZLACKZUCLUDBDKZLIGHCDMABEFIGHCDNTAOUAUDUCTACPQUBBOUCUEUDUBBDPR
|
|
S $.
|
|
$}
|
|
|
|
${
|
|
opkelxpk.1 $e |- A e. _V $.
|
|
opkelxpk.2 $e |- B e. _V $.
|
|
$( Kuratowski ordered pair membership in a Kuratowski cross product.
|
|
(Contributed by SF, 13-Jan-2015.) $)
|
|
opkelxpk $p |- ( << A , B >> e. ( C X._k D ) <-> ( A e. C /\ B e. D ) ) $=
|
|
( cvv wcel copk cxpk wa wb opkelxpkg mp2an ) AGHBGHABICDJHACHBDHKLEFABCDG
|
|
GMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $.
|
|
$( Kuratowski ordered pair membership in a Kuratowski converse.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
opkelcnvkg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. `'_k C <-> << B , A >> e. C ) ) $=
|
|
( vy vx vz copk wcel ccnvk df-cnvk wceq opkeq2 eleq1d opkeq1 opkelopkabg
|
|
cv ) FRZGRZIZCJSAIZCJBAIZCJHGFCKABDEHGFCLTAMUAUBCTASNOSBMUBUCCSBAPOQ $.
|
|
$}
|
|
|
|
${
|
|
opkelcnvk.1 $e |- A e. _V $.
|
|
opkelcnvk.2 $e |- B e. _V $.
|
|
$( Kuratowski ordered pair membership in a Kuratowski converse.
|
|
(Contributed by SF, 14-Jan-2015.) $)
|
|
opkelcnvk $p |- ( << A , B >> e. `'_k C <-> << B , A >> e. C ) $=
|
|
( cvv wcel copk ccnvk wb opkelcnvkg mp2an ) AFGBFGABHCIGBAHCGJDEABCFFKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w t u $. $d B x y z w t u $. $d C x y z w t u $.
|
|
$( Kuratowski ordered pair membership in Kuratowski insertion operator.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
opkelins2kg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. Ins2_k C <->
|
|
E. x E. y E. z ( A = { { x } } /\ B = << y , z >> /\
|
|
<< x , z >> e. C ) ) ) $=
|
|
( vw vu vt cv csn wceq copk wcel w3a wex eqeq1 3exbidv cins2k opkelopkabg
|
|
df-ins2k 3anbi1d 3anbi2d ) ILZALZMMZNZJLZBLCLZOZNZUGUKOFPZQZCRBRARDUHNZUM
|
|
UNQZCRBRARUPEULNZUNQZCRBRARKIJFUADEGHKIJCBAFUCUFDNZUOUQABCUTUIUPUMUNUFDUH
|
|
SUDTUJENZUQUSABCVAUMURUPUNUJEULSUETUB $.
|
|
|
|
$( Kuratowski ordered pair membership in Kuratowski insertion operator.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
opkelins3kg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. Ins3_k C <->
|
|
E. x E. y E. z ( A = { { x } } /\ B = << y , z >> /\
|
|
<< x , y >> e. C ) ) ) $=
|
|
( vw vu vt cv csn wceq copk wcel w3a wex eqeq1 3exbidv cins3k opkelopkabg
|
|
df-ins3k 3anbi1d 3anbi2d ) ILZALZMMZNZJLZBLZCLOZNZUGUKOFPZQZCRBRARDUHNZUM
|
|
UNQZCRBRARUPEULNZUNQZCRBRARKIJFUADEGHKIJCBAFUCUFDNZUOUQABCUTUIUPUMUNUFDUH
|
|
SUDTUJENZUQUSABCVAUMURUPUNUJEULSUETUB $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $. $d T x y z $.
|
|
$( Kuratowski ordered triple membership in Kuratowski insertion operator.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
otkelins2kg $p |- ( ( A e. V /\ B e. W /\ C e. T ) ->
|
|
( << { { A } } , << B , C >> >> e. Ins2_k D <->
|
|
<< A , C >> e. D ) ) $=
|
|
( vx vy vz csn copk wcel cv wceq wa wex cvv wb bitri w3a snex opkelins2kg
|
|
cins2k opkex mp2an 3anass eqcom sneqb anbi1i 2exbii 19.42vv opkeq1 eleq1d
|
|
exbii anbi2d 2exbidv ceqsexgv 3ad2ant1 opkthg mp3an12 syl5bb anbi1d anass
|
|
vex 19.42v syl6bb adantl biidd opkeq2 sylan9bb bitrd 3adant1 ) AKZKZBCLZL
|
|
DUDMZHNZAOZVPINZJNZLZOZVRWALZDMZPZJQIQZPZHQZAFMZBGMZCEMZUAZACLZDMZVQVOVRK
|
|
ZKZOZWCWEUAZJQIQZHQZWIVORMVPRMVQXASVNUBBCUEHIJVOVPDRRUCUFWTWHHWTVSWFPZJQI
|
|
QWHWSXBIJWSWRWFPXBWRWCWEUGWRVSWFWRWQVOOZVSVOWQUHXCWPVNOVSWPVNVRUBUIVRAHVE
|
|
UITTUJTUKVSWFIJULTUOTWMWIWCAWALZDMZPZJQIQZWOWJWKWIXGSWLWGXGHAFVSWFXFIJVSW
|
|
EXEWCVSWDXDDVRAWAUMUNUPUQURUSWKWLXGWOSWJWKWLPXGVTBOZWACOZXEPZJQZPZIQZWOWL
|
|
XGXMSWKWLXGXHXIPZXEPZJQZIQXMWLXFXOIJWLWCXNXEWCWBVPOZWLXNVPWBUHVTRMWARMWLX
|
|
QXNSIVEJVEVTWABCERRUTVAVBVCUQXPXLIXPXHXJPZJQXLXOXRJXHXIXEVDUOXHXJJVFTUOVG
|
|
VHWKXMXKWLWOXKXKIBGXHXKVIURXEWOJCEXIXDWNDWACAVJUNURVKVLVMVLVB $.
|
|
|
|
$( Kuratowski ordered triple membership in Kuratowski insertion operator.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
otkelins3kg $p |- ( ( A e. V /\ B e. W /\ C e. T ) ->
|
|
( << { { A } } , << B , C >> >> e. Ins3_k D <->
|
|
<< A , B >> e. D ) ) $=
|
|
( vx vy vz csn copk wcel cv wceq wex cvv wb wa bitri w3a snex opkelins3kg
|
|
cins3k opkex mp2an 3anass eqcom sneqb anbi1i 2exbii 19.42vv opkeq1 eleq1d
|
|
exbii anbi2d 2exbidv ceqsexgv syl5bb 3ad2ant1 opkthg mp3an12 anbi1d anass
|
|
vex syl6bb exdistr adantl opkeq2 exbidv biidd sylan9bb bitrd 3adant1 ) AK
|
|
ZKZBCLZLDUDMZVPHNZKZKZOZVQINZJNZLZOZVSWCLZDMZUAZJPIPZHPZAFMZBGMZCEMZUAZAB
|
|
LZDMZVPQMVQQMVRWKRVOUBBCUEHIJVPVQDQQUCUFWOWKWFAWCLZDMZSZJPIPZWQWLWMWKXARW
|
|
NWKVSAOZWFWHSZJPIPZSZHPWLXAWJXEHWJXBXCSZJPIPXEWIXFIJWIWBXCSXFWBWFWHUGWBXB
|
|
XCWBWAVPOZXBVPWAUHXGVTVOOXBVTVOVSUBUIVSAHVEUITTUJTUKXBXCIJULTUOXDXAHAFXBX
|
|
CWTIJXBWHWSWFXBWGWRDVSAWCUMUNUPUQURUSUTWMWNXAWQRWLWMWNSXAWCBOZWDCOZWSSZJP
|
|
ZSIPZWQWNXAXLRWMWNXAXHXJSZJPIPXLWNWTXMIJWNWTXHXISZWSSXMWNWFXNWSWFWEVQOZWN
|
|
XNVQWEUHWCQMWDQMWNXOXNRIVEJVEWCWDBCEQQVAVBUSVCXHXIWSVDVFUQXHXJIJVGVFVHWMX
|
|
LXIWQSZJPZWNWQXKXQIBGXHXJXPJXHWSWQXIXHWRWPDWCBAVIUNUPVJURWQWQJCEXIWQVKURV
|
|
LVMVNVMUS $.
|
|
$}
|
|
|
|
${
|
|
otkelinsk.1 $e |- A e. _V $.
|
|
otkelinsk.2 $e |- B e. _V $.
|
|
otkelinsk.3 $e |- C e. _V $.
|
|
$( Kuratowski ordered triple membership in Kuratowski insertion operator.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
otkelins2k $p |- ( << { { A } } , << B , C >> >> e. Ins2_k D <->
|
|
<< A , C >> e. D ) $=
|
|
( cvv wcel csn copk cins2k wb otkelins2kg mp3an ) AHIBHICHIAJJBCKKDLIACKD
|
|
IMEFGABCDHHHNO $.
|
|
|
|
$( Kuratowski ordered triple membership in Kuratowski insertion operator.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
otkelins3k $p |- ( << { { A } } , << B , C >> >> e. Ins3_k D <->
|
|
<< A , B >> e. D ) $=
|
|
( cvv wcel csn copk cins3k wb otkelins3kg mp3an ) AHIBHICHIAJJBCKKDLIABKD
|
|
IMEFGABCDHHHNO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $.
|
|
$( Membership in a Kuratowski image. (Contributed by SF, 13-Jan-2015.) $)
|
|
elimakg $p |- ( C e. V ->
|
|
( C e. ( A "_k B ) <-> E. y e. B << y , C >> e. A ) ) $=
|
|
( vx cv copk wcel wrex cimak wceq opkeq2 eleq1d rexbidv df-imak elab2g )
|
|
AGZFGZHZBIZACJRDHZBIZACJFDBCKESDLZUAUCACUDTUBBSDRMNOFABCPQ $.
|
|
|
|
$( Membership in a Kuratowski image under ` _V ` . (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
elimakvg $p |- ( C e. V ->
|
|
( C e. ( A "_k _V ) <-> E. y << y , C >> e. A ) ) $=
|
|
( wcel cvv cimak cv copk wrex wex elimakg rexv syl6bb ) CDECBFGEAHCIBEZAF
|
|
JOAKABFCDLOAMN $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d C y $.
|
|
elimak.1 $e |- C e. _V $.
|
|
$( Membership in a Kuratowski image. (Contributed by SF, 13-Jan-2015.) $)
|
|
elimak $p |- ( C e. ( A "_k B ) <-> E. y e. B << y , C >> e. A ) $=
|
|
( cvv wcel cimak cv copk wrex wb elimakg ax-mp ) DFGDBCHGAIDJBGACKLEABCDF
|
|
MN $.
|
|
|
|
$( Membership in a Kuratowski image under ` _V ` . (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
elimakv $p |- ( C e. ( A "_k _V ) <-> E. y << y , C >> e. A ) $=
|
|
( cvv wcel cimak cv copk wex wb elimakvg ax-mp ) CEFCBEGFAHCIBFAJKDABCELM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $. $d B x y z w $. $d C x y z w $. $d D x y z w $.
|
|
$( Membership in a Kuratowski composition. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
opkelcokg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. ( C o._k D ) <->
|
|
E. x ( << A , x >> e. D /\ << x , B >> e. C ) ) ) $=
|
|
( vz vw vy wcel cvv copk cv wa wex wb wceq vex exbii ccomk elex csn ccnvk
|
|
cins3k cins2k cin cimak df-cok eleq2i opkex elimakv w3a opkelins2kg mp2an
|
|
3anass 2exbii 19.42vv bitri anbi1i elin 19.41v 3bitr4i excom anass opkeq1
|
|
snex eleq1d anbi2d 3bitri ancom otkelins3kg mp3an1 opkelcnvkg mpan adantr
|
|
ceqsexv bitrd eqcom opkthg mp3an12 anbi1d syl6bb 2exbidv adantl biantrurd
|
|
syl5bb elisset bicomd opkeq2 ceqsexgv sylan9bb anbi12d exbidv syl2an
|
|
eeanv ) BFKBLKZCLKZBCMZDEUAZKZBANZMEKZXBCMZDKZOZAPZQCGKBFUBCGUBXAWSHNZINZ
|
|
MZRZXBXIMZDKZOZIPHPZXBUCZUCZWSMZEUDZUEZKZOZAPZWQWROZXGXAWSDUFZXTUGZLUHZKJ
|
|
NZWSMZYFKZJPZYCWTYGWSDEUIUJJYFWSBCUKZULYKYHXQRZXOOZYIXTKZOZAPZJPYPJPZAPYC
|
|
YJYQJYIYEKZYOOYNAPZYOOYJYQYSYTYOYSYMXKXMUMZIPHPZAPZYTYHLKWSLKYSUUCQJSYLAH
|
|
IYHWSDLLUNUOUUBYNAUUBYMXNOZIPHPYNUUAUUDHIYMXKXMUPUQYMXNHIURUSTUSUTYIYEXTV
|
|
AYNYOAVBVCTYPJAVDYRYBAYRYMXOYOOZOZJPYBYPUUFJYMXOYOVETUUEYBJXQXPVGYMYOYAXO
|
|
YMYIXRXTYHXQWSVFVHVIVQUSTVJVJYDYBXFAYBYAXOOYDXFXOYAVKYDYAXCXOXEYDYAXBBMXS
|
|
KZXCXBLKZWQWRYAUUGQASZXBBCXSLLLVLVMWQUUGXCQZWRUUHWQUUJUUIXBBELLVNVOVPVRYD
|
|
XOXHBRZHPZXICRZXMOZIPZOZXEXOXJWSRZXMOZIPHPZYDUUPXNUURHIXKUUQXMWSXJVSUTUQY
|
|
DUUSUUKUUNOZIPHPZUUPWRUUSUVAQWQWRUURUUTHIWRUURUUKUUMOZXMOUUTWRUUQUVBXMXHL
|
|
KXILKWRUUQUVBQHSISXHXIBCLLLVTWAWBUUKUUMXMVEWCWDWEUUKUUNHIWPWCWGWQUUPUUOWR
|
|
XEWQUUOUUPWQUULUUOHBLWHWFWIXMXEICLUUMXLXDDXICXBWJVHWKWLVRWMWGWNWGWO $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $. $d D x $.
|
|
opkelcok.1 $e |- A e. _V $.
|
|
opkelcok.2 $e |- B e. _V $.
|
|
$( Membership in a Kuratowski composition. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
opkelcok $p |- ( << A , B >> e. ( C o._k D ) <->
|
|
E. x ( << A , x >> e. D /\ << x , B >> e. C ) ) $=
|
|
( cvv wcel copk ccomk cv wa wex wb opkelcokg mp2an ) BHICHIBCJDEKIBALZJEI
|
|
RCJDIMANOFGABCDEHHPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Membership in the P6 operator. (Contributed by SF, 13-Jan-2015.) $)
|
|
elp6 $p |- ( A e. V -> ( A e. P6 B <-> A. x << x , { A } >> e. B ) ) $=
|
|
( vy wcel cp6 cvv csn cxpk wss cv copk wal wceq sneq wi vex albii bitri
|
|
sneqd xpkeq2d sseq1d df-p6 elab2g xpkssvvk ssrelk ax-mp opkelxpk biantrur
|
|
wb wa df-sn abeq2i 3bitr2i imbi1i snex opkeq2 eleq1d ceqsalv syl6bb ) BDF
|
|
BCGZFHBIZIZJZCKZALZVCMZCFZANZHELZIZIZJZCKVFEBVBDVKBOZVNVECVOVMVDHVOVLVCVK
|
|
BPUAUBUCECUDUEVFVGVKMZVEFZVPCFZQZENZANZVJVEHHJKVFWAUKHVDUFAEVECUGUHVTVIAV
|
|
TVKVCOZVRQZENVIVSWCEVQWBVRVQVGHFZVKVDFZULWEWBVGVKHVDARZERUIWDWEWFUJWBEVDE
|
|
VCUMUNUOUPSVRVIEVCBUQWBVPVHCVKVCVGURUSUTTSTVA $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z t u $. $d B x y z t u $. $d C x y z t u $.
|
|
$( Membership in Kuratowski singleton image. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
opkelsikg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. SI_k C <->
|
|
E. x E. y ( A = { x } /\ B = { y } /\ << x , y >> e. C ) ) ) $=
|
|
( vt vu vz cv csn wceq copk wcel w3a wex csik eqeq1 2exbidv opkelopkabg
|
|
df-sik 3anbi1d 3anbi2d ) HKZAKZLZMZIKZBKZLZMZUFUJNEOZPZBQAQCUGMZULUMPZBQA
|
|
QUODUKMZUMPZBQAQJHIERCDFGJHIBAEUBUECMZUNUPABUSUHUOULUMUECUGSUCTUIDMZUPURA
|
|
BUTULUQUOUMUIDUKSUDTUA $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $.
|
|
opksnelsik.1 $e |- A e. _V $.
|
|
opksnelsik.2 $e |- B e. _V $.
|
|
$( Membership of an ordered pair of singletons in a Kuratowski singleton
|
|
image. (Contributed by SF, 13-Jan-2015.) $)
|
|
opksnelsik $p |- ( << { A } , { B } >> e. SI_k C <-> << A , B >> e. C ) $=
|
|
( vx vy csn copk wcel cv wceq w3a wex cvv snex eqcom vex sneqb bitri csik
|
|
wb opkelsikg mp2an biid 3anbi123i 2exbii opkeq1 eleq1d opkeq2 ceqsex2v )
|
|
AHZBHZICUAJZULFKZHZLZUMGKZHZLZUOURIZCJZMZGNFNZABIZCJZULOJUMOJUNVDUBAPBPFG
|
|
ULUMCOOUCUDVDUOALZURBLZVBMZGNFNVFVCVIFGUQVGUTVHVBVBUQUPULLVGULUPQUOAFRSTU
|
|
TUSUMLVHUMUSQURBGRSTVBUEUFUGVBAURIZCJVFFGABDEVGVAVJCUOAURUHUIVHVJVECURBAU
|
|
JUIUKTT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z t u $.
|
|
$( A Kuratowski singleton image is a Kuratowski relationship. (Contributed
|
|
by SF, 13-Jan-2015.) $)
|
|
sikssvvk $p |- SI_k A C_ ( _V X._k _V ) $=
|
|
( vy vt vz vu vx cv csn wceq copk wcel w3a wex csik df-sik opkabssvvki )
|
|
BGCGZHIDGEGZHIQRJAKLEMCMFBDANFBDECAOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w t a b $.
|
|
$( A Kuratowski singleton image is a subset of ` ( 1c X._k 1c ) ` .
|
|
(Contributed by SF, 13-Jan-2015.) $)
|
|
sikss1c1c $p |- SI_k A C_ ( 1c X._k 1c ) $=
|
|
( vx vy va vb vz vw vt c1c cxpk wss cv copk wcel wal csn wceq w3a wex vex
|
|
csik wi df-sik eqeq1 3anbi1d 2exbidv 3anbi2d opkelopkab wa opkeq12 snel1c
|
|
wb opkelxpkg mp2an mpbir2an syl6eqel 3adant3 exlimivv sylbi gen2 sikssvvk
|
|
weq cvv ssrelk ax-mp mpbir ) AUAZIIJZKZBLZCLZMZVGNZVLVHNZUBZCOBOZVOBCVMVJ
|
|
DLZPZQZVKELZPZQZVQVTMANZRZESDSZVNFLZVRQZGLZWAQZWCRZESDSVSWIWCRZESDSWEHFGV
|
|
GVJVKHFGEDAUCFBVBZWJWKDEWLWGVSWIWCWFVJVRUDUEUFGCVBZWKWDDEWMWIWBVSWCWHVKWA
|
|
UDUGUFBTCTUHWDVNDEVSWBVNWCVSWBUIVLVRWAMZVHVJVKVRWAUJWNVHNZVRINZWAINZVQDTU
|
|
KZVTETUKZWPWQWOWPWQUIULWRWSVRWAIIIIUMUNUOUPUQURUSUTVGVCVCJKVIVPULAVABCVGV
|
|
HVDVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Membership in the Kuratowski subset relationship. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
opkelssetkg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. _S_k <-> A C_ B ) ) $=
|
|
( vy vz vx cv wss cssetk df-ssetk sseq1 sseq2 opkelopkabg ) EHZFHZIAPIABI
|
|
GEFJABCDGEFKOAPLPBAMN $.
|
|
$}
|
|
|
|
$( Membership via the Kuratowski subset relationship. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
elssetkg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << { A } , B >> e. _S_k <-> A e. B ) ) $=
|
|
( wcel csn copk cssetk wss cvv snex opkelssetkg mpan snssg bicomd sylan9bbr
|
|
wb ) BDEZAFZBGHEZSBIZACEZABEZSJERTUAQAKSBJDLMUBUCUAABCNOP $.
|
|
|
|
${
|
|
elssetk.1 $e |- A e. _V $.
|
|
elssetk.2 $e |- B e. _V $.
|
|
$( Membership via the Kuratowski subset relationship. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
elssetk $p |- ( << { A } , B >> e. _S_k <-> A e. B ) $=
|
|
( cvv wcel csn copk cssetk wb elssetkg mp2an ) AEFBEFAGBHIFABFJCDABEEKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y $. $d C x y z $.
|
|
$( Membership in the Kuratowski image functor. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
opkelimagekg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. Image_k C <-> B = ( C "_k A ) ) ) $=
|
|
( vx vy vz wcel cvv copk wb wa cssetk c1c wn wex csn exbii bitri cimak cv
|
|
cimagek wceq elex cxpk cins2k csik ccnvk ccomk csymdif cpw1 wal opkelxpkg
|
|
cins3k ibir biantrurd exnal wrex opkex elimak df-rex anbi1i 19.41v bitr4i
|
|
elpw121c excom snex opkeq1 eleq1d ceqsexv 3bitri elsymdif otkelins2kg vex
|
|
mp3an1 elssetkg mpan adantl otkelins3kg opkelcokg anbi1d exbidv opkelcnvk
|
|
bitrd sikss1c1c sseli opkelxpk el1c biimpi adantr sylbi syl anass anbi12d
|
|
pm4.71ri opksnelsik anbi2i syl6bb 3bitr4g syl6bbr bibi12d syl5rbb syl5bbr
|
|
ancom notbid syl5bb con1bid bitr3d df-imagek eleq2i eldif dfcleq syl2an
|
|
cdif ) ADIAJIZBJIZABKZCUCZIZBCAUAZUDZLBEIADUEBEUEXPXQMZXRJJUFZIZXRNUGZNCU
|
|
HZUIZUJZUOZUKZOULULZUAZIZPZMZFUBZBIZYQYAIZLZFUMZXTYBYCYOYPUUAYCYEYOYCYEAB
|
|
JJJJUNUPUQYCUUAYNUUAPYTPZFQZYCYNYTFURYNYQRZRZRZXRKZYKIZFQZYCUUCYNGUBZXRKZ
|
|
YKIZGYLUSUUJYLIZUULMZGQZUUIGYKYLXRABUTVAUULGYLVBUUOUUJUUFUDZUULMZFQZGQUUQ
|
|
GQZFQUUIUUNUURGUUNUUPFQZUULMUURUUMUUTUULFUUJVFVCUUPUULFVDVESUUQGFVGUUSUUH
|
|
FUULUUHGUUFUUEVHUUPUUKUUGYKUUJUUFXRVIVJVKSVLVLYCUUHUUBFUUHUUGYFIZUUGYJIZL
|
|
ZPYCUUBUUGYFYJVMYCUVCYTYCUVAYRUVBYSYCUVAUUDBKNIZYRUUDJIZXPXQUVAUVDLYQVHZU
|
|
UDABNJJJVNVPXQUVDYRLZXPYQJIXQUVGFVOZYQBJJVQVRVSWEYCUVBUUDAKYIIZYSUVEXPXQU
|
|
VBUVILUVFUUDABYIJJJVTVPXPUVIYSLXQXPUVIUUJYQKCIZGAUSZYSXPUVIUUDHUBZKYHIZUV
|
|
LAKZNIZMZHQZUVKUVEXPUVIUVQLUVFHUUDANYHJJWAVRXPUUJRZAKZNIZUVJMZGQZUUJAIZUV
|
|
JMZGQUVQUVKXPUWAUWDGXPUVTUWCUVJUUJJIXPUVTUWCLGVOZUUJAJJVQVRWBWCUVQUVLUVRU
|
|
DZUVLUUDKZYGIZUVOMZMZGQZHQUWJHQZGQUWBUVPUWKHUVPUWFGQZUWHMZUVOMZUWKUVMUWNU
|
|
VOUVMUWHUWNUUDUVLYGUVFHVOZWDUWHUWMUWHUWGOOUFZIZUWMYGUWQUWGCWFWGUWRUVLOIZU
|
|
UDOIZMUWMUVLUUDOOUWPUVFWHUWSUWMUWTUWSUWMGUVLWIWJWKWLWMWPTVCUWOUWMUWIMUWKU
|
|
WMUWHUVOWNUWFUWIGVDVETSUWJHGVGUWLUWAGUWIUWAHUVRUUJVHUWFUWIUVRUUDKZYGIZUVT
|
|
MZUWAUWFUWHUXBUVOUVTUWFUWGUXAYGUVLUVRUUDVIVJUWFUVNUVSNUVLUVRAVIVJWOUXCUVT
|
|
UXBMUWAUXBUVTXEUXBUVJUVTUUJYQCUWEUVHWQWRTWSVKSVLUVJGAVBWTWEGCAYQUVHVAXAWK
|
|
WEXBXFXGWCXCXDXHXIXTXRYDYMXOZIYPXSUXDXRCXJXKXRYDYMXLTFBYAXMWTXN $.
|
|
$}
|
|
|
|
${
|
|
opkelimagek.1 $e |- A e. _V $.
|
|
opkelimagek.2 $e |- B e. _V $.
|
|
$( Membership in the Kuratowski image functor. (Contributed by SF,
|
|
20-Jan-2015.) $)
|
|
opkelimagek $p |- ( << A , B >> e. Image_k C <-> B = ( C "_k A ) ) $=
|
|
( cvv wcel copk cimagek cimak wceq wb opkelimagekg mp2an ) AFGBFGABHCIGBC
|
|
AJKLDEABCFFMN $.
|
|
$}
|
|
|
|
$( The Kuratowski image functor is a relationship. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
imagekrelk $p |- Image_k A C_ ( _V X._k _V ) $=
|
|
( cimagek cvv cxpk cssetk cins2k csik ccnvk ccomk cins3k csymdif cpw1 cimak
|
|
c1c cdif df-imagek difss eqsstri ) ABCCDZEFEAGHIJKNLLMZOSAPSTQR $.
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Membership in the Kuratowski identity relationship. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
opkelidkg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. _I_k <-> A = B ) ) $=
|
|
( vx vy vz weq cv wceq cidk df-idk eqeq1 eqeq2 opkelopkabg ) EFHAFIZJABJG
|
|
EFKABCDGEFLEIAPMPBANO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( A Kuratowski converse is a Kuratowski relationship. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
cnvkssvvk $p |- `'_k A C_ ( _V X._k _V ) $=
|
|
( vz vy vx cv copk wcel ccnvk df-cnvk opkabssvvki ) BECEFAGDCBAHDCBAIJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The converse of a Kuratowski cross product. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
cnvkxpk $p |- `'_k ( A X._k B ) = ( B X._k A ) $=
|
|
( vx vy cxpk ccnvk cnvkssvvk xpkssvvk cv wcel wa copk ancom vex opkelcnvk
|
|
opkelxpk bitri 3bitr4i eqrelkriiv ) CDABEZFZBAEZTGBAHDIZAJZCIZBJZKZUFUDKU
|
|
EUCLZUAJZUHUBJUDUFMUIUCUELTJUGUEUCTCNZDNZOUCUEABUKUJPQUEUCBAUJUKPRS $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $. $d D x y $.
|
|
$( The intersection of two Kuratowski cross products. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
inxpk $p |- ( ( A X._k B ) i^i ( C X._k D ) ) =
|
|
( ( A i^i C ) X._k ( B i^i D ) ) $=
|
|
( vx vy cxpk cin cvv inss1 xpkssvvk sstri cv wcel wa vex opkelxpk anbi12i
|
|
elin bitri copk an4 3bitr4i eqrelkriiv ) EFABGZCDGZHZACHZBDHZGZUGUEIIGUEU
|
|
FJABKLUHUIKEMZANZFMZBNZOZUKCNZUMDNZOZOZULUPOZUNUQOZOZUKUMUAZUGNZVCUJNZULU
|
|
NUPUQUBVDVCUENZVCUFNZOUSVCUEUFSVFUOVGURUKUMABEPZFPZQUKUMCDVHVIQRTVEUKUHNZ
|
|
UMUINZOVBUKUMUHUIVHVIQVJUTVKVAUKACSUMBDSRTUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The Kuratowski subset relationship is a Kuratowski relationship.
|
|
(Contributed by SF, 13-Jan-2015.) $)
|
|
ssetkssvvk $p |- _S_k C_ ( _V X._k _V ) $=
|
|
( vy vz vx cv wss cssetk df-ssetk opkabssvvki ) ADBDECABFCABGH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z t u w $.
|
|
$( Subset law for ` Ins2_k A ` . (Contributed by SF, 14-Jan-2015.) $)
|
|
ins2kss $p |- Ins2_k A C_ ( ~P1 1c X._k ( _V X._k _V ) ) $=
|
|
( vy vz vw vt vu vx c1c cvv cxpk wss copk wcel wal csn wceq w3a wex vex
|
|
cv cins2k cpw1 wi wb opkelins2kg mp2an wa opkeq12 snel1c snelpw1 opkelxpk
|
|
mpbir mpbir2an snex opkex syl6eqel 3adant3 exlimiv exlimivv gen2 df-ins2k
|
|
sylbi opkabssvvki ssrelk ax-mp ) AUAZHUBZIIJZJZKZBTZCTZLZVFMZVMVIMZUCZCNB
|
|
NZVPBCVNVKDTZOZOZPZVLETZFTZLZPZVRWCLAMZQZFRZERDRZVOVKIMVLIMVNWIUDBSCSDEFV
|
|
KVLAIIUEUFWHVODEWGVOFWAWEVOWFWAWEUGVMVTWDLZVIVKVLVTWDUHWJVIMVTVGMZWDVHMZW
|
|
KVSHMVRDSUIVSHUJULWLWBIMWCIMESZFSZWBWCIIWMWNUKUMVTWDVGVHVSUNWBWCUOUKUMUPU
|
|
QURUSVBUTVFVHKVJVQUDVKWBOOPVLWCVRLPWBVRLAMQDRFRERGBCVFGBCDFEAVAVCBCVFVIVD
|
|
VEUL $.
|
|
|
|
$( Subset law for ` Ins3_k A ` . (Contributed by SF, 14-Jan-2015.) $)
|
|
ins3kss $p |- Ins3_k A C_ ( ~P1 1c X._k ( _V X._k _V ) ) $=
|
|
( vy vz vt vu vw vx c1c cvv cxpk wss cv copk wcel wal csn wceq wex wb vex
|
|
cins3k cpw1 wi opkelins3kg mp2an wa opkeq12 snel1c snelpw1 mpbir opkelxpk
|
|
mpbir2an snex opkex syl6eqel 3adant3 exlimiv exlimivv sylbi gen2 df-ins3k
|
|
w3a opkabssvvki ssrelk ax-mp ) AUAZHUBZIIJZJZKZBLZCLZMZVFNZVMVINZUCZCOBOZ
|
|
VPBCVNVKDLZPZPZQZVLELZFLZMZQZVRWBMANZVBZFRZERDRZVOVKINVLINVNWISBTCTDEFVKV
|
|
LAIIUDUEWHVODEWGVOFWAWEVOWFWAWEUFVMVTWDMZVIVKVLVTWDUGWJVINVTVGNZWDVHNZWKV
|
|
SHNVRDTUHVSHUIUJWLWBINWCINETZFTZWBWCIIWMWNUKULVTWDVGVHVSUMWBWCUNUKULUOUPU
|
|
QURUSUTVFVHKVJVQSWIGBCVFGBCFEDAVAVCBCVFVIVDVEUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The Kuratowski identity relationship is a Kuratowski relationship.
|
|
(Contributed by SF, 14-Jan-2015.) $)
|
|
idkssvvk $p |- _I_k C_ ( _V X._k _V ) $=
|
|
( vy vz vx weq cidk df-idk opkabssvvki ) ABDCABECABFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $.
|
|
$( Image under a composition. (Contributed by SF, 4-Feb-2015.) $)
|
|
imacok $p |- ( ( A o._k B ) "_k C ) = ( A "_k ( B "_k C ) ) $=
|
|
( vz vx vy ccomk cimak cv copk wcel wrex wa wex vex rexbii rexcom4 df-rex
|
|
opkelcok elimak anbi1i r19.41v bitr4i exbii bitr2i 3bitri 3bitr4i eqriv )
|
|
DABGZCHZABCHZHZEIZDIZJUIKZECLZFIZUNJAKZFUKLZUNUJKUNULKUPUMUQJBKZURMZFNZEC
|
|
LVAECLZFNZUSUOVBECFUMUNABEODOZSPVAEFCQUSUQUKKZURMZFNVDURFUKRVGVCFVGUTECLZ
|
|
URMVCVFVHUREBCUQFOTUAUTURECUBUCUDUEUFEUICUNVETFAUKUNVETUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
elimaksn.1 $e |- B e. _V $.
|
|
elimaksn.2 $e |- C e. _V $.
|
|
$( Membership in a Kuratowski image of a singleton. (Contributed by SF,
|
|
4-Feb-2015.) $)
|
|
elimaksn $p |- ( C e. ( A "_k { B } ) <-> << B , C >> e. A ) $=
|
|
( vx csn cimak wcel cv copk wrex elimak wceq opkeq1 eleq1d rexsn bitri )
|
|
CABGZHIFJZCKZAIZFSLBCKZAIZFASCEMUBUDFBDTBNUAUCATBCOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d A x y a b c $. $d B x y a b c $.
|
|
$( A Kuratowski composition is a Kuratowski relationship. (Contributed by
|
|
SF, 4-Feb-2015.) $)
|
|
cokrelk $p |- ( A o._k B ) C_ ( _V X._k _V ) $=
|
|
( vx vy va vb vc ccomk cvv cxpk wcel copk cins2k wex vex csn wceq exlimiv
|
|
cv sylbi ccnvk cins3k cimak df-cok eleq2i elimakv bitri inss1 opkelins2kg
|
|
cin sseli wb mp2an opkelxpk mpbir2an eleq1 mpbiri 3ad2ant2 exlimivv ssriv
|
|
w3a syl ) CABHZIIJZCSZVCKZDSZVELZAMZBUAUBZUJZKZDNZVEVDKZVFVEVKIUCZKVMVCVO
|
|
VEABUDUEDVKVECOZUFUGVLVNDVLVHVIKZVNVKVIVHVIVJUHUKVQVGESZPPQZVEFSZGSZLZQZV
|
|
RWALAKZVAZGNZFNENZVNVGIKVEIKVQWGULDOVPEFGVGVEAIIUIUMWFVNEFWEVNGWCVSVNWDWC
|
|
VNWBVDKZWHVTIKWAIKFOZGOZVTWAIIWIWJUNUOVEWBVDUPUQURRUSTVBRTUT $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Kuratowski existence theorems
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d A a b x y z $.
|
|
$( The Kuratowski cross product of ` _V ` with a set is a set.
|
|
(Contributed by SF, 13-Jan-2015.) $)
|
|
xpkvexg $p |- ( A e. V -> ( _V X._k A ) e. _V ) $=
|
|
( vx vz vy va vb cvv cv cxpk wcel wceq xpkeq2 eleq1d wel wa wex wal bitri
|
|
wb copk ax-xp isset dfcleq elxpk vex biantrur anbi2i 2exbii bitr4i bibi2i
|
|
albii exbii mpbir vtoclg ) HCIZJZHKZHAJZHKCABUPALUQUSHUPAHMNURDEOZDIZFIZG
|
|
IUALZGCOZPZGQFQZTZDRZEQZCEDFGUBUREIZUQLZEQVIEUQUCVKVHEVKUTVAUQKZTZDRVHDVJ
|
|
UQUDVMVGDVLVFUTVLVCVBHKZVDPZPZGQFQVFFGVAHUPUEVEVPFGVDVOVCVNVDFUFUGUHUIUJU
|
|
KULSUMSUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $.
|
|
$( The Kuratowski converse of a set is a set. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
cnvkexg $p |- ( A e. V -> `'_k A e. _V ) $=
|
|
( vx vz vw vy cv ccnvk cvv wcel wceq cnvkeq eleq1d copk wal wex wss ax-mp
|
|
wb vex ax-cnv cxpk inss1 cnvkssvvk eqrelk mp2an opkelxpk mpbir2an mpbiran
|
|
cin elin opkelcnvk bibi12i 2albii bitri biimpri vvex xpkvexg inex exlimiv
|
|
syl6eqelr vtoclg ) CGZHZIJZAHZIJCABVCAKVDVFIVCALMDGZEGZNZFGZJZVHVGNVCJZSZ
|
|
EODOZFPVECFDEUAVNVEFVNVDIIUBZVJUJZIVPVDKZVNVQVIVPJZVIVDJZSZEODOZVNVPVOQVD
|
|
VOQVQWASVOVJUCVCUDDEVPVDUEUFVTVMDEVRVKVSVLVRVIVOJZVKWBVGIJVHIJDTZETZVGVHI
|
|
IWCWDUGUHVIVOVJUKUIVGVHVCWCWDULUMUNUOUPVOVJIIJVOIJUQIIURRFTUSVAUTRVB $.
|
|
$}
|
|
|
|
${
|
|
cnvkex.1 $e |- A e. _V $.
|
|
$( The Kuratowski converse of a set is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
cnvkex $p |- `'_k A e. _V $=
|
|
( cvv wcel ccnvk cnvkexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
$( The Kuratowski cross product of two sets is a set. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
xpkexg $p |- ( ( A e. V /\ B e. W ) -> ( A X._k B ) e. _V ) $=
|
|
( wcel cvv cxpk ccnvk cnvkxpk xpkvexg cnvkexg syl syl5eqelr cin inxpk incom
|
|
wa inv1 eqtri xpkeq12i inexg syl2an ) ACEZAFGZFEZFBGZFEZABGZFEBDEUCUDFAGZHZ
|
|
FFAIUCUIFEUJFEACJUIFKLMBDJUEUGQUHUDUFNZFUKAFNZFBNZGUHAFFBOULAUMBARUMBFNBFBP
|
|
BRSTSUDUFFFUAMUB $.
|
|
|
|
${
|
|
xpkex.1 $e |- A e. _V $.
|
|
xpkex.2 $e |- B e. _V $.
|
|
$( The Kuratowski cross product of two sets is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
xpkex $p |- ( A X._k B ) e. _V $=
|
|
( cvv wcel cxpk xpkexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $.
|
|
$( The P6 operator applied to a set yields a set. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
p6exg $p |- ( A e. V -> P6 A e. _V ) $=
|
|
( vx vz vy vw cv cp6 cvv wcel wceq p6eq eleq1d wel csn copk wal vex ax-mp
|
|
wb ax-typlower dfcleq bibi2i albii bitri biimpri syl6eqelr exlimiv vtoclg
|
|
wex elp6 ) CGZHZIJZAHZIJCABULAKUMUOIULALMDENZFGDGZOPULJFQZTZDQZEUJUNCEDFU
|
|
AUTUNEUTUMEGZIVAUMKZUTVBUPUQUMJZTZDQUTDVAUMUBVDUSDVCURUPUQIJVCURTDRFUQULI
|
|
UKSUCUDUEUFERUGUHSUI $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A z $. $d x z $.
|
|
$( Alternate definition of unit union. (Contributed by SF,
|
|
15-Mar-2015.) $)
|
|
dfuni12 $p |- U.1 A = P6 ( _V X._k A ) $=
|
|
( vx vz cuni1 cvv cxpk cp6 cv csn wcel copk wal 19.27v vex opkelxpk albii
|
|
wa snex ax-gen biantrur 3bitr4ri eluni1 wb elp6 ax-mp 3bitr4i eqriv ) BAD
|
|
ZEAFZGZBHZIZAJZCHZULKUIJZCLZUKUHJUKUJJZUNEJZUMQZCLURCLZUMQUPUMURUMCMUOUSC
|
|
UNULEACNZUKROPUTUMURCVASTUAUKABNZUBUKEJUQUPUCVBCUKUIEUDUEUFUG $.
|
|
$}
|
|
|
|
$( The unit union operator preserves sethood. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
uni1exg $p |- ( A e. V -> U.1 A e. _V ) $=
|
|
( wcel cuni1 cvv cxpk cp6 dfuni12 vvex xpkexg mpan p6exg syl syl5eqel ) ABC
|
|
ZADEAFZGZEAHOPECZQECEECORIEAEBJKPELMN $.
|
|
|
|
${
|
|
uni1ex.1 $e |- A e. _V $.
|
|
$( The unit union operator preserves sethood. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
uni1ex $p |- U.1 A e. _V $=
|
|
( cvv wcel cuni1 uni1exg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( The Kuratowski subset relationship is a set. (Contributed by SF,
|
|
13-Jan-2015.) $)
|
|
ssetkex $p |- _S_k e. _V $=
|
|
( vy vz vx vw cv copk wcel wel wi wal wb wex cssetk cvv ax-sset wss mp2an
|
|
vex bitri ax-mp cxpk cin wceq ssetkssvvk eqrelk opkelxpk mpbir2an mpbiran
|
|
inss1 elin opkelssetkg bibi12i 2albii biimpri vvex xpkvexg inex syl6eqelr
|
|
dfss2 exlimiv ) AEZBEZFZCEZGZDAHDBHIDJZKZBJAJZCLMNGZCABDOVHVICVHMNNUAZVDU
|
|
BZNVKMUCZVHVLVCVKGZVCMGZKZBJAJZVHVKVJPMVJPVLVPKVJVDUIUDABVKMUEQVOVGABVMVE
|
|
VNVFVMVCVJGZVEVQVANGZVBNGZARZBRZVAVBNNVTWAUFUGVCVJVDUJUHVNVAVBPZVFVRVSVNW
|
|
BKVTWAVAVBNNUKQDVAVBUSSULUMSUNVJVDNNGVJNGUONNUPTCRUQURUTT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w t $. $d B x y z w t $.
|
|
sikexlem.1 $e |- A C_ ( 1c X._k 1c ) $.
|
|
sikexlem.2 $e |- B C_ ( 1c X._k 1c ) $.
|
|
$( Lemma for ~ sikexg . Equality for two subsets of ` 1c ` squared .
|
|
(Contributed by SF, 14-Jan-2015.) $)
|
|
sikexlem $p |- ( A = B <->
|
|
A. x A. y ( << { x } , { y } >> e. A <->
|
|
<< { x } , { y } >> e. B ) ) $=
|
|
( vz vw vt wceq cv wcel wb c1c copk wal wex wa bitr4i bitri cxpk wral csn
|
|
wss ssofeq mp2an df-ral w3a elxpk el1c anbi12i eeanv anbi2i df-3an 2exbii
|
|
wi ancom 19.42vv exrot4 snex opkeq1 eqeq2d opkeq2 ceqsex2v 3bitri 19.23vv
|
|
imbi1i albii alrot3 opkex eleq1 bibi12d ceqsalv 2albii ) CDJZGKZCLZVPDLZM
|
|
ZGNNUAZUBZAKZUCZBKZUCZOZCLZWFDLZMZBPAPZCVTUDDVTUDVOWAMEFGCDVTUEUFWAVPVTLZ
|
|
VSUPZGPZVPWFJZVSUPZGPZBPAPZWJVSGVTUGWMWOBPAPZGPWQWLWRGWLWNBQAQZVSUPWRWKWS
|
|
VSWKVPHKZIKZOZJZWTNLZXANLZRZRZIQHQZWTWCJZXAWEJZXCUHZIQHQZBQAQZWSHIVPNNUIX
|
|
HXKBQAQZIQHQXMXGXNHIXGXCXIXJRZBQAQZRZXNXFXPXCXFXIAQZXJBQZRXPXDXRXEXSAWTUJ
|
|
BXAUJUKXIXJABULSUMXNXCXORZBQAQXQXKXTABXKXOXCRXTXIXJXCUNXOXCUQTUOXCXOABURT
|
|
SUOXKABHIUSSXLWNABXCVPWCXAOZJWNHIWCWEWBUTWDUTXIXBYAVPWTWCXAVAVBXJYAWFVPXA
|
|
WEWCVCVBVDUOVEVGWNVSABVFSVHWOGABVITWPWIABVSWIGWFWCWEVJWNVQWGVRWHVPWFCVKVP
|
|
WFDVKVLVMVNVET $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $.
|
|
$( The Kuratowski singleton image of a set is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
sikexg $p |- ( A e. V -> SI_k A e. _V ) $=
|
|
( vx vz vw vy cv csik cvv wcel wceq csn copk wal c1c vex snel1c snex 1cex
|
|
wb sikeq eleq1d wex ax-si cxpk inss1 sikss1c1c sikexlem opkelxpk mpbir2an
|
|
elin mpbiran opksnelsik bibi12i 2albii bitri biimpri xpkex inex syl6eqelr
|
|
cin exlimiv ax-mp vtoclg ) CGZHZIJZAHZIJCABVEAKVFVHIVEAUAUBDGZLZEGZLZMZFG
|
|
ZJZVIVKMVEJZTZENDNZFUCVGCFDEUDVRVGFVRVFOOUEZVNVAZIVTVFKZVRWAVMVTJZVMVFJZT
|
|
ZENDNVRDEVTVFVSVNUFVEUGUHWDVQDEWBVOWCVPWBVMVSJZVOWEVJOJVLOJVIDPZQVKEPZQVJ
|
|
VLOOVIRVKRUIUJVMVSVNUKULVIVKVEWFWGUMUNUOUPUQVSVNOOSSURFPUSUTVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
sikex.1 $e |- A e. _V $.
|
|
$( The Kuratowski singleton image of a set is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
sikex $p |- SI_k A e. _V $=
|
|
( cvv wcel csik sikexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Alternate definition of Kuratowski image. This is the first of a series
|
|
of definitions throughout the file designed to prove existence of
|
|
various operations. (Contributed by SF, 14-Jan-2015.) $)
|
|
dfimak2 $p |- ( A "_k B ) =
|
|
~ P6 ( ~ ( 1c X._k _V ) u. SI_k ~ ( A i^i ( B X._k _V ) ) ) $=
|
|
( vx vy vz c1c cvv cxpk ccompl cv copk wcel wn wex csn vex elcompl 3bitri
|
|
wa bitri cimak cin csik cun cp6 wrex df-rex exancom wceq wb elp6 ax-mp wo
|
|
elun opkex snex opkelxpk mpbiran2 xchbinx orbi1i wi iman imor el1c anbi1i
|
|
19.41v bitr4i notbii 3bitr3i albii opkeq1 eleq1d opksnelsik syl6bb notbid
|
|
alnex excom ceqsexv elin notnot anbi2i exbii con2bii elimak 3bitr4i eqriv
|
|
wal ) CABUAZFGHZIZABGHZUBZIZUCZUDZUEZIZDJZCJZKZALZDBUFZWSWPLZMZWSWHLWSWQL
|
|
XBWRBLZXASDNXAXESZDNZXDXADBUGXEXADUHXCXGXCEJZWSOZKZWOLZEWGZXHWROZUIZXJWNL
|
|
ZMZSZDNZMZEWGZXGMWSGLZXCXLUJCPZEWSWOGUKULXKXSEXKXJWJLZXOUMXHFLZMZXOUMZXSX
|
|
JWJWNUNYCYEXOYCXJWILZYDXJWIXHXIUOQYGYDXIGLWSUPZXHXIFGEPYHUQURUSUTYDXOVAYD
|
|
XPSZMYFXSYDXOVBYDXOVCYIXRYIXNDNZXPSXRYDYJXPDXHVDVEXNXPDVFVGVHVIRVJXTXRENZ
|
|
XGXREVPYKXQENZDNXGXQEDVQYLXFDYLWTWLLZMZMZXFXPYOEXMWRUPXNXOYNXNXOXMXIKZWNL
|
|
ZYNXNXJYPWNXHXMXIVKVLYQWTWMLYNWRWSWMDPZYBVMWTWLWRWSUOQTVNVOVRYMXAWTWKLZSY
|
|
OXFWTAWKVSYMVTYSXEXAYSXEYAYBWRWSBGYRYBUQURWAVITWBTUSRWCRDABWSYBWDWSWPYBQW
|
|
EWF $.
|
|
$}
|
|
|
|
$( The image of a set under a set is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
imakexg $p |- ( ( A e. V /\ B e. W ) -> ( A "_k B ) e. _V ) $=
|
|
( wcel wa cimak c1c cvv cxpk ccompl cin csik cun dfimak2 1cex vvex complexg
|
|
cp6 3syl xpkex complex xpkexg mpan2 inexg sylan2 sikexg unexg sylancr p6exg
|
|
syl5eqel ) ACEZBDEZFZABGHIJZKZABIJZLZKZMZNZSZKZIABOUNVAIEZVBIEVCIEUNUPIEUTI
|
|
EZVDUOHIPQUAUBUNURIEZUSIEVEUMULUQIEZVFUMIIEVGQBIDIUCUDAUQCIUEUFURIRUSIUGTUP
|
|
UTIIUHUIVAIUJVBIRTUK $.
|
|
|
|
${
|
|
imakex.1 $e |- A e. _V $.
|
|
imakex.2 $e |- B e. _V $.
|
|
$( The image of a set under a set is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
imakex $p |- ( A "_k B ) e. _V $=
|
|
( cvv wcel cimak imakexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $.
|
|
$( Alternate expression for unit power classes. (Contributed by SF,
|
|
26-Jan-2015.) $)
|
|
dfpw12 $p |- ~P1 A = ( SI_k ( A X._k A ) "_k _V ) $=
|
|
( vx vy vz vw cpw1 cvv cv wcel csn wceq wex vex wb exbii bitr4i wa 3bitri
|
|
copk weq cxpk csik cimak wrex elpw1 elimakv opkelsikg mp2an exrot3 df-3an
|
|
w3a opkelxpk anbi2i an4 2exbii 19.41vv eqeq12 sylan2 eleq1 adantl anbi12d
|
|
sneq spc2ev pm4.71ri ancom bitr3i df-rex eqriv ) BAFZAAUAZUBZGUCZBHZVIIVM
|
|
CHZJZKZCAUDZVMVLIZCVMAUEVRDHZVMSVKIZDLZVSEHZJZKZVPWBVNSVJIZUKZELDLZCLZVQD
|
|
VKVMBMZUFWAWFCLELZDLWHVTWJDVSGIVMGIVTWJNDMWIECVSVMVJGGUGUHOWFCDEUIPWHVNAI
|
|
ZVPQZCLVQWGWLCWGWDWBAIZQZVPWKQZQZELDLWNELDLZWOQZWLWFWPDEWFWDVPQZWEQWSWMWK
|
|
QZQWPWDVPWEUJWEWTWSWBVNAAEMCMZULUMWDVPWMWKUNRUOWNWODEUPWRWOWLWOWQWNWODEVM
|
|
VNWIXADBTZECTZQWDVPWMWKXCXBWCVOKWDVPNWBVNVBVSVMWCVOUQURXCWMWKNXBWBVNAUSUT
|
|
VAVCVDVPWKVEVFROVPCAVGPRPVH $.
|
|
$}
|
|
|
|
$( The unit power class preserves sethood. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
pw1exg $p |- ( A e. V -> ~P1 A e. _V ) $=
|
|
( wcel cpw1 cxpk csik cvv cimak dfpw12 xpkexg anidms sikexg imakexg sylancl
|
|
syl vvex syl5eqel ) ABCZADAAEZFZGHZGAIRTGCZGGCUAGCRSGCZUBRUCAABBJKSGLOPTGGG
|
|
MNQ $.
|
|
|
|
${
|
|
pw1ex.1 $e |- A e. _V $.
|
|
$( The unit power class preserves sethood. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
pw1ex $p |- ~P1 A e. _V $=
|
|
( cvv wcel cpw1 pw1exg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w t u $. $d B x y z w t u $.
|
|
insklem.1 $e |- A C_ ( ~P1 1c X._k ( _V X._k _V ) ) $.
|
|
insklem.2 $e |- B C_ ( ~P1 1c X._k ( _V X._k _V ) ) $.
|
|
$( Lemma for ~ ins2kexg and ~ ins3kexg . Equality for subsets of
|
|
` ( ~P1 1c X._k ( _V X._k _V ) ) ` . (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
insklem $p |- ( A = B <->
|
|
A. x A. y A. z ( << { { x } } , << y , z >> >> e. A <->
|
|
<< { { x } } , << y , z >> >> e. B ) ) $=
|
|
( vw vt vu wceq cv wcel copk wal wi wex albii wa bitri c1c cpw1 cxpk wral
|
|
wb cvv csn wss ssofeq mp2an 19.23v 19.23vv 19.42vv anbi2i 3bitr4ri 2exbii
|
|
elvvk elxpk exrot3 exancom elpw11c anbi1i 19.41v bitr4i exbii ancom opkex
|
|
an12 opkeq2 eqeq2d anbi1d ceqsexv snex opkeq1 excom bitr3i exrot4 3bitr4i
|
|
imbi1i df-ral alcom alrot3 eleq1 bibi12d ceqsalv 2albii 3bitrri ) DEKZHLZ
|
|
DMZWIEMZUEZHUAUBZUFUFUCZUCZUDZALUGZUGZBLZCLZNZNZDMZXBEMZUEZCOZBOAOZDWOUHE
|
|
WOUHWHWPUEFGHDEWOUIUJWIWOMZWLPZHOWIXBKZWLPZCOBOZAOZHOZWPXGXIXMHXJCQBQZWLP
|
|
ZAOXOAQZWLPXMXIXOWLAUKXLXPAXJWLBCULRXHXQWLWIILZJLZNZKZXRWMMZXSWNMZSZSZJQI
|
|
QYAYBXSXAKZSZSZCQBQZJQIQZXHXQYEYIIJYAYGCQBQZSYAYBYFCQBQZSZSYIYEYKYMYAYBYF
|
|
BCUMUNYAYGBCUMYDYMYAYCYLYBBCXSUQUNUNUOUPIJWIWMWNURXQXJAQZCQBQZYJXJABCUSYO
|
|
YHJQIQZCQBQYJYNYPBCWIXRXANZKZYBSZIQZXRWRKZYRSZAQZIQZYPYNYTYBYRSZIQUUDYRYB
|
|
IUTUUEUUCIUUEUUAAQZYRSUUCYBUUFYRAXRVAVBUUAYRAVCVDVETYPYFYAYBSZSZJQZIQYTYH
|
|
UUHIJYHYAYFYBSZSUUHYGUUJYAYBYFVFUNYAYFYBVHTUPUUIYSIUUGYSJXAWSWTVGYFYAYRYB
|
|
YFXTYQWIXSXAXRVIVJVKVLVETYNUUBIQZAQUUDUUKXJAYRXJIWRWQVMUUAYQXBWIXRWRXAVNV
|
|
JVLVEUUBAIVOVPUOUPYHBCIJVQTTVRVSUORWLHWOVTXNXLHOZAOXKHOZCOZBOZAOXGXLHAWAU
|
|
ULUUOAXKHBCWBRUUNXFABUUMXECWLXEHXBWRXAVGXJWJXCWKXDWIXBDWCWIXBEWCWDWERWFWG
|
|
VRT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w t $.
|
|
$( ` Ins2_k ` preserves sethood. (Contributed by SF, 14-Jan-2015.) $)
|
|
ins2kexg $p |- ( A e. V -> Ins2_k A e. _V ) $=
|
|
( vx vz vw vt vy cv cins2k cvv wcel wceq csn copk wb wal c1c vex opkelxpk
|
|
cxpk ins2keq eleq1d wex ax-ins2 cpw1 inss1 ins2kss insklem snel1c snelpw1
|
|
cin mpbir mpbir2an snex opkex elin mpbiran otkelins2k bibi12i albii bitri
|
|
2albii biimpri 1cex pw1ex vvex xpkex inex syl6eqelr exlimiv ax-mp vtoclg
|
|
) CHZIZJKZAIZJKCABVMALVNVPJVMAUAUBDHZMZMZEHZFHZNZNZGHZKZVQWANVMKZOZFPZEPD
|
|
PZGUCVOCGDEFUDWIVOGWIVNQUEZJJTZTZWDUKZJWMVNLZWIWNWCWMKZWCVNKZOZFPZEPDPWID
|
|
EFWMVNWLWDUFVMUGUHWRWHDEWQWGFWOWEWPWFWOWCWLKZWEWSVSWJKZWBWKKZWTVRQKVQDRZU
|
|
IVRQUJULXAVTJKWAJKERZFRZVTWAJJXCXDSUMVSWBWJWKVRUNVTWAUOSUMWCWLWDUPUQVQVTW
|
|
AVMXBXCXDURUSUTVBVAVCWLWDWJWKQVDVEJJVFVFVGVGGRVHVIVJVKVL $.
|
|
|
|
$( ` Ins3_k ` preserves sethood. (Contributed by SF, 14-Jan-2015.) $)
|
|
ins3kexg $p |- ( A e. V -> Ins3_k A e. _V ) $=
|
|
( vx vz vw vt vy cv cins3k cvv wcel wceq csn copk wb wal c1c vex opkelxpk
|
|
cxpk ins3keq eleq1d wex ax-ins3 cpw1 inss1 ins3kss insklem snel1c snelpw1
|
|
cin mpbir mpbir2an snex opkex elin mpbiran otkelins3k bibi12i albii bitri
|
|
2albii biimpri 1cex pw1ex vvex xpkex inex syl6eqelr exlimiv ax-mp vtoclg
|
|
) CHZIZJKZAIZJKCABVMALVNVPJVMAUAUBDHZMZMZEHZFHZNZNZGHZKZVQVTNVMKZOZFPEPZD
|
|
PZGUCVOCGDEFUDWIVOGWIVNQUEZJJTZTZWDUKZJWMVNLZWIWNWCWMKZWCVNKZOZFPEPZDPWID
|
|
EFWMVNWLWDUFVMUGUHWRWHDWQWGEFWOWEWPWFWOWCWLKZWEWSVSWJKZWBWKKZWTVRQKVQDRZU
|
|
IVRQUJULXAVTJKWAJKERZFRZVTWAJJXCXDSUMVSWBWJWKVRUNVTWAUOSUMWCWLWDUPUQVQVTW
|
|
AVMXBXCXDURUSVBUTVAVCWLWDWJWKQVDVEJJVFVFVGVGGRVHVIVJVKVL $.
|
|
$}
|
|
|
|
${
|
|
inskex.1 $e |- A e. _V $.
|
|
$( ` Ins2_k ` preserves sethood. (Contributed by SF, 14-Jan-2015.) $)
|
|
ins2kex $p |- Ins2_k A e. _V $=
|
|
( cvv wcel cins2k ins2kexg ax-mp ) ACDAECDBACFG $.
|
|
|
|
$( ` Ins3_k ` preserves sethood. (Contributed by SF, 14-Jan-2015.) $)
|
|
ins3kex $p |- Ins3_k A e. _V $=
|
|
( cvv wcel cins3k ins3kexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
$( The Kuratowski composition of two sets is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
cokexg $p |- ( ( A e. V /\ B e. W ) -> ( A o._k B ) e. _V ) $=
|
|
( wcel wa ccomk cins2k ccnvk cins3k cin cvv cimak ins2kexg cnvkexg ins3kexg
|
|
df-cok syl inexg syl2an vvex imakexg sylancl syl5eqel ) ACEZBDEZFZABGAHZBIZ
|
|
JZKZLMZLABQUGUKLEZLLEULLEUEUHLEUJLEZUMUFACNUFUILEUNBDOUILPRUHUJLLSTUAUKLLLU
|
|
BUCUD $.
|
|
|
|
${
|
|
cokex.1 $e |- A e. _V $.
|
|
cokex.2 $e |- B e. _V $.
|
|
$( The Kuratowski composition of two sets is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
cokex $p |- ( A o._k B ) e. _V $=
|
|
( cvv wcel ccomk cokexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
$( The Kuratowski image functor preserves sethood. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
imagekexg $p |- ( A e. V -> Image_k A e. _V ) $=
|
|
( wcel cimagek cvv cxpk cssetk cins2k csik ccnvk ccomk csymdif c1c cpw1 syl
|
|
cins3k ssetkex mpan pw1ex vvex cimak cdif df-imagek sikexg cnvkexg ins3kexg
|
|
cokexg ins2kex symdifexg 1cex imakexg mpan2 xpkex difexg syl5eqel ) ABCZADE
|
|
EFZGHZGAIZJZKZPZLZMNZNZUAZUBZEAUCUPVFECZVGECZUPVCECZVHUPVBECZVJUPVAECZVKUPU
|
|
TECZVLUPUSECVMABUDUSEUEOGECVMVLQGUTEEUGROVAEUFOURECVKVJGQUHURVBEEUIROVJVEEC
|
|
VHVDMUJSSVCVEEEUKULOUQECVHVIEETTUMUQVFEEUNROUO $.
|
|
|
|
${
|
|
imagekex.1 $e |- A e. _V $.
|
|
$( The Kuratowski image functor preserves sethood. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
imagekex $p |- Image_k A e. _V $=
|
|
( cvv wcel cimagek imagekexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Definition of ` _I_k ` in terms of ` _S_k ` . (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
dfidk2 $p |- _I_k = ( _S_k i^i `'_k _S_k ) $=
|
|
( vx vy cidk cssetk ccnvk cin idkssvvk cvv cxpk inss1 cv wss wa copk wcel
|
|
wb vex mp2an opkelssetkg bitri ssetkssvvk weq eqss opkelidkg elin anbi12i
|
|
sstri opkelcnvk 3bitr4i eqrelkriiv ) ABCDDEZFZGULDHHIDUKJUAUGABUBZAKZBKZL
|
|
ZUOUNLZMZUNUONZCOZUSULOZUNUOUCUNHOZUOHOZUTUMPAQZBQZUNUOHHUDRVAUSDOZUSUKOZ
|
|
MURUSDUKUEVFUPVGUQVBVCVFUPPVDVEUNUOHHSRVGUOUNNDOZUQUNUODVDVEUHVCVBVHUQPVE
|
|
VDUOUNHHSRTUFTUIUJ $.
|
|
$}
|
|
|
|
$( The Kuratowski identity relationship is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
idkex $p |- _I_k e. _V $=
|
|
( cidk cssetk ccnvk cin cvv dfidk2 ssetkex cnvkex inex eqeltri ) ABBCZDEFBK
|
|
GBGHIJ $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Alternate definition of class union for existence proof. (Contributed
|
|
by SF, 14-Jan-2015.) $)
|
|
dfuni3 $p |- U. A = U.1 ( `'_k _S_k "_k A ) $=
|
|
( vx vy cuni cssetk ccnvk cimak cuni1 cv csn copk wcel wrex wel opkelcnvk
|
|
vex snex elssetk bitri rexbii eluni1 elimak eluni2 3bitr4ri eqriv ) BADZE
|
|
FZAGZHZCIZBIZJZKUGLZCAMZBCNZCAMUKUILZUKUFLUMUOCAUMULUJKELUOUJULECPZUKQZOU
|
|
KUJBPZUQRSTUPULUHLUNUKUHUSUACUGAULURUBSCUKAUCUDUE $.
|
|
$}
|
|
|
|
$( The sum class of a set is a set. (Contributed by SF, 14-Jan-2015.) $)
|
|
uniexg $p |- ( A e. V -> U. A e. _V ) $=
|
|
( wcel cuni cssetk ccnvk cimak cuni1 cvv dfuni3 ssetkex cnvkex imakexg mpan
|
|
uni1exg syl syl5eqel ) ABCZADEFZAGZHZIAJRTICZUAICSICRUBEKLSAIBMNTIOPQ $.
|
|
|
|
${
|
|
uniex.1 $e |- A e. _V $.
|
|
$( The sum class of a set is a set. (Contributed by SF, 14-Jan-2015.) $)
|
|
uniex $p |- U. A e. _V $=
|
|
( cvv wcel cuni uniexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Alternate definition of class intersection for the existence proof.
|
|
(Contributed by SF, 14-Jan-2015.) $)
|
|
dfint3 $p |- |^| A = ~ U.1 ( `'_k ~ _S_k "_k A ) $=
|
|
( vx vy cint cssetk ccompl ccnvk cimak cuni1 wel wral cv wcel wn csn copk
|
|
wrex vex elcompl 3bitri eluni1 snex elimak bitri opkelcnvk elssetk notbii
|
|
opkex rexbii rexnal con2bii elint2 3bitr4i eqriv ) BADZEFZGZAHZIZFZBCJZCA
|
|
KZBLZUSMZNVCUOMVCUTMVDVBVDCLZVCOZPUQMZCAQZVANZCAQVBNVDVFURMVHVCURBRZUACUQ
|
|
AVFVCUBZUCUDVGVICAVGVFVEPZUPMVLEMZNVIVEVFUPCRZVKUEVLEVFVEUHSVMVAVCVEVJVNU
|
|
FUGTUIVACAUJTUKCVCAVJULVCUSVJSUMUN $.
|
|
$}
|
|
|
|
$( The intersection of a set is a set. (Contributed by SF, 14-Jan-2015.) $)
|
|
intexg $p |- ( A e. V -> |^| A e. _V ) $=
|
|
( wcel cint cssetk ccompl ccnvk cimak dfint3 ssetkex complex cnvkex imakexg
|
|
cuni1 cvv mpan uni1exg complexg 3syl syl5eqel ) ABCZADEFZGZAHZNZFZOAIUAUDOC
|
|
ZUEOCUFOCUCOCUAUGUBEJKLUCAOBMPUDOQUEORST $.
|
|
|
|
${
|
|
intex.1 $e |- A e. _V $.
|
|
$( The intersection of a set is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
intex $p |- |^| A e. _V $=
|
|
( cvv wcel cint intexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Two ways to express the class of all sets that contain ` A ` .
|
|
(Contributed by SF, 14-Jan-2015.) $)
|
|
setswith $p |- { x | A e. x } =
|
|
if ( A e. _V , ( _S_k "_k { { A } } ) , (/) ) $=
|
|
( vy cvv wcel cv cab cssetk csn cimak c0 cif wceq copk wrex opkeq1 eleq1d
|
|
snex eqtr4d wn rexsn wb vex elssetkg mpan2 syl5rbb abbidv df-imak syl6eqr
|
|
iftrue wal elex con3i alrimiv ab0 sylibr iffalse pm2.61i ) BDEZBAFZEZAGZU
|
|
SHBIZIZJZKLZMUSVBVEVFUSVBCFZUTNZHEZCVDOZAGVEUSVAVJAVJVCUTNZHEZUSVAVIVLCVC
|
|
BRVGVCMVHVKHVGVCUTPQUAUSUTDEVLVAUBAUCBUTDDUDUEUFUGACHVDUHUIUSVEKUJSUSTZVB
|
|
KVFVMVATZAUKVBKMVMVNAVAUSBUTULUMUNVAAUOUPUSVEKUQSUR $.
|
|
|
|
$( The class of all sets that contain ` A ` exist. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
setswithex $p |- { x | A e. x } e. _V $=
|
|
( wcel cab cvv cssetk csn cimak cif setswith ssetkex snex imakex 0ex ifex
|
|
cv c0 eqeltri ) BAPCADBECZFBGZGZHZQIEABJSUBQFUAKTLMNOR $.
|
|
$}
|
|
|
|
${
|
|
$d A x t $. $d B x t $.
|
|
ndisjrelk.1 $e |- A e. _V $.
|
|
ndisjrelk.2 $e |- B e. _V $.
|
|
$( Membership in a particular Kuratowski relationship is equivalent to
|
|
non-disjointedness. (Contributed by SF, 15-Jan-2015.) $)
|
|
ndisjrelk $p |- (
|
|
<< A , B >> e. ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c )
|
|
<-> ( A i^i B ) =/= (/) ) $=
|
|
( vt vx cv csn copk cssetk cin wcel wa cpw1 snex elin elssetk bitri exbii
|
|
wex wceq cins3k cins2k c1c cimak wne opkeq1 eleq1d ceqsexv otkelins3k vex
|
|
otkelins2k anbi12i wrex opkex elimak elpw121c anbi1i 19.41v bitr4i df-rex
|
|
c0 excom 3bitr4i n0 ) EGZFGZHZHZHZUAZVFABIZIZJUBZJUCZKZLZMZETZFTZVGALZVGB
|
|
LZMZFTZVLVPUDNNZUELZABKZVBUFZVSWCFVSVJVLIZVPLZWCVQWJEVJVIOVKVMWIVPVFVJVLU
|
|
GUHUIWJWIVNLZWIVOLZMWCWIVNVOPWKWAWLWBWKVHAIJLWAVHABJVGOZCDUJVGAFUKZCQRWLV
|
|
HBIJLWBVHABJWMCDULVGBWNDQRUMRRSWFVQEWEUNZVTEVPWEVLABUOUPVFWELZVQMZETVRFTZ
|
|
ETWOVTWQWREWQVKFTZVQMWRWPWSVQFVFUQURVKVQFUSUTSVQEWEVAVRFEVCVDRWHVGWGLZFTW
|
|
DFWGVEWTWCFVGABPSRVD $.
|
|
$}
|
|
|
|
${
|
|
$d ph x $.
|
|
$( When ` x ` does not occur in ` ph ` , ` { x | ph } ` is a set.
|
|
(Contributed by SF, 17-Jan-2015.) $)
|
|
abexv $p |- { x | ph } e. _V $=
|
|
( cab cvv wceq c0 wo wcel abvor0 vvex eleq1 mpbiri 0ex jaoi ax-mp ) ABCZD
|
|
EZPFEZGPDHZABIQSRQSDDHJPDDKLRSFDHMPFDKLNO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( The union of a unit power class is the original set. (Contributed by
|
|
SF, 20-Jan-2015.) $)
|
|
unipw1 $p |- U. ~P1 A = A $=
|
|
( vx vy vz cpw1 cuni cv wcel wel wa wex csn wceq eluni elpw1 anbi1i ancom
|
|
wrex weq 3bitri r19.41v 3bitr4i exbii risset ceqsexv abeq2i equcom rexbii
|
|
snex eleq2 df-sn rexcom4 3bitr2ri eqriv ) BAEZFZABGZUPHBCIZCGZUOHZJZCKUSD
|
|
GZLZMZURJZDARZCKZUQAHZCUQUONVAVFCUTURJVDDARZURJVAVFUTVIURDUSAOPURUTQVDURD
|
|
AUAUBUCVHDBSZDARVECKZDARVGDUQAUDVKVJDAVKUQVCHZBDSZVJURVLCVCVBUIUSVCUQUJUE
|
|
VMBVCBVBUKUFBDUGTUHVEDCAULUMTUN $.
|
|
$}
|
|
|
|
$( Biconditional existence for unit power class. (Contributed by SF,
|
|
20-Jan-2015.) $)
|
|
pw1exb $p |- ( ~P1 A e. _V <-> A e. _V ) $=
|
|
( cpw1 cvv wcel cuni unipw1 uniexg syl5eqelr pw1exg impbii ) ABZCDZACDLAKEC
|
|
AFKCGHACIJ $.
|
|
|
|
${
|
|
$d A x y t $.
|
|
$( Definition of power set for existence proof. (Contributed by SF,
|
|
21-Jan-2015.) $)
|
|
dfpw2 $p |- ~P A = ~ ( ( _S_k \ ( ~P1 A X._k _V ) ) "_k 1c ) $=
|
|
( vx vy vt cpw cssetk cpw1 cvv c1c cv wcel wn copk wa wex vex exbii bitri
|
|
3bitr4i 3bitri cxpk cdif cimak ccompl wel wal csn wceq wrex elimak anbi1i
|
|
el1c 19.41v bitr4i df-rex excom snex opkeq1 eleq1d ceqsexv eldif opkelxpk
|
|
wi elssetk mpbiran2 snelpw1 notbii anbi12i annim exnal con2bii elpw dfss2
|
|
wss elcompl eqriv ) BAEZFAGZHUAZUBZIUCZUDZCBUEZCJZAKZVCZCUFZBJZWAKZLWHVQK
|
|
ZWHWBKWIWGWIDJZWDUGZUHZWKWHMZVTKZNZDOZCOZWFLZCOWGLWIWODIUIZWRDVTIWHBPZUJW
|
|
KIKZWONZDOWPCOZDOWTWRXCXDDXCWMCOZWONXDXBXEWOCWKULUKWMWOCUMUNQWODIUOWPCDUP
|
|
SRWQWSCWQWLWHMZVTKZWSWOXGDWLWDUQZWMWNXFVTWKWLWHURUSUTXGXFFKZXFVSKZLZNWCWE
|
|
LZNWSXFFVSVAXIWCXKXLWDWHCPXAVDXJWEXJWLVRKZWEXJXMWHHKXAWLWHVRHXHXAVBVEWDAV
|
|
FRVGVHWCWEVITRQWFCVJTVKWJWHAVNWGWHAXAVLCWHAVMRWHWAXAVOSVP $.
|
|
$}
|
|
|
|
$( The power class of a set is a set. (Contributed by SF, 21-Jan-2015.) $)
|
|
pwexg $p |- ( A e. V -> ~P A e. _V ) $=
|
|
( wcel cpw cssetk cpw1 cvv cxpk cdif cimak ccompl dfpw2 ssetkex pw1exg vvex
|
|
c1c xpkexg sylancl difexg sylancr 1cex imakexg complexg syl syl5eqel ) ABCZ
|
|
ADEAFZGHZIZPJZKZGALUFUJGCZUKGCUFUIGCZPGCULUFEGCUHGCZUMMUFUGGCGGCUNABNOUGGGG
|
|
QREUHGGSTUAUIPGGUBRUJGUCUDUE $.
|
|
|
|
${
|
|
pwex.1 $e |- A e. _V $.
|
|
$( The power class of a set is a set. (Contributed by SF, 21-Jan-2015.) $)
|
|
pwex $p |- ~P A e. _V $=
|
|
( cvv wcel cpw pwexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( A class of singletons is equal to the unit power class of its union.
|
|
(Contributed by SF, 26-Jan-2015.) $)
|
|
eqpw1uni $p |- ( A C_ 1c -> A = ~P1 U. A ) $=
|
|
( vx vy vz c1c wss cuni cpw1 cv wcel ssel wi csn wceq wb el1c eleq2 sylbi
|
|
wex eleq1 pw1ss1c sseli a1i wel wrex vex snid rspcev weq elsn sneq eleq1d
|
|
mpan2 biimprcd imbi1d imbi12d mpbiri exlimiv syli rexlimdv impbid2 eluni2
|
|
syl6bbr snelpw1 syl6bb bibi12d syl5ibrcom exlimdv syl5bi pm5.21ndd eqrdv
|
|
) AEFZBAAGZHZVLBIZEJZVOAJZVOVNJZAEVOKZVRVPLVLVNEVOVMUAUBUCVPVOCIZMZNZCSVL
|
|
VQVROZCVOPVLWBWCCVLWCWBWAAJZVTVMJZOVLWDCBUDZBAUEZWEVLWDWGWDVTWAJZWGVTCUFU
|
|
GWFWHBWAAVOWAVTQUHUMVLWFWDBAVQVLVPWFWDLZVSVPVODIZMZNZDSVQWILZDVOPWLWMDWLW
|
|
MWKAJZVTWKJZWDLZLWOWDWNWOCDUIZWDWNOCWJUJWQWAWKAVTWJUKULRUNWLVQWNWIWPVOWKA
|
|
TWLWFWOWDVOWKVTQUOUPUQURRUSUTVABVTAVBVCWBVQWDVRWEVOWAATWBVRWAVNJWEVOWAVNT
|
|
VTVMVDVEVFVGVHVIVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d C x $. $d C y $. $d x y $.
|
|
pw1equn.1 $e |- A e. _V $.
|
|
pw1equn.2 $e |- B e. _V $.
|
|
$( A condition for a unit power class to equal a union. (Contributed by
|
|
SF, 26-Jan-2015.) $)
|
|
pw1equn $p |- ( ~P1 C = ( A u. B ) <->
|
|
E. x E. y ( C = ( x u. y ) /\ A = ~P1 x /\ B = ~P1 y ) ) $=
|
|
( cpw1 cun wceq cv w3a wex cuni c1c wss mpbiri eqeq2d wb pw1eq unipw1 syl
|
|
unieq syl5eqr ssun1 sseq2 pw1ss1c syl6ss eqpw1uni ssun2 wa uneq12 syl6eqr
|
|
uniex uniun adantr adantl 3anbi123d spc2ev pw1un eqeqan12d 3impb exlimivv
|
|
syl3anc impbii ) EHZCDIZJZEAKZBKZIZJZCVIHZJZDVJHZJZLZBMAMZVHEVGNZJZCCNZHZ
|
|
JZDDNZHZJZVRVHEVFNVSEUAVFVGUCUDVHCOPWCVHCVFOVHCVFPCVGPCDUEVFVGCUFQEUGZUHC
|
|
UIUBVHDOPWFVHDVFOVHDVFPDVGPDCUJVFVGDUFQWGUHDUIUBVQVTWCWFLABWAWDCFUNDGUNVI
|
|
WAJZVJWDJZUKZVLVTVNWCVPWFWJVKVSEWJVKWAWDIVSVIWAVJWDULCDUOUMRWHVNWCSWIWHVM
|
|
WBCVIWATRUPWIVPWFSWHWIVOWEDVJWDTRUQURUSVDVQVHABVQVHVKHZVMVOIZJZVIVJUTVLVN
|
|
VPVHWMSVLVNVPUKVFWKVGWLEVKTCVMDVOULVAVBQVCVE $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $.
|
|
pw1eqadj.1 $e |- A e. _V $.
|
|
pw1eqadj.2 $e |- B e. _V $.
|
|
$( A condition for a unit power class to work out to an adjunction.
|
|
(Contributed by SF, 26-Jan-2015.) $)
|
|
pw1eqadj $p |- ( ~P1 C = ( A u. { B } ) <->
|
|
E. x E. y ( C = ( x u. { y } ) /\ A = ~P1 x /\ B = { y } ) ) $=
|
|
( cpw1 csn cun wceq cv wex cuni c1c wcel mpbiri wss eqeq2d wb unieq uniun
|
|
w3a unipw1 3eqtr3g unisn pw1ss1c ssun2 snid sselii eleq2 sseldi vex sneqi
|
|
eqcomi id sneqd 3eqtr4a exlimiv sylbi syl syl5eq uneq2d eqtrd ssun1 sseq2
|
|
el1c syl6ss eqpw1uni uniex wa uneq12 sylan2 pw1eq adantr adantl 3anbi123d
|
|
spc2ev syl3anc pw1un pw1sn uneq2i eqtri eqeqan12d 3impb exlimivv impbii
|
|
sneq ) EHZCDIZJZKZEALZBLZIZJZKZCWMHZKZDWOKZUCZBMAMZWLECNZDNZIZJZKZCXCHZKZ
|
|
DXEKZXBWLEXCWJNZJZXFWLWINWKNEXLWIWKUAEUDCWJUBUEWLXKXEXCWLXKDXEDGUFWLDOPZX
|
|
JWLWIODEUGZWLDWIPDWKPWJWKDWJCUHDGUIUJWIWKDUKQULXMDWMIZKZAMXJADVGXPXJAXPXO
|
|
XONZIZDXEXRXOXQWMWMAUMUFUNUOXPUPXPXDXQDXOUAUQURUSUTVAZVBVCVDWLCORXIWLCWIO
|
|
WLCWIRCWKRCWJVEWIWKCVFQXNVHCVIVAXSXAXGXIXJUCABXCXDCFVJDGVJWMXCKZWNXDKZVKZ
|
|
WQXGWSXIWTXJYBWPXFEYAXTWOXEKWPXFKWNXDWHZWMXCWOXEVLVMSXTWSXITYAXTWRXHCWMXC
|
|
VNSVOYAWTXJTXTYAWOXEDYCSVPVQVRVSXAWLABXAWLWPHZWRWOIZJZKZYDWRWOHZJYFWMWOVT
|
|
YHYEWRWNBUMWAWBWCWQWSWTWLYGTWQWSWTVKWIYDWKYFEWPVNWTWSWJYEKWKYFKDWOWHCWRWJ
|
|
YEVLVMWDWEQWFWG $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Alternate definition of existential uniqueness in terms of abstraction.
|
|
(Contributed by SF, 29-Jan-2015.) $)
|
|
dfeu2 $p |- ( E! x ph <-> { x | ph } e. 1c ) $=
|
|
( vy weq wb wal wex cab cv csn wceq weu c1c wcel abbi df-sn eqeq2i bitr4i
|
|
exbii df-eu el1c 3bitr4i ) ABCDZEBFZCGABHZCIZJZKZCGABLUEMNUDUHCUDUEUCBHZK
|
|
UHAUCBOUGUIUEBUFPQRSABCTCUEUAUB $.
|
|
$}
|
|
|
|
$( If there is a unique object satisfying a property ` ph ` , then the set of
|
|
all elements that satisfy ` ph ` exists. (Contributed by SF,
|
|
16-Jan-2015.) $)
|
|
euabex $p |- ( E! x ph -> { x | ph } e. _V ) $=
|
|
( weu cab c1c wcel cvv dfeu2 elex sylbi ) ABCABDZEFKGFABHKEIJ $.
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
sspw1.1 $e |- A e. _V $.
|
|
$( A condition for being a subclass of a unit power class. Corollary 2 of
|
|
theorem IX.6.14 of [Rosser] p. 255. (Contributed by SF, 3-Feb-2015.) $)
|
|
sspw1 $p |- ( A C_ ~P1 B <-> E. x ( x C_ B /\ A = ~P1 x ) ) $=
|
|
( cpw1 wss cv wceq wa wex cuni uniss unipw1 syl6sseq c1c pw1ss1c eqpw1uni
|
|
sstr mpan2 sseq1 uniex pw1eq eqeq2d anbi12d spcev syl2anc syl5ibr exlimiv
|
|
syl pw1ss impcom impbii ) BCEZFZAGZCFZBUOEZHZIZAJZUNBKZCFZBVAEZHZUTUNVAUM
|
|
KCBUMLCMNUNBOFZVDUNUMOFVECPBUMORSBQUIUSVBVDIAVABDUAUOVAHZUPVBURVDUOVACTVF
|
|
UQVCBUOVAUBUCUDUEUFUSUNAURUPUNUPUNURUQUMFUOCUJBUQUMTUGUKUHUL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
sspw12.1 $e |- A e. _V $.
|
|
$( A set is a subset of cardinal one iff it is the unit power class of some
|
|
other set. (Contributed by SF, 17-Mar-2015.) $)
|
|
sspw12 $p |- ( A C_ 1c <-> E. x A = ~P1 x ) $=
|
|
( c1c wss cv cpw1 wceq wex cuni eqpw1uni uniex pw1eq eqeq2d spcev pw1ss1c
|
|
syl sseq1 mpbiri exlimiv impbii ) BDEZBAFZGZHZAIZUBBBJZGZHZUFBKUEUIAUGBCL
|
|
UCUGHUDUHBUCUGMNOQUEUBAUEUBUDDEUCPBUDDRSTUA $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Definite description binder (inverted iota)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c iota $.
|
|
$( Extend class notation with Russell's definition description binder
|
|
(inverted iota). $)
|
|
cio $a class ( iota x ph ) $.
|
|
|
|
${
|
|
$d w x z $. $d ph w z $. $d ph w y $. $d x y $.
|
|
$( Soundness justification theorem for ~ df-iota . (Contributed by Andrew
|
|
Salmon, 29-Jun-2011.) $)
|
|
iotajust $p |- U. { y | { x | ph } = { y } } =
|
|
U. { z | { x | ph } = { z } } $=
|
|
( vw cab cv csn wceq sneq eqeq2d cbvabv eqtri unieqi ) ABFZCGZHZIZCFZODGZ
|
|
HZIZDFZSOEGZHZIZEFUCRUFCEPUDIQUEOPUDJKLUFUBEDUDTIUEUAOUDTJKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y ph $.
|
|
$( Define Russell's definition description binder, which can be read as
|
|
"the unique ` x ` such that ` ph ` ," where ` ph ` ordinarily contains
|
|
` x ` as a free variable. Our definition is meaningful only when there
|
|
is exactly one ` x ` such that ` ph ` is true (see ~ iotaval );
|
|
otherwise, it evaluates to the empty set (see ~ iotanul ). Russell used
|
|
the inverted iota symbol ` iota ` to represent the binder. (Contributed
|
|
by SF, 12-Jan-2015.) $)
|
|
df-iota $a |- ( iota x ph ) = U. { y | { x | ph } = { y } } $.
|
|
$}
|
|
|
|
${
|
|
$d y x $. $d y ph $.
|
|
$( Alternate definition for descriptions. Definition 8.18 in [Quine]
|
|
p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) $)
|
|
dfiota2 $p |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) } $=
|
|
( cio cab cv csn wceq cuni wb wal df-iota df-sn eqeq2i abbi bitr4i unieqi
|
|
abbii eqtri ) ABDABEZCFZGZHZCEZIABFUAHZJBKZCEZIABCLUDUGUCUFCUCTUEBEZHUFUB
|
|
UHTBUAMNAUEBOPRQS $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y ph $.
|
|
$( Bound-variable hypothesis builder for the ` iota ` class. (Contributed
|
|
by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
nfiota1 $p |- F/_ x ( iota x ph ) $=
|
|
( vy cio cv wceq wb wal cab cuni dfiota2 nfaba1 nfuni nfcxfr ) BABDABECEF
|
|
GZBHCIZJABCKBPOBCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d z ps $. $d z ph $. $d x z $. $d y z $.
|
|
nfiotad.1 $e |- F/ y ph $.
|
|
nfiotad.2 $e |- ( ph -> F/ x ps ) $.
|
|
$( Deduction version of ~ nfiota . (Contributed by NM, 18-Feb-2013.) $)
|
|
nfiotad $p |- ( ph -> F/_ x ( iota y ps ) ) $=
|
|
( vz cio cv wceq wb wal cab cuni dfiota2 nfv wn wa wnf adantr wnfc adantl
|
|
nfcvf nfcvd nfeqd nfbid nfald2 nfabd nfunid nfcxfrd ) ACBDHBDIZGIZJZKZDLZ
|
|
GMZNBDGOACUPAUOCGAGPAUNCDEACIUKJCLQZRZBUMCABCSUQFTURCUKULUQCUKUAACDUCUBUR
|
|
CULUDUEUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
nfiota.1 $e |- F/ x ph $.
|
|
$( Bound-variable hypothesis builder for the ` iota ` class. (Contributed
|
|
by NM, 23-Aug-2011.) $)
|
|
nfiota $p |- F/_ x ( iota y ph ) $=
|
|
( cio wnfc wtru nftru wnf a1i nfiotad trud ) BACEFGABCCHABIGDJKL $.
|
|
$}
|
|
|
|
${
|
|
$d z w x $. $d z w y $. $d z w ph $. $d z w ps $.
|
|
cbviota.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
cbviota.2 $e |- F/ y ph $.
|
|
cbviota.3 $e |- F/ x ps $.
|
|
$( Change bound variables in a description binder. (Contributed by Andrew
|
|
Salmon, 1-Aug-2011.) $)
|
|
cbviota $p |- ( iota x ph ) = ( iota y ps ) $=
|
|
( vw vz cv wceq wb wal cab cuni cio wsb nfv nfbi equequ1 nfs1v cbval nfsb
|
|
sbequ12 bibi12d sbequ sbie syl6bb bitri abbii unieqi dfiota2 3eqtr4i ) AC
|
|
JZHJZKZLZCMZHNZOBDJZUOKZLZDMZHNZOACPBDPUSVDURVCHURACIQZIJZUOKZLZIMVCUQVHC
|
|
IUQIRVEVGCACIUAVGCRSUNVFKAVEUPVGACIUDCIHTUEUBVHVBIDVEVGDACIDFUCVGDRSVBIRV
|
|
FUTKZVEBVGVAVIVEACDQBAIDCUFABCDGEUGUHIDHTUEUBUIUJUKACHULBDHULUM $.
|
|
$}
|
|
|
|
${
|
|
$d ph y $. $d ps x $.
|
|
cbviotav.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
$( Change bound variables in a description binder. (Contributed by Andrew
|
|
Salmon, 1-Aug-2011.) $)
|
|
cbviotav $p |- ( iota x ph ) = ( iota y ps ) $=
|
|
( nfv cbviota ) ABCDEADFBCFG $.
|
|
$}
|
|
|
|
${
|
|
$d w z ph $. $d w z x $. $d w z y $.
|
|
sb8iota.1 $e |- F/ y ph $.
|
|
$( Variable substitution in description binder. Compare ~ sb8eu .
|
|
(Contributed by NM, 18-Mar-2013.) $)
|
|
sb8iota $p |- ( iota x ph ) = ( iota y [ y / x ] ph ) $=
|
|
( vz vw cv wceq wb wal cab cuni wsb cio nfv sb8 sbbi nfsb nfxfr dfiota2
|
|
eqsb3 nfbi sbequ cbval equsb3 sblbis albii 3bitri abbii unieqi 3eqtr4i )
|
|
ABGEGZHZIZBJZEKZLABCMZCGULHZIZCJZEKZLABNUQCNUPVAUOUTEUOUNBFMZFJUNBCMZCJUT
|
|
UNBFUNFOPVBVCFCVBABFMZUMBFMZICAUMBFQVDVECABFCDRVEFGULHZCFBULUAVFCOSUBSVCF
|
|
OUNFCBUCUDVCUSCUMURABCCBEUEUFUGUHUIUJABETUQCETUK $.
|
|
$}
|
|
|
|
${
|
|
$d y z $. $d x z $. $d ph z $.
|
|
$( Equality theorem for descriptions. (Contributed by Andrew Salmon,
|
|
30-Jun-2011.) $)
|
|
iotaeq $p |- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) ) $=
|
|
( vz cv wceq wal cab csn cuni cio wcel drsb1 df-clab 3bitr4g eqrdv eqeq1d
|
|
wsb abbidv df-iota unieqd 3eqtr4g ) BECEFBGZABHZDEZIZFZDHZJACHZUFFZDHZJAB
|
|
KACKUCUHUKUCUGUJDUCUDUIUFUCDUDUIUCABDRACDRUEUDLUEUILABCDMADBNADCNOPQSUAAB
|
|
DTACDTUB $.
|
|
$}
|
|
|
|
${
|
|
$d ph z $. $d ps z $. $d x y z $.
|
|
$( Equivalence theorem for descriptions. (Contributed by Andrew Salmon,
|
|
30-Jun-2011.) $)
|
|
iotabi $p |- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) ) $=
|
|
( vz wb wal cab cv csn wceq cuni abbi biimpi eqeq1d abbidv unieqd df-iota
|
|
cio 3eqtr4g ) ABECFZACGZDHIZJZDGZKBCGZUBJZDGZKACRBCRTUDUGTUCUFDTUAUEUBTUA
|
|
UEJABCLMNOPACDQBCDQS $.
|
|
|
|
$( Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma
|
|
for the fundamental property of iota. (Contributed by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
uniabio $p |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y ) $=
|
|
( cv wceq wb wal cab cuni csn abbi biimpi df-sn syl6eqr unieqd vex syl6eq
|
|
unisn ) ABDCDZEZFBGZABHZISJZISUAUBUCUAUBTBHZUCUAUBUDEATBKLBSMNOSCPRQ $.
|
|
|
|
$( Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property
|
|
of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) $)
|
|
iotaval $p |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y ) $=
|
|
( vz cv wceq wb wal cio cab cuni dfiota2 cvv wcel wa sbeqalb equcomi syl6
|
|
wi vex ax-mp ex equequ2 eqcoms bibi2d biimpd alimdv com12 alrimiv uniabio
|
|
impbid syl syl5eq ) ABEZCEZFZGZBHZABIAUNDEZFZGZBHZDJKZUOABDLURVBUSUOFZGZD
|
|
HVCUOFURVEDURVBVDURVBVDUOMNZURVBOZVDSCTVFVGUOUSFVDABUOUSMPCDQRUAUBVDURVBV
|
|
DUQVABVDUQVAVDUPUTAUPUTGUOUSCDBUCUDUEUFUGUHUKUIVBDCUJULUM $.
|
|
|
|
$( Equivalence between two different forms of ` iota ` . (Contributed by
|
|
Andrew Salmon, 12-Jul-2011.) $)
|
|
iotauni $p |- ( E! x ph -> ( iota x ph ) = U. { x | ph } ) $=
|
|
( vz weu cv wceq wb wal wex cio cuni df-eu iotaval uniabio eqtr4d exlimiv
|
|
cab sylbi ) ABDABECEZFGBHZCIABJZABQKZFZABCLTUCCTUASUBABCMABCNOPR $.
|
|
|
|
$( Equivalence between two different forms of ` iota ` . (Contributed by
|
|
Mario Carneiro, 24-Dec-2016.) $)
|
|
iotaint $p |- ( E! x ph -> ( iota x ph ) = |^| { x | ph } ) $=
|
|
( weu cio cab cuni cint iotauni wceq uniintab biimpi eqtrd ) ABCZABDABEZF
|
|
ZNGZABHMOPIABJKL $.
|
|
|
|
$( Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario
|
|
Carneiro, 23-Dec-2016.) $)
|
|
iota1 $p |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) ) $=
|
|
( vz weu cv wceq wb wal wex cio df-eu iotaval eqeq2d bitr4d eqcom exlimiv
|
|
sp syl6bb sylbi ) ABDABEZCEZFZGZBHZCIAABJZTFZGZABCKUDUGCUDATUEFZUFUDAUBUH
|
|
UCBQUDUEUATABCLMNTUEORPS $.
|
|
|
|
$( Theorem 8.22 in [Quine] p. 57. This theorem is the result if there
|
|
isn't exactly one ` x ` that satisfies ` ph ` . (Contributed by Andrew
|
|
Salmon, 11-Jul-2011.) $)
|
|
iotanul $p |- ( -. E! x ph -> ( iota x ph ) = (/) ) $=
|
|
( vz weu cv wceq wb wal wex cio c0 df-eu wn cuni dfiota2 alnex ax-1 eqidd
|
|
cab impbid1 con2bid alimi abbi dfnul2 syl6eqr sylbir unieqd syl6eq syl5eq
|
|
sylib uni0 sylnbi ) ABDABECEZFGBHZCIZABJZKFABCLUOMZUPUNCSZNZKABCOUQUSKNKU
|
|
QURKUQUNMZCHZURKFUNCPVAURUMUMFZMZCSZKVAUNVCGZCHURVDFUTVECUTVBUNUTVBUTUTVB
|
|
QUTUMRTUAUBUNVCCUCUJCUDUEUFUGUKUHUIUL $.
|
|
|
|
$( The ` iota ` class is a subset of the union of all elements satisfying
|
|
` ph ` . (Contributed by Mario Carneiro, 24-Dec-2016.) $)
|
|
iotassuni $p |- ( iota x ph ) C_ U. { x | ph } $=
|
|
( weu cio cab cuni wss iotauni eqimss syl wn c0 0ss iotanul sseq1d mpbiri
|
|
wceq pm2.61i ) ABCZABDZABEFZGZSTUAQUBABHTUAIJSKZUBLUAGUAMUCTLUAABNOPR $.
|
|
|
|
$( Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the
|
|
` iota ` class under our definition. (Contributed by Andrew Salmon,
|
|
11-Jul-2011.) $)
|
|
iotaex $p |- ( iota x ph ) e. _V $=
|
|
( vz weu cio cvv wcel cv wceq wb wal wex iotaval eqcomd eximi df-eu isset
|
|
3imtr4i wn c0 iotanul 0ex syl6eqel pm2.61i ) ABDZABEZFGZABHCHZIJBKZCLUHUF
|
|
IZCLUEUGUIUJCUIUFUHABCMNOABCPCUFQRUESUFTFABUAUBUCUD $.
|
|
|
|
$( Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew
|
|
Salmon, 12-Jul-2011.) $)
|
|
iota4 $p |- ( E! x ph -> [. ( iota x ph ) / x ]. ph ) $=
|
|
( vz weu weq wb wal wex cio wsbc df-eu wsb bi2 alimi sb2 syl wceq iotaval
|
|
wi cv eqcomd dfsbcq2 mpbid exlimiv sylbi ) ABDABCEZFZBGZCHABABIZJZABCKUHU
|
|
JCUHABCLZUJUHUFASZBGUKUGULBAUFMNABCOPUHCTZUIQUKUJFUHUIUMABCRUAABCUIUBPUCU
|
|
DUE $.
|
|
$}
|
|
|
|
$( Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew
|
|
Salmon, 12-Jul-2011.) $)
|
|
iota4an $p |- ( E! x ( ph /\ ps )
|
|
-> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) $=
|
|
( wa weu cio wsbc iota4 wi cvv wcel iotaex simpl sbcth ax-mp wb sbcimg mpbi
|
|
syl ) ABDZCETCTCFZGZACUAGZTCHTAIZCUAGZUBUCIZUAJKZUETCLZUDCUAJABMNOUGUEUFPUH
|
|
TACUAJQORS $.
|
|
|
|
${
|
|
$d x y A $. $d x V $. $d x ph $. $d y ps $.
|
|
iota5.1 $e |- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) ) $.
|
|
$( A method for computing iota. (Contributed by NM, 17-Sep-2013.) $)
|
|
iota5 $p |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A ) $=
|
|
( vy wcel wa cv wceq wb wal cio alrimiv wi eqeq2 bibi2d albidv imbi12d
|
|
iotaval vtoclg adantl mpd ) ADEHZIZBCJZDKZLZCMZBCNZDKZUFUICFOUEUJULPZABUG
|
|
GJZKZLZCMZUKUNKZPUMGDEUNDKZUQUJURULUSUPUICUSUOUHBUNDUGQRSUNDUKQTBCGUAUBUC
|
|
UD $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
iotabidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Formula-building deduction rule for iota. (Contributed by NM,
|
|
20-Aug-2011.) $)
|
|
iotabidv $p |- ( ph -> ( iota x ps ) = ( iota x ch ) ) $=
|
|
( wb wal cio wceq alrimiv iotabi syl ) ABCFZDGBDHCDHIAMDEJBCDKL $.
|
|
$}
|
|
|
|
${
|
|
iotabii.1 $e |- ( ph <-> ps ) $.
|
|
$( Formula-building deduction rule for iota. (Contributed by Mario
|
|
Carneiro, 2-Oct-2015.) $)
|
|
iotabii $p |- ( iota x ph ) = ( iota x ps ) $=
|
|
( wb cio wceq iotabi mpg ) ABEACFBCFGCABCHDI $.
|
|
$}
|
|
|
|
$( Membership law for descriptions.
|
|
|
|
This can useful for expanding an unbounded iota-based definition (see
|
|
~ df-iota ). If you have a bounded iota-based definition, riotacl2 in
|
|
set.mm may be useful.
|
|
|
|
(Contributed by Andrew Salmon, 1-Aug-2011.) $)
|
|
iotacl $p |- ( E! x ph -> ( iota x ph ) e. { x | ph } ) $=
|
|
( weu cio wsbc cab wcel iota4 df-sbc sylib ) ABCABABDZEKABFGABHABKIJ $.
|
|
|
|
$( Membership law for descriptions. (Contributed by SF, 21-Aug-2011.) $)
|
|
reiotacl2 $p |- ( E! x e. A ph
|
|
-> ( iota x ( x e. A /\ ph ) ) e. { x e. A | ph } ) $=
|
|
( wreu cv wcel wa cio cab crab weu df-reu iotacl sylbi df-rab syl6eleqr ) A
|
|
BCDZBECFAGZBHZRBIZABCJQRBKSTFABCLRBMNABCOP $.
|
|
|
|
${
|
|
$d x A $.
|
|
$( Membership law for descriptions. (Contributed by SF, 21-Aug-2011.) $)
|
|
reiotacl $p |- ( E! x e. A ph
|
|
-> ( iota x ( x e. A /\ ph ) ) e. A ) $=
|
|
( wreu crab cv wcel wa cio wss ssrab2 a1i reiotacl2 sseldd ) ABCDZABCEZCB
|
|
FCGAHBIPCJOABCKLABCMN $.
|
|
$}
|
|
|
|
${
|
|
iota2df.1 $e |- ( ph -> B e. V ) $.
|
|
iota2df.2 $e |- ( ph -> E! x ps ) $.
|
|
iota2df.3 $e |- ( ( ph /\ x = B ) -> ( ps <-> ch ) ) $.
|
|
${
|
|
iota2df.4 $e |- F/ x ph $.
|
|
iota2df.5 $e |- ( ph -> F/ x ch ) $.
|
|
iota2df.6 $e |- ( ph -> F/_ x B ) $.
|
|
$( A condition that allows us to represent "the unique element such that
|
|
` ph ` " with a class expression ` A ` . (Contributed by NM,
|
|
30-Dec-2014.) $)
|
|
iota2df $p |- ( ph -> ( ch <-> ( iota x ps ) = B ) ) $=
|
|
( wnfc cio wceq wb wnf cv wal alrimi wi wcel nfiota1 a1i nfeqd nfbid wa
|
|
simpr eqeq2d bibi12d ex weu iota1 syl vtoclgft syl221anc ) ADEMCBDNZEOZ
|
|
PZDQDRZEOZBUQUTOZPZUSPZUAZDSVCDSEFUBUSLACURDKADUQEDUQMABDUCUDLUEUFAVEDJ
|
|
AVAVDAVAUGZBCVBURIVFUTEUQAVAUHUIUJUKTAVCDJABDULVCHBDUMUNTGVCUSDEFUOUP
|
|
$.
|
|
$}
|
|
|
|
$d x B $. $d x ph $. $d x ch $.
|
|
$( A condition that allows us to represent "the unique element such that
|
|
` ph ` " with a class expression ` A ` . (Contributed by NM,
|
|
30-Dec-2014.) $)
|
|
iota2d $p |- ( ph -> ( ch <-> ( iota x ps ) = B ) ) $=
|
|
( nfv nfvd nfcvd iota2df ) ABCDEFGHIADJACDKADELM $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d ps x $.
|
|
iota2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
$( The unique element such that ` ph ` . (Contributed by Jeff Madsen,
|
|
1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) $)
|
|
iota2 $p |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) ) $=
|
|
( wcel cvv weu cio wceq wb elex wa simpl simpr cv adantl nfv nfeu1 nfcvd
|
|
nfan nfvd iota2df sylan ) DEGDHGZACIZBACJDKLDEMUFUGNZABCDHUFUGOUFUGPCQDKA
|
|
BLUHFRUFUGCUFCSACTUBUHBCUCUHCDUAUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x ps $.
|
|
reiota2.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
|
|
$( A condition allowing us to represent "the unique element in ` A ` such
|
|
that ` ph ` " with a class expression ` B ` . (Contributed by Scott
|
|
Fenton, 7-Jan-2018.) $)
|
|
reiota2 $p |- ( ( B e. A /\ E! x e. A ph ) ->
|
|
( ps <-> ( iota x ( x e. A /\ ph ) ) = B ) ) $=
|
|
( wcel wreu wa cv cio wceq simpl biantrurd weu df-reu eleq1 anbi12d iota2
|
|
wb sylan2b bitrd ) EDGZACDHZIZBUCBIZCJZDGZAIZCKELZUEUCBUCUDMNUDUCUICOUFUJ
|
|
TACDPUIUFCEDUGELUHUCABUGEDQFRSUAUB $.
|
|
$}
|
|
|
|
$( A class abstraction with a unique member can be expressed as a singleton.
|
|
(Contributed by Mario Carneiro, 23-Dec-2016.) $)
|
|
sniota $p |- ( E! x ph -> { x | ph } = { ( iota x ph ) } ) $=
|
|
( weu cv cab wcel cio csn wb wal wceq nfeu1 iota1 eqcom syl6bb abid 3bitr4g
|
|
vex elsnc alrimi nfab1 nfiota1 nfsn cleqf sylibr ) ABCZBDZABEZFZUGABGZHZFZI
|
|
ZBJUHUKKUFUMBABLUFAUGUJKZUIULUFAUJUGKUNABMUJUGNOABPUGUJBRSQTBUHUKABUABUJABU
|
|
BUCUDUE $.
|
|
|
|
$( The ` iota ` operation using the ` if ` operator. (Contributed by Scott
|
|
Fenton, 6-Oct-2017.) $)
|
|
dfiota3 $p |- ( iota x ph ) = if ( E! x ph , U. { x | ph } , (/) ) $=
|
|
( weu cio cab cuni c0 wceq iotauni iftrue eqtr4d wn iotanul iffalse pm2.61i
|
|
cif ) ABCZABDZQABEFZGPZHQRSTABIQSGJKQLRGTABMQSGNKO $.
|
|
|
|
${
|
|
$d A y z $. $d x y z $. $d ph z $.
|
|
$( Class substitution within a description binder. (Contributed by Scott
|
|
Fenton, 6-Oct-2017.) $)
|
|
csbiotag $p |- ( A e. V ->
|
|
[_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) ) $=
|
|
( vz cv cio csb wsb wceq csbeq1 dfsbcq2 iotabidv eqeq12d vex nfs1v nfiota
|
|
wsbc weq sbequ12 csbief vtoclg ) BFGZACHZIZABFJZCHZKBDUEIZABDSZCHZKFDEUDD
|
|
KZUFUIUHUKBUDDUELULUGUJCABFDMNOBUDUEUHFPUGBCABFQRBFTAUGCABFUANUBUC $.
|
|
$}
|
|
|
|
$( Alternate definition of iota in terms of ` 1c ` . (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
dfiota4 $p |- ( iota x ph ) = U. U. ( 1c i^i { { x | ph } } ) $=
|
|
( weu cio c1c cab csn cin cuni wceq iotauni wss wcel dfeu2 unieqd eqtr4d wn
|
|
cvv c0 uni0 snssi sylbi df-ss incom eqeq1i bitri sylib euabex eqtrd iotanul
|
|
unisng syl notbii disjsn bitr4i biimpi unieqi eqtri syl6eq pm2.61i ) ABCZAB
|
|
DZEABFZGZHZIZIZJVAVBVCIVGABKVAVFVCVAVFVDIZVCVAVEVDVAVDELZVEVDJZVAVCEMZVIABN
|
|
ZVCEUAUBVIVDEHZVDJVJVDEUCVMVEVDVDEUDUEUFUGOVAVCRMVHVCJABUHVCRUKULUIOPVAQZVB
|
|
SVGABUJVNVGSIZIZSVNVFVOVNVESVNVESJZVNVKQVQVAVKVLUMEVCUNUOUPOOVPVOSVOSTUQTUR
|
|
USPUT $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Finite cardinals
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c Nn $. $( Natural cardinals $)
|
|
$c 0c $. $( Cardinal zero $)
|
|
$c +c $. $( Cardinal addition. $)
|
|
$c Fin $. $( Finite sets. $)
|
|
|
|
$( Extend the definition of a class to include the set of finite
|
|
cardinals. $)
|
|
cnnc $a class Nn $.
|
|
|
|
$( Extend the definition of a class to include cardinal zero. $)
|
|
c0c $a class 0c $.
|
|
|
|
$( Extend the definition of a class to include cardinal addition. $)
|
|
cplc $a class ( A +c B ) $.
|
|
|
|
$( Extend the definition of a class to include the set of all finite sets. $)
|
|
cfin $a class Fin $.
|
|
|
|
$( Define cardinal zero. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-0c $a |- 0c = { (/) } $.
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Define cardinal addition. Definition from [Rosser] p. 275.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
df-addc $a |- ( A +c B ) =
|
|
{ x | E. y e. A E. z e. B ( ( y i^i z ) = (/) /\ x = ( y u. z ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d b y $.
|
|
$( Define the finite cardinals. Definition from [Rosser] p. 275.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
df-nnc $a |- Nn = |^| { b | ( 0c e. b /\ A. y e. b ( y +c 1c ) e. b ) } $.
|
|
$}
|
|
|
|
$( Define the set of all finite sets. Definition from [Rosser], p. 417.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
df-fin $a |- Fin = U. Nn $.
|
|
|
|
${
|
|
$d A x y z t w $. $d B x y z t w $.
|
|
$( Alternate definition of cardinal addition to establish stratification.
|
|
(Contributed by SF, 15-Jan-2015.) $)
|
|
dfaddc2 $p |- ( A +c B ) =
|
|
( ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 B ) "_k A ) $=
|
|
( vy vz vx vt vw cv wceq wa wrex cssetk cpw1 wcel copk csn wn snex 3bitri
|
|
wex cplc cin c0 cun cab cins3k cins2k c1c cimak csik csymdif cdif df-addc
|
|
ccompl vex elimak opkex elpw12 anbi1i r19.41v bitr4i exbii df-rex rexcom4
|
|
3bitr4i opkeq1 eleq1d ceqsexv eldif ndisjrelk necon2bbii otkelins3k incom
|
|
elcompl eqeq1i wel wb wal dfcleq elpw141c 19.41v excom otkelins2k elssetk
|
|
elsymdif wo opksnelsik bitri orbi12i bibi12i notbii exnal con2bii anbi12i
|
|
elun bitr2i rexbii abbi2i eqtr4i ) ABUACHZDHZUBZUCIZEHZWTXAUDZIZJZDBKZCAK
|
|
ZEUELUFZLUGZUBUHMMZUIZUNZUFZXKUGZXJUGZLUJZUJZUFZUDZUKZXLMMZUIZULZBMMZUIZA
|
|
UIZECDABUMXIEYHXDYHNWTXDOZYGNZCAKXICYGAXDEUOZUPYJXHCAYJFHZYIOZYENZFYFKZYL
|
|
XAPZPZIZYNJZFTZDBKZXHFYEYFYIWTXDUQZUPYLYFNZYNJZFTYSDBKZFTYOUUAUUDUUEFUUDY
|
|
RDBKZYNJUUEUUCUUFYNDYLBURUSYRYNDBUTVAVBYNFYFVCYSDFBVDVEYTXGDBYTYQYIOZYENZ
|
|
UUGXONZUUGYDNZQZJXGYNUUHFYQYPRZYRYMUUGYEYLYQYIVFVGVHUUGXOYDVIUUIXCUUKXFXA
|
|
WTOZXNNZXAWTUBZUCIZUUIXCUUNUUMXMNZQUUPUUMXMXAWTUQVNUUQUUOUCXAWTDUOZCUOZVJ
|
|
VKVAXAWTXDXNUURUUSYKVLXBUUOUCWTXAVMVOVEXFGEVPZGHZXENZVQZGVRZUUKGXDXEVSUUJ
|
|
UVDUUJYLUVAPZPZPZPZPZIZYLUUGOZYBNZJZFTZGTZUVCQZGTUVDQUUJUVLFYCKYLYCNZUVLJ
|
|
ZFTZUVOFYBYCUUGYQYIUQUPUVLFYCVCUVSUVMGTZFTUVOUVRUVTFUVRUVJGTZUVLJUVTUVQUW
|
|
AUVLGYLVTUSUVJUVLGWAVAVBUVMGFWBVASUVNUVPGUVNUVIUUGOZYBNZUWBXPNZUWBYANZVQZ
|
|
QUVPUVLUWCFUVIUVHRUVJUVKUWBYBYLUVIUUGVFVGVHUWBXPYAWEUWFUVCUWDUUTUWEUVBUWD
|
|
UVGYIOZXKNUVEXDOLNUUTUVGYQYIXKUVFRZUULUUBWCUVEWTXDLUVARZUUSYKWCUVAXDGUOZY
|
|
KWDSUWBXQNZUWBXTNZWFGCVPZGDVPZWFUWEUVBUWKUWMUWLUWNUWKUWGXJNUVEWTOLNUWMUVG
|
|
YQYIXJUWHUULUUBWCUVEWTXDLUWIUUSYKVLUVAWTUWJUUSWDSUWLUVGYQOXSNUVFYPOXRNZUW
|
|
NUVGYQYIXSUWHUULUUBVLUVFYPXRUVERXARWGUWOUVEXAOLNUWNUVEXALUWIUURWGUVAXAUWJ
|
|
UURWDWHSWIUWBXQXTWOUVAWTXAWOVEWJWKSVBUVCGWLSWMWPWNSWQSWQWHWRWS $.
|
|
$}
|
|
|
|
$( The expression at the heart of ~ dfaddc2 is a set. (Contributed by SF,
|
|
17-Jan-2015.) $)
|
|
addcexlem $p |- ( Ins3_k
|
|
~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) ) e. _V $=
|
|
( cssetk cins3k cins2k cin c1c cpw1 ccompl csik cun csymdif ssetkex ins3kex
|
|
cimak ins2kex inex 1cex pw1ex imakex complex sikex unex symdifex difex ) AB
|
|
ZACZDZEFZFZMZGZBUECZUDCZAHZHZBZIZJZUHFZFZMUJUIUFUHUDUEAKLZAKNZOUGEPQQZRSLUQ
|
|
USUKUPUEVANULUOUDUTNUNUMAKTTLUAUBURUHVBQQRUC $.
|
|
|
|
$( Equality law for cardinal addition. (Contributed by SF, 15-Jan-2015.) $)
|
|
addceq1 $p |- ( A = B -> ( A +c C ) = ( B +c C ) ) $=
|
|
( wceq cssetk cins3k cins2k cin c1c cpw1 cimak ccompl csik cun csymdif cdif
|
|
cplc imakeq2 dfaddc2 3eqtr4g ) ABDEFZEGZHIJJZKLFUBGUAGEMMFNOUCJJKPCJJKZAKUD
|
|
BKACQBCQABUDRACSBCST $.
|
|
|
|
$( Equality law for cardinal addition. (Contributed by SF, 15-Jan-2015.) $)
|
|
addceq2 $p |- ( A = B -> ( C +c A ) = ( C +c B ) ) $=
|
|
( wceq cssetk cins3k cins2k cin c1c cpw1 cimak ccompl csik cun csymdif cdif
|
|
cplc pw1eq syl dfaddc2 imakeq2d imakeq1d 3eqtr4g ) ABDZEFZEGZHIJJZKLFUFGUEG
|
|
EMMFNOUGJJKPZAJZJZKZCKUHBJZJZKZCKCAQCBQUDUKUNCUDUJUMUHUDUIULDUJUMDABRUIULRS
|
|
UAUBCATCBTUC $.
|
|
|
|
$( Equality law for cardinal addition. (Contributed by SF, 15-Jan-2015.) $)
|
|
addceq12 $p |- ( ( A = C /\ B = D ) -> ( A +c B ) = ( C +c D ) ) $=
|
|
( wceq cplc addceq1 addceq2 sylan9eq ) ACEBDEABFCBFCDFACBGBDCHI $.
|
|
|
|
${
|
|
addceqi.1 $e |- A = B $.
|
|
$( Equality inference for cardinal addition. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
addceq1i $p |- ( A +c C ) = ( B +c C ) $=
|
|
( wceq cplc addceq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for cardinal addition. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
addceq2i $p |- ( C +c A ) = ( C +c B ) $=
|
|
( wceq cplc addceq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
addceqi.2 $e |- C = D $.
|
|
$( Equality inference for cardinal addition. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
addceq12i $p |- ( A +c C ) = ( B +c D ) $=
|
|
( wceq cplc addceq12 mp2an ) ABGCDGACHBDHGEFACBDIJ $.
|
|
$}
|
|
|
|
${
|
|
addceqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for cardinal addition. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
addceq1d $p |- ( ph -> ( A +c C ) = ( B +c C ) ) $=
|
|
( wceq cplc addceq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for cardinal addition. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
addceq2d $p |- ( ph -> ( C +c A ) = ( C +c B ) ) $=
|
|
( wceq cplc addceq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
addceqd.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for cardinal addition. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
addceq12d $p |- ( ph -> ( A +c C ) = ( B +c D ) ) $=
|
|
( wceq cplc addceq12 syl2anc ) ABCHDEHBDICEIHFGBDCEJK $.
|
|
$}
|
|
|
|
$( Cardinal zero is a set. (Contributed by SF, 14-Jan-2015.) $)
|
|
0cex $p |- 0c e. _V $=
|
|
( c0c c0 csn cvv df-0c snex eqeltri ) ABCDEBFG $.
|
|
|
|
$( The cardinal sum of two sets is a set. (Contributed by SF,
|
|
15-Jan-2015.) $)
|
|
addcexg $p |- ( ( A e. V /\ B e. W ) -> ( A +c B ) e. _V ) $=
|
|
( wcel wa cplc cssetk cins3k cins2k cin c1c cpw1 ccompl csik cun cvv pw1exg
|
|
cimak imakexg csymdif cdif dfaddc2 addcexlem mpan sylan ancoms syl5eqel
|
|
3syl ) ACEZBDEZFABGHIZHJZKLMMZSNIUMJULJHOOIPUAUNMMSUBZBMZMZSZASZQABUCUKUJUS
|
|
QEZUKURQEZUJUTUKUPQEUQQEZVABDRUPQRUOQEVBVAUDUOUQQQTUEUIURAQCTUFUGUH $.
|
|
|
|
${
|
|
addcex.1 $e |- A e. _V $.
|
|
addcex.2 $e |- B e. _V $.
|
|
$( The cardinal sum of two sets is a set. (Contributed by SF,
|
|
25-Jan-2015.) $)
|
|
addcex $p |- ( A +c B ) e. _V $=
|
|
( cvv wcel cplc addcexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w t $.
|
|
$( Definition of the finite cardinals for existence theorem. (Contributed
|
|
by SF, 14-Jan-2015.) $)
|
|
dfnnc2 $p |- Nn =
|
|
|^| ( { x | 0c e. x } \
|
|
( ( _S_k \ ( _S_k o._k SI_k
|
|
Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k
|
|
~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k 1c ) ) $=
|
|
( vy vz vt vw cv wcel c1c wa cssetk cpw1 cimak wn vex wceq copk wex bitri
|
|
cvv exbii cnnc c0c cplc wral cab cint cins3k cins2k cin csik csymdif cdif
|
|
ccompl cun cimagek ccomk df-nnc eldif eleq2 elab wrex csn wel snex opkeq1
|
|
eleq1d ceqsexv elssetk opkeq2 opksnelsik syl6bb opkelimagekg mp2an eqeq2i
|
|
anbi12d dfaddc2 bitr4i anbi12i opkelcok el1c anbi1i 19.41v cxpk sikss1c1c
|
|
wb sseli opkelxpk simprbi syl pm4.71ri anass excom 3bitr4i df-clel notbii
|
|
elimak df-rex rexnal bitr2i con1bii abbi2i inteqi eqtr4i ) UAUBBFZGZCFZHU
|
|
CZXDGZCXDUDZIZBUEZUFUBAFZGZAUEZJJJUGZJUHZUIHKKZLUMUGXPUHXOUHJUJUJUGUNUKXQ
|
|
KKLULXQLZUOZUJZUPZULZHLZULZUFCBUQYDXKXJBYDXDYDGXDXNGZXDYCGZMZIXJXDXNYCURY
|
|
EXEYGXIXMXEAXDBNZXLXDUBUSUTXIYFYFXHMZCXDVAZXIMDFZXFVBZOZYKXDPZYBGZIZDQZCQ
|
|
ZCBVCZYIIZCQYFYJYQYTCYQYLXDPZYBGZYTYOUUBDYLXFVDZYMYNUUAYBYKYLXDVEVFVGUUBU
|
|
UAJGZUUAYAGZMZIYTUUAJYAURUUDYSUUFYIXFXDCNZYHVHUUEXHYKEFZVBZOZYLYKPZXTGZYN
|
|
JGZIZIZDQZEQZUUHXGOZEBVCZIZEQUUEXHUUPUUTEUUPXFUUHPXSGZUUIXDPZJGZIZUUTUUNU
|
|
VDDUUIUUHVDUUJUULUVAUUMUVCUUJUULYLUUIPZXTGUVAUUJUUKUVEXTYKUUIYLVIVFXFUUHX
|
|
SUUGENZVJVKUUJYNUVBJYKUUIXDVEVFVOVGUVAUURUVCUUSUVAUUHXRXFLZOZUURXFSGUUHSG
|
|
UVAUVHWEUUGUVFXFUUHXRSSVLVMXGUVGUUHXFHVPVNVQUUHXDUVFYHVHVRRTUUEUUNDQZUUQD
|
|
YLXDJXTUUCYHVSYKHGZUUNIZDQUUOEQZDQUVIUUQUVKUVLDUVKUUJEQZUUNIUVLUVJUVMUUNE
|
|
YKVTWAUUJUUNEWBVQTUUNUVKDUUNUVJUULIZUUMIUVKUULUVNUUMUULUVJUULUUKHHWCZGZUV
|
|
JXTUVOUUKXSWDWFUVPYLHGUVJYLYKHHUUCDNWGWHWIWJWAUVJUULUUMWKRTUUOEDWLWMREXGX
|
|
DWNWMWOVRRRTYFYODHVAZYRDYBHXDYHWPUVJYOIZDQYPCQZDQUVQYRUVRUVSDUVRYMCQZYOIU
|
|
VSUVJUVTYOCYKVTWAYMYOCWBVQTYODHWQYPCDWLWMRYICXDWQWMXHCXDWRWSWTVRRXAXBXC
|
|
$.
|
|
$}
|
|
|
|
$( The class of all finite cardinals is a set. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
nncex $p |- Nn e. _V $=
|
|
( vx cnnc c0c cv wcel cab cssetk cins3k cins2k cin c1c cpw1 cimak csik cdif
|
|
ssetkex 1cex pw1ex imakex difex ccompl cun csymdif cimagek ccomk cvv dfnnc2
|
|
cint setswithex addcexlem imagekex sikex cokex intex eqeltri ) BCADEAFZGGGH
|
|
ZGIZJKLZLZMUAHURIUQIGNNHUBUCUTLLMOZUTMZUDZNZUEZOZKMZOZUHUFAUGVHUPVGACUIVFKG
|
|
VEPGVDPVCVBVAUTUJUSKQRRSUKULUMTQSTUNUO $.
|
|
|
|
$( The class of all finite sets is a set. (Contributed by SF,
|
|
19-Jan-2015.) $)
|
|
finex $p |- Fin e. _V $=
|
|
( cfin cnnc cuni cvv df-fin nncex uniex eqeltri ) ABCDEBFGH $.
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d b c $. $d M a $.
|
|
$d M b $. $d M c $. $d N a $. $d N b $. $d N c $.
|
|
|
|
$( Membership in cardinal addition. Theorem X.1.1 of [Rosser] p. 275.
|
|
(Contributed by SF, 16-Jan-2015.) $)
|
|
eladdc $p |- ( A e. ( M +c N ) <->
|
|
E. b e. M E. c e. N ( ( b i^i c ) = (/) /\ A = ( b u. c ) ) ) $=
|
|
( va cplc wcel cvv cv cin c0 wceq cun wa wrex elex id vex rexlimivw eqeq1
|
|
unex syl6eqel adantl anbi2d 2rexbidv df-addc elab2g pm5.21nii ) ABCGZHAIH
|
|
ZDJZEJZKLMZAULUMNZMZOZECPZDBPZAUJQURUKDBUQUKECUPUKUNUPAUOIUPRULUMDSESUBUC
|
|
UDTTUNFJZUOMZOZECPDBPUSFAUJIUTAMZVBUQDEBCVCVAUPUNUTAUOUAUEUFFDEBCUGUHUI
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B a b $. $d M a b $. $d N a b $.
|
|
$( Inference form of membership in cardinal addition. (Contributed by SF,
|
|
26-Jan-2015.) $)
|
|
eladdci $p |- ( ( A e. M /\ B e. N /\ ( A i^i B ) = (/) ) ->
|
|
( A u. B ) e. ( M +c N ) ) $=
|
|
( va vb wcel cin c0 wceq w3a cv cun wrex cplc eqid eqeq1d eqeq2d anbi12d
|
|
wa ineq1 uneq1 ineq2 uneq2 rspc2ev 3expa mpanr2 3impa eladdc sylibr ) ACG
|
|
ZBDGZABHZIJZKELZFLZHZIJZABMZUOUPMZJZTZFDNECNZUSCDOGUKULUNVCUKULTUNUSUSJZV
|
|
CUSPUKULUNVDTZVCVBVEAUPHZIJZUSAUPMZJZTEFABCDUOAJZURVGVAVIVJUQVFIUOAUPUAQV
|
|
JUTVHUSUOAUPUBRSUPBJZVGUNVIVDVKVFUMIUPBAUCQVKVHUSUSUPBAUDRSUEUFUGUHUSCDEF
|
|
UIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A n m $.
|
|
$( The empty class is not a member of a successor. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
0nelsuc $p |- -. (/) e. ( A +c 1c ) $=
|
|
( vn vm c0 c1c cplc wcel cv cin wceq cun wa wrex csn el1c simpr nsyl nrex
|
|
wn wex vex snid ax-mp eqeq1 mtbiri exlimiv sylbi eqcom bitri notbii sylib
|
|
n0i un00 a1i eladdc mtbir ) DAEFGBHZCHZIDJZDUQURKZJZLZCEMZBAMVCBAVCSUQAGV
|
|
BCEUREGZVAVBVDUQDJZURDJZLZSVASVDVFVGVDURUQNZJZBTVFSZBUROVIVJBVIVFVHDJZUQV
|
|
HGVKSUQBUAUBVHUQULUCURVHDUDUEUFUGVEVFPQVGVAVGUTDJVAUQURUMUTDUHUIUJUKUSVAP
|
|
QRUNRDAEBCUOUP $.
|
|
$}
|
|
|
|
$( Cardinal zero is not a successor. Compare Theorem X.1.2 of [Rosser]
|
|
p. 275. (Contributed by SF, 14-Jan-2015.) $)
|
|
0cnsuc $p |- ( A +c 1c ) =/= 0c $=
|
|
( c1c cplc c0c wne wceq wn wcel 0nelsuc csn 0ex df-0c eleqtrri eleq2 mpbiri
|
|
c0 snid mto df-ne mpbir ) ABCZDEUADFZGUBPUAHZAIUBUCPDHPPJDPKQLMUADPNORUADST
|
|
$.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Cardinal zero is a finite cardinal. Theorem X.1.4 of [Rosser] p. 276.
|
|
(Contributed by SF, 14-Jan-2015.) $)
|
|
peano1 $p |- 0c e. Nn $=
|
|
( vx vy c0c cnnc wcel cv c1c cplc wral wa cab cint wal df-nnc eleq2i 0cex
|
|
wi elintab bitri simpl mpgbir ) CDEZCAFZEZBFGHUCEBUCIZJZUDQZAUBCUFAKLZEUG
|
|
AMDUHCBANOUFACPRSUDUETUA $.
|
|
$}
|
|
|
|
${
|
|
$d A a x y $.
|
|
$( The finite cardinals are closed under addition of one. Theorem X.1.5 of
|
|
[Rosser] p. 276. (Contributed by SF, 14-Jan-2015.) $)
|
|
peano2 $p |- ( A e. Nn -> ( A +c 1c ) e. Nn ) $=
|
|
( va vx vy cv c1c cplc cnnc wcel wceq addceq1 eleq1d c0c wa wi wal eleq2i
|
|
wral elintab bitri wel weq rspccv adantl a2i alimi cab cint df-nnc addcex
|
|
vex 1cex 3imtr4i vtoclga ) BEZFGZHIZAFGZHIBAHUOAJUPURHUOAFKLMCEZIZDEZFGZU
|
|
SIZDUSRZNZBCUAZOZCPZVEUPUSIZOZCPZUOHIZUQVGVJCVEVFVIVDVFVIOUTVCVIDUOUSDBUB
|
|
VBUPUSVAUOFKLUCUDUEUFVLUOVECUGUHZIVHHVMUODCUIZQVECUOBUKZSTUQUPVMIVKHVMUPV
|
|
NQVECUPUOFVOULUJSTUMUN $.
|
|
$}
|
|
|
|
$( The successor of a finite cardinal is not zero. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
peano3 $p |- ( A e. Nn -> ( A +c 1c ) =/= 0c ) $=
|
|
( c1c cplc c0c wne cnnc wcel 0cnsuc a1i ) ABCDEAFGAHI $.
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Cardinal zero is a fixed point for cardinal addition. Theorem X.1.8 of
|
|
[Rosser] p. 276. (Contributed by SF, 16-Jan-2015.) $)
|
|
addcid1 $p |- ( A +c 0c ) = A $=
|
|
( vx vy vz c0c cplc c0 csn df-0c addceq2i cv cin wceq cun wa wrex weq 0ex
|
|
wcel ineq2 eqeq1d uneq2 eqeq2d anbi12d in0 biantrur syl6bbr eqeq2i equcom
|
|
rexsn un0 3bitri rexbii eladdc risset 3bitr4i eqriv eqtri ) AEFAGHZFZAEUS
|
|
AIJBUTACKZDKZLZGMZBKZVAVBNZMZOZDUSPZCAPCBQZCAPVEUTSVEASVIVJCAVIVEVAGNZMZB
|
|
CQVJVHVLDGRVBGMZVHVAGLZGMZVLOVLVMVDVOVGVLVMVCVNGVBGVATUAVMVFVKVEVBGVAUBUC
|
|
UDVOVLVAUEUFUGUJVKVAVEVAUKUHBCUIULUMVEAUSCDUNCVEAUOUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Cardinal sum commutes. Theorem X.1.9 of [Rosser] p. 276. (Contributed
|
|
by SF, 15-Jan-2015.) $)
|
|
addccom $p |- ( A +c B ) = ( B +c A ) $=
|
|
( vy vz vx cv cin c0 wceq cun wa wrex cab cplc incom eqeq1i uncom anbi12i
|
|
eqeq2i df-addc 2rexbii rexcom bitri abbii 3eqtr4i ) CFZDFZGZHIZEFZUFUGJZI
|
|
ZKZDBLCALZEMUGUFGZHIZUJUGUFJZIZKZCALDBLZEMABNBANUNUTEUNUSDBLCALUTUMUSCDAB
|
|
UIUPULURUHUOHUFUGOPUKUQUJUFUGQSRUAUSCDABUBUCUDECDABTEDCBATUE $.
|
|
$}
|
|
|
|
$( Cardinal zero is a fixed point for cardinal addition. Theorem X.1.8 of
|
|
[Rosser] p. 276. (Contributed by SF, 16-Jan-2015.) $)
|
|
addcid2 $p |- ( 0c +c A ) = A $=
|
|
( c0c cplc addccom addcid1 eqtri ) BACABCABADAEF $.
|
|
|
|
$( Cardinal one is a finite cardinal. Theorem X.1.12 of [Rosser] p. 277.
|
|
(Contributed by SF, 16-Jan-2015.) $)
|
|
1cnnc $p |- 1c e. Nn $=
|
|
( c1c c0c cplc cnnc addcid1 addccom eqtr3i wcel peano1 peano2 ax-mp eqeltri
|
|
) ABACZDABCAMAEABFGBDHMDHIBJKL $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The principle of mathematical induction: a set containing cardinal zero
|
|
and closed under the successor operator is a superset of the finite
|
|
cardinals. Theorem X.1.6 of [Rosser] p. 276. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
peano5 $p |- ( ( A e. V /\ 0c e. A /\
|
|
A. x e. Nn ( x e. A -> ( x +c 1c ) e. A ) ) ->
|
|
Nn C_ A ) $=
|
|
( vy wcel cnnc cin cvv c0c cv c1c wi wral wss mpan elin biimpri syl eleq2
|
|
wa cplc nncex inexg peano1 imbi1i impexp bitri inss1 sseli peano2 a1i a2i
|
|
mpand sylbir ralimi2 w3a cab cint df-nnc raleqbi1dv anbi12d elabg biimprd
|
|
wceq 3impib intss1 syl5eqss inss2 syl6ss syl3an ) BCEZFBGZHEZIBEZIVLEZAJZ
|
|
BEZVPKUAZBEZLZAFMVRVLEZAVLMZFBNFHEVKVMUBFBHCUCOIFEZVNVOUDVOWCVNTIFBPQOVTW
|
|
AAFVLVPFEZVTLZVPVLEZVSLZWFWALWGWDVQTZVSLWEWFWHVSVPFBPUEWDVQVSUFUGWFVSWAWF
|
|
VRFEZVSWAWFWDWIVLFVPFBUHUIVPUJRWIVSTZWALWFWAWJVRFBPQUKUMULUNUOVMVOWBUPZFV
|
|
LBWKFIDJZEZVRWLEZAWLMZTZDUQZURZVLADUSWKVLWQEZWRVLNVMVOWBWSVMWSVOWBTZWPWTD
|
|
VLHWLVLVDWMVOWOWBWLVLISWNWAAWLVLWLVLVRSUTVAVBVCVEVLWQVFRVGFBVHVIVJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d ch x $. $d ph y $. $d ps x $. $d ta x $. $d th x $.
|
|
$d x y $. $d et y $.
|
|
findsd.1 $e |- ( et -> { x | ph } e. V ) $.
|
|
findsd.2 $e |- ( x = 0c -> ( ph <-> ps ) ) $.
|
|
findsd.3 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
findsd.4 $e |- ( x = ( y +c 1c ) -> ( ph <-> th ) ) $.
|
|
findsd.5 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
findsd.6 $e |- ( et -> ps ) $.
|
|
findsd.7 $e |- ( ( y e. Nn /\ et ) -> ( ch -> th ) ) $.
|
|
$( Principle of finite induction over the finite cardinals, using implicit
|
|
substitutions. The first hypothesis ensures stratification of ` ph ` ,
|
|
the next four set up the substitutions, and the last two set up the base
|
|
case and induction hypothesis. This version allows for an extra
|
|
deduction clause that may make proving stratification simpler. Compare
|
|
Theorem X.1.13 of [Rosser] p. 277. (Contributed by SF, 31-Jul-2019.) $)
|
|
findsd $p |- ( ( A e. Nn /\ et ) -> ta ) $=
|
|
( cnnc wcel elab wa cab c0c c1c cplc wral wss 0cex sylibr vex 1cex addcex
|
|
cv wi 3imtr4g ancoms ralrimiva peano5 syl3anc sseld impcom wb elabg mpbid
|
|
adantr ) IRSZFUAIAGUBZSZEFVFVHFRVGIFVGJSUCVGSZHUMZVGSZVJUDUEZVGSZUNZHRUFR
|
|
VGUGKFBVIPABGUCUHLTUIFVNHRVJRSZFVNVOFUACDVKVMQACGVJHUJZMTADGVLVJUDVPUKULN
|
|
TUOUPUQHVGJURUSUTVAVFVHEVBFAEGIROVCVEVD $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d ch x $. $d ph y $. $d ps x $. $d ta x $. $d th x $.
|
|
$d x y $.
|
|
finds.1 $e |- { x | ph } e. _V $.
|
|
finds.2 $e |- ( x = 0c -> ( ph <-> ps ) ) $.
|
|
finds.3 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
finds.4 $e |- ( x = ( y +c 1c ) -> ( ph <-> th ) ) $.
|
|
finds.5 $e |- ( x = A -> ( ph <-> ta ) ) $.
|
|
finds.6 $e |- ps $.
|
|
finds.7 $e |- ( y e. Nn -> ( ch -> th ) ) $.
|
|
$( Principle of finite induction over the finite cardinals, using implicit
|
|
substitutions. The first hypothesis ensures stratification of ` ph ` ,
|
|
the next four set up the substitutions, and the last two set up the base
|
|
case and induction hypothesis. Compare Theorem X.1.13 of [Rosser]
|
|
p. 277. (Contributed by SF, 14-Jan-2015.) $)
|
|
finds $p |- ( A e. Nn -> ta ) $=
|
|
( cnnc wcel wtru cvv a1i tru cab cv wi adantr findsd mpan2 ) HPQREUAABCDE
|
|
RFGHSAFUBSQRITJKLMBRNTGUCPQCDUDROUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A x n m $.
|
|
$( All naturals are either zero or a successor. Theorem X.1.7 of [Rosser]
|
|
p. 276. (Contributed by SF, 14-Jan-2015.) $)
|
|
nnc0suc $p |- ( A e. Nn <->
|
|
( A = 0c \/ E. x e. Nn A = ( x +c 1c ) ) ) $=
|
|
( vn vm cnnc wcel c0c wceq cv c1c wrex cssetk cins3k cpw1 cimak cvv eqeq1
|
|
wo rexbidv orbi12d cplc csn cins2k cin ccompl csik cun csymdif cdif df-sn
|
|
cimagek cab vex elimak wb opkelimagekg mp2an dfaddc2 eqeq2i bitr4i rexbii
|
|
copk bitri abbi2i uneq12i unab eqtri snex addcexlem pw1ex imakex imagekex
|
|
1cex nncex unex eqeltrri eqid orci addceq1 eqeq2d rspcev mpan2 olcd finds
|
|
weq a1d peano1 eleq1 mpbiri peano2 syl5ibrcom rexlimiv jaoi impbii ) BEFZ
|
|
BGHZBAIZJUAZHZAEKZRZCIZGHZXBWRHZAEKZRZGGHZGWRHZAEKZRDIZGHZXJWRHZAEKZRZXJJ
|
|
UAZGHZXOWRHZAEKZRZXACDBGUBZLMZLUCZUDJNZNZOUEMYBUCYAUCLUFUFMUGUHYDNNOUIZYD
|
|
OZUKZEOZUGZXFCULZPYIXCCULZXECULZUGYJXTYKYHYLCGUJXECYHXBYHFWQXBVBYGFZAEKXE
|
|
AYGEXBCUMZUNYMXDAEYMXBYFWQOZHZXDWQPFXBPFYMYPUOAUMYNWQXBYFPPUPUQWRYOXBWQJU
|
|
RUSUTVAVCVDVEXCXECVFVGXTYHGVHYGEYFYEYDVIYCJVMVJVJVKVLVNVKVOVPXCXCXGXEXIXB
|
|
GGQXCXDXHAEXBGWRQSTCDWEZXCXKXEXMXBXJGQYQXDXLAEXBXJWRQSTXBXOHZXCXPXEXRXBXO
|
|
GQYRXDXQAEXBXOWRQSTXBBHZXCWPXEWTXBBGQYSXDWSAEXBBWRQSTXGXIGVQVRXJEFZXSXNYT
|
|
XRXPYTXOXOHZXRXOVQXQUUAAXJEADWEWRXOXOWQXJJVSVTWAWBWCWFWDWPWOWTWPWOGEFWGBG
|
|
EWHWIWSWOAEWQEFWOWSWREFWQWJBWREWHWKWLWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d A b $. $d A x $. $d A y $. $d b x $. $d b y $. $d M b $. $d M y $.
|
|
$d x y $.
|
|
$( Membership in a successor. Theorem X.1.16 of [Rosser] p. 279.
|
|
(Contributed by SF, 16-Jan-2015.) $)
|
|
elsuc $p |- ( A e. ( M +c 1c ) <->
|
|
E. b e. M E. x e. ~ b A = ( b u. { x } ) ) $=
|
|
( vy c1c wcel cv cin c0 wceq cun wa wrex bitr4i anbi1i bitri exbii df-rex
|
|
wex cplc csn ccompl eladdc snex ineq2 eqeq1d uneq2 eqeq2d anbi12d ceqsexv
|
|
wel wn disjsn vex elcompl el1c 19.41v excom 3bitr4i rexbii ) BCFUAGDHZEHZ
|
|
IZJKZBVBVCLZKZMZEFNZDCNBVBAHZUBZLZKZAVBUCZNZDCNBCFDEUDVIVODCVCVKKZVHMZETZ
|
|
ATZVJVNGZVMMZATVIVOVRWAAVRVBVKIZJKZVMMZWAVHWDEVKVJUEVPVEWCVGVMVPVDWBJVCVK
|
|
VBUFUGVPVFVLBVCVKVBUHUIUJUKWCVTVMWCADULUMVTVBVJUNVJVBAUOUPOPQRVIVCFGZVHMZ
|
|
ETZVSVHEFSWGVQATZETVSWFWHEWFVPATZVHMWHWEWIVHAVCUQPVPVHAURORVQEAUSQQVMAVNS
|
|
UTVAQ $.
|
|
$}
|
|
|
|
${
|
|
$d N a x $. $d A a x $. $d X a x $.
|
|
elsuci.1 $e |- X e. _V $.
|
|
$( Lemma for ~ ncfinraise . Take a natural and a disjoint union and
|
|
compute membership in the successor natural. (Contributed by SF,
|
|
22-Jan-2015.) $)
|
|
elsuci $p |- ( ( A e. N /\ -. X e. A ) ->
|
|
( A u. { X } ) e. ( N +c 1c ) ) $=
|
|
( va vx wcel wn wa csn cun cv wceq ccompl wrex cplc elcompl eqeq2d rspcev
|
|
c1c eqid uneq2d mpan2 sylbir compleq uneq1 rexeqbidv sylan2 elsuc sylibr
|
|
sneq ) ABGZCAGHZIACJZKZELZFLZJZKZMZFUPNZOZEBOZUOBTPGUMULUOAURKZMZFANZOZVC
|
|
UMCVFGZVGCADQVHUOUOMZVGUOUAVEVIFCVFUQCMZVDUOUOVJURUNAUQCUKUBRSUCUDVBVGEAB
|
|
UPAMZUTVEFVAVFUPAUEVKUSVDUOUPAURUFRUGSUHFUOBEUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. $d A d $.
|
|
$d a e $. $d A e $. $d a x $. $d A x $. $d B a $. $d B b $. $d b c $.
|
|
$d B c $. $d b d $. $d B d $. $d b e $. $d B e $. $d b x $. $d B x $.
|
|
$d C a $. $d C b $. $d C c $. $d c d $. $d C d $. $d c e $. $d C e $.
|
|
$d c x $. $d C x $. $d d x $. $d e x $.
|
|
|
|
$( Cardinal addition is associative. Theorem X.1.11, corollary 1 of
|
|
[Rosser] p. 277. (Contributed by SF, 17-Jan-2015.) $)
|
|
addcass $p |- ( ( A +c B ) +c C ) = ( A +c ( B +c C ) ) $=
|
|
( va vb vd vc ve cv cin c0 wceq cun wex wrex bitri exbii rexbii rexcom4
|
|
wa vx cplc wcel ancom anbi2i an12 indir eqeq1i un00 bitr4i 3bitr4i eqeq2i
|
|
indi unass anbi12i anass 3bitr3i ineq1 eqeq1d uneq1 eqeq2d anbi12d anbi2d
|
|
vex unex ceqsexv ineq2 uneq2 2rexbii eladdc df-rex r19.42v r19.41v anbi1i
|
|
3bitr4ri eqriv ) UAABUBZCUBZABCUBZUBZDIZEIZJZKLZFIZWAWBMZLZTZWEGIZJZKLZUA
|
|
IZWEWIMZLZTZTZFNZGCOZEBOZDAOZWBWIJZKLZHIZWBWIMZLZTZWAXCJZKLZWLWAXCMZLZTZT
|
|
ZHNZGCOZEBOZDAOZWLVRUCZWLVTUCZWRXNDEABWQXMGCWDWFWIJZKLZWLWFWIMZLZTZTZXBWA
|
|
XDJZKLZWLWAXDMZLZTZTZWQXMWDXTTZYBTXBYFTZYHTYDYJYKYLYBYHWDWAWIJZKLZXBTZTZX
|
|
BWDYNTZTZYKYLYPWDXBYNTZTYRYOYSWDYNXBUDUEWDXBYNUFPXTYOWDXTYMXAMZKLYOXSYTKW
|
|
AWBWIUGUHYMXAUIUJUEYFYQXBYFWCYMMZKLYQYEUUAKWAWBWIUMUHWCYMUIUJUEUKYAYGWLWA
|
|
WBWIUNULUOWDXTYBUPXBYFYHUPUQWQWGWDWOTZTZFNYDWPUUCFWPWDWGWOTTUUCWDWGWOUPWD
|
|
WGWOUFPQUUBYDFWFWAWBDVDEVDZVEWGWOYCWDWGWKXTWNYBWGWJXSKWEWFWIURUSWGWMYAWLW
|
|
EWFWIUTVAVBVCVFPXMXEXBXKTZTZHNYJXLUUFHXLXBXEXKTTUUFXBXEXKUPXBXEXKUFPQUUEY
|
|
JHXDWBWIUUDGVDVEXEXKYIXBXEXHYFXJYHXEXGYEKXCXDWAVGUSXEXIYGWLXCXDWAVHVAVBVC
|
|
VFPUKRVIXQWOGCOZFVQOZWTWLVQCFGVJUUHWEVQUCZUUGTZFNZWTUUGFVQVKWHUUGTZEBOZFN
|
|
ZDAOUUMDAOZFNWTUUKUUMDFASWSUUNDAWSUULFNZEBOUUNWRUUPEBWRWPGCOZFNUUPWPGFCSU
|
|
UQUULFWHWOGCVLQPRUULEFBSPRUUJUUOFWHEBOZUUGTZDAOUURDAOZUUGTUUOUUJUURUUGDAV
|
|
MUUMUUSDAWHUUGEBVMRUUIUUTUUGWEABDEVJVNVOQVOPPXRXKHVSOZDAOXPWLAVSDHVJUVAXO
|
|
DAUVAXCVSUCZXKTZHNZXOXKHVSVKXLGCOZHNZEBOUVEEBOZHNXOUVDUVEEHBSXNUVFEBXLGHC
|
|
SRUVCUVGHXFGCOZXKTZEBOUVHEBOZXKTUVGUVCUVHXKEBVMUVEUVIEBXFXKGCVMRUVBUVJXKX
|
|
CBCEGVJVNVOQVOPRPUKVP $.
|
|
$}
|
|
|
|
$( Swap arguments two and three in cardinal addition. (Contributed by SF,
|
|
22-Jan-2015.) $)
|
|
addc32 $p |- ( ( A +c B ) +c C ) = ( ( A +c C ) +c B ) $=
|
|
( cplc addccom addceq2i addcass 3eqtr4i ) ABCDZDACBDZDABDCDACDBDIJABCEFABCG
|
|
ACBGH $.
|
|
|
|
$( Swap arguments two and three in quadruple cardinal addition. (Contributed
|
|
by SF, 25-Jan-2015.) $)
|
|
addc4 $p |- ( ( A +c B ) +c ( C +c D ) ) = ( ( A +c C ) +c ( B +c D ) ) $=
|
|
( cplc addc32 addceq1i addcass 3eqtr3i ) ABEZCEZDEACEZBEZDEJCDEELBDEEKMDABC
|
|
FGJCDHLBDHI $.
|
|
|
|
$( Rearrange cardinal summation of six arguments. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
addc6 $p |- ( ( ( A +c B ) +c ( C +c D ) ) +c ( E +c F ) ) =
|
|
( ( ( A +c C ) +c E ) +c ( ( B +c D ) +c F ) ) $=
|
|
( cplc addc4 addceq1i addc32 eqtri addcass 3eqtr3i ) ABGCDGGZEGZFGACGZEGZBD
|
|
GZGZFGNEFGGQRFGGOSFOPRGZEGSNTEABCDHIPREJKINEFLQRFLM $.
|
|
|
|
${
|
|
$d A a b c x $. $d B a b c x $.
|
|
$( The finite cardinals are closed under addition. Theorem X.1.14 of
|
|
[Rosser] p. 278. (Contributed by SF, 17-Jan-2015.) $)
|
|
nncaddccl $p |- ( ( A e. Nn /\ B e. Nn ) -> ( A +c B ) e. Nn ) $=
|
|
( va vb vc vx cnnc wcel cplc cv wceq eleq1d imbi2d c1c cab cpw1 cimak cvv
|
|
wi addceq2 addceq1 c0c wn cssetk cins3k cins2k cin ccompl csymdif cimagek
|
|
csik cun cdif ccnvk wo unab copk wrex wb vex opkelimagekg mp2an opkelcnvk
|
|
addccom dfaddc2 eqeq2i 3bitr4i rexbii elimak risset abbi2i uneq2i 3eqtr4i
|
|
eqtri imor abbii abexv addcexlem pw1ex imakex imagekex nncex eqeltrri weq
|
|
cnvkex unex addcid1 id syl5eqel addcass peano2 syl5eqelr imim2i a1i finds
|
|
com12 vtoclga imp ) AGHBGHZABIZGHZWSCJZBIZGHZSWSXASCAGXBAKZXDXAWSXEXCWTGX
|
|
BABUALMWSXBGHZXDXFXBDJZIZGHZSZXFXBUBIZGHZSXFXBEJZIZGHZSZXFXBXMNIZIZGHZSZX
|
|
FXDSDEBXFUCZDOZUDUEZUDUFZUGNPPZQUHUEYDUFYCUFUDUKUKUEULUIYEPPQUMZXBPZPZQZU
|
|
JZUNZGQZULZXJDOZRYBXIDOZULYAXIUOZDOYMYNYAXIDUPYLYOYBXIDYLFJZXGUQYKHZFGURY
|
|
QXHKZFGURXGYLHXIYRYSFGXGYQUQYJHZYQYIXGQZKZYRYSXGRHYQRHYTUUBUSDUTZFUTZXGYQ
|
|
YIRRVAVBYQXGYJUUDUUCVCXHUUAYQXHXGXBIUUAXBXGVDXGXBVEVNVFVGVHFYKGXGUUCVIFXH
|
|
GVJVGVKVLXJYPDXFXIVOVPVMYBYLYADVQYKGYJYIYFYHVRYGXBCUTVSVSVTWAWEWBVTWFWCXG
|
|
UBKZXIXLXFUUEXHXKGXGUBXBTLMDEWDZXIXOXFUUFXHXNGXGXMXBTLMXGXQKZXIXSXFUUGXHX
|
|
RGXGXQXBTLMXGBKZXIXDXFUUHXHXCGXGBXBTLMXFXKXBGXBWGXFWHWIXPXTSXMGHXOXSXFXOX
|
|
RXNNIGXBXMNWJXNWKWLWMWNWOWPWQWR $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Membership in the set of finite sets. (Contributed by SF,
|
|
19-Jan-2015.) $)
|
|
elfin $p |- ( A e. Fin <-> E. x e. Nn A e. x ) $=
|
|
( cfin wcel cnnc cuni cv wrex df-fin eleq2i eluni2 bitri ) BCDBEFZDBAGDAE
|
|
HCMBIJABEKL $.
|
|
$}
|
|
|
|
$( Membership in cardinal zero. (Contributed by SF, 22-Jan-2015.) $)
|
|
el0c $p |- ( A e. 0c <-> A = (/) ) $=
|
|
( c0c wcel c0 csn wceq df-0c eleq2i 0ex elsnc2 bitri ) ABCADEZCADFBLAGHADIJ
|
|
K $.
|
|
|
|
$( The empty set is a member of cardinal zero. (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
nulel0c $p |- (/) e. 0c $=
|
|
( c0 c0c wcel wceq eqid el0c mpbir ) ABCAADAEAFG $.
|
|
|
|
$( The empty set is finite. (Contributed by SF, 19-Jan-2015.) $)
|
|
0fin $p |- (/) e. Fin $=
|
|
( vn c0 cfin wcel cv cnnc wrex c0c peano1 wceq eqid el0c mpbir eleq2 rspcev
|
|
mp2an elfin ) BCDBAEZDZAFGZHFDBHDZTIUABBJBKBLMSUAAHFRHBNOPABQM $.
|
|
|
|
${
|
|
$d a e $. $d a m $. $d a t $. $d a w $. $d a x $. $d e m $. $d e t $.
|
|
$d e w $. $d e x $. $d m t $. $d m w $. $d m x $. $d t w $. $d t x $.
|
|
$d w x $.
|
|
$( Lemma for ~ nnsucelr . Establish stratification for the inductive
|
|
hypothesis. (Contributed by SF, 15-Jan-2015.) $)
|
|
nnsucelrlem1 $p |- { m |
|
|
A. a A. x ( ( -. x e. a /\ ( a u. { x } ) e. ( m +c 1c ) ) -> a e. m ) }
|
|
e. _V $=
|
|
( vt vw ve cssetk c1c cvv wn csn wcel wa wex wceq copk bitr4i exbii bitri
|
|
snex csik cins3k cins2k cidk cun csymdif cpw1 cimak ccompl cin cdif ccnvk
|
|
cimagek ccomk cxpk wel cv cplc wal cab wrex vex elimak df-rex el1c anbi1i
|
|
19.41v excom opkeq1 eleq1d ceqsexv opkex elpw131c eldif elpw141c elpw171c
|
|
wi elin wb elsymdif otkelins3k opksnelsik elssetk otkelins2k wo weq sneqb
|
|
opkelidkg mp2an elsnc 3bitr4i orbi12i bibi12i notbii elcompl dfcleq ancom
|
|
elun opkelcnvk opkelimagekg dfaddc2 eqeq2i bicomi anbi12i opkelcok addcex
|
|
alex 1cex clel3 df-clel opkelxpk mpbiran annim exnal abbi2i ssetkex sikex
|
|
ins3kex ins2kex idkex unex symdifex pw1ex imakex complex addcexlem cnvkex
|
|
imagekex cokex inex difex vvex xpkex eqeltrri ) GUAZUAZUAZUAZUAZUBZYOUBZU
|
|
CZUDUBZUEZUCZUFZHUGZUGZUGZUGZUGZUGZUGZUHZUIZGUBZGUCZUJUUHUHUIUBUUQUCUUPUC
|
|
YPUBUEUFUUJUHUKZUUHUHZUMZULZGUNZUCZUCZUJZUUJUHZUUAUKZIGUOZUKZUUIUHZHUHZUI
|
|
ZACUPZJZCUQZAUQZKZUEZBUQZHURZLZMZCBUPZVQZAUSZCUSZBUTIUWFBUVLUVSUVKLZJUWEJ
|
|
ZCNZJUVSUVLLUWFUWGUWIUWGDUQZUVOKZOZUWJUVSPZUVJLZMZDNZCNZUWIUWGUWNDHVAZUWQ
|
|
DUVJHUVSBVBZVCUWRUWJHLZUWNMZDNZUWQUWNDHVDUXBUWOCNZDNUWQUXAUXCDUXAUWLCNZUW
|
|
NMUXCUWTUXDUWNCUWJVEVFUWLUWNCVGQRUWOCDVHQSSUWPUWHCUWPUWKUVSPZUVJLZUWHUWNU
|
|
XFDUWKUVOTZUWLUWMUXEUVJUWJUWKUVSVIVJVKUXFUWDJZANZUWHUXFUWJUVQKZKZKZOZUWJU
|
|
XEPZUVILZMZDNZANZUXIUXFUXODUUIVAZUXRDUVIUUIUXEUWKUVSVLZVCUXSUWJUUILZUXOMZ
|
|
DNZUXRUXODUUIVDUYCUXPANZDNUXRUYBUYDDUYBUXMANZUXOMUYDUYAUYEUXOAUWJVMVFUXMU
|
|
XOAVGQRUXPADVHQSSUXQUXHAUXQUXLUXEPZUVILZUXHUXOUYGDUXLUXKTZUXMUXNUYFUVIUWJ
|
|
UXLUXEVIVJVKUYGUWBUWCJZMZUXHUYGUYFUVGLZUYFUVHLZJZMUYJUYFUVGUVHVNUYKUWBUYM
|
|
UYIUYKUWAUVNMZUWBUYKUYFUVFLZUYFUUALZJZMUYNUYFUVFUUAVNUYOUWAUYQUVNUYOEUQZU
|
|
VROZUYRUVTLZMZENZUWAUYOUWJUYFPZUVELZDUUJVAZVUBDUVEUUJUYFUXLUXEVLZVCVUEUWJ
|
|
UYRKZKZKZKZKZOZVUDMZDNZENZVUBVUEUWJUUJLZVUDMZDNZVUOVUDDUUJVDVURVUMENZDNVU
|
|
OVUQVUSDVUQVULENZVUDMVUSVUPVUTVUDEUWJVOVFVULVUDEVGQRVUMEDVHQSVUNVUAEVUNVU
|
|
KUYFPZUVELZVUAVUDVVBDVUKVUJTZVULVUCVVAUVEUWJVUKUYFVIVJVKVVBVVAUUOLZVVAUVD
|
|
LZMVUAVVAUUOUVDVRVVDUYSVVEUYTVVAUUNLZJFEUPZFUQZUVRLZVSZJZFNZJZVVDUYSVVFVV
|
|
LVVFUWJVVHKZKZKZKZKZKZKZKZOZUWJVVAPZUUFLZMZDNZFNZVVLVVFVWDDUUMVAZVWGDUUFU
|
|
UMVVAVUKUYFVLZVCVWHUWJUUMLZVWDMZDNZVWGVWDDUUMVDVWLVWEFNZDNVWGVWKVWMDVWKVW
|
|
BFNZVWDMVWMVWJVWNVWDFUWJVPVFVWBVWDFVGQRVWEFDVHQSSVWFVVKFVWFVWAVVAPZUUFLZV
|
|
VKVWDVWPDVWAVVTTVWBVWCVWOUUFUWJVWAVVAVIVJVKVWPVWOYTLZVWOUUELZVSZJVVKVWOYT
|
|
UUEVTVWSVVJVWQVVGVWRVVIVWQVVSVUKPYSLZVVGVVSVUKUYFYSVVRTZVVCVUFWAVWTVVRVUJ
|
|
PYRLZVVGVVRVUJYRVVQTVUITWBVXBVVQVUIPYQLZVVGVVQVUIYQVVPTZVUHTZWBVXCVVPVUHP
|
|
YPLZVVGVVPVUHYPVVOTZVUGTWBVXFVVOVUGPYOLZVVGVVOVUGYOVVNTZUYRTZWBVXHVVNUYRP
|
|
GLVVGVVNUYRGVVHTZEVBZWBVVHUYRFVBZVXLWCSSSSSSVWRVVSUYFPZUUDLZVVIVVSVUKUYFU
|
|
UDVXAVVCVUFWDVXNUUBLZVXNUUCLZWEFCUPZVVHUVQLZWEVXOVVIVXPVXRVXQVXSVXPVVQUXE
|
|
PUUALZVXRVVQUXLUXEUUAVXDUYHUXTWDVXTVVOUWKPYOLZVXRVVOUWKUVSYOVXIUXGUWSWAVY
|
|
AVVNUVOPGLVXRVVNUVOGVXKCVBZWBVVHUVOVXMVYBWCSSSVXQVVQUXLPUDLZVXSVVQUXLUXEU
|
|
DVXDUYHUXTWAVVQUXLOZFAWFZVYCVXSVYDVVPUXKOZVYEVVPUXKVXGWGVYFVVOUXJOZVYEVVO
|
|
UXJVXIWGVYGVVNUVQOVYEVVNUVQVXKWGVVHUVPVXMWGSSSVVQILUXLILZVYCVYDVSVXDUYHVV
|
|
QUXLIIWHWIVVHUVPVXMWJWKSWLVXNUUBUUCWRVVHUVOUVQWRWKSWMWNSSRSWNVVAUUNVWIWOU
|
|
YSVVJFUSVVMFUYRUVRWPVVJFXGSWKVVEVUIUXEPUVCLZUYTVUIUXLUXEUVCVXEUYHUXTWDVYI
|
|
VUGUVSPUVBLZUYTVUGUWKUVSUVBVXJUXGUWSWDVUGUWJPGLZUWMUVALZMZDNUWJUVTOZEDUPZ
|
|
MZDNVYJUYTVYMVYPDVYMVYLVYKMVYPVYKVYLWQVYLVYNVYKVYOVYLUVSUWJPUUTLZVYNUWJUV
|
|
SUUTDVBZUWSWSVYQUWJUUSUVSUHZOZVYNUVSILUWJILVYQVYTVSUWSVYRUVSUWJUUSIIWTWIV
|
|
YNVYTUVTVYSUWJUVSHXAXBXCSSUYRUWJVXLVYRWCXDSRDVUGUVSUVAGVXJUWSXEDUYRUVTUVS
|
|
HUWSXHXFXIWKSSXDSSRSSEUVRUVTXJQUYPUVMUYPUXJUWKPYOLZUVMUXJUWKUVSYOUVQTUXGU
|
|
WSWAWUAUVQUVOPGLUVMUVQUVOGUVPTVYBWBUVPUVOAVBVYBWCSSWNXDSUWAUVNWQSUYLUWCUY
|
|
LUXEGLZUWCUYLVYHWUBUYHUXLUXEIGUYHUXTXKXLUVOUVSVYBUWSWCSWNXDSUWBUWCXMSSRSU
|
|
WDAXNSSRSWNUVSUVKUWSWOUWECXGWKXOUVKUVJHUVIUUIUVGUVHUVFUUAUVEUUJUUOUVDUUNU
|
|
UFUUMYTUUEYSYRYQYPYOGXPXQZXQXQXQXQXRUUDUUBUUCUUAYOWUCXRZXSUDXTXRYAXSYBUUL
|
|
UUKUUJUUIUUHUUGHXHYCYCZYCZYCZYCYCYCYDYEUVCUVBUVAGUUTUUSUURUUHYFWUEYDYHYGX
|
|
PYIXSXSYJWUGYDWUDYKIGYLXPYMYKWUFYDXHYDYEYN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Lemma for ~ nnsucelr . Subtracting a non-element from a set adjoined
|
|
with the non-element retrieves the original set. (Contributed by SF,
|
|
15-Jan-2015.) $)
|
|
nnsucelrlem2 $p |- ( -. B e. A -> ( ( A u. { B } ) \ { B } ) = A ) $=
|
|
( vx wcel wn csn cun cdif cv wceq wa wne wo eldifsn elun elsn bitri df-ne
|
|
orbi2i anbi12i pm5.61 3bitri ancom eleq1 biimpcd con3d com12 bicomd eqrdv
|
|
pm4.71rd syl5bb ) BADZEZCABFZGZUNHZACIZUPDZUQBJZEZUQADZKZUMVAURVAUTKZVBUR
|
|
UQUODZUQBLZKVAUSMZUTKVCUQUOBNVDVFVEUTVDVAUQUNDZMVFUQAUNOVGUSVACBPSQUQBRTV
|
|
AUSUAUBVAUTUCQUMVAVBUMVAUTVAUMUTVAUSULUSVAULUQBAUDUEUFUGUJUHUKUI $.
|
|
$}
|
|
|
|
${
|
|
nnsucelrlem3.1 $e |- X e. _V $.
|
|
$( Lemma for ~ nnsucelr . Rearrange union and difference for a particular
|
|
group of classes. (Contributed by SF, 15-Jan-2015.) $)
|
|
nnsucelrlem3 $p |-
|
|
( ( X =/= Y /\ ( A u. { X } ) = ( B u. { Y } ) /\
|
|
-. Y e. B ) -> B = ( ( A \ { Y } ) u. { X } ) ) $=
|
|
( wne csn cun wceq wcel wn w3a ccompl cin cdif indir df-dif eqtri sylib
|
|
c0 eqcomi incompl uneq12i un0 difsn 3ad2ant3 syl5req simp2 df-ne 3ad2ant1
|
|
wss biimpi elcompl elsnc xchbinx snss bitr3i ssequn2 ineq12d eqtr4d undir
|
|
uneq1i syl6eqr ) CDFZACGZHZBDGZHZIZDBJKZLZBVFVGMZVEHZNZAVGOZVEHZVKBVHVLNZ
|
|
VNVKVQBVGOZBVQBVLNZVGVLNZHZVRBVGVLPWAVRTHVRVSVRVTTVRVSBVGQUAVGUBUCVRUDRRV
|
|
JVDVRBIVIDBUEUFUGVKVFVHVMVLVDVIVJUHVKVEVLUKZVMVLIVKCDIZKZWBVDVIWDVJVDWDCD
|
|
UIULUJWDCVLJZWBWECVGJWCCVGEUMCDEUNUOCVLEUPUQSVEVLURSUSUTVPAVLNZVEHVNVOWFV
|
|
EAVGQVBAVLVEVARVC $.
|
|
$}
|
|
|
|
$( Lemma for ~ nnsucelr . Remove and re-adjoin an element to a set.
|
|
(Contributed by SF, 15-Jan-2015.) $)
|
|
nnsucelrlem4 $p |- ( A e. B -> ( ( B \ { A } ) u. { A } ) = B ) $=
|
|
( wcel csn cdif cun undif1 wss wceq snssi ssequn2 sylib syl5eq ) ABCZBADZEO
|
|
FBOFZBBOGNOBHPBIABJOBKLM $.
|
|
|
|
${
|
|
$d A a $. $d a c $. $d a m $. $d a n $. $d a x $. $d a y $. $d a z $.
|
|
$d c m $. $d c n $. $d c x $. $d c z $. $d M a $. $d M m $. $d m n $.
|
|
$d m x $. $d M x $. $d m y $. $d m z $. $d n x $. $d n z $. $d X a $.
|
|
$d X x $. $d x y $. $d x z $. $d a b $. $d b c $. $d b n $. $d b x $.
|
|
$d b y $. $d b z $. $d c y $. $d n y $. $d y z $. $d a d $. $d a w $.
|
|
$d d n $. $d d w $. $d d y $. $d w y $.
|
|
nnsucelr.1 $e |- A e. _V $.
|
|
nnsucelr.2 $e |- X e. _V $.
|
|
$( Transfer membership in the successor of a natural into membership of the
|
|
natural itself. Theorem X.1.17 of [Rosser] p. 525. (Contributed by SF,
|
|
14-Jan-2015.) $)
|
|
nnsucelr $p |- ( ( M e. Nn /\
|
|
( -. X e. A /\ ( A u. { X } ) e. ( M +c 1c ) ) ) ->
|
|
A e. M ) $=
|
|
( vx va vy vz vc vw wcel wn cun c1c wa cv wi wal wceq vm vn cnnc csn cplc
|
|
vb vd wel wex c0 nnsucelrlem1 addceq1 addcid2 syl6eq eleq2d syl6bb anbi2d
|
|
c0c el1c eleq2 df-0c eleq2i vex elsnc bitri imbi12d 2albidv weq wb eleq12
|
|
ancoms notbid sneq uneq12 sylan2 eleq1d anbi12d eleq1 adantr unsneqsn ord
|
|
cbval2v snid mpbiri syl6 con1d exlimiv impcom ccompl elsuc elcompl anbi2i
|
|
w3a simprrl difeq12d simprrr nnsucelrlem2 simprlr 3eqtr3d simprll eqeltrd
|
|
gen2 wrex cdif syl 3adantr1 ex simpl simpr3l simpr2r nnsucelrlem3 syl3anc
|
|
wne simp22r difsn uneq1d eqeq2d biimpcd 3ad2ant3 simp23l wss snss ssequn2
|
|
bitr2i biimpi eqcoms syl6bi syld mt3d nnsucelrlem4 simpl3r difss nsyl cvv
|
|
sseli mp2an uneq2d rspcev uneq1 spcv simp2l biimpd mpan9 snex difex eldif
|
|
simpl1 spc2gv mp2and 3adant1 simprbi mpbir eqid compleq rexeqbidv sylancl
|
|
mt2 sylibr eqeltrrd mpd3an3 pm2.61ine 3expa exp32 rexlimdvva syl5bi com23
|
|
sylan2b imp3a alrimivv a1i finds imbi1d alimi 3syl imp ) BUCLZCALZMZACUDZ
|
|
NZBOUEZLZPZABLZUVPFGUHZMZGQZFQZUDZNZUWALZPZUWGBLZRZFSZGSZCUWGLZMZUWGUVSNZ
|
|
UWALZPZUWMRZGSUWCUWDRZUWFUWJUAQZOUEZLZPZGUAUHZRZFSGSZUWFUWJHQZUDZTZHUIZPZ
|
|
UWGUJTZRZFSGSIJUHZMZJQZIQZUDZNZUBQZOUEZLZPZJUBUHZRZISJSZUWFUWJUYEOUEZLZPZ
|
|
UWGUYELZRZFSGSZUWPUAUBBFUAGUKUXDURTZUXIUXQGFUYQUXGUXOUXHUXPUYQUXFUXNUWFUY
|
|
QUXFUWJOLUXNUYQUXEOUWJUYQUXEUROUEOUXDUROULOUMUNUOHUWJUSUPUQUYQUXHUWGURLZU
|
|
XPUXDURUWGUTUYRUWGUJUDZLUXPURUYSUWGVAVBUWGUJGVCZVDVEUPVFVGUAUBVHZUXJUWFUW
|
|
JUYELZPZGUBUHZRZFSGSUYJVUAUXIVUEGFVUAUXGVUCUXHVUDVUAUXFVUBUWFVUAUXEUYEUWJ
|
|
UXDUYDOULUOUQUXDUYDUWGUTVFVGVUEUYIGFJIGJVHZFIVHZPZVUCUYGVUDUYHVUHUWFUXSVU
|
|
BUYFVUHUWEUXRVUGVUFUWEUXRVIUWHUYAUWGUXTVJVKVLVUHUWJUYCUYEVUGVUFUWIUYBTUWJ
|
|
UYCTUWHUYAVMUWGUXTUWIUYBVNVOVPVQVUFVUDUYHVIVUGUWGUXTUYDVRVSVFWBUPUXDUYETZ
|
|
UXIUYOGFVUIUXGUYMUXHUYNVUIUXFUYLUWFVUIUXEUYKUWJUXDUYEOULUOUQUXDUYEUWGUTVF
|
|
VGUXDBTZUXIUWNGFVUJUXGUWLUXHUWMVUJUXFUWKUWFVUJUXEUWAUWJUXDBOULUOUQUXDBUWG
|
|
UTVFVGUXQGFUXNUWFUXPUXMUWFUXPRHUXMUXPUWEUXMUXPMUWGUWITZUWEUXMUXPVUKUWGUWH
|
|
UXKFVCZVTWAVUKUWEUWHUWILUWHVULWCUWGUWIUWHUTWDWEWFWGWHXBUYJUYPRUYDUCLUYJUY
|
|
OGFUYJUWFUYLUYNUYJUYLUWFUYNUYLUWJUFQZUXLNZTZHVUMWIZXCUFUYEXCUYJUWFUYNRZHU
|
|
WJUYEUFWJUYJVUOVUQUFHUYEVUPVUMUYELZUXKVUPLZPUYJVURHUFUHZMZPZVUOVUQRVUSVVA
|
|
VURUXKVUMHVCZWKWLUYJVVBPVUOUWFUYNUYJVVBVUOUWFPZUYNUYJVVBVVDWMZUYNRUWHUXKF
|
|
HVHZVVEUYNVVFVVBVVDUYNUYJVVFVVBVVDPZPZUWGVUMUYEVVHUWJUWIXDZVUNUXLXDZUWGVU
|
|
MVVHUWJVUNUWIUXLVVFVVBVUOUWFWNVVFUWIUXLTVVGUWHUXKVMVSWOVVHUWFVVIUWGTVVFVV
|
|
BVUOUWFWPUWGUWHWQXEVVHVVAVVJVUMTVVFVURVVAVVDWRVUMUXKWQXEWSVVFVURVVAVVDWTX
|
|
AXFXGUWHUXKXMZVVEUYNVVKVVEVUMUWGUXLXDZUWINZTZUYNVVKVVEPVVKVUOVVAVVNVVKVVE
|
|
XHVUOUWFUYJVVBVVKXIVURVVAUYJVVDVVKXJUWGVUMUWHUXKVULXKXLVVKVVEVVNWMZVVLUXL
|
|
NZUWGUYEVVOHGUHZVVPUWGTVVOVVQVUTVURVVAUYJVVDVVKVVNXNVVOVVQMZVUMUWJTZVUTVV
|
|
NVVKVVRVVSRVVEVVRVVNVVSVVRVVMUWJVUMVVRVVLUWGUWIUXKUWGXOXPXQXRXSVVOVVSVUMV
|
|
UNTVUTVVOUWJVUNVUMVUOUWFUYJVVBVVKVVNXTXQVUTVUNVUMVUNVUMTZVUTVUTUXLVUMYAVV
|
|
TUXKVUMVVCYBUXLVUMYCYDYEYFYGYHYIUXKUWGYJXEVVOVVLUYDLZVVPVVLKQZUDZNZTZKVVL
|
|
WIZXCZVVPUYELZVVEVVNVWAVVKVVEVVNPZUWHVVLLZMZVVMUYELZVWAVWIUWEVWJVUOUWFUYJ
|
|
VVBVVNYKVVLUWGUWHUWGUXLYLYOYMVVEVURVVNVWLUYJVURVVAVVDUUAVVNVURVWLVUMVVMUY
|
|
EVRUUBUUCVWIUYJVWKVWLPZVWARZUYJVVBVVDVVNUUGVVLYNLUWHYNLUYJVWNRUWGUXLUYTUX
|
|
KUUDUUEVULUYIVWNJIVVLUWHYNYNUXTVVLTZIFVHZPZUYGVWMUYHVWAVWQUXSVWKUYFVWLVWQ
|
|
UXRVWJVWPVWOUXRVWJVIUYAUWHUXTVVLVJVKVLVWQUYCVVMUYEVWPVWOUYBUWITUYCVVMTUYA
|
|
UWHVMUXTVVLUYBUWIVNVOVPVQVWOUYHVWAVIVWPUXTVVLUYDVRVSVFUUHYPXEUUIUUJUXKVWF
|
|
LZVVPVVPTZVWGVWRUXKVVLLZMVWTUXKUXLLZUXKVVCWCVWTVVQVXAMUXKUWGUXLUUFUUKUUQU
|
|
XKVVLVVCWKUULVVPUUMVWEVWSKUXKVWFKHVHZVWDVVPVVPVXBVWCUXLVVLVWBUXKVMYQXQYRY
|
|
PVWAVWGPVVPUGQZVWCNZTZKVXCWIZXCZUGUYDXCVWHVXGVWGUGVVLUYDVXCVVLTZVXEVWEKVX
|
|
FVWFVXCVVLUUNVXHVXDVWDVVPVXCVVLVWCYSXQUUOYRKVVPUYDUGWJUURUUPUUSUUTXGUVAUV
|
|
BUVCUVGUVDUVEUVFUVHUVIUVJUVKUWOUXBGUWNUXBFCEUWHCTZUWLUXAUWMVXIUWFUWRUWKUW
|
|
TVXIUWEUWQUWHCUWGVRVLVXIUWJUWSUWAVXIUWIUVSUWGUWHCVMYQVPVQUVLYTUVMUXBUXCGA
|
|
DUWGATZUXAUWCUWMUWDVXJUWRUVRUWTUWBVXJUWQUVQUWGACUTVLVXJUWSUVTUWAUWGAUVSYS
|
|
VPVQUWGABVRVFYTUVNUVO $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a m $. $d a n $. $d a q $. $d a x $. $d b m $. $d b n $.
|
|
$d b q $. $d b x $. $d m n $. $d M n $. $d m p $. $d M p $. $d m q $.
|
|
$d m x $. $d N n $. $d n p $. $d n q $. $d n x $. $d p q $. $d q x $.
|
|
$d b p $. $d p x $.
|
|
|
|
$( Either two naturals are disjoint or they are the same natural. Theorem
|
|
X.1.18 of [Rosser] p. 526. (Contributed by SF, 17-Jan-2015.) $)
|
|
nndisjeq $p |- ( ( M e. Nn /\ N e. Nn ) -> ( ( M i^i N ) = (/) \/ M = N ) )
|
|
$=
|
|
( vn vp vm vq vb cnnc wcel cin c0 wceq wo cv wi weq wn c0c eqeq1d wa wral
|
|
va vx c1c cplc cssetk cins3k cins2k cpw1 cimak ccompl cun cab cvv elcompl
|
|
cidk vex copk wrex elimak opkex elun wne ndisjrelk notbii con2bii 3bitr4i
|
|
df-ne wb opkelidkg mp2an orbi12i incom eqeq1i eqcom 3bitri xchbinx rexbii
|
|
rexnal bitr4i abbi2i ssetkex ins3kex ins2kex inex 1cex pw1ex imakex idkex
|
|
complex nncex eqeltrri csn df-0c eqeq2i biimpi ineq1d disjsn bitri syl6bb
|
|
eqeq1 orbi12d ralbidv ineq1 ineq2 equequ2 cbvralv nnc0suc 0nelsuc biimpcd
|
|
unex eleq2 mtoi adantr orel2 syl com12 sylbi imor sylib rgen wex neq0 wel
|
|
nrexdv elin elsuc anbi2i w3a simp1r n0i rspccv syl6 com23 imp com3l eqeq2
|
|
ex syld syl5bi ssun2 sselii ax-mp eleq2i elsnc mtbir adantl orel1 simpr3r
|
|
snid snex simpll simpr nnsucelr syl12anc sylbir pm2.53 syl5 exp3a adantrr
|
|
3adant1 impcom addceq1 anbi2d imbi2d imbi12d mpbiri rexlimdv 3expa imbi1d
|
|
mpid eleq1 syl5ibrcom sylan2b rexlimdvva imp3a orrd exp31 ralrimdv finds
|
|
exlimdv ) AHIZBHIZABJZKLZABLZMZUWBACNZJZKLZAUWHLZMZCHUAZUWCUWGODNZUWHJZKL
|
|
ZDCPZMZCHUAZKUWHIZQZUWHRLZMZCHUAENZFNZJZKLZEFPZMZFHUAZUXDUDUEZUWHJZKLZUXK
|
|
UWHLZMZCHUAUWMDEAUFUGZUFUHZJZUDUIZUIZUJZUKZUPULZUKZHUJZUKZUWSDUMUNUWSDUYF
|
|
UWNUYFIUWNUYEIZQUWSUWNUYEDUQZUOUYGUWSUYGUWHUWNURZUYDIZCHUSUWRQZCHUSUWSQCU
|
|
YDHUWNUYHUTUYJUYKCHUYJUYIUYCIZUWRUYIUYCUWHUWNVAZUOUYLUYIUYBIZUYIUPIZMUWHU
|
|
WNJZKLZCDPZMUWRUYIUYBUPVBUYNUYQUYOUYRUYIUYAIZQUYPKVCZQUYNUYQUYSUYTUWHUWNC
|
|
UQZUYHVDVEUYIUYAUYMUOUYTUYQUYPKVHVFVGUWHUNIUWNUNIUYOUYRVIVUAUYHUWHUWNUNUN
|
|
VJVKVLUYQUWPUYRUWQUYPUWOKUWHUWNVMVNUWHUWNVOVLVPVQVRUWRCHVSVPVFVTWAUYEUYDH
|
|
UYCUYBUPUYAUXRUXTUXPUXQUFWBWCUFWBWDWEUXSUDWFWGWGWHWJWIXKWJWKWHWJWLUWNRLZU
|
|
WRUXCCHVUBUWPUXAUWQUXBVUBUWPKWMZUWHJZKLZUXAVUBUWOVUDKVUBUWNVUCUWHVUBUWNVU
|
|
CLRVUCUWNWNWOWPWQSVUEUWHVUCJZKLUXAVUDVUFKVUCUWHVMVNUWHKWRWSWTVUBUWQRUWHLU
|
|
XBUWNRUWHXARUWHVOWTXBXCDEPZUWSUXDUWHJZKLZECPZMZCHUAUXJVUGUWRVUKCHVUGUWPVU
|
|
IUWQVUJVUGUWOVUHKUWNUXDUWHXDSUWNUXDUWHXAXBXCVUKUXICFHCFPZVUIUXGVUJUXHVULV
|
|
UHUXFKUWHUXEUXDXESCFEXFXBXGWTUWNUXKLZUWRUXOCHVUMUWPUXMUWQUXNVUMUWOUXLKUWN
|
|
UXKUWHXDSUWNUXKUWHXAXBXCUWNALZUWRUWLCHVUNUWPUWJUWQUWKVUNUWOUWIKUWNAUWHXDS
|
|
UWNAUWHXAXBXCUXCCHUWHHIZUWTUXBOZUXCVUOUXBUWHUXKLZEHUSZMZVUPEUWHXHUWTVUSUX
|
|
BUWTVURQVUSUXBOUWTVUQEHUWTVUQQUXDHIZUWTVUQKUXKIZUXDXIVUQUWTVVAUWHUXKKXLXJ
|
|
XMXNYEVURUXBXOXPXQXRUWTUXBXSXTYAVUTUXJUXOCHVUTVUOUXJUXOVUTVUOUXJUXOVUTVUO
|
|
TZUXJTZUXMUXNUXMQUBNZUXLIZUBYBVVCUXNUBUXLYCVVCVVEUXNUBVVEVVDUXKIZUBCYDZTV
|
|
VCUXNVVDUXKUWHYFVVCVVFVVGUXNVVFVVDGNZUCNZWMZULZLZUCVVHUKZUSGUXDUSVVCVVGUX
|
|
NOZUCVVDUXDGYGVVCVVLVVNGUCUXDVVMGEYDZVVIVVMIZTVVCVVOUCGYDQZTZVVLVVNOVVPVV
|
|
QVVOVVIVVHUCUQZUOYHVVCVVRTVVNVVLVVKUWHIZUXNOZVVBUXJVVRVWAVVBUXJVVRYIZVVTU
|
|
XBUWHUWNUDUEZLZDHUSZMZUXNVWBVUOVWFVUTVUOUXJVVRYJDUWHXHXTVWBVVTVWFUXNOVWBV
|
|
VTTZVWFVWEUXNVWGUXBQZVWFVWEOVVTVWHVWBVVTUXBVVKRIZVWIVVKKLZVVIVVKIVWJQVVJV
|
|
VKVVIVVJVVHUUAVVIVVSUUJUUBVVKVVIYKUUCVWIVVKVUCIVWJRVUCVVKWNUUDVVKKVVHVVJG
|
|
UQZVVIUUKXKUUEWSUUFUXBVVTVWIUWHRVVKXLXJXMUUGUXBVWEUUHXPVWGVWDUXNDHVWDVWGU
|
|
WNHIZUXNVWDVWGVWLUXNOZOVWBVVKVWCIZTZVWLUXKVWCLZOZOVWOVWLEDPZVWPVWBVWNVWLV
|
|
WROVWLVWBVWNVWRVWLVWBVWNVWROVWLVWBTZVWNGDYDZVWRVWSVWNVWTVWSVWNTVWLVVQVWNV
|
|
WTVWLVWBVWNUULVWSVVQVWNVVOVVQVVBUXJVWLUUIXNVWSVWNUUMVVHUWNVVIVWKVVSUUNUUO
|
|
YRVWBVWLVWTVWROZUXJVVRVWLVXAOZVVBUXJVVOVXBVVQUXJVVOVXBUXJVWLVVOVXAUXJVWLU
|
|
XDUWNJZKLZVWRMZVVOVXAOUXIVXEFUWNHFDPZUXGVXDUXHVWRVXFUXFVXCKUXEUWNUXDXESFD
|
|
EXFXBYLVXEVVOVWTVWRVVOVWTTZVXDQZVXEVWRVXGVVHVXCIVXHVVHUXDUWNYFVXCVVHYKUUP
|
|
VXDVWRUUQUURUUSYMYNYOUUTUVAUVBYSYRYPYOUXDUWNUDUVCYMVWDVWGVWOVWMVWQVWDVVTV
|
|
WNVWBUWHVWCVVKXLUVDVWDUXNVWPVWLUWHVWCUXKYQUVEUVFUVGYPUVHYSYRUVKUVIVVLVVGV
|
|
VTUXNVVDVVKUWHUVLUVJUVMUVNUVOYTUVPYTUWAYTUVQUVRYNUVSUVTUWLUWGCBHUWHBLZUWJ
|
|
UWEUWKUWFVXIUWIUWDKUWHBAXESUWHBAYQXBYLXPYO $.
|
|
$}
|
|
|
|
$( If two naturals have an element in common, then they are equal.
|
|
(Contributed by SF, 13-Feb-2015.) $)
|
|
nnceleq $p |- ( ( ( M e. Nn /\ N e. Nn ) /\ ( A e. M /\ A e. N ) ) ->
|
|
M = N ) $=
|
|
( cnnc wcel wa cin c0 wceq wn wo elin n0i sylbir adantl nndisjeq orel1 sylc
|
|
adantr ) BDECDEFZABEACEFZFBCGZHIZJZUCBCIZKZUEUAUDTUAAUBEUDABCLUBAMNOTUFUABC
|
|
PSUCUEQR $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( A singleton is finite. (Contributed by SF, 23-Feb-2015.) $)
|
|
snfi $p |- { A } e. Fin $=
|
|
( vx cvv wcel csn cfin cv cnnc wrex c1c 1cnnc snel1cg eleq2 sylancr elfin
|
|
rspcev sylibr wn c0 wceq snprc 0fin eleq1 mpbiri sylbi pm2.61i ) ACDZAEZF
|
|
DZUGUHBGZDZBHIZUIUGJHDUHJDZULKACLUKUMBJHUJJUHMPNBUHOQUGRUHSTZUIAUAUNUISFD
|
|
UBUHSFUCUDUEUF $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Deriving infinity
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c <_[fin] $.
|
|
$c <[fin] $.
|
|
$c Nc[fin] $.
|
|
$c _T[fin] $.
|
|
$c Even[fin] $.
|
|
$c Odd[fin] $.
|
|
$c _S[fin] $.
|
|
$c Sp[fin] $.
|
|
|
|
$( Extend class notation to include the less than or equal to relationship
|
|
for finite cardinals. $)
|
|
clefin $a class <_[fin] $.
|
|
|
|
$( Extend class notation to include the less than relationship for finite
|
|
cardinals. $)
|
|
cltfin $a class <[fin] $.
|
|
|
|
$( Extend class notation to include the finite cardinal function. $)
|
|
cncfin $a class Nc[fin] A $.
|
|
|
|
$( Extend class notation to include the finite T operation. $)
|
|
ctfin $a class _T[fin] A $.
|
|
|
|
$( Extend class notation to include the (temporary) set of all even
|
|
numbers. $)
|
|
cevenfin $a class Even[fin] $.
|
|
|
|
$( Extend class notation to include the (temporary) set of all odd
|
|
numbers. $)
|
|
coddfin $a class Odd[fin] $.
|
|
|
|
$( Extend wff notation to include the finite S relationship. $)
|
|
wsfin $a wff _S[fin] ( A , B ) $.
|
|
|
|
$( Extend class notation to include the finite Sp set. $)
|
|
cspfin $a class Sp[fin] $.
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( Define the less than or equal to relationship for finite cardinals.
|
|
Definition from Ex. X.1.4 of [Rosser] p. 279. (Contributed by SF,
|
|
12-Jan-2015.) $)
|
|
df-lefin $a |- <_[fin] = { x | E. y E. z ( x = << y , z >> /\
|
|
E. w e. Nn z = ( y +c w ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d m n p x $.
|
|
$( Define the less than relationship for finite cardinals. Definition from
|
|
[Rosser] p. 527. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-ltfin $a |- <[fin] = { x | E. m E. n ( x = << m , n >> /\
|
|
( m =/= (/) /\ E. p e. Nn n = ( ( m +c p ) +c 1c ) ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Define the finite cardinal function. Definition from [Rosser] p. 527.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
df-ncfin $a |- Nc[fin] A = ( iota x ( x e. Nn /\ A e. x ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d M n a $.
|
|
$( Define the finite T operator. Definition from [Rosser] p. 528.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
df-tfin $a |- _T[fin] M = if ( M = (/) , (/) ,
|
|
( iota n ( n e. Nn /\ E. a e. M ~P1 a e. n ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d x n $.
|
|
$( Define the temporary set of all even numbers. This differs from the
|
|
final definition due to the non-null condition. Definition from
|
|
[Rosser] p. 529. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-evenfin $a |- Even[fin] = { x |
|
|
( E. n e. Nn x = ( n +c n ) /\ x =/= (/) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x n $.
|
|
$( Define the temporary set of all odd numbers. This differs from the
|
|
final definition due to the non-null condition. Definition from
|
|
[Rosser] p. 529. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-oddfin $a |- Odd[fin] = { x |
|
|
( E. n e. Nn x = ( ( n +c n ) +c 1c ) /\
|
|
x =/= (/) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d M a $. $d N a $.
|
|
$( Define the finite S relationship. This relationship encapsulates the
|
|
idea of ` M ` being a "smaller" number than ` N ` . Definition from
|
|
[Rosser] p. 530. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-sfin $a |- ( _S[fin] ( M , N ) <-> ( M e. Nn /\ N e. Nn /\
|
|
E. a ( ~P1 a e. M /\ ~P a e. N ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d a x z $.
|
|
$( Define the finite Sp set. Definition from [Rosser] p. 533.
|
|
(Contributed by SF, 12-Jan-2015.) $)
|
|
df-spfin $a |- Sp[fin] =
|
|
|^| { a | ( Nc[fin] _V e. a /\
|
|
A. x e. a A. z ( _S[fin] ( z , x ) -> z e. a ) ) } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $. $d B x y z w $.
|
|
$( Kuratowski ordered pair membership in finite less than or equal to.
|
|
(Contributed by SF, 18-Jan-2015.) $)
|
|
opklefing $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. <_[fin] <-> E. x e. Nn B = ( A +c x ) ) ) $=
|
|
( vz vy vw cv cplc wceq cnnc clefin df-lefin addceq1 eqeq2d rexbidv eqeq1
|
|
wrex opkelopkabg ) FIZGIZAIZJZKZALSUABUCJZKZALSCUFKZALSHGFMBCDEHGFANUBBKZ
|
|
UEUGALUIUDUFUAUBBUCOPQUACKUGUHALUACUFRQT $.
|
|
$}
|
|
|
|
${
|
|
$d A w $. $d A x $. $d A y $. $d A z $. $d B w $. $d B x $. $d B y $.
|
|
$d B z $. $d w x $. $d w y $. $d w z $. $d x y $. $d x z $. $d y z $.
|
|
|
|
$( Kuratowski ordered pair membership in finite less than. (Contributed by
|
|
SF, 27-Jan-2015.) $)
|
|
opkltfing $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. <[fin] <->
|
|
( A =/= (/) /\ E. x e. Nn B = ( ( A +c x ) +c 1c ) ) ) ) $=
|
|
( vz vw vy cv c0 wne cplc c1c wceq cnnc wrex wa cltfin df-ltfin rexbidv
|
|
neeq1 addceq1 addceq1d eqeq2d anbi12d eqeq1 anbi2d opkelopkabg ) FIZJKZGI
|
|
ZUIAIZLZMLZNZAOPZQBJKZUKBULLZMLZNZAOPZQUQCUSNZAOPZQHFGRBCDEHFGASUIBNZUJUQ
|
|
UPVAUIBJUAVDUOUTAOVDUNUSUKVDUMURMUIBULUBUCUDTUEUKCNZVAVCUQVEUTVBAOUKCUSUF
|
|
TUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d A n $. $d N n $.
|
|
$( Cardinal sum always yields a larger set. (Contributed by SF,
|
|
27-Jan-2015.) $)
|
|
lefinaddc $p |- ( ( A e. V /\ N e. Nn ) ->
|
|
<< A , ( A +c N ) >> e. <_[fin] ) $=
|
|
( vn wcel cnnc wa cplc copk clefin cv wceq wrex eqid addceq2 eqeq2d mpan2
|
|
rspcev adantl cvv wb addcexg opklefing syldan mpbird ) ACEZBFEZGAABHZIJEZ
|
|
UHADKZHZLZDFMZUGUMUFUGUHUHLZUMUHNULUNDBFUJBLUKUHUHUJBAOPRQSUFUGUHTEUIUMUA
|
|
ABCFUBDAUHCTUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d M a b x $. $d N a b x $.
|
|
$( Assuming a non-null successor, cardinal successor is one-to-one.
|
|
Theorem X.1.19 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.) $)
|
|
prepeano4 $p |- ( ( ( M e. Nn /\ N e. Nn ) /\
|
|
( ( M +c 1c ) = ( N +c 1c ) /\ ( M +c 1c ) =/= (/) ) ) -> M = N ) $=
|
|
( va vb vx cnnc wcel wa c1c cplc wceq c0 wne cv wex csn wrex vex syl5bi
|
|
n0 cun ccompl elsuc simplll simpllr simprl wn simprr elcompl sylib elsuci
|
|
wel sylan2b adantl eleqtrd nnsucelr syl12anc nnceleq syl22anc a1d exlimdv
|
|
simplr rexlimdvva impr ) AFGZBFGZHZAIJZBIJZKZVHLMZABKZVKCNZVHGZCOVGVJHZVL
|
|
CVHTVOVNVLCVNVMDNZENZPUAZKZEVPUBZQDAQVOVLEVMADUCVOVSVLDEAVTVOVPAGZVQVTGZH
|
|
ZHZVLVSWDVEVFWAVPBGZVLVEVFVJWCUDVEVFVJWCUEZVOWAWBUFWDVFEDULUGZVRVIGWEWFWD
|
|
WBWGVOWAWBUHVQVPERZUIZUJWDVRVHVIWCVRVHGZVOWBWAWGWJWIVPAVQWHUKUMUNVGVJWCVB
|
|
UOVPBVQDRWHUPUQVPABURUSUTVCSVASVD $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c $.
|
|
$( Cardinal addition with the empty set. Theorem X.1.20, corollary 1 of
|
|
[Rosser] p. 526. (Contributed by SF, 18-Jan-2015.) $)
|
|
addcnul1 $p |- ( A +c (/) ) = (/) $=
|
|
( va vb vc c0 cplc wceq cv wcel wn eq0 cin cun wrex rex0 a1i eladdc mtbir
|
|
wa nrex mpgbir ) AEFZEGBHZUBIZJBBUBKUDCHZDHZLEGUCUEUFMGSZDENZCANUHCAUHJUE
|
|
AIUGDOPTUCAECDQRUA $.
|
|
$}
|
|
|
|
$( If cardinal addition is non-empty, then both addends are non-empty.
|
|
Theorem X.1.20 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.) $)
|
|
addcnnul $p |- ( ( A +c B ) =/= (/) -> ( A =/= (/) /\ B =/= (/) ) ) $=
|
|
( cplc wne wceq addceq1 addccom addcnul1 eqtri syl6eq necon3i addceq2 jca
|
|
c0 ) ABCZNDANDBNDANONANEONBCZNANBFPBNCNNBGBHIJKBNONBNEOANCNBNALAHJKM $.
|
|
|
|
${
|
|
$d M m n p k t $. $d N m n p k t $. $d P m n p k t $.
|
|
$( Lemma for ~ preaddccan2 . Establish stratification for the induction
|
|
step. (Contributed by SF, 30-Mar-2021.) $)
|
|
preaddccan2lem1 $p |- ( ( N e. Nn /\ P e. Nn ) ->
|
|
{ m | ( ( ( m +c N ) =/= (/) /\ ( m +c N ) = ( m +c P ) ) -> N = P ) }
|
|
e. _V ) $=
|
|
( vt cv cplc c0 wceq wa cab cvv wcel cpw1 cimak copk 3bitr4i bitri pw1ex
|
|
wn vex vn vp wne cnnc addceq2 neeq1d eqeq1d anbi12d imbi1d abbidv eleq1d
|
|
wi eqeq2d anbi2d cun imor abbii unab eqtr4i cssetk cins3k cins2k cin c1c
|
|
wo ccompl csik csymdif cdif cimagek ccnvk csn elcompl elin 0ex opkelcnvk
|
|
elimaksn dfaddc2 eqeq2i eqcom opkelimagek notbii df-ne wrex rexv anbi12i
|
|
wex bitr4i exbii elimak addcex eqvinc abbi2i imakex imagekex cnvkex snex
|
|
addcexlem complex inex vvex eqeltrri abexv unex eqeltri vtocl2g ) BEZUAE
|
|
ZFZGUCZXIXGUBEZFZHZIZCAHZULZBJZKLXGCFZGUCZXRXLHZIZXOULZBJZKLXSXRXGAFZHZI
|
|
ZXOULZBJZKLUAUBCAUDUDXHCHZXQYCKYIXPYBBYIXNYAXOYIXJXSXMXTYIXIXRGXHCXGUEZU
|
|
FYIXIXRXLYJUGUHUIUJUKXKAHZYCYHKYKYBYGBYKYAYFXOYKXTYEXSYKXLYDXRXKAXGUEUMU
|
|
NUIUJUKXQXNSZBJZXOBJZUOZKXQYLXOVEZBJYOXPYPBXNXOUPUQYLXOBURUSYMYNUTVAZUTV
|
|
BZVCVDMMZNVFVAYRVBYQVBUTVGVGVAUOVHYSMMNVIZXHMZMZNZVJZVKZGVLZNZVFZUUDYTXK
|
|
MZMZNZVJZVCZVKZKNZVCZVFZYMKYLBUUQXGUUQLXGUUPLZSYLXGUUPBTZVMUURXNUURXGUUH
|
|
LZXGUUOLZIXNXGUUHUUOVNUUTXJUVAXMXGUUGLZSXIGHZSUUTXJUVBUVCGXGOUUELXGGOUUD
|
|
LZUVBUVCGXGUUDVOUUSVPUUEGXGVOUUSVQGXIHGUUCXGNZHUVCUVDXIUVEGXGXHVRZVSXIGV
|
|
TXGGUUCUUSVOWAPPWBXGUUGUUSVMXIGWCPDEZXGOUUNLZDKWDZUVGXIHZUVGXLHZIZDWGZUV
|
|
AXMUVIUVHDWGUVMUVHDWEUVHUVLDUVHXGUVGOZUUMLZUVLUVGXGUUMDTZUUSVPUVOUVNUUDL
|
|
ZUVNUULLZIUVLUVNUUDUULVNUVQUVJUVRUVKUVQUVGUVEHUVJXGUVGUUCUUSUVPWAXIUVEUV
|
|
GUVFVSWHUVRUVGUUKXGNZHUVKXGUVGUUKUUSUVPWAXLUVSUVGXGXKVRVSWHWFQQWIQDUUNKX
|
|
GUUSWJDXIXLXGXHUUSUATZWKWLPWFQWBQWMUUPUUHUUOUUGUUEUUFUUDUUCYTUUBWRUUAXHU
|
|
VTRRWNWOZWPGWQWNWSUUNKUUMUUDUULUWAUUKYTUUJWRUUIXKUBTRRWNWOWTWPXAWNWTWSXB
|
|
XOBXCXDXEXF $.
|
|
|
|
$( Cancellation law for natural addition with a non-null condition.
|
|
(Contributed by SF, 29-Jan-2015.) $)
|
|
preaddccan2 $p |- ( ( ( M e. Nn /\ N e. Nn /\ P e. Nn ) /\
|
|
( M +c N ) =/= (/) ) -> ( ( M +c N ) = ( M +c P ) <-> N = P ) ) $=
|
|
( vm vk cnnc wcel cplc c0 wne wa wceq wi c0c c1c addceq1 eqeq12d anbi12d
|
|
neeq1d imbi1d w3a cv cvv preaddccan2lem1 weq addc32 syl6eq biimpi adantl
|
|
addcid2 eqeq12i addcnnul simpld simpll simplrl nncaddccl syl2anc simplrr
|
|
a1i ad2antrl simprr simprl prepeano4 syl22anc jca ex imim1d findsd 3impb
|
|
expdimp addceq2 impbid1 ) BFGZCFGZAFGZUAZBCHZIJZKVQBAHZLZCALZVPVRVTWAVMV
|
|
NVOVRVTKZWAMZDUBZCHZIJZWEWDAHZLZKZWAMNCHZIJZWJNAHZLZKZWAMZEUBZCHZIJZWQWP
|
|
AHZLZKZWAMWQOHZIJZXBWSOHZLZKZWAMWCVNVOKZDEBUCADCUDWDNLZWIWNWAXHWFWKWHWMX
|
|
HWEWJIWDNCPZSXHWEWJWGWLXIWDNAPQRTDEUEZWIXAWAXJWFWRWHWTXJWEWQIWDWPCPZSXJW
|
|
EWQWGWSXKWDWPAPQRTWDWPOHZLZWIXFWAXMWFXCWHXEXMWEXBIXMWEXLCHXBWDXLCPWPOCUF
|
|
UGZSXMWEXBWGXDXNXMWGXLAHXDWDXLAPWPOAUFUGQRTWDBLZWIWBWAXOWFVRWHVTXOWEVQIW
|
|
DBCPZSXOWEVQWGVSXPWDBAPQRTWOXGWMWAWKWMWAWJCWLACUJAUJUKUHUIUSWPFGZXGKZXFX
|
|
AWAXRXFXAXRXFKZWRWTXCWRXRXEXCWROIJWQOULUMUTXSWQFGZWSFGZXEXCWTXSXQVNXTXQX
|
|
GXFUNZXQVNVOXFUOWPCUPUQXSXQVOYAYBXQVNVOXFURWPAUPUQXRXCXEVAXRXCXEVBWQWSVC
|
|
VDVEVFVGVHVIVJCABVKVL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( If the empty set is a finite cardinal, then it is a maximal element.
|
|
(Contributed by SF, 19-Jan-2015.) $)
|
|
nulge $p |- ( ( (/) e. Nn /\ A e. V ) -> << A , (/) >> e. <_[fin] ) $=
|
|
( vx c0 cnnc wcel wa copk clefin cv cplc wceq wrex addcnul1 eqcomi eqeq2d
|
|
addceq2 rspcev mpan2 adantr wb opklefing ancoms mpbird ) DEFZABFZGADHIFZD
|
|
ACJZKZLZCEMZUEUKUFUEDADKZLZUKULDANOUJUMCDEUHDLUIULDUHDAQPRSTUFUEUGUKUACAD
|
|
BEUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Irreflexive law for finite less than. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
ltfinirr $p |- ( A e. Nn -> -. << A , A >> e. <[fin] ) $=
|
|
( vx cnnc wcel copk cltfin c0 wne cv cplc c1c wceq wa wn wi 0cnsuc necomi
|
|
wrex c0c wb df-ne addcid1 eqcomi addcass eqeq12i simpll peano1 a1i peano2
|
|
mpbi adantl neeq1i biimpri ad2antlr preaddccan2 syl31anc syl5bb mtbiri ex
|
|
nrexdv imnan sylib opkltfing anidms mtbird ) ACDZAAEFDZAGHZAABIZJKJZLZBCR
|
|
ZMZVFVHVLNZOVMNVFVHVNVFVHMZVKBCVOVICDZMZVKSVIKJZLZSVRHVSNVRSVIPQSVRUAUJVK
|
|
ASJZAVRJZLZVQVSAVTVJWAVTAAUBZUCAVIKUDUEVQVFSCDZVRCDZVTGHZWBVSTVFVHVPUFWDV
|
|
QUGUHVPWEVOVIUIUKVHWFVFVPWFVHVTAGWCULUMUNVRASUOUPUQURUTUSVHVLVAVBVFVGVMTB
|
|
AACCVCVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $.
|
|
$( Transitivity law for finite less than and less than or equal.
|
|
(Contributed by SF, 2-Feb-2015.) $)
|
|
leltfintr $p |- ( ( A e. Nn /\ B e. Nn /\ C e. Nn ) ->
|
|
( ( << A , B >> e. <_[fin] /\ << B , C >> e. <[fin] ) ->
|
|
<< A , C >> e. <[fin] ) ) $=
|
|
( vx vy vz cnnc wcel copk cltfin cv cplc wceq wrex wi wb wa c0 wne c1c
|
|
w3a clefin opklefing 3adant3 simpld a1i nncaddccl 3adant1 addcass addceq1
|
|
addcnnul ax-mp addceq2 syl eqeq2d rspcev syl2anc eqeq1 rexbidv syl5ibrcom
|
|
3expa adantllr rexlimdva anim12d addcexg adantlr simplr opkltfing 3imtr4d
|
|
cvv adantr opkeq1 eleq1d imbi1d 3adant2 sylbid imp3a ) AGHZBGHZCGHZUAZABI
|
|
UBHZBCIZJHZACIJHZWAWBBADKZLZMZDGNZWDWEOZVRVSWBWIPVTDABGGUCUDVRVTWIWJOVSVR
|
|
VTQZWHWJDGWKWFGHZQZWJWHWGCIZJHZWEOWMWGRSZCWGEKZLZTLZMZEGNZQZARSZCAFKZLZTL
|
|
ZMZFGNZQZWOWEWMWPXCXAXHWPXCOWMWPXCWFRSAWFUKUEUFWMWTXHEGVRWLWQGHZWTXHOZVTV
|
|
RWLXJXKVRWLXJUAZXHWTWSXFMZFGNZXLWFWQLZGHZWSAXOLZTLZMZXNWLXJXPVRWFWQUGUHXS
|
|
XLWRXQMXSAWFWQUIWRXQTUJULUFXMXSFXOGXDXOMZXFXRWSXTXEXQMXFXRMXDXOAUMXEXQTUJ
|
|
UNUOUPUQWTXGXMFGCWSXFURUSUTVAVBVCVDWMWGVJHZVTWOXBPVRWLYAVTAWFGGVEVFVRVTWL
|
|
VGEWGCVJGVHUQWKWEXIPWLFACGGVHVKVIWHWDWOWEWHWCWNJBWGCVLVMVNUTVCVOVPVQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $.
|
|
$( Transitivity law for finite less than. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
ltfintr $p |- ( ( A e. Nn /\ B e. Nn /\ C e. Nn ) ->
|
|
( ( << A , B >> e. <[fin] /\ << B , C >> e. <[fin] ) ->
|
|
<< A , C >> e. <[fin] ) ) $=
|
|
( vx vy vz cnnc wcel c0 cv cplc c1c wceq wrex wa cltfin addceq1 opkltfing
|
|
copk wb w3a wne an4 wi simpl a1i reeanv addccom peano2 syl5eqel nncaddccl
|
|
sylan2 adantl syl eqeq2d biimpa addceq2 addcass syl6eqr rspcev rexlimdvva
|
|
ex syl2im syl5bir anim12d syl5bi 3adant3 3adant1 anbi12d 3adant2 3imtr4d
|
|
eqtri ) AGHZBGHZCGHZUAZAIUBZBADJZKZLKZMZDGNZOZBIUBZCBEJZKZLKZMZEGNZOZOZVQ
|
|
CAFJZKZLKZMZFGNZOZABSPHZBCSPHZOACSPHZWKVQWDOZWBWIOZOVPWQVQWBWDWIUCVPXAVQX
|
|
BWPXAVQUDVPVQWDUEUFXBWAWHOZEGNDGNVPWPWAWHDEGGUGVPXCWPDEGGVPVRGHZWEGHZOZOV
|
|
RLWEKZKZGHZXCCVTWEKZLKZMZWPXFXIVPXEXDXGGHXIXEXGWELKGLWEUHWEUIUJVRXGUKULUM
|
|
WAWHXLWAWGXKCWAWFXJMWGXKMBVTWEQWFXJLQUNUOUPXIXLWPWOXLFXHGWLXHMZWNXKCXMWMX
|
|
JMWNXKMXMWMAXHKZXJWLXHAUQXJVSXGKXNVSLWEURAVRXGURVLUSWMXJLQUNUOUTVBVCVAVDV
|
|
EVFVPWRWCWSWJVMVNWRWCTVODABGGRVGVNVOWSWJTVMEBCGGRVHVIVMVOWTWQTVNFACGGRVJV
|
|
K $.
|
|
$}
|
|
|
|
$( Asymmetry law for finite less than. (Contributed by SF, 29-Jan-2015.) $)
|
|
ltfinasym $p |- ( ( A e. Nn /\ B e. Nn ) ->
|
|
( << A , B >> e. <[fin] -> -. << B , A >> e. <[fin] ) ) $=
|
|
( cnnc wcel wa copk cltfin wn ltfinirr ad2antrr wi ltfintr 3anidm13 expdimp
|
|
mtod ex ) ACDZBCDZEZABFGDZBAFGDZHSTEUAAAFGDZQUBHRTAIJSTUAUBQRTUAEUBKABALMNO
|
|
P $.
|
|
|
|
$( Cardinal zero is a minimal element for finite less than or equal.
|
|
(Contributed by SF, 29-Jan-2015.) $)
|
|
0cminle $p |- ( A e. Nn -> << 0c , A >> e. <_[fin] ) $=
|
|
( cnnc wcel c0c copk cplc clefin addcid2 opkeq2i peano1 lefinaddc syl5eqelr
|
|
mpan ) ABCZDAEDDAFZEZGOADAHIDBCNPGCJDABKML $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( One plus a finite cardinal is strictly greater. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
ltfinp1 $p |- ( ( A e. V /\ A =/= (/) ) ->
|
|
<< A , ( A +c 1c ) >> e. <[fin] ) $=
|
|
( vx wcel c0 wne wa c1c cplc copk cltfin wceq cnnc wrex simpr c0c addcid1
|
|
cv peano1 cvv addceq1i eqcomi addceq2 addceq1d eqeq2d mp2an jctir wb 1cex
|
|
rspcev addcexg mpan2 opkltfing mpdan adantr mpbird ) ABDZAEFZGZAAHIZJKDZU
|
|
RUTACRZIZHIZLZCMNZGZUSURVFUQUROPMDUTAPIZHIZLZVFSVIUTVHAHAQUAUBVEVJCPMVBPL
|
|
ZVDVIUTVKVCVHHVBPAUCUDUEUJUFUGUQVAVGUHZURUQUTTDZVLUQHTDVMUIAHBTUKULCAUTBT
|
|
UMUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Transfer from less than or equal to less than. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
lefinlteq $p |- ( ( A e. V /\ B e. W /\ A =/= (/) ) ->
|
|
( << A , B >> e. <_[fin] <-> ( << A , B >> e. <[fin] \/ A = B ) ) ) $=
|
|
( vx vy wcel cv cplc wceq cnnc wrex c1c wo wb c0c addceq2 eqeq1 rexbidv
|
|
wa c0 wne w3a copk clefin nnc0suc addcid1 syl6req addcass syl6eqr orim12i
|
|
cltfin reximi sylbi orcomd eqeq2 syl5ibrcom rexlimiv eqeq2i peano2 eqeq2d
|
|
orbi12d rspcev sylan sylan2b rexlimiva peano1 eqcomi mp2an mpbii jaoi a1i
|
|
impbii opklefing 3adant3 opkltfing adantr ibar adantl bitr4d orbi1d 3impa
|
|
3bitr4d ) ACGZBDGZAUAUBZUCZBAEHZIZJZEKLZBAFHZIMIZJZFKLZABJZNZABUDZUEGZWRU
|
|
LGZWPNZWKWQOWGWKWQWJWQEKWHKGZWQWJWIWMJZFKLZAWIJZNXBXEXDXBWHPJZWHWLMIZJZFK
|
|
LZNXEXDNFWHUFXFXEXIXDXFWIAPIZAWHPAQZAUGZUHXHXCFKXHWIAXGIZWMWHXGAQZAWLMUIZ
|
|
UJUMUKUNUOWJWOXDWPXEWJWNXCFKBWIWMRSBWIAUPVBUQURWOWKWPWNWKFKWNWLKGZBXMJZWK
|
|
WMXMBXOUSXPXGKGXQWKWLUTWJXQEXGKXHWIXMBXNVAVCVDVEVFWPXEEKLZWKPKGAXJJZXRVGX
|
|
JAXLVHXEXSEPKXFWIXJAXKVAVCVIWPXEWJEKABWIRSVJVKVMVLWDWEWSWKOWFEABCDVNVOWDW
|
|
EWFXAWQOWDWETZWFTZWTWOWPYAWTWFWOTZWOXTWTYBOWFFABCDVPVQWFWOYBOXTWFWOVRVSVT
|
|
WAWBWC $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a c $. $d a d $. $d a t $. $d a w $. $d a y $. $d a z $.
|
|
$d b c $. $d b d $. $d b t $. $d b w $. $d b y $. $d b z $. $d c d $.
|
|
$d c t $. $d c w $. $d c x $. $d c y $. $d c z $. $d d t $. $d d w $.
|
|
$d d y $. $d d z $. $d t w $. $d t x $. $d t y $. $d t z $. $d w x $.
|
|
$d w y $. $d w z $. $d x y $. $d x z $. $d y z $.
|
|
|
|
$( Finite less than is stratified. (Contributed by SF, 29-Jan-2015.) $)
|
|
ltfinex $p |- <[fin] e. _V $=
|
|
( vx vt vc va vb vd cssetk csik cpw1 csn copk wceq wrex wa wex opksnelsik
|
|
wcel snex 3bitri sikex vy vz cltfin cvv cxpk cins3k cins2k cin c1c ccompl
|
|
vw cimak cun csymdif cdif cimagek cnnc c0 cv wne cplc df-ltfin elin elvvk
|
|
cab anbi1i 19.41vv eleq1 opkex elimak elpw12 r19.41v bitr4i exbii rexcom4
|
|
wn df-rex 3bitr4i opkeq1 eleq1d ceqsexv elpw131c 19.41v excom wb elpw161c
|
|
wel elsymdif otkelins3k vex elssetk otkelins2k elpw181c ndisjrelk elcompl
|
|
bitri notbii df-ne con2bii elpw1111c orbi12i elun bibi12i wal dfcleq alex
|
|
wo anbi12i rexcom df-addc eqeq2i abeq2 opkelimagek dfaddc2 addcex addceq1
|
|
eqeq2d rexbii opkelxpk mpbiran2 elsnc anbi2i syl6bb pm5.32i 2exbii bitr3i
|
|
eldif ancom abbi2i eqtr4i vvex xpkex ssetkex ins3kex ins2kex pw1ex imakex
|
|
inex complex symdifex 1cex unex addcexlem imagekex nncex difex eqeltri )
|
|
UCUDUDUEZGHZHZHZHZUFZUUJUFZUGZGUFZUGZUGZUGZUUPGUGZUHZUIIZIZULZUJZHZHZHZHZ
|
|
HZHZHZUFZUULHZUFZUGZUGZUVNHZHZHZHZUFZUVSUFZUGZUMZUNZUVCIZIZIZIZIZIZIZIZIZ
|
|
ULZUJZUHZUHZUWLULZUHZUWJULZUGZUNZUWJULZUJZUVEUFUUTUGUUQUUNUMUNUWHULUOZUVC
|
|
ULZUPZUGZUGZUHZUWGULZUQIZIZULZURJZUDUEZUOZUHZUDUCAUSZUAUSZUBUSZKZLZUYBURU
|
|
TZUYCUYBUKUSZVAZUIVAZLZUKUQMZNZNZUBOUAOZAVEUXTAUAUBUKVBUYNAUXTUYAUXTQUYAU
|
|
UHQZUYAUXSQZNUYEUBOUAOZUYPNZUYNUYAUUHUXSVCUYOUYQUYPUAUBUYAVDVFUYRUYEUYPNZ
|
|
UBOUAOUYNUYEUYPUAUBVGUYSUYMUAUBUYEUYPUYLUYEUYPUYDUXSQZUYLUYAUYDUXSVHUYDUX
|
|
PQZUYDUXRQZVPZNUYKUYBURLZVPZNZUYTUYLVUAUYKVUCVUEVUABUSZUYDKZUXMQZBUXOMZVU
|
|
GUYGJZJZLZVUINZBOZUKUQMZUYKBUXMUXOUYDUYBUYCVIZVJVUGUXOQZVUINZBOVUNUKUQMZB
|
|
OVUJVUPVUSVUTBVUSVUMUKUQMZVUINVUTVURVVAVUIUKVUGUQVKVFVUMVUIUKUQVLVMVNVUIB
|
|
UXOVQVUNUKBUQVOVRVUOUYJUKUQVUOVULUYDKZUXMQZUYAUYHLZUYCUYAUIVAZLZNZAOZUYJV
|
|
UIVVCBVULVUKRZVUMVUHVVBUXMVUGVULUYDVSVTWAVVCVUGVVBKZUXLQZBUWGMZVUGUYAJZJZ
|
|
JZJZLZVVKNZBOZAOZVVHBUXLUWGVVBVULUYDVIZVJVUGUWGQZVVKNZBOVVRAOZBOVVLVVTVWC
|
|
VWDBVWCVVQAOZVVKNVWDVWBVWEVVKAVUGWBVFVVQVVKAWCVMVNVVKBUWGVQVVRABWDVRVVSVV
|
|
GAVVSVVPVVBKZUXLQZVWFUXFQZVWFUXKQZNVVGVVKVWGBVVPVVORZVVQVVJVWFUXLVUGVVPVV
|
|
BVSVTWAVWFUXFUXKVCVWHVVDVWIVVFVWFUXEQZVPCAWGZDUSZEUSZUHZURLZCUSZVWMVWNUMZ
|
|
LZNZEUYGMDUYBMZWEZVPZCOZVPZVWHVVDVWKVXDVWKVUGVWFKZUXDQZBUWJMZVUGVWQJZJZJZ
|
|
JZJZJZJZLZVXGNZBOZCOZVXDBUXDUWJVWFVVPVVBVIZVJVUGUWJQZVXGNZBOVXQCOZBOVXHVX
|
|
SVYBVYCBVYBVXPCOZVXGNVYCVYAVYDVXGCVUGWFVFVXPVXGCWCVMVNVXGBUWJVQVXQCBWDVRV
|
|
XRVXCCVXRVXOVWFKZUXDQZVYEUUMQZVYEUXCQZWEZVPVXCVXGVYFBVXOVXNRVXPVXFVYEUXDV
|
|
UGVXOVWFVSVTWAVYEUUMUXCWHVYIVXBVYGVWLVYHVXAVYGVXMVVPKUULQVXLVVOKUUKQZVWLV
|
|
XMVVPVVBUULVXLRZVWJVWAWIVXLVVOUUKVXKRZVVNRPVYJVXKVVNKUUJQVXJVVMKUUIQZVWLV
|
|
XKVVNUUJVXJRZVVMRZPVXJVVMUUIVXIRZUYARPVYMVXIUYAKGQVWLVXIUYAGVWQRZAWJZPVWQ
|
|
UYACWJZVYRWKWPSSVXMVVBKZUXBQZEUKWGZVWTDUYBMZNZEOZVYHVXAWUAVUGVYTKZUXAQZBU
|
|
WJMZVUGVWNJZJZJZJZJZJZJZLZWUGNZBOZEOZWUEBUXAUWJVYTVXMVVBVIZVJVYAWUGNZBOWU
|
|
QEOZBOWUHWUSWVAWVBBWVAWUPEOZWUGNWVBVYAWVCWUGEVUGWFVFWUPWUGEWCVMVNWUGBUWJV
|
|
QWUQEBWDVRWURWUDEWURWUOVYTKZUXAQZWVDUUOQZWVDUWTQZNWUDWUGWVEBWUOWUNRZWUPWU
|
|
FWVDUXAVUGWUOVYTVSVTWAWVDUUOUWTVCWVFWUBWVGWUCWVFWUMVVBKUUNQWUKVULKUUJQZWU
|
|
BWUMVXMVVBUUNWULRZVYKVWAWLWUKVULUYDUUJWUJRZVVIVUQWIWVIWUJVUKKUUIQWUIUYGKG
|
|
QWUBWUJVUKUUIWUIRZUYGRPWUIUYGGVWNRZUKWJZPVWNUYGEWJZWVNWKSSWVGDUAWGZVWTNZD
|
|
OZWUCWVGVUGWVDKZUWSQZBUWLMZVUGVWMJZJZJZJZJZJZJZJZJZLZWVTNZBOZDOZWVRBUWSUW
|
|
LWVDWUOVYTVIZVJVUGUWLQZWVTNZBOWWLDOZBOWWAWWNWWQWWRBWWQWWKDOZWVTNWWRWWPWWS
|
|
WVTDVUGWMVFWWKWVTDWCVMVNWVTBUWLVQWWLDBWDVRWWMWVQDWWMWWJWVDKZUWSQZWWTUUSQZ
|
|
WWTUWRQZNWVQWVTWXABWWJWWIRZWWKWVSWWTUWSVUGWWJWVDVSVTWAWWTUUSUWRVCWXBWVPWX
|
|
CVWTWXBWWHVYTKUURQWWFVVBKUUQQZWVPWWHWUOVYTUURWWGRZWVHWUTWLWWFVXMVVBUUQWWE
|
|
RZVYKVWAWLWXEWWDUYDKUUPQWWBUYBKGQWVPWWDVULUYDUUPWWCRZVVIVUQWLWWBUYBUYCGVW
|
|
MRZUAWJZUBWJZWIVWMUYBDWJZWXJWKSSWXCWWTUVMQZWWTUWQQZNVWTWWTUVMUWQVCWXMVWPW
|
|
XNVWSWXMWWHWUOKUVLQWWGWUNKUVKQZVWPWWHWUOVYTUVLWXFWVHWUTWIWWGWUNUVKWWFRZWU
|
|
MRZPWXOWWFWUMKUVJQWWEWULKUVIQZVWPWWFWUMUVJWXGWVJPWWEWULUVIWWDRZWUKRZPWXRW
|
|
WDWUKKUVHQWWCWUJKUVGQZVWPWWDWUKUVHWXHWVKPWWCWUJUVGWWBRZWVLPWYAWWBWUIKUVFQ
|
|
VWMVWNKZUVEQZVWPWWBWUIUVFWXIWVMPVWMVWNUVEWXLWVOPWYCUVDQZVPVWOURUTZVPWYDVW
|
|
PWYEWYFVWMVWNWXLWVOWNWQWYCUVDVWMVWNVIWOWYFVWPVWOURWRWSVRSSSSWWTUWPQZVPFCW
|
|
GZFUSZVWRQZWEZVPZFOZVPZWXNVWSWYGWYMWYGVUGWWTKZUWFQZBUWOMZVUGWYIJZJZJZJZJZ
|
|
JZJZJZJZJZJZJZLZWYPNZBOZFOZWYMBUWFUWOWWTWWJWVDVIZVJVUGUWOQZWYPNZBOXUKFOZB
|
|
OWYQXUMXUPXUQBXUPXUJFOZWYPNXUQXUOXURWYPFVUGWTVFXUJWYPFWCVMVNWYPBUWOVQXUKF
|
|
BWDVRXULWYLFXULXUIWWTKZUWFQZXUSUVQQZXUSUWEQZWEZVPWYLWYPXUTBXUIXUHRXUJWYOX
|
|
USUWFVUGXUIWWTVSVTWAXUSUVQUWEWHXVCWYKXVAWYHXVBWYJXVAXUGWVDKZUVPQXUEVYTKUV
|
|
OQZWYHXUGWWJWVDUVPXUFRZWXDWWOWLXUEWUOVYTUVOXUDRZWVHWUTWLXVEXUCVXMKUVNQXUB
|
|
VXLKUULQZWYHXUCVXMVVBUVNXUBRZVYKVWAWIXUBVXLUULXUARZVYLPXVHXUAVXKKUUKQWYTV
|
|
XJKUUJQZWYHXUAVXKUUKWYTRZVYNPWYTVXJUUJWYSRZVYPPXVKWYSVXIKUUIQWYRVWQKGQWYH
|
|
WYSVXIUUIWYRRZVYQPWYRVWQGWYIRZVYSPWYIVWQFWJZVYSWKSSSSXUSUWBQZXUSUWDQZXGFD
|
|
WGZFEWGZXGXVBWYJXVQXVSXVRXVTXVQXUGWWJKUWAQXUFWWIKUVTQZXVSXUGWWJWVDUWAXVFW
|
|
XDWWOWIXUFWWIUVTXUERWWHRPXWAXUEWWHKUVSQXUDWWGKUVRQZXVSXUEWWHUVSXVGWXFPXUD
|
|
WWGUVRXUCRZWXPPXWBXUCWWFKUVNQXUBWWEKUULQZXVSXUCWWFUVNXVIWXGPXUBWWEUULXVJW
|
|
XSPXWDXUAWWDKUUKQWYTWWCKUUJQZXVSXUAWWDUUKXVLWXHPWYTWWCUUJXVMWYBPXWEWYSWWB
|
|
KUUIQWYRVWMKGQXVSWYSWWBUUIXVNWXIPWYRVWMGXVOWXLPWYIVWMXVPWXLWKSSSSSXVRXVDU
|
|
WCQXUEWUOKUVSQZXVTXUGWWJWVDUWCXVFWXDWWOWLXUEWUOVYTUVSXVGWVHWUTWIXWFXUDWUN
|
|
KUVRQXUCWUMKUVNQZXVTXUDWUNUVRXWCWXQPXUCWUMUVNXVIWVJPXWGXUBWULKUULQXUAWUKK
|
|
UUKQZXVTXUBWULUULXVJWXTPXUAWUKUUKXVLWVKPXWHWYTWUJKUUJQWYSWUIKUUIQZXVTWYTW
|
|
UJUUJXVMWVLPWYSWUIUUIXVNWVMPXWIWYRVWNKGQXVTWYRVWNGXVOWVOPWYIVWNXVPWVOWKWP
|
|
SSSSXAXUSUWBUWDXBWYIVWMVWNXBVRXCWQSVNSWQWWTUWPXUNWOVWSWYKFXDWYNFVWQVWRXEW
|
|
YKFXFWPVRXHWPXHSVNSVWTDUYBVQVMXHSVNSVXMVVPVVBUXBVYKVWJVWAWLVXAWUCEUYGMWUE
|
|
VWTDEUYBUYGXIWUCEUYGVQWPVRXCWQSVNSWQVWFUXEVXTWOVVDUYAVXACVEZLVXBCXDVXEUYH
|
|
XWJUYACDEUYBUYGXJXKVXACUYAXLVXBCXFSVRVWIVVNUYDKUXJQUYAUYCKUXIQZVVFVVNVULU
|
|
YDUXJVYOVVIVUQWLUYAUYBUYCUXIVYRWXJWXKWLXWKUYCUXHUYAULZLVVFUYAUYCUXHVYRWXK
|
|
XMVVEXWLUYCUYAUIXNXKVMSXHSVNSVVFUYJAUYHUYBUYGWXJWVNXOVVDVVEUYIUYCUYAUYHUI
|
|
XPXQWASXRSVUBVUDVUBUYBUXQQZVUDVUBXWMUYCUDQWXKUYBUYCUXQUDWXJWXKXSXTUYBURWX
|
|
JYAWPWQXHUYDUXPUXRYGUYLUYKUYFNVUFUYFUYKYHUYFVUEUYKUYBURWRYBWPVRYCYDYEYFSY
|
|
IYJUUHUXSUDUDYKYKYLUXPUXRUXMUXOUXLUWGUXFUXKUXEUXDUWJUUMUXCUULUUKUUJUUIGYM
|
|
TTZTTZYNUXBUXAUWJUUOUWTUUNUUJXWNYNYOUWSUWLUUSUWRUURUUQUUPGYMYNZYOYOYOUVMU
|
|
WQUVLUVKUVJUVIUVHUVGUVFUVEUVDUVAUVCUUPUUTXWPGYMYOYRUVBUIUUAYPYPZYQYSTTTTT
|
|
TTYNUWPUWFUWOUVQUWEUVPUVOUVNUULXWOTZYNYOYOUWBUWDUWAUVTUVSUVRUVNXWRTTZTTYN
|
|
UWCUVSXWSYNYOUUBYTUWNUWMUWLUWKUWJUWIUWHUWGUVCXWQYPZYPYPYPZYPYPZYPYPYPYQYS
|
|
YRYRXXBYQYRXXAYQYOYTXXAYQYSUXJUXIUXHUXGUVCUUCXWQYQUUDYOYOYRXWTYQUXNUQUUEY
|
|
PYPYQUXQUDURRYKYLUUFYRUUG $.
|
|
$}
|
|
|
|
${
|
|
$d m n t $.
|
|
$( Lemma for ~ ltfintri . Establish stratification for induction.
|
|
(Contributed by SF, 29-Jan-2015.) $)
|
|
ltfintrilem1 $p |- { m | ( n e. Nn -> ( m =/= (/) ->
|
|
( << m , n >> e. <[fin] \/ m = n \/ << n , m >> e. <[fin] ) )
|
|
) } e. _V $=
|
|
( vt cv wcel wn cab c0 csn cltfin cimak cun copk weq wi wo unab elun unex
|
|
bitri cnnc ccnvk wne w3o cvv wceq wrex vex elimak opkeq1 eleq1d opkelcnvk
|
|
df-sn rexsn elsnc orbi12i df-3or 3bitr4i abbi2i uneq12i eqtri uneq2i imor
|
|
df-ne imbi1i df-or bitr4i orbi2i abbii 3eqtr4i snex ltfinex cnvkex imakex
|
|
abexv eqeltrri ) BDZUAEZFZAGZHIZJUBZVQIZKZWCLZJWCKZLZLZLZVRADZHUCZWJVQMJE
|
|
ZABNZVQWJMZJEZUDZOZOZAGZUEVTWJHUFZWPPZAGZLVSXAPZAGWIWSVSXAAQWHXBVTWHWTAGZ
|
|
WPAGZLXBWAXDWGXEAHUMWPAWGWJWEEZWJWFEZPWLWMPZWOPWJWGEWPXFXHXGWOXFWJWDEZWJW
|
|
CEZPXHWJWDWCRXIWLXJWMXIWNWBEZWLXICDZWJMZWBEZCWCUGXKCWBWCWJAUHZUIXNXKCVQBU
|
|
HZCBNZXMWNWBXLVQWJUJZUKUNTVQWJJXPXOULTWJVQXOUOUPTXGXMJEZCWCUGWOCJWCWJXOUI
|
|
XSWOCVQXPXQXMWNJXRUKUNTUPWJWEWFRWLWMWOUQURUSUTWTWPAQVAVBWRXCAWRVSWQPXCVRW
|
|
QVCWQXAVSWQWTFZWPOXAWKXTWPWJHVDVEWTWPVFVGVHTVIVJVTWHVSAVOWAWGHVKWEWFWDWCW
|
|
BWCJVLVMVQVKZVNYASJWCVLYAVNSSSVP $.
|
|
$}
|
|
|
|
${
|
|
$d M m n k p $. $d N m n k p $.
|
|
$( Trichotomy law for finite less than. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
ltfintri $p |- ( ( M e. Nn /\ N e. Nn /\
|
|
M =/= (/) ) ->
|
|
( << M , N >> e. <[fin] \/ M = N \/ << N , M >> e. <[fin] ) ) $=
|
|
( vn vk cnnc wcel wne copk cltfin wceq w3o opkeq2 eleq1d opkeq1 3orbi123d
|
|
vp c0 wi imbi2d c0c vm cv eqeq2 weq cplc ltfintrilem1 neeq1 eqeq1 imbi12d
|
|
c1c wa wo clefin 0cminle adantr cvv wb 0cex lefinlteq mp3an1 orcom syl6bb
|
|
mpbid 3mix2 3mix1 jaoi syl ex addcnnul simpld 3ad2ant3 wrex addc32 eqeq2i
|
|
w3a rexbii biimpi adantl a1i opkltfing 3adant3 simp1 peano2 simp2 syl2anc
|
|
opklefing 3imtr4d syl3an1 sylibd ltfinp1 sylan2 3adant2 syl5ibcom ltfintr
|
|
syl6 3mix3 syl3anc mpan2d 3jaod embantd 3expia com23 a2d finds com12 3imp
|
|
vtoclga ) AEFZBEFZAQGZABHZIFZABJZBAHZIFZKZXIXHXJXPRZXHXJACUBZHZIFZAXRJZXR
|
|
AHZIFZKZRZRXHXQRCBEXRBJZYEXQXHYFYDXPXJYFXTXLYAXMYCXOYFXSXKIXRBALMXRBAUCYF
|
|
YBXNIXRBANMOSSXHXREFZYEYGDUBZQGZYHXRHZIFZDCUDZXRYHHZIFZKZRZRYGTQGZTXRHZIF
|
|
ZTXRJZXRTHZIFZKZRZRYGUAUBZQGZUUEXRHZIFZUACUDZXRUUEHZIFZKZRZRYGUUEUJUEZQGZ
|
|
UUNXRHZIFZUUNXRJZXRUUNHZIFZKZRZRYGYERDUAADCUFYHTJZYPUUDYGUVCYIYQYOUUCYHTQ
|
|
UGUVCYKYSYLYTYNUUBUVCYJYRIYHTXRNMYHTXRUHUVCYMUUAIYHTXRLMOUISDUAUDZYPUUMYG
|
|
UVDYIUUFYOUULYHUUEQUGUVDYKUUHYLUUIYNUUKUVDYJUUGIYHUUEXRNMYHUUEXRUHUVDYMUU
|
|
JIYHUUEXRLMOUISYHUUNJZYPUVBYGUVEYIUUOYOUVAYHUUNQUGUVEYKUUQYLUURYNUUTUVEYJ
|
|
UUPIYHUUNXRNMYHUUNXRUHUVEYMUUSIYHUUNXRLMOUISYHAJZYPYEYGUVFYIXJYOYDYHAQUGU
|
|
VFYKXTYLYAYNYCUVFYJXSIYHAXRNMYHAXRUHUVFYMYBIYHAXRLMOUISYGYQUUCYGYQUKZYTYS
|
|
ULZUUCUVGYRUMFZUVHYGUVIYQXRUNUOUVGUVIYSYTULZUVHTUPFYGYQUVIUVJUQURTXRUPEUS
|
|
UTYSYTVAVBVCYTUUCYSYTYSUUBVDYSYTUUBVEVFVGVHUUEEFZYGUUMUVBUVKYGUUMUVBRUVKY
|
|
GUKUUOUUMUVAUVKYGUUOUUMUVARUVKYGUUOVOZUUFUULUVAUUOUVKUUFYGUUOUUFUJQGUUEUJ
|
|
VIVJZVKUVLUUHUVAUUIUUKUVLUUHUUQUURULZUVAUVLUUHUUPUMFZUVNUVLUUFXRUUEPUBZUE
|
|
UJUEZJZPEVLZUKZXRUUNUVPUEZJZPEVLZUUHUVOUVTUWCRUVLUVSUWCUUFUVSUWCUVRUWBPEU
|
|
VQUWAXRUUEUVPUJVMVNVPVQVRVSUVKYGUUHUVTUQUUOPUUEXREEVTWAUVLUUNEFZYGUVOUWCU
|
|
QUVLUVKUWDUVKYGUUOWBZUUEWCZVGZUVKYGUUOWDZPUUNXREEWFWEWGUVKUWDYGUUOUVOUVNU
|
|
QUWFUUNXREEUSWHWIUUQUVAUURUUQUURUUTVEUURUUQUUTVDVFWOUVLUUIUUTUVAUVLUUEUUN
|
|
HZIFZUUIUUTUVKUUOUWJYGUUOUVKUUFUWJUVMUUEEWJWKWLZUUIUWIUUSIUUEXRUUNNMWMUUT
|
|
UUQUURWPZWOUVLUUKUUTUVAUVLUUKUWJUUTUWKUVLYGUVKUWDUUKUWJUKUUTRUWHUWEUWGXRU
|
|
UEUUNWNWQWRUWLWOWSWTXAXBVHXCXDXEXGXEXF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Less than or equal to is reflexive. (Contributed by SF, 2-Feb-2015.) $)
|
|
lefinrflx $p |- ( A e. V -> << A , A >> e. <_[fin] ) $=
|
|
( vx wcel copk clefin cv cplc wceq cnnc c0c peano1 addcid1 eqcomi addceq2
|
|
wrex eqeq2d rspcev mp2an wb opklefing anidms mpbiri ) ABDZAAEFDZAACGZHZIZ
|
|
CJPZKJDAAKHZIZUILUJAAMNUHUKCKJUFKIUGUJAUFKAOQRSUDUEUITCAABBUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Less than implies less than or equal. (Contributed by SF,
|
|
2-Feb-2015.) $)
|
|
ltlefin $p |- ( ( A e. V /\ B e. W ) ->
|
|
( << A , B >> e. <[fin] -> << A , B >> e. <_[fin] ) ) $=
|
|
( vx vy wcel wa c0 wne cv cplc c1c wceq cnnc wrex copk cltfin clefin wi
|
|
addcass eqeq2i peano2 addceq2 eqeq2d rspcev sylan rexlimiva a1i opkltfing
|
|
sylan2b adantl opklefing 3imtr4d ) ACGBDGHZAIJZBAEKZLMLZNZEOPZHZBAFKZLZNZ
|
|
FOPZABQZRGVFSGVAVETUOUTVEUPUSVEEOUSUQOGZBAUQMLZLZNZVEURVIBAUQMUAUBVGVHOGV
|
|
JVEUQUCVDVJFVHOVBVHNVCVIBVBVHAUDUEUFUGUKUHULUIEABCDUJFABCDUMUN $.
|
|
$}
|
|
|
|
$( Less than or equal is the same as negated less than. (Contributed by SF,
|
|
2-Feb-2015.) $)
|
|
lenltfin $p |- ( ( A e. Nn /\ B e. Nn ) ->
|
|
( << A , B >> e. <_[fin] <-> -. << B , A >> e. <[fin] ) ) $=
|
|
( cnnc wcel wa copk clefin cltfin wn ltfinirr wi leltfintr 3anidm13 expdimp
|
|
adantr c0 wceq opkeq2 eleq1d wo ex nulge ancoms eleq1 anbi2d imbi12d mpbiri
|
|
mtod a1dd wne w3o simplr simpll simpr ltfintri syl3anc 3orass ord lefinrflx
|
|
sylib syl5ibrcom ltlefin jaod syld expcom pm2.61ine impbid ) ACDZBCDZEZABFZ
|
|
GDZBAFHDZIZVJVLVNVJVLEVMAAFZHDZVJVPIZVLVHVQVIAJOOVJVLVMVPVHVIVLVMEVPKABALMN
|
|
UHUAVJVNVLKZKBPBPQZVJVLVNVSVJVLKVHPCDZEZAPFZGDZKVTVHWCACUBUCVSVJWAVLWCVSVIV
|
|
TVHBPCUDUEVSVKWBGBPARSUFUGUIVJBPUJZVRVJWDEZVNBAQZVKHDZTZVLWEVMWHWEVMWFWGUKZ
|
|
VMWHTWEVIVHWDWIVHVIWDULVHVIWDUMVJWDUNBAUOUPVMWFWGUQUTURWEWFVLWGVJWFVLKWDVJV
|
|
LWFVOGDZVHWJVIACUSOWFVKVOGBAARSVAOVJWGVLKWDABCCVBOVCVDVEVFVG $.
|
|
|
|
${
|
|
$d A a $. $d a b $. $d a c $. $d a d $. $d a k $. $d a m $. $d a n $.
|
|
$d a x $. $d B a $. $d B b $. $d b c $. $d b d $. $d b k $. $d b m $.
|
|
$d b n $. $d b t $. $d b x $. $d c d $. $d c k $. $d c m $. $d c x $.
|
|
$d d k $. $d d m $. $d d x $. $d k m $. $d k x $. $d m n $. $d m t $.
|
|
$( A subset of a finite set is itself finite. Theorem X.1.21 of [Rosser]
|
|
p. 527. (Contributed by SF, 19-Jan-2015.) $)
|
|
ssfin $p |- ( ( A e. V /\ B e. Fin /\ A C_ B ) -> A e. Fin ) $=
|
|
( va vb vn vm vt vx wcel cfin wss cv wi wceq wa wal cssetk cvv wex imbi2d
|
|
vd vk vc sseq1 eleq1 imbi12d sseq2 imbi1d wel cnnc wrex elfin c0 c1c cplc
|
|
ccompl cimak cpw1 cxpk cin cab wn elcompl alcom impexp albii 19.21v bitri
|
|
vex copk elimak csn df-rex anbi1i 19.41v bitr4i exbii excom opkeq1 eleq1d
|
|
el1c ceqsexv elssetk opkelxpk mpbiran2 snelpw1 ancom wb opkelssetkg mp2an
|
|
snex elin anbi12i exanali 3bitri con2bii abbi2i ssetkex finex imakex vvex
|
|
pw1ex xpkex inex 1cex eqeltrri c0c eleq2 df-0c eleq2i elsnc syl6bb anbi1d
|
|
complex 2albidv weq elequ2 adantl sseq12 anbi12d adantr biimpa ss0b sylib
|
|
cbval2v 0fin syl6eqel wo w3a cdif cun 3ad2ant2 syl5eq df-dif df-ss sylibr
|
|
com23 imp3a ex gen2 wpss sspss dfpss4 orbi1i simp1 snid eldif simprbi mt2
|
|
a1i undif1 snssi ssequn2 simp3r eqeltrd nnsucelr syl12anc ineq2i 3eqtr4ri
|
|
difex inass biimpi 3ad2ant3 difeq1d difsn eqtrd jca 3adant1r 3ad2ant1 mpd
|
|
spc2gv 3exp exp5c rexlimdv peano2 rspcev syl biimprd syl9 syl5bi alrimivv
|
|
jaod finds 19.21bbi exp3a rexlimiv sylbi vtoclga vtoclg 3imp ) ACJBKJZABL
|
|
ZAKJZUWLDMZBLZUWOKJZNZNUWLUWMUWNNZNDACUWOAOZUWRUWSUWLUWTUWPUWMUWQUWNUWOAB
|
|
UEUWOAKUFUGUAUWOEMZLZUWQNZUWREBKUXABOUXBUWPUWQUXABUWOUHUIUXAKJZEFUJZFUKUL
|
|
UXCFUXAUMUXEUXCFUKFMZUKJZUXEUXBUWQUXGUXEUXBPZUWQNZDEEGUJZUXBPZUWQNZEQDQZU
|
|
XAUNOZUXBPZUWQNZEQDQUBUCUJZUDMZUBMZLZPZUXRKJZNZUBQUDQZUXAUCMZUOUPZJZUXBPZ
|
|
UWQNZEQDQZUXIEQDQGUCUXFRRKUQZURZUSZSUTZVAZUOURZUQZUXMGVBSUXMGUYQGMZUYQJUY
|
|
RUYPJZVCZUXMUYRUYPGVJZVDUXMUXLDQZEQUXJUXCDQZNZEQZUYTUXLDEVEVUBVUDEVUBUXJU
|
|
XCNZDQVUDUXLVUFDUXJUXBUWQVFVGUXJUXCDVHVIVGUYSVUEUYSHMZUYRVKZUYOJZHUOULZUX
|
|
JVUCVCZPZETZVUEVCHUYOUOUYRVUAVLVUJVUGUOJZVUIPZHTZVUGUXAVMZOZVUIPZHTZETZVU
|
|
MVUIHUOVNVUPVUSETZHTVVAVUOVVBHVUOVURETZVUIPVVBVUNVVCVUIEVUGWBVOVURVUIEVPV
|
|
QVRVUSEHVSVQVUTVULEVUTVUQUYRVKZUYOJZVVDRJZVVDUYNJZPVULVUIVVEHVUQUXAWLZVUR
|
|
VUHVVDUYOVUGVUQUYRVTWAWCVVDRUYNWMVVFUXJVVGVUKUXAUYREVJZVUAWDVVGVUQUYMJZUX
|
|
AUYLJZVUKVVGVVJUYRSJVUAVUQUYRUYMSVVHVUAWEWFUXAUYLWGVVKUWOUXAVKRJZDUYKULZU
|
|
XBUWQVCZPZDTZVUKDRUYKUXAVVIVLVVMUWOUYKJZVVLPZDTVVPVVLDUYKVNVVRVVODVVRVVLV
|
|
VQPVVOVVQVVLWHVVLUXBVVQVVNUWOSJZUXASJVVLUXBWIDVJZVVIUWOUXASSWJWKUWOKVVTVD
|
|
WNVIVRVIUXBUWQDWOWPWPWNWPVRWPUXJVUCEWOWPWQWPVQWRUYPUYOUORUYNWSUYMSUYLRUYK
|
|
WSKWTXOXAXCXBXDXEXFXAXOXGUYRXHOZUXLUXPDEVWAUXKUXOUWQVWAUXJUXNUXBVWAUXJUXA
|
|
XHJZUXNUYRXHUXAXIVWBUXAUNVMZJUXNXHVWCUXAXJXKUXAUNVVIXLVIXMXNUIXPGUCXQZUXM
|
|
EUCUJZUXBPZUWQNZEQDQUYDVWDUXLVWGDEVWDUXKVWFUWQVWDUXJVWEUXBGUCEXRXNUIXPVWG
|
|
UYCDEUDUBDUDXQZEUBXQZPZVWFUYAUWQUYBVWJVWEUXQUXBUXTVWIVWEUXQWIVWHUXAUXSUYE
|
|
UFXSUWOUXRUXAUXSXTYAVWHUWQUYBWIVWIUWOUXRKUFYBUGYFXMUYRUYFOZUXLUYIDEVWKUXK
|
|
UYHUWQVWKUXJUYGUXBUYRUYFUXAXIXNUIXPGFXQZUXLUXIDEVWLUXKUXHUWQVWLUXJUXEUXBG
|
|
FEXRXNUIXPUXPDEUXOUWOUNKUXOUWOUNLZUWOUNOUXNUXBVWMUXAUNUWOUHYCUWOYDYEYGYHU
|
|
UAUYEUKJZUYDUYJVWNUYDPZUYIDEVWOUYGUXBUWQVWOUXBUYGUWQUXBUXBIDUJVCZIUXAULZP
|
|
ZDEXQZYIZVWOUYGUWQNZUXBUWOUXAUUBZVWSYIVWTUWOUXAUUCVXBVWRVWSIUWOUXAUUDUUEV
|
|
IVWOVWRVXAVWSVWOUXBVWQVXAVWOVWQUXBVXAVWOVWPUXBVXANIUXAVWOIEUJZVWPUXBUYGUW
|
|
QVWOVXCVWPPZUXBUYGPZUWQVWOVXDVXEYJUXAIMZVMZYKZUYEJZUWOVXHLZPZUWQVWNVXDVXE
|
|
VXKUYDVWNVXDVXEYJZVXIVXJVXLVWNVXFVXHJZVCZVXHVXGYLZUYFJVXIVWNVXDVXEUUFVXNV
|
|
XLVXMVXFVXGJZVXFIVJZUUGVXMVXCVXPVCVXFUXAVXGUUHUUIUUJUUKVXLVXOUXAUYFVXLVXO
|
|
UXAVXGYLZUXAUXAVXGUULVXDVWNVXRUXAOZVXEVXCVXSVWPVXCVXGUXALVXSVXFUXAUUMVXGU
|
|
XAUUNYEYBYMYNVWNVXDUXBUYGUUOUUPVXHUYEVXFUXAVXGVVIVXFWLUVAZVXQUUQUURVXLUWO
|
|
VXHVAZUWOOVXJVXLVYAUWOUXAVAZVXGYKZUWOVYBVXGUQZVAUWOUXAVYDVAZVAVYCVYAUWOUX
|
|
AVYDUVBVYBVXGYOVXHVYEUWOUXAVXGYOUUSUUTVXLVYCUWOVXGYKZUWOVXLVYBUWOVXGVXEVW
|
|
NVYBUWOOZVXDUXBVYGUYGUXBVYGUWOUXAYPUVCYBUVDUVEVXDVWNVYFUWOOZVXEVWPVYHVXCV
|
|
XFUWOUVFXSYMUVGYNUWOVXHYPYQUVHUVIVWOVXDVXKUWQNZVXEUYDVYIVWNVVSVXHSJUYDVYI
|
|
NVVTVXTUYCVYIUDUBUWOVXHSSUDDXQZUXSVXHOZPZUYAVXKUYBUWQVYLUXQVXIUXTVXJVYKUX
|
|
QVXIWIVYJUXSVXHUYEUFXSUXRUWOUXSVXHXTYAVYJUYBUWQWIVYKUXRUWOKUFYBUGUVLWKXSU
|
|
VJUVKUVMUVNUVOYRYSVWNVWSVXANUYDVWNUYGUXDVWSUWQVWNUYFUKJZUYGUXDNUYEUVPVYMU
|
|
YGUXDVYMUYGPEIUJZIUKULUXDVYNUYGIUYFUKVXFUYFUXAXIUVQIUXAUMYQYTUVRVWSUWQUXD
|
|
UWOUXAKUFUVSUVTYBUWCUWAYRYSUWBYTUWDUWEUWFUWGUWHUWIUWJUWK $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( If the universe is finite, then there is a unique natural containing any
|
|
set. Theorem X.1.22 of [Rosser] p. 527. (Contributed by SF,
|
|
19-Jan-2015.) $)
|
|
vfinnc $p |- ( ( A e. V /\ _V e. Fin ) -> E! x e. Nn A e. x ) $=
|
|
( vy wcel cvv cfin wa cv cnnc wrex weq wi wral wss ssv ssfin mp3an3 elfin
|
|
wreu sylib nnceleq ex rgen2a a1i eleq2 reu4 sylanbrc ) BCEZFGEZHZBAIZEZAJ
|
|
KZUMBDIZEZHZADLZMZDJNAJNZUMAJTUKBGEZUNUIUJBFOVABPBFCQRABSUAUTUKUSADJULJEU
|
|
OJEHUQURBULUOUBUCUDUEUMUPADJULUOBUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( The finite cardinality of a set exists. (Contributed by SF,
|
|
27-Jan-2015.) $)
|
|
ncfinex $p |- Nc[fin] A e. _V $=
|
|
( vx cncfin cv cnnc wcel wa cio cvv df-ncfin iotaex eqeltri ) ACBDZEFAMFG
|
|
ZBHIBAJNBKL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $.
|
|
$( Equality theorem for finite cardinality. (Contributed by SF,
|
|
20-Jan-2015.) $)
|
|
ncfineq $p |- ( A = B -> Nc[fin] A = Nc[fin] B ) $=
|
|
( vx wceq cv cnnc wcel cio cncfin eleq1 anbi2d iotabidv df-ncfin 3eqtr4g
|
|
wa ) ABDZCEZFGZAQGZOZCHRBQGZOZCHAIBIPTUBCPSUARABQJKLCAMCBMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Properties of finite cardinal number. Theorem X.1.23 of [Rosser] p. 527
|
|
(Contributed by SF, 20-Jan-2015.) $)
|
|
ncfinprop $p |- ( ( _V e. Fin /\ A e. V ) ->
|
|
( Nc[fin] A e. Nn /\ A e. Nc[fin] A ) ) $=
|
|
( vx wcel cvv cfin cncfin cnnc cio df-ncfin wreu vfinnc reiotacl syl5eqel
|
|
wa cv syl wceq eqcomi wb eleq2 reiota2 syl2anc mpbiri jca ancoms ) ABDZEF
|
|
DZAGZHDZAUIDZOUGUHOZUJUKULUICPZHDAUMDZOCIZHCAJZULUNCHKZUOHDCABLZUNCHMQNZU
|
|
LUKUOUIRZUIUOUPSULUJUQUKUTTUSURUNUKCHUIUMUIAUAUBUCUDUEUF $.
|
|
$}
|
|
|
|
$( Distribution law for finite cardinality. (Contributed by SF,
|
|
30-Jan-2015.) $)
|
|
ncfindi $p |- ( ( ( _V e. Fin /\ A e. V ) /\ B e. W /\ ( A i^i B ) = (/) )
|
|
-> Nc[fin] ( A u. B ) = ( Nc[fin] A +c Nc[fin] B ) ) $=
|
|
( cvv cfin wcel wa cin c0 wceq w3a cun cncfin cnnc syl2anc ncfinprop simpld
|
|
cplc simprd simp1l simp1r simp2 nncaddccl eladdci syl3anc nnceleq syl22anc
|
|
unexg simp3 ) EFGZACGZHZBDGZABIJKZLZABMZNZOGZANZBNZSZOGZUQURGZUQVBGZURVBKUP
|
|
USVDUPUKUQEGZUSVDHUKULUNUOUAZUPULUNVFUKULUNUOUBZUMUNUOUCZABCDUIPUQEQPZRUPUT
|
|
OGZVAOGZVCUPVKAUTGZUPUKULVKVMHVGVHACQPZRUPVLBVAGZUPUKUNVLVOHVGVIBDQPZRUTVAU
|
|
DPUPUSVDVJTUPVMVOUOVEUPVKVMVNTUPVLVOVPTUMUNUOUJABUTVAUEUFUQURVBUGUH $.
|
|
|
|
$( If the universe is finite, then the cardinality of a singleton is ` 1c ` .
|
|
(Contributed by SF, 30-Jan-2015.) $)
|
|
ncfinsn $p |- ( ( _V e. Fin /\ A e. V ) -> Nc[fin] { A } = 1c ) $=
|
|
( cvv cfin wcel csn cncfin cnnc c1c wceq snex ncfinprop mpan2 adantr simpld
|
|
wa 1cnnc a1i simprd snel1cg adantl nnceleq syl22anc ) CDEZABEZPZAFZGZHEZIHE
|
|
ZUGUHEZUGIEZUHIJUFUIUKUDUIUKPZUEUDUGCEUMAKUGCLMNZOUJUFQRUFUIUKUNSUEULUDABTU
|
|
AUGUHIUBUC $.
|
|
|
|
$( Equality law for finite cardinality. Theorem X.1.24 of [Rosser] p. 527.
|
|
(Contributed by SF, 20-Jan-2015.) $)
|
|
ncfineleq $p |- ( ( _V e. Fin /\ A e. V /\ B e. W ) ->
|
|
( A e. Nc[fin] B <-> Nc[fin] A = Nc[fin] B ) ) $=
|
|
( cvv cfin wcel w3a cncfin wceq wa cnnc simpl ncfinprop 3adant3 3syl simpld
|
|
3adant2 adantr simprd simpr nnceleq syl22anc ex eleq2 syl5ibcom impbid ) EF
|
|
GZACGZBDGZHZABIZGZAIZULJZUKUMUOUKUMKZUNLGZULLGZAUNGZUMUOUPUKUQUSKZUQUKUMMUH
|
|
UIUTUJACNOZUQUSMPUKURUMUKURBULGZUHUJURVBKUIBDNRQSUKUSUMUKUQUSVATZSUKUMUAAUN
|
|
ULUBUCUDUKUSUOUMVCUNULAUEUFUG $.
|
|
|
|
${
|
|
$d A x t $. $d B x t $.
|
|
eqpwrelk.1 $e |- A e. _V $.
|
|
eqpwrelk.2 $e |- B e. _V $.
|
|
$( Represent equality to power class via a Kuratowski relationship.
|
|
(Contributed by SF, 26-Jan-2015.) $)
|
|
eqpwrelk $p |- ( << { A } , B >> e.
|
|
~ ( ( Ins2_k _S_k (+) Ins3_k SI_k _S_k ) "_k
|
|
~P1 ~P1 1c ) <-> B = ~P A ) $=
|
|
( vx vt csn copk cssetk cpw1 wcel wn cv wb wex wceq wa snex cvv 3bitri
|
|
cins2k csik cins3k csymdif c1c cimak wss ccompl cpw opkex elimak elpw121c
|
|
anbi1i 19.41v bitr4i exbii df-rex excom 3bitr4i opkeq1 ceqsexv otkelins2k
|
|
eleq1d elsymdif vex elssetk bitri otkelins3k opksnelsik opkelssetkg mp2an
|
|
wrex bibi12i notbii elcompl cab wal df-pw eqeq2i abeq2 alex ) AGZBHZIUAZI
|
|
UBZUCZUDZUEJJZUFZKZLEMZBKZWKAUGZNZLZEOZLZWCWIUHKBAUIZPZWJWPWJFMZWCHZWGKZF
|
|
WHVLZWTWKGZGZGZPZXBQZFOZEOZWPFWGWHWCWBBUJZUKWTWHKZXBQZFOXHEOZFOXCXJXMXNFX
|
|
MXGEOZXBQXNXLXOXBEWTULUMXGXBEUNUOUPXBFWHUQXHEFURUSXIWOEXIXFWCHZWGKZXPWDKZ
|
|
XPWFKZNZLWOXBXQFXFXERXGXAXPWGWTXFWCUTVCVAXPWDWFVDXTWNXRWLXSWMXRXDBHIKWLXD
|
|
WBBIWKRZARZDVBWKBEVEZDVFVGXSXDWBHWEKWKAHIKZWMXDWBBWEYAYBDVHWKAIYCCVIWKSKA
|
|
SKYDWMNYCCWKASSVJVKTVMVNTUPTVNWCWIXKVOWSBWMEVPZPWNEVQWQWRYEBEAVRVSWMEBVTW
|
|
NEWATUS $.
|
|
$}
|
|
|
|
${
|
|
$d A x t $. $d B x t $.
|
|
eqpw1relk.1 $e |- A e. _V $.
|
|
eqpw1relk.2 $e |- B e. _V $.
|
|
$( Represent equality to unit power class via a Kuratowski relationship.
|
|
(Contributed by SF, 21-Jan-2015.) $)
|
|
eqpw1relk $p |- ( << A , { B } >> e. ( ( ~P 1c X._k _V ) \
|
|
( ( Ins3_k _S_k (+) Ins2_k SI_k _S_k )
|
|
"_k ~P1 ~P1 ~P1 1c ) ) <-> A = ~P1 B ) $=
|
|
( vx vt csn copk c1c cvv wcel cssetk cpw1 wn wa cv wb snex bitri wex cxpk
|
|
cpw cins3k csik cins2k csymdif cimak wss cdif wceq opkelxpk mpbiran2 elpw
|
|
wal opkex elimak elpw131c anbi1i 19.41v bitr4i exbii df-rex excom 3bitr4i
|
|
wrex opkeq1 ceqsexv elsymdif otkelins3k elssetk otkelins2k opksnelsik vex
|
|
eleq1d bibi12i xchbinx exnal 3bitrri con1bii anbi12i eldif eqpw1 ) ABGZHZ
|
|
IUBZJUAZKZWDLUCZLUDZUEZUFZIMMMZUGZKZNZOAIUHZEPZGZAKZWQBKZQZEUNZOWDWFWMUIK
|
|
ABMUJWGWPWOXBWGAWEKZWPWGXCWCJKBRZAWCWEJCXDUKULAICUMSXBWNWNFPZWRGZGZGZUJZX
|
|
EWDHZWKKZOZFTZETZXANZETXBNWNXKFWLVEZXNFWKWLWDAWCUOUPXEWLKZXKOZFTXLETZFTXP
|
|
XNXRXSFXRXIETZXKOXSXQXTXKEXEUQURXIXKEUSUTVAXKFWLVBXLEFVCVDSXMXOEXMXHWDHZW
|
|
KKZXOXKYBFXHXGRXIXJYAWKXEXHWDVFVNVGYBYAWHKZYAWJKZQXAYAWHWJVHYCWSYDWTYCXFA
|
|
HLKWSXFAWCLWRRZCXDVIWRAWQRZCVJSYDXFWCHWIKZWTXFAWCWIYECXDVKYGWRBHLKWTWRBLY
|
|
FDVLWQBEVMDVJSSVOVPSVAXAEVQVRVSVTWDWFWMWAEABWBVD $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a m $. $d a n $. $d a t $. $d a x $. $d b m $. $d b n $.
|
|
$d b t $. $d b x $. $d m t $. $d n t $. $d n x $. $d t x $.
|
|
|
|
$( Lemma for ~ ncfinraise . Show stratification for induction.
|
|
(Contributed by SF, 22-Jan-2015.) $)
|
|
ncfinraiselem2 $p |- { m | A. a e. m A. b e. m E. n e. Nn
|
|
( ~P1 a e. n /\ ~P1 b e. n ) } e. _V $=
|
|
( vt vx cssetk cpw1 cvv wcel wa wrex snex copk bitr4i exbii df-rex 3bitri
|
|
csn wex csik cins2k cnnc cxpk c1c cpw cins3k csymdif cimak cdif cin cuni1
|
|
ccompl cv wral wn elcompl wceq elimak elpw11c anbi1i 19.41v excom 3bitr4i
|
|
cab wel opkeq1 eleq1d ceqsexv vex opksnelsik elssetk bitri opkex elpw131c
|
|
elin eldif otkelins2k opkelxpk mpbiran2 snelpw1 eqpw1relk anbi12i df-clel
|
|
elpw121c notbii rexnal xchbinx eluni1 dfral2 abbi2i ssetkex sikex ins2kex
|
|
otkelins3k nncex pw1ex vvex xpkex 1cex pwex ins3kex symdifex imakex difex
|
|
inex complex uni1ex eqeltrri ) GUAZXJUBZUCHZHZIUDZUEUFZIUDZGUGZXKUHZUEHZH
|
|
ZHZUIZUJZUAZUBZXQUKZXTUIZUBZYGUGZUKZUKZXSUIZUGZUJZYAUIZUKZXSUIZUMZULZCUNZ
|
|
HZBUNZJZDUNZHZUUBJZKZBUCLZDAUNZUOZCUUIUOZAVEIUUKAYSUUISZYRJZUUJUPZCUUILZU
|
|
PUUIYSJUUKUUMUULYQJZUUOUULYQUUIMZUQUUPEUNZUULNZYPJZEXSLZUURYTSZSZURZUUTKZ
|
|
ETZCTZUUOEYPXSUULUUQUSUURXSJZUUTKZETUVECTZETUVAUVGUVIUVJEUVIUVDCTZUUTKUVJ
|
|
UVHUVKUUTCUURUTVAUVDUUTCVBOPUUTEXSQUVECEVCVDUVGCAVFZUUNKZCTUUOUVFUVMCUVFU
|
|
VCUULNZYPJZUVNXJJZUVNYOJZKUVMUUTUVOEUVCUVBMZUVDUUSUVNYPUURUVCUULVGVHVIUVN
|
|
XJYOVPUVPUVLUVQUUNUVPUVBUUINGJUVLUVBUUIGYTMZAVJZVKYTUUICVJZUVTVLVMUVQUURU
|
|
UDSZSZSZSZURZUURUVNNZYNJZKZETZDTZUUHUPZDUUILZUUNUVQUWHEYALUURYAJZUWHKZETZ
|
|
UWKEYNYAUVNUVCUULVNUSUWHEYAQUWPUWIDTZETUWKUWOUWQEUWOUWFDTZUWHKUWQUWNUWRUW
|
|
HDUURVOVAUWFUWHDVBOPUWIDEVCORUWKDAVFZUWLKZDTUWMUWJUWTDUWJUWEUVNNZYNJZUXAX
|
|
KJZUXAYMJZUPZKUWTUWHUXBEUWEUWDMUWFUWGUXAYNUURUWEUVNVGVHVIUXAXKYMVQUXCUWSU
|
|
XEUWLUXCUWCUULNXJJUWBUUINGJUWSUWCUVCUULXJUWBMZUVRUUQVRUWBUUIGUUDMZUVTVKUU
|
|
DUUIDVJZUVTVLRUXDUUHUXDUWCUVCNZYLJZUURUUBSZSZURZUURUXINZYKJZKZETZBTZUUHUW
|
|
CUVCUULYLUXFUVRUUQWOUXOEXSLZUXPBTZETZUXJUXRUXSUVHUXOKZETUYAUXOEXSQUYBUXTE
|
|
UYBUXMBTZUXOKUXTUVHUYCUXOBUURUTVAUXMUXOBVBOPVMEYKXSUXIUWCUVCVNZUSUXPBEVCV
|
|
DUXRUUBUCJZUUGKZBTUUHUXQUYFBUXQUXLUXINZYKJZUYGXNJZUYGYJJZKUYFUXOUYHEUXLUX
|
|
KMZUXMUXNUYGYKUURUXLUXIVGVHVIUYGXNYJVPUYIUYEUYJUUGUYIUXLXMJZUXKXLJUYEUYIU
|
|
YLUXIIJUYDUXLUXIXMIUYKUYDVSVTUXKXLWAUUBUCWARUYJUYGYHJZUYGYIJZKUUGUYGYHYIV
|
|
PUYMUUCUYNUUFUYMUUBUVCNZYGJZUURFUNZSZSZSZURZUURUYONZYFJZKZETZFTZUUCUUBUWC
|
|
UVCYGBVJZUXFUVRVRUYPVUCEXTLUURXTJZVUCKZETZVUFEYFXTUYOUUBUVCVNUSVUCEXTQVUJ
|
|
VUDFTZETVUFVUIVUKEVUIVUAFTZVUCKVUKVUHVULVUCFUURWEZVAVUAVUCFVBOPVUDFEVCORV
|
|
UFUYQUUAURZFBVFZKZFTUUCVUEVUPFVUEUYTUYONZYFJZVUQYEJZVUQXQJZKVUPVUCVUREUYT
|
|
UYSMZVUAVUBVUQYFUURUYTUYOVGVHVIVUQYEXQVPVUSVUNVUTVUOVUSUYRUVCNYDJUYQUVBNY
|
|
CJVUNUYRUUBUVCYDUYQMZVUGUVRVRUYQUVBYCFVJZUVSVKUYQYTVVCUWAWBRVUTUYRUUBNGJZ
|
|
VUOUYRUUBUVCGVVBVUGUVRWOUYQUUBVVCVUGVLZVMWCRPFUUAUUBWDORUUBUWCNZYGJZUYQUU
|
|
EURZVUOKZFTZUYNUUFVVGUURVVFNZYFJZEXTLZVUAVVLKZETZFTZVVJEYFXTVVFUUBUWCVNUS
|
|
VUHVVLKZETVVNFTZETVVMVVPVVQVVREVVQVULVVLKVVRVUHVULVVLVUMVAVUAVVLFVBOPVVLE
|
|
XTQVVNFEVCVDVVOVVIFVVOUYTVVFNZYFJZVVSYEJZVVSXQJZKVVIVVLVVTEUYTVVAVUAVVKVV
|
|
SYFUURUYTVVFVGVHVIVVSYEXQVPVWAVVHVWBVUOVWAUYRUWCNYDJUYQUWBNYCJVVHUYRUUBUW
|
|
CYDVVBVUGUXFVRUYQUWBYCVVCUXGVKUYQUUDVVCUXHWBRVWBVVDVUOUYRUUBUWCGVVBVUGUXF
|
|
WOVVEVMWCRPRUUBUWCUVCYGVUGUXFUVRWOFUUEUUBWDVDWCVMWCRPUUGBUCQORWFWCRPUWLDU
|
|
UIQOUUHDUUIWGRWCRPUUNCUUIQORWHUUIYRUVTWIUUJCUUIWJVDWKYRYQYPXSXJYOGWLWMZYN
|
|
YAXKYMXJVWCWNZYLYKXSXNYJXMIXLUCWPWQWQWRWSYHYIYGYFXTYEXQYDYCXPYBXOIUEWTXAW
|
|
RWSXRYAXQXKGWLXBZVWDXCXTXSUEWTWQZWQZWQZXDXEWMWNVWEXFVWGXDZWNYGVWIXBXFXFVW
|
|
FXDXBXEVWHXDXFVWFXDXGXHXI $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a m $. $d a n $. $d A n $. $d B b $.
|
|
$d b m $. $d b n $. $d B n $. $d M a $. $d M b $. $d a c $. $d a d $.
|
|
$d a k $. $d a x $. $d a y $. $d b c $. $d b d $. $d b k $. $d b x $.
|
|
$d b y $. $d c d $. $d c k $. $d c x $. $d c y $. $d d k $. $d d x $.
|
|
$d d y $. $d k m $. $d k n $. $d M m $. $d m n $. $d x y $. $d c n $.
|
|
$d d n $. $d k x $. $d k y $. $d n x $. $d n y $. $d c m $. $d d m $.
|
|
$d m x $. $d m y $.
|
|
|
|
$( If two sets are in a particular finite cardinal, then their unit power
|
|
sets are in the same natural. Theorem X.1.25 of [Rosser] p. 527.
|
|
(Contributed by SF, 21-Jan-2015.) $)
|
|
ncfinraise $p |- ( ( M e. Nn /\ A e. M /\ B e. M ) ->
|
|
E. n e. Nn ( ~P1 A e. n /\ ~P1 B e. n ) ) $=
|
|
( va vb vc vx vd vy cnnc wcel cpw1 cv wa wrex wral c0c c0 wceq vm vk cplc
|
|
ncfinraiselem2 raleq raleqbi1dv el0c peano1 nulel0c pm3.2i anbi12d rspcev
|
|
c1c eleq2 mp2an pw10 syl6eq eleq1d bi2anan9 rexbidv mpbiri syl2anb rgen2a
|
|
pw1eq nfv nfra1 nfan nfra2 csn cun ccompl reeanv 2rexbii anbi12i 3bitr4ri
|
|
elsuc wel wi weq anbi1d anbi2d rspc2v vex elcompl anbi2i peano2 ad3antrrr
|
|
com12 simplrl adantr simprrl snelpw1 sylnibr snex syl2anc simplrr simprrr
|
|
wn elsuci syl12anc rexlimiva cbvrexv syl8ib imp31 pw1un uneq2i syl5ibrcom
|
|
ex pw1sn eqtri sylan2b expr anasss rexlimdvv exp32 sylan9r pm2.43d syl5bi
|
|
exp3a ralrimd ralrimi finds syl5com 3impib ) DKLZADLZBDLZAMZCNZLZBMZYILZO
|
|
ZCKPZYEENZMZYILZFNZMZYILZOZCKPZFDQZEDQZYFYGOYNUUBFUANZQZEUUEQUUBFRQZERQUU
|
|
BFUBNZQZEUUHQZUUBFUUHUMUCZQZEUUKQZUUDUAUBDUACEFUDUUFUUGEUUERUUBFUUERUEUFU
|
|
UFUUIEUUEUUHUUBFUUEUUHUEUFUUFUULEUUEUUKUUBFUUEUUKUEUFUUFUUCEUUEDUUBFUUEDU
|
|
EUFUUBEFRYORLYOSTZYRSTZUUBYRRLYOUGYRUGUUNUUOOZUUBSYILZUUQOZCKPZRKLSRLZUUT
|
|
OZUUSUHUUTUUTUIUIUJUURUVACRKYIRTUUQUUTUUQUUTYIRSUNZUVBUKULUOUUPUUAUURCKUU
|
|
NYQUUQUUOYTUUQUUNYPSYIUUNYPSMZSYOSVDUPUQURUUOYSSYIUUOYSUVCSYRSVDUPUQURUSU
|
|
TVAVBVCUUHKLZUUJUUMUVDUUJOZUULEUUKUVDUUJEUVDEVEUUIEUUHVFVGUVEYOUUKLZUUBFU
|
|
UKUVDUUJFUVDFVEUUBEFUUHUUHVHVGUVFFVEUVEUVFYRUUKLZUUBUVFUVGOZYOGNZHNZVIZVJ
|
|
ZTZYRINZJNZVIZVJZTZOZJUVNVKZPHUVIVKZPZIUUHPGUUHPZUVEUUBUVMHUWAPZUVRJUVTPZ
|
|
OZIUUHPGUUHPUWDGUUHPZUWEIUUHPZOUWCUVHUWDUWEGIUUHUUHVLUWBUWFGIUUHUUHUVMUVR
|
|
HJUWAUVTVLVMUVFUWGUVGUWHHYOUUHGVPJYRUUHIVPVNVOUVEUWBUUBGIUUHUUHUVEGUBVQIU
|
|
BVQOZUWBUUBVRZUUJUWIUVIMZYILZUVNMZYILZOZCKPZUVDUWIUWJVRUWIUUJUWPUUBUWPUWL
|
|
YTOZCKPEFUVIUVNUUHUUHEGVSZUUAUWQCKUWRYQUWLYTUWRYPUWKYIYOUVIVDURVTUTFIVSZU
|
|
WQUWOCKUWSYTUWNUWLUWSYSUWMYIYRUVNVDURWAUTWBWHUVDUWPUWIUWJUVDUWPUWIOOUVSUU
|
|
BHJUWAUVTUVDUWPUWIUVJUWALZUVOUVTLZOZUVSUUBVRZVRUVDUWPOZUWIUXBUXCUWIUXBOUX
|
|
DUWIHGVQZWRZJIVQZWRZOZOZUXCUXBUXIUWIUWTUXFUXAUXHUVJUVIHWCZWDUVOUVNJWCZWDV
|
|
NWEUXDUXJOUUBUVSUWKUVKVIZVJZYILZUWMUVPVIZVJZYILZOZCKPZUVDUWPUXJUXTUWPUVDU
|
|
XJUXTVRUWPUVDUXJUXNUUELZUXQUUELZOZUAKPZUXTUWOUVDUXJUYDVRZVRCKYIKLZUWOOZUV
|
|
DUYEUYGUVDOZUXJUYDUYHUXJOZYIUMUCZKLZUXNUYJLZUXQUYJLZUYDUYFUYKUWOUVDUXJYIW
|
|
FWGUYIUWLUVKUWKLZWRUYLUYHUWLUXJUYFUWLUWNUVDWIWJUYIUXEUYNUYHUWIUXFUXHWKUVJ
|
|
UVIWLWMUWKYIUVKUVJWNWSWOUYIUWNUVPUWMLZWRUYMUYHUWNUXJUYFUWLUWNUVDWPWJUYIUX
|
|
GUYOUYHUWIUXFUXHWQUVOUVNWLWMUWMYIUVPUVOWNWSWOUYCUYLUYMOUAUYJKUUEUYJTUYAUY
|
|
LUYBUYMUUEUYJUXNUNUUEUYJUXQUNUKULWTXHXHXAUYCUXSUACKUACVSUYAUXOUYBUXRUUEYI
|
|
UXNUNUUEYIUXQUNUKXBXCWHXDUVSUUAUXSCKUVMYQUXOUVRYTUXRUVMYPUXNYIUVMYPUVLMZU
|
|
XNYOUVLVDUYPUWKUVKMZVJUXNUVIUVKXEUYQUXMUWKUVJUXKXIXFXJUQURUVRYSUXQYIUVRYS
|
|
UVQMZUXQYRUVQVDUYRUWMUVPMZVJUXQUVNUVPXEUYSUXPUWMUVOUXLXIXFXJUQURUSUTXGXKX
|
|
LXMXNXOXPXQXNXRXSXTYAXHYBUUBYNYJYTOZCKPEFABDDYOATZUUAUYTCKVUAYQYJYTVUAYPY
|
|
HYIYOAVDURVTUTYRBTZUYTYMCKVUBYTYLYJVUBYSYKYIYRBVDURWAUTWBYCYD $.
|
|
$}
|
|
|
|
${
|
|
$d m n a b t x $.
|
|
$( Lemma for ~ ncfinlower . Set up stratification for induction.
|
|
(Contributed by SF, 22-Jan-2015.) $)
|
|
ncfinlowerlem1 $p |- { m | A. a A. b ( ( ~P1 a e. m /\ ~P1 b
|
|
e. m ) -> E. n e. Nn ( a e. n /\ b e. n ) ) } e. _V $=
|
|
( vt vx cvv cssetk cpw1 cnnc wcel wa wrex wex copk csn exbii 3bitr4i snex
|
|
3bitri c1c cpw cxpk cins3k csik cins2k csymdif cimak cdif ccnvk ccompl cv
|
|
cin wel wi wal cab wn wceq vex elimak elpw11c anbi1i 19.41v bitr4i df-rex
|
|
excom opkeq1 eleq1d ceqsexv opkex elpw131c eldif elin elpw121c otkelins3k
|
|
opksnelsik eqpw1relk elssetk bitri anbi12i mpbiran df-clel elpw12 r19.41v
|
|
opkelxpk rexcom4 opkelcnvk rexbii notbii exanali elcompl alex abbi2i vvex
|
|
otkelins2k 1cex xpkex ssetkex ins3kex sikex ins2kex symdifex pw1ex imakex
|
|
pwex difex inex cnvkex nncex complex eqeltrri ) GUAUBZGUCZHUDZHUEZUFZUGZU
|
|
AIZIZIZUHZUIZUEZUDZHUFZUMZXTUHZUCZYHUFZUMZHUJZUFZYLUDZUMZJIZIZUHZUEZUDZUI
|
|
ZYAUHZXSUHZUKZCULZIZAULZKZDULZIZUUGKZLZCBUNZDBUNZLZBJMZUODUPZCUPZAUQGUURA
|
|
UUDUUGUUCKZURUUQURZCNZURUUGUUDKUURUUSUVAUUSEULZUUGOZUUBKZEXSMZUVBUUEPZPZU
|
|
SZUVDLZENZCNZUVAEUUBXSUUGAUTZVAUVBXSKZUVDLZENUVICNZENUVEUVKUVNUVOEUVNUVHC
|
|
NZUVDLUVOUVMUVPUVDCUVBVBVCUVHUVDCVDVEQUVDEXSVFUVICEVGRUVJUUTCUVJUVGUUGOZU
|
|
UBKZUULUUPURZLZDNZUUTUVDUVREUVGUVFSZUVHUVCUVQUUBUVBUVGUUGVHVIVJUVRUVBUVQO
|
|
ZUUAKZEYAMZUVBUUIPZPZPZPZUSZUWDLZENZDNZUWAEUUAYAUVQUVGUUGVKZVAUVBYAKZUWDL
|
|
ZENUWKDNZENUWEUWMUWPUWQEUWPUWJDNZUWDLUWQUWOUWRUWDDUVBVLVCUWJUWDDVDVEQUWDE
|
|
YAVFUWKDEVGRUWLUVTDUWLUWIUVQOZUUAKZUWSYKKZUWSYTKZURZLUVTUWDUWTEUWIUWHSZUW
|
|
JUWCUWSUUAUVBUWIUVQVHVIVJUWSYKYTVMUXAUULUXCUVSUXAUWSYIKZUWSYJKZLUULUWSYIY
|
|
JVNUXEUUHUXFUUKUVQYHKZFULZUUFUSZFAUNZLZFNZUXEUUHUXGUWCYGKZEXTMZUVBUXHPZPZ
|
|
PZUSZUXMLZENZFNZUXLEYGXTUVQUWNVAUVBXTKZUXMLZENUXSFNZENUXNUYAUYCUYDEUYCUXR
|
|
FNZUXMLUYDUYBUYEUXMFUVBVOZVCUXRUXMFVDVEQUXMEXTVFUXSFEVGRUXTUXKFUXTUXQUVQO
|
|
ZYGKZUYGYEKZUYGYFKZLUXKUXMUYHEUXQUXPSZUXRUWCUYGYGUVBUXQUVQVHVIVJUYGYEYFVN
|
|
UYIUXIUYJUXJUYIUXOUVGOYDKUXHUVFOYCKUXIUXOUVGUUGYDUXHSZUWBUVLVPUXHUVFYCFUT
|
|
ZUUESZVQUXHUUEUYMCUTZVRTUYJUXOUUGOHKZUXJUXOUVGUUGHUYLUWBUVLWPUXHUUGUYMUVL
|
|
VSZVTWATQTUXEUWIGKUXGUXDUWIUVQGYHUXDUWNWFWBFUUFUUGWCRUWGUUGOZYHKZUXHUUJUS
|
|
ZUXJLZFNZUXFUUKUYSUVBUYROZYGKZEXTMZUXRVUDLZENZFNZVUBEYGXTUYRUWGUUGVKVAUYB
|
|
VUDLZENVUFFNZENVUEVUHVUIVUJEVUIUYEVUDLVUJUYBUYEVUDUYFVCUXRVUDFVDVEQVUDEXT
|
|
VFVUFFEVGRVUGVUAFVUGUXQUYROZYGKZVUKYEKZVUKYFKZLVUAVUDVULEUXQUYKUXRVUCVUKY
|
|
GUVBUXQUYRVHVIVJVUKYEYFVNVUMUYTVUNUXJVUMUXOUWGOYDKUXHUWFOYCKUYTUXOUWGUUGY
|
|
DUYLUWFSZUVLVPUXHUWFYCUYMUUISZVQUXHUUIUYMDUTZVRTVUNUYPUXJUXOUWGUUGHUYLVUO
|
|
UVLWPUYQVTWATQTUWGUVGUUGYHVUOUWBUVLWPFUUJUUGWCRWAVTUXBUUPUXBUWGUVGOYSKUWF
|
|
UVFOZYRKZUUPUWGUVGUUGYSVUOUWBUVLVPUWFUVFYRVUPUYNVQVUSUVBVUROZYOKZEYQMZUVB
|
|
BULZPZPZUSZVVALZENZBJMZUUPEYOYQVURUWFUVFVKVAUVBYQKZVVALZENVVGBJMZENVVBVVI
|
|
VVKVVLEVVKVVFBJMZVVALVVLVVJVVMVVABUVBJWDVCVVFVVABJWEVEQVVAEYQVFVVGBEJWGRV
|
|
VHUUOBJVVHVVEVUROZYOKZVVNYMKZVVNYNKZLUUOVVAVVOEVVEVVDSVVFVUTVVNYOUVBVVEVU
|
|
RVHVIVJVVNYMYNVNVVPUUMVVQUUNVVPVVCUVFOYLKUVFVVCOHKUUMVVCUWFUVFYLBUTZVUPUY
|
|
NWPVVCUVFHVVRUYNWHUUEVVCUYOVVRVSTVVQVVCUWFOYLKUWFVVCOHKUUNVVCUWFUVFYLVVRV
|
|
UPUYNVPVVCUWFHVVRVUPWHUUIVVCVUQVVRVSTWATWITTWJWATQTUULUUPDWKTQTWJUUGUUCUV
|
|
LWLUUQCWMRWNUUCUUBXSUUAYAYKYTYIYJGYHWOYGXTYEYFYDYCXNYBXMGUAWQXFWOWRXRYAXO
|
|
XQHWSWTXPHWSXAXBXCXTXSUAWQXDZXDZXDZXEXGXAWTHWSXBXHVVTXEZWRYHVWBXBXHYSYRYO
|
|
YQYMYNYLHWSXIZXBYLVWCWTXHYPJXJXDXDXEXAWTXGVWAXEVVSXEXKXL $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a m $. $d a n $. $d A n $. $d B a $.
|
|
$d B b $. $d b m $. $d b n $. $d B n $. $d M a $. $d M b $. $d a k $.
|
|
$d b k $. $d k m $. $d k n $. $d M m $. $d m n $. $d a c $. $d a d $.
|
|
$d a x $. $d a y $. $d b c $. $d b d $. $d b x $. $d b y $. $d c d $.
|
|
$d c k $. $d c x $. $d c y $. $d d k $. $d d x $. $d d y $. $d x y $.
|
|
$d a e $. $d a f $. $d a w $. $d a z $. $d b e $. $d b f $. $d b w $.
|
|
$d b z $. $d c e $. $d c f $. $d c n $. $d c w $. $d c z $. $d d e $.
|
|
$d d f $. $d d n $. $d d w $. $d d z $. $d e f $. $d e w $. $d e x $.
|
|
$d e y $. $d e z $. $d f w $. $d f x $. $d f y $. $d f z $. $d k x $.
|
|
$d k y $. $d n x $. $d n y $. $d w x $. $d w y $. $d w z $. $d x z $.
|
|
$d y z $. $d e k $. $d e m $. $d e n $. $d f k $. $d f m $. $d f n $.
|
|
$d k w $. $d k z $. $d m w $. $d m z $. $d n w $. $d n z $. $d c m $.
|
|
$d d m $. $d m x $. $d m y $.
|
|
$( If the unit power classes of two sets are in the same natural, then so
|
|
are the sets themselves. Theorem X.1.26 of [Rosser] p. 527.
|
|
(Contributed by SF, 22-Jan-2015.) $)
|
|
ncfinlower $p |- ( ( M e. Nn /\ ~P1 A e. M /\ ~P1 B e. M ) ->
|
|
E. n e. Nn ( A e. n /\ B e. n ) ) $=
|
|
( va vb vm vc vd ve vf cnnc wcel cv wa wrex wel wi wceq eleq2 vk vx vy vz
|
|
vw cpw1 wal c0c c1c cplc ncfinlowerlem1 anbi12d imbi1d 2albidv el0c pw10b
|
|
weq bitri peano1 nulel0c rspcev mp2an eleq1 bi2anan9 anidm syl6bb rexbidv
|
|
c0 mpbiri syl2anb gen2 nfv nfa1 nfan nfa2 csn ccompl reeanv 2rexbii elsuc
|
|
cun anbi12i 3bitr4ri w3a wex wn cvv vex pw1eq eleq1d elequ1 imbi12d com12
|
|
spc2gv ad2antrl peano2 simprrl adantr syl2anc simprrr expr rexlimdva syld
|
|
elsuci syl12anc imp an32s wb 3ad2ant2 compleq eleq12 snex elcompl snelpw1
|
|
sylan2 xchbinx ancoms 3adant1 anbi2d 3ad2ant1 eeanv 2exbii elequ2 cbvrexv
|
|
exlimdvv pw1eqadj 3imtr4g rexlimdvv rexlimdvva syl5bi alrimi finds pw1exb
|
|
ex elex sylib syl2an pm2.43b syl 3impib ) DLMZAUFZDMZBUFZDMZACNZMZBUUFMZO
|
|
ZCLPZUUAENZUFZDMZFNZUFZDMZOZECQZFCQZOZCLPZRZFUGEUGZUUCUUEOZUUJRZUULGNZMZU
|
|
UOUVFMZOZUVARZFUGEUGUULUHMZUUOUHMZOZUVARZFUGEUGUULUANZMZUUOUVOMZOZUVARZFU
|
|
GZEUGZUULUVOUIUJZMZUUOUWBMZOZUVARZFUGZEUGZUVCGUADGCEFUKUVFUHSZUVJUVNEFUWI
|
|
UVIUVMUVAUWIUVGUVKUVHUVLUVFUHUULTUVFUHUUOTULUMUNGUAUQZUVJUVSEFUWJUVIUVRUV
|
|
AUWJUVGUVPUVHUVQUVFUVOUULTUVFUVOUUOTULUMUNUVFUWBSZUVJUWFEFUWKUVIUWEUVAUWK
|
|
UVGUWCUVHUWDUVFUWBUULTUVFUWBUUOTULUMUNUVFDSZUVJUVBEFUWLUVIUUQUVAUWLUVGUUM
|
|
UVHUUPUVFDUULTUVFDUUOTULUMUNUVNEFUVKUUKVHSZUUNVHSZUVAUVLUVKUULVHSUWMUULUO
|
|
UUKUPURUVLUUOVHSUWNUUOUOUUNUPURUWMUWNOZUVAVHUUFMZCLPZUHLMVHUHMZUWQUSUTUWP
|
|
UWRCUHLUUFUHVHTVAVBUWOUUTUWPCLUWOUUTUWPUWPOUWPUWMUURUWPUWNUUSUWPUUKVHUUFV
|
|
CUUNVHUUFVCVDUWPVEVFVGVIVJVKUVOLMZUWAUWHUWSUWAOZUWGEUWSUWAEUWSEVLUVTEVMVN
|
|
UWTUWFFUWSUWAFUWSFVLUVSFEVOVNUWEUULHNZUBNZVPWASZUUOINZUCNZVPWASZOZUCUXDVQ
|
|
ZPUBUXAVQZPZIUVOPHUVOPZUWTUVAUXCUBUXIPZUXFUCUXHPZOZIUVOPHUVOPUXLHUVOPZUXM
|
|
IUVOPZOUXKUWEUXLUXMHIUVOUVOVRUXJUXNHIUVOUVOUXCUXFUBUCUXIUXHVRVSUWCUXOUWDU
|
|
XPUBUULUVOHVTUCUUOUVOIVTWBWCUWTUXJUVAHIUVOUVOUWTHUAQZIUAQZOZOUXGUVAUBUCUX
|
|
IUXHUWTUXSUXBUXIMZUXEUXHMZOZUXGUVARUWTUXSUYBOZOZUUKJNZUDNZVPZWAZSZUXAUYEU
|
|
FZSZUXBUYGSZWDZUUNKNZUENZVPZWAZSZUXDUYNUFZSZUXEUYPSZWDZOZUEWEUDWEZKWEJWEZ
|
|
EGQZFGQZOZGLPZUXGUVAUYDVUDVUIJKUYDVUCVUIUDUEVUCUYDVUIVUCUYDVUIRUWTUYJUVOM
|
|
ZUYSUVOMZOZUDJQZWFZUEKQZWFZOZOZOZUYHUVFMZUYQUVFMZOZGLPZRUWSVURUWAVVCUWSVU
|
|
ROZUWAVVCVVDUWAJCQZKCQZOZCLPZVVCVULUWAVVHRUWSVUQUWAVULVVHUYEWGMUYNWGMUWAV
|
|
ULVVHRZRJWHKWHUVSVVIEFUYEUYNWGWGEJUQZFKUQZOZUVRVULUVAVVHVVJUVPVUJVVKUVQVU
|
|
KVVJUULUYJUVOUUKUYEWIWJVVKUUOUYSUVOUUNUYNWIWJVDVVLUUTVVGCLVVJUURVVEVVKUUS
|
|
VVFEJCWKFKCWKVDVGWLWNVBWMWOVVDVVGVVCCLVVDUUFLMZVVGVVCVVDVVMVVGOZOZUUFUIUJ
|
|
ZLMZUYHVVPMZUYQVVPMZVVCVVMVVQVVDVVGUUFWPWOVVOVVEVUNVVRVVDVVMVVEVVFWQVVDVU
|
|
NVVNUWSVULVUNVUPWQWRUYEUUFUYFUDWHXDWSVVOVVFVUPVVSVVDVVMVVEVVFWTVVDVUPVVNU
|
|
WSVULVUNVUPWTWRUYNUUFUYOUEWHXDWSVVBVVRVVSOGVVPLUVFVVPSVUTVVRVVAVVSUVFVVPU
|
|
YHTUVFVVPUYQTULVAXEXAXBXCXFXGVUCUYDVUSVUIVVCVUCUYCVURUWTVUCUXSVULUYBVUQUY
|
|
MUXQVUJVUBUXRVUKUYKUYIUXQVUJXHUYLUXAUYJUVOVCXIUYTUYRUXRVUKXHVUAUXDUYSUVOV
|
|
CXIVDUYMUXTVUNVUBUYAVUPUYKUYLUXTVUNXHZUYIUYLUYKVVTUYLUYKOUXTUYGUYJVQZMZVU
|
|
NUYKUYLUXIVWASUXTVWBXHUXAUYJXJUXBUYGUXIVWAXKXOVWBUYGUYJMVUMUYGUYJUYFXLXMU
|
|
YFUYEXNXPVFXQXRUYTVUAUYAVUPXHZUYRVUAUYTVWCVUAUYTOUYAUYPUYSVQZMZVUPUYTVUAU
|
|
XHVWDSUYAVWEXHUXDUYSXJUXEUYPUXHVWDXKXOVWEUYPUYSMVUOUYPUYSUYOXLXMUYOUYNXNX
|
|
PVFXQXRVDULXSVUCVUHVVBGLUYMVUFVUTVUBVUGVVAUYIUYKVUFVUTXHUYLUUKUYHUVFVCXTU
|
|
YRUYTVUGVVAXHVUAUUNUYQUVFVCXTVDVGWLVIWMYEYEUYMUDWEZVUBUEWEZOZKWEJWEVWFJWE
|
|
ZVWGKWEZOVUEUXGVWFVWGJKYAVUDVWHJKUYMVUBUDUEYAYBUXCVWIUXFVWJJUDUXAUXBUUKHW
|
|
HUBWHYFKUEUXDUXEUUNIWHUCWHYFWBWCUUTVUHCGLCGUQUURVUFUUSVUGCGEYCCGFYCULYDYG
|
|
XAYHYIYJYKYKYNYLUVCUVDUUJUUCAWGMZBWGMZUVCUVERUUEUUCUUBWGMVWKUUBDYOAYMYPUU
|
|
EUUDWGMVWLUUDDYOBYMYPUVBUVEEFABWGWGUUKASZUUNBSZOZUUQUVDUVAUUJVWMUUMUUCVWN
|
|
UUPUUEVWMUULUUBDUUKAWIWJVWNUUOUUDDUUNBWIWJVDVWOUUTUUICLVWMUURUUGVWNUUSUUH
|
|
UUKAUUFVCUUNBUUFVCVDVGWLWNYQYRYSYT $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a n $. $d a p $. $d b n $. $d b p $. $d M a $. $d M b $.
|
|
$d M n $. $d M p $. $d n p $. $d a q $. $d b q $. $d M q $. $d n q $.
|
|
$d p q $.
|
|
$( For any non-empty finite cardinal, there is a unique natural containing
|
|
a unit power class of one of its elements. Theorem X.1.27 of [Rosser]
|
|
p. 528. (Contributed by SF, 22-Jan-2015.) $)
|
|
nnpw1ex $p |- ( ( M e. Nn /\ M =/= (/) ) ->
|
|
E! n e. Nn E. a e. M ~P1 a e. n ) $=
|
|
( vp vb vq cnnc wcel cpw1 wrex weq wral wex w3a ncfinraise anbi2i nnceleq
|
|
wa cv syl22anc c0 wne wi wreu anidm rexbii sylib 3anidm23 ex ancld eximdv
|
|
n0 rexcom df-rex bitri 3imtr4i pw1eq eleq1d cbvrexv reeanv bitr4i simplll
|
|
imp simprll simprlr syl3anc simp1rl simp3l simp2rl simp3rl simp1rr eqtr4d
|
|
simp2rr simp3rr 3expa exp32 mpd rexlimdvv syl5bi ralrimivva eleq2 rexbidv
|
|
rexlimdv reu4 sylanbrc ) BGHZBUAUBZRZCSZIZASZHZCBJZAGJZWMWJDSZHZCBJZRZADK
|
|
ZUCZDGLAGLWMAGUDWFWIBHZCMZRXAWLAGJZRZCMZWHWNWFXBXEWFXAXDCWFXAXCWFXAXCWFXA
|
|
XCWFXAXANWLWLRZAGJXCWIWIABOXFWLAGWLUEUFUGUHUIUJUKVCWGXBWFCBULPWNXCCBJXEWL
|
|
ACGBUMXCCBUNUOUPWHWTADGGWRWLESZIZWOHZRZEBJCBJZWHWKGHZWOGHZRZRZWSWRWMXIEBJ
|
|
ZRXKWQXPWMWPXICEBCEKWJXHWOWIXGUQURUSPWLXICEBBUTVAXOXJWSCEBBXOXAXGBHZRZXJW
|
|
SXOXRXJRZRZWJFSZHZXHYAHZRZFGJZWSXTWFXAXQYEWFWGXNXSVBXOXAXQXJVDXOXAXQXJVEW
|
|
IXGFBOVFXTYDWSFGXTYAGHZYDWSXOXSYFYDRZWSXOXSYGNZWKYAWOYHXLYFWLYBAFKXLXMWHX
|
|
SYGVGXOXSYFYDVHZWLXIXRXOYGVIYBYCYFXOXSVJWJWKYAQTYHXMYFXIYCDFKXLXMWHXSYGVK
|
|
YIWLXIXRXOYGVMYBYCYFXOXSVNXHWOYAQTVLVOVPWCVQVPVRVSVTWMWQADGWSWLWPCBWKWOWJ
|
|
WAWBWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The finite T operator is always a set. (Contributed by SF,
|
|
26-Jan-2015.) $)
|
|
tfinex $p |- _T[fin] A e. _V $=
|
|
( vx vy ctfin c0 wceq cv cnnc wcel cpw1 wa cio cif cvv df-tfin 0ex iotaex
|
|
wrex ifex eqeltri ) ADAEFZEBGZHICGJUBICARKZBLZMNBACOUAEUDPUCBQST $.
|
|
$}
|
|
|
|
${
|
|
$d M n $. $d M t $. $d M y $. $d M z $. $d n t $. $d n x $. $d n y $.
|
|
$d n z $. $d t x $. $d t y $. $d t z $. $d X t $. $d x y $. $d X z $.
|
|
$d y z $. $d a n $. $d a t $. $d a x $. $d a y $. $d a z $. $d M a $.
|
|
eqtfinrelk.1 $e |- M e. _V $.
|
|
eqtfinrelk.2 $e |- X e. _V $.
|
|
$( Equality to a T raising expressed via a Kuratowski relationship.
|
|
(Contributed by SF, 29-Jan-2015.) $)
|
|
eqtfinrelk $p |- ( << { M } , X >> e. ( ( { { (/) } } X._k { (/) } ) u.
|
|
( ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k
|
|
( ( Ins3_k `'_k _S_k \
|
|
Ins2_k
|
|
( ( Ins2_k ( ( Nn X._k _V ) i^i
|
|
( ( Ins2_k SI_k _S_k i^i
|
|
Ins3_k
|
|
( ( Ins3_k SI_k ( ( ~P 1c X._k _V ) \
|
|
( ( Ins3_k _S_k (+) Ins2_k SI_k _S_k )
|
|
"_k ~P1 ~P1 ~P1 1c ) ) i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 ~P1 1c ) ) (+)
|
|
Ins3_k _I_k ) "_k ~P1 1c ) ) "_k ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c )
|
|
\ ( { { (/) } } X._k _V ) ) ) <-> X = _T[fin] M ) $=
|
|
( vy va vz vt vn vx csn copk c0 cssetk cvv wcel wceq wa snex wex cxpk c1c
|
|
cins2k ccnvk cins3k cnnc csik cpw csymdif cpw1 cimak cdif cin cidk ccompl
|
|
cun cv wrex cio ctfin wb wo snid opkelxpk mpbiran elsnc bitri orbi1i elun
|
|
cif wn mpbir2an notnoti intnan eldif mtbir biorfi 3bitr4i a1i sneq eleq1d
|
|
opkeq1d iftrue eqeq2d 3bitr4d iffalse opkex elimak elpw121c anbi1i 19.41v
|
|
bitr4i exbii df-rex excom ceqsexv elsymdif otkelins2k vex elssetk wel weq
|
|
opkeq1 wal otkelins3k opkelcnvk 3bitri elin mpbiran2 opksnelsik eqpw1relk
|
|
elpw11c anbi12i df-clel opkelidkg mp2an bibi12i xchbinx exnal 3bitrri cab
|
|
elpw131c con1bii cuni dfiota2 eleq2i eluniab notbii elcompl biantru sneqb
|
|
dfcleq alex 3bitr2i biimpi con3i biantrud simpl biorf syl6bbr syl orbi12i
|
|
bitrd 3bitr2rd pm2.61i df-tfin eqeq2i ) AKZBLZMKZKZUUJUAZNUCZNUDZUEZUFOUA
|
|
ZNUGZUCZUBUHOUANUEUURUIUBUJZUJZUJZUKULZUGZUEZUUMUMZUUTUKZUEZUMZUVAUKZUMZU
|
|
CZUNUEZUIZUUSUKZUCZULZUUSUKZUEZUIZUUTUKZUOZUUKOUAZULZUPZPZBAMQZMEUQZUFPZF
|
|
UQZUJZUWGPZFAURZRZEUSZVJZQZBAUTZQUWFUWEUWPVAUWFUUJBLZUWDPZBMQZUWEUWPUWSUW
|
|
TVAUWFUWRUULPZUWRUWCPZVBUWTUXBVBUWSUWTUXAUWTUXBUXABUUJPZUWTUXAUUJUUKPZUXC
|
|
UUJMSZVCZUUJBUUKUUJUXEDVDVEBMDVFZVGVHUWRUULUWCVIUXBUWTUXBUWRUWAPZUWRUWBPZ
|
|
VKZRUXJUXHUXIUXIUXDBOPZUXFDUUJBUUKOUXEDVDVLVMVNUWRUWAUWBVOVPVQVRVSUWFUUIU
|
|
WRUWDUWFUUHUUJBAMVTWBWAUWFUWOMBUWFMUWNWCWDWEUWFVKZUWPBUWNQZUUIUWAPZUWEUXL
|
|
UWOUWNBUWFMUWNWFWDUXNUXMVAUXLUUIUVTPZVKGUQZBPZUXPUWNPZVAZVKZGTZVKZUXNUXMU
|
|
XOUYAUXOHUQZUUILZUVSPZHUUTURZUYCUXPKZKZKZQZUYERZHTZGTZUYAHUVSUUTUUIUUHBWG
|
|
ZWHUYCUUTPZUYERZHTUYKGTZHTUYFUYMUYPUYQHUYPUYJGTZUYERUYQUYOUYRUYEGUYCWIWJU
|
|
YJUYEGWKWLWMUYEHUUTWNUYKGHWOVRUYLUXTGUYLUYIUUILZUVSPZUXTUYEUYTHUYIUYHSUYJ
|
|
UYDUYSUVSUYCUYIUUIXCWAWPUYTUYSUUMPZUYSUVRPZVAUXSUYSUUMUVRWQVUAUXQVUBUXRVU
|
|
AUYGBLNPUXQUYGUUHBNUXPSZASZDWRUXPBGWSZDWTVGUYCIUQZKZKZQZUYCUYGUUHLZLZUVPP
|
|
ZRZHTZITZGIXAZUWMEIXBZVAZEXDZRZITZVUBUXRVUNVUTIVUNVUHVUJLZUVPPZVVBUUOPZVV
|
|
BUVOPZVKZRVUTVULVVCHVUHVUGSVUIVUKVVBUVPUYCVUHVUJXCWAWPVVBUUOUVOVOVVDVUPVV
|
|
FVUSVVDVUFUYGLUUNPUYGVUFLNPVUPVUFUYGUUHUUNIWSZVUCVUDXEVUFUYGNVVGVUCXFUXPV
|
|
UFVUEVVGWTXGVUSVVEVVEVUFUUHLZUVNPZVURVKZETZVUSVKVUFUYGUUHUVNVVGVUCVUDWRVV
|
|
IUYCVVHLZUVMPZHUUSURZUYCUWGKZKZQZVVMRZHTZETZVVKHUVMUUSVVHVUFUUHWGWHUYCUUS
|
|
PZVVMRZHTVVRETZHTVVNVVTVWBVWCHVWBVVQETZVVMRVWCVWAVWDVVMEUYCXLWJVVQVVMEWKW
|
|
LWMVVMHUUSWNVVREHWOVRVVSVVJEVVSVVPVVHLZUVMPZVVJVVMVWFHVVPVVOSVVQVVLVWEUVM
|
|
UYCVVPVVHXCWAWPVWFVWEUVKPZVWEUVLPZVAVURVWEUVKUVLWQVWGUWMVWHVUQVWGUWGUUHLZ
|
|
UVJPVWIUUPPZVWIUVIPZRUWMUWGVUFUUHUVJEWSZVVGVUDWRVWIUUPUVIXHVWJUWHVWKUWLVW
|
|
JUWHUUHOPVUDUWGUUHUFOVWLVUDVDXIUYCUWIKZKZKZKZQZUYCVWILZUVHPZRZHTZFTZUWIAP
|
|
ZUWKRZFTVWKUWLVXAVXDFVXAVWPVWILZUVHPZVXEUURPZVXEUVGPZRVXDVWSVXFHVWPVWOSVW
|
|
QVWRVXEUVHUYCVWPVWIXCWAWPVXEUURUVGXHVXGVXCVXHUWKVXGVWNUUHLUUQPVWMALNPVXCV
|
|
WNUWGUUHUUQVWMSZVWLVUDWRVWMANUWISZCXJUWIAFWSZCWTXGVWNUWGLZUVFPZJUQZUWJQZJ
|
|
EXAZRZJTZVXHUWKVXMUYCVXLLZUVEPZHUUTURZUYCVXNKZKZKZQZVXTRZHTZJTZVXRHUVEUUT
|
|
VXLVWNUWGWGWHUYOVXTRZHTVYFJTZHTVYAVYHVYIVYJHVYIVYEJTZVXTRVYJUYOVYKVXTJUYC
|
|
WIWJVYEVXTJWKWLWMVXTHUUTWNVYFJHWOVRVYGVXQJVYGVYDVXLLZUVEPZVYLUVDPZVYLUUMP
|
|
ZRVXQVXTVYMHVYDVYCSVYEVXSVYLUVEUYCVYDVXLXCWAWPVYLUVDUUMXHVYNVXOVYOVXPVYNV
|
|
YBVWNLUVCPVXNVWMLUVBPVXOVYBVWNUWGUVCVXNSZVXIVWLXEVXNVWMUVBJWSZVXJXJVXNUWI
|
|
VYQVXKXKXGVYOVYBUWGLNPVXPVYBVWNUWGNVYPVXIVWLWRVXNUWGVYQVWLWTVGXMXGWMXGVWN
|
|
UWGUUHUVFVXIVWLVUDXEJUWJUWGXNVRXMXGWMVWSHUVAURZVWTFTZHTZVWKVXBVYRUYCUVAPZ
|
|
VWSRZHTVYTVWSHUVAWNWUBVYSHWUBVWQFTZVWSRVYSWUAWUCVWSFUYCYBWJVWQVWSFWKWLWMV
|
|
GHUVHUVAVWIUWGUUHWGWHVWTFHWOVRUWKFAWNVRXMXGVWHUWGVUFLUNPZVUQUWGVUFUUHUNVW
|
|
LVVGVUDXEUWGOPVUFOPWUDVUQVAVWLVVGUWGVUFOOXOXPVGXQXRVGWMXGVUREXSXTYCXMXGWM
|
|
VUBVUJUVQPVULHUUSURZVUOUYGUUHBUVQVUCVUDDXEHUVPUUSVUJUYGUUHWGWHVWAVULRZHTV
|
|
UMITZHTWUEVUOWUFWUGHWUFVUIITZVULRWUGVWAWUHVULIUYCXLWJVUIVULIWKWLWMVULHUUS
|
|
WNVUMIHWOVRXGUXRUXPVUSIYAYDZPVVAUWNWUIUXPUWMEIYEYFVUSIUXPYGVGVRXQXRVGWMXG
|
|
YHUUIUVTUYNYIUXMUXSGXDUYBGBUWNYLUXSGYMVGVRVSUXLUXNUUIUULPZUUIUWCPZVBZUWEU
|
|
XLUXNUWFUWTRZUXNUUIUWBPZVKZRZVBZWULUXLUXNWUPWUQUXLWUOUXNWUNUWFWUNUWFWUNUU
|
|
HUUKPZUXKRWURUWFUUHBUUKOVUDDVDUXKWURDYJWURUUHUUJQUWFUUHUUJVUDVFAMCYKVGZYN
|
|
YOYPYQUXLWUMVKWUPWUQVAWUMUWFUWFUWTYRYPWUMWUPYSUUAUUCWUJWUMWUKWUPWUJWURUXC
|
|
RWUMUUHBUUKUUJVUDDVDWURUWFUXCUWTWUSUXGXMVGUUIUWAUWBVOUUBYTUUIUULUWCVIYTUU
|
|
DUUEUWQUWOBEAFUUFUUGWL $.
|
|
$}
|
|
|
|
$( The expression at the core of ~ eqtfinrelk exists. (Contributed by SF,
|
|
30-Jan-2015.) $)
|
|
tfinrelkex $p |- ( ( { { (/) } } X._k { (/) } ) u.
|
|
( ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k
|
|
( ( Ins3_k `'_k _S_k \
|
|
Ins2_k
|
|
( ( Ins2_k ( ( Nn X._k _V ) i^i
|
|
( ( Ins2_k SI_k _S_k i^i
|
|
Ins3_k
|
|
( ( Ins3_k SI_k ( ( ~P 1c X._k _V ) \
|
|
( ( Ins3_k _S_k (+) Ins2_k SI_k _S_k )
|
|
"_k ~P1 ~P1 ~P1 1c ) ) i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 ~P1 1c ) ) (+)
|
|
Ins3_k _I_k ) "_k ~P1 1c ) ) "_k ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c )
|
|
\ ( { { (/) } } X._k _V ) ) ) e. _V $=
|
|
( cxpk cssetk cins2k cins3k cvv c1c csymdif cpw1 cimak cdif ssetkex ins2kex
|
|
cin xpkex ins3kex vvex symdifex pw1ex imakex difex csn ccnvk cnnc csik cidk
|
|
c0 cpw ccompl snex cnvkex nncex sikex 1cex pwex inex idkex complex unex ) U
|
|
FUAZUAZUSABCZBUBZDZUCEAZBUDZCZFUGZEAZBDZVFGZFHZHZHZIZJZUDZDZVAMZVLIZDZMZVMI
|
|
ZMZCZUEDZGZVKIZCZJZVKIZDZGZVLIZUHZUTEAZJUTUSUSUIZUFUINWNWOWMWLVLVAWKBKLZWJW
|
|
IVKVCWHVBBKUJOWGWFVKWDWEWCVDWBUCEUKPNWAVMVFVTVEBKULLZVSVRVLVQVAVPVOVHVNVGEF
|
|
UMUNPNVJVMVIVFBKOWRQVLVKFUMRZRZRZSTULOWQUOWTSOUOXASUOLUEUPOQWSSLTWSSOQWTSUQ
|
|
UTEWPPNTUR $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Equality theorem for the finite T operator. (Contributed by SF,
|
|
24-Jan-2015.) $)
|
|
tfineq $p |- ( A = B -> _T[fin] A = _T[fin] B ) $=
|
|
( vx vy wceq c0 cv cnnc wcel cpw1 wrex cio cif ctfin eqeq1 rexeq iotabidv
|
|
wa anbi2d df-tfin ifbieq2d 3eqtr4g ) ABEZAFEZFCGZHIZDGJUEIZDAKZRZCLZMBFEZ
|
|
FUFUGDBKZRZCLZMANBNUCUDUKUJUNFABFOUCUIUMCUCUHULUFUGDABPSQUACADTCBDTUB $.
|
|
$}
|
|
|
|
${
|
|
$d a n $. $d M a $. $d M n $.
|
|
|
|
$( Properties of the finite T operator for a non-empty natural. Theorem
|
|
X.1.28 of [Rosser] p. 528. (Contributed by SF, 22-Jan-2015.) $)
|
|
tfinprop $p |- ( ( M e. Nn /\ M =/= (/) ) ->
|
|
( _T[fin] M e. Nn /\ E. a e. M ~P1 a e. _T[fin] M ) ) $=
|
|
( vn cnnc wcel c0 wne wa ctfin cv cpw1 wrex wceq cio cif df-tfin wn df-ne
|
|
syl jca iffalse sylbi adantl nnpw1ex reiotacl eqeltrd syl5eqel syl5req wb
|
|
wreu eleq2 rexbidv reiota2 mpbird ) ADEZAFGZHZAIZDEZBJKZUREZBALZUQURAFMZF
|
|
CJZDEUTVDEZBALZHCNZOZDCABPZUQVHVGDUPVHVGMZUOUPVCQVJAFRVCFVGUAUBUCZUQVFCDU
|
|
JZVGDECABUDZVFCDUESUFUGZUQVBVGURMZUQURVHVGVIVKUHUQUSVLHVBVOUIUQUSVLVNVMTV
|
|
FVBCDURVDURMVEVABAVDURUTUKULUMSUNT $.
|
|
$}
|
|
|
|
${
|
|
$d M x $.
|
|
$( If ` M ` is a non-empty natural, then ` _T[fin] M ` is also non-empty.
|
|
Corollary 1 of Theorem X.1.28 of [Rosser] p. 528. (Contributed by SF,
|
|
23-Jan-2015.) $)
|
|
tfinnnul $p |- ( ( M e. Nn /\ M =/= (/) ) -> _T[fin] M =/= (/) ) $=
|
|
( vx cnnc wcel c0 wne wa ctfin cv cpw1 wrex tfinprop rexlimivw adantl syl
|
|
ne0i ) ACDAEFGAHZCDZBIJZQDZBAKZGQEFZABLUAUBRTUBBAQSPMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The finite T operator applied to the empty set is empty. Theorem X.1.29
|
|
of [Rosser] p. 528. (Contributed by SF, 22-Jan-2015.) $)
|
|
tfinnul $p |- _T[fin] (/) = (/) $=
|
|
( vx vy c0 ctfin wceq cv cnnc wcel cpw1 wrex cio cif df-tfin iftrue ax-mp
|
|
wa eqid eqtri ) CDCCEZCAFZGHBFITHBCJPAKZLZCACBMSUBCECQSCUANOR $.
|
|
$}
|
|
|
|
${
|
|
$d N a $.
|
|
$( Closure law for finite T operation. (Contributed by SF, 2-Feb-2015.) $)
|
|
tfincl $p |- ( N e. Nn -> _T[fin] N e. Nn ) $=
|
|
( va cnnc wcel ctfin wi c0 wceq tfinnul tfineq 3eqtr4a eleq1d biimprd wne
|
|
id wa cv cpw1 wrex tfinprop simpld expcom pm2.61ine ) ACDZAEZCDZFAGAGHZUF
|
|
UDUGUEACUGGEGUEAIAGJUGOKLMUDAGNZUFUDUHPUFBQRUEDBASABTUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a p $. $d b p $. $d M a $. $d M b $. $d M p $. $d N a $.
|
|
$d N b $. $d N p $.
|
|
|
|
$( The finite T operator is one-to-one over the naturals. Theorem X.1.30
|
|
of [Rosser] p. 528. (Contributed by SF, 24-Jan-2015.) $)
|
|
tfin11 $p |- ( ( M e. Nn /\ N e. Nn /\ _T[fin] M = _T[fin] N ) ->
|
|
M = N ) $=
|
|
( va vb vp cnnc wcel ctfin wceq w3a wi c0 wa tfinnnul ex necon4d 3ad2ant1
|
|
wne cv wrex impcom eqeq1 adantl adantr sylbid 3adant1 eqtr4d neeq1 biimpd
|
|
wb ancld tfineq tfinnul syl6eq necon3i anim12i 3ad2ant3 tfinprop 3ad2ant2
|
|
syl6 reeanv wel simp31 tfincl syl simp2l simp2r simp33 ncfinlower syl3anc
|
|
eleqtrrd simpl31 simpl1l simprrl nnceleq syl22anc simpl32 simpl1r simprrr
|
|
cpw1 simprl expr rexlimdva mpd 3exp rexlimivv ad2ant2l com12 syl2and syld
|
|
sylbir pm2.61ine ) AFGZBFGZAHZBHZIZJZABIZKZWOLWOLIZWRWSXAWRMALBWRXAALIZWM
|
|
WNXAXBKWQWMALWOLWMALRZWOLRZANOPQUAWRXABLIZWNWQXAXEKWMWNWQMXAWPLIZXEWQXAXF
|
|
UJWNWOWPLUBUCWNXFXEKWQWNBLWPLWNBLRZWPLRZBNOPUDUEUFUAUGOWRXDWSWRXDXCXGMZWS
|
|
WQWMXDXIKWNWQXDXDXHMXIWQXDXHWQXDXHWOWPLUHUIUKXDXCXHXGALWOLXBWOLHZLALULUMU
|
|
NUOBLWPLXEWPXJLBLULUMUNUOUPUTUQWRXCWOFGZCSZVTWOGZCATZMZXGWPFGZDSZVTZWPGZD
|
|
BTZMZWSWMWNXCXOKWQWMXCXOACUROQWNWMXGYAKWQWNXGYABDUROUSXOYAMWRWSXNXTWTXKXP
|
|
XNXTMXMXSMZDBTCATWTXMXSCDABVAYBWTCDABXLAGZXQBGZMZYBWRWSYEYBWRJZCEVBZDEVBZ
|
|
MZEFTZWSYFXKXMXRWOGYJYFWMXKYEYBWMWNWQVCAVDVEYEXMXSWRVFYFXRWPWOYEXMXSWRVGY
|
|
EYBWMWNWQVHVKXLXQEWOVIVJYFYIWSEFYFESZFGZYIWSYFYLYIMZMZAYKBYNWMYLYCYGAYKIW
|
|
MWNWQYEYBYMVLYFYLYIWAZYCYDYBWRYMVMYFYLYGYHVNXLAYKVOVPYNWNYLYDYHBYKIWMWNWQ
|
|
YEYBYMVQYOYCYDYBWRYMVRYFYLYGYHVSXQBYKVOVPUGWBWCWDWEWFWKWGWHWIWJWHWL $.
|
|
$}
|
|
|
|
${
|
|
$d A b $. $d A n $. $d b n $. $d M b $. $d M n $.
|
|
$( The finite T operator on a natural contains the unit power class of any
|
|
element of the natural. Theorem X.1.31 of [Rosser] p. 528.
|
|
(Contributed by SF, 24-Jan-2015.) $)
|
|
tfinpw1 $p |- ( ( M e. Nn /\ A e. M ) -> ~P1 A e. _T[fin] M ) $=
|
|
( vb vn cnnc wcel wa ctfin cv cpw1 wrex c0 wne ne0i tfinprop sylan2 3expa
|
|
expr rexlimdva mpd ncfinraise adantrr w3a simp3rl wceq simp3l syl simp3rr
|
|
simp1l tfincl simp2r nnceleq syl22anc eleqtrd adantld ) BEFZABFZGZBHZEFZC
|
|
IZJZUSFZCBKZGZAJZUSFZUQUPBLMVEBANBCOPURVDVGUTURVCVGCBURVABFZVCVGURVHVCGZG
|
|
ZVFDIZFZVBVKFZGZDEKZVGURVHVOVCUPUQVHVOAVADBUAQUBVJVNVGDEVJVKEFZVNVGURVIVP
|
|
VNGZVGURVIVQUCZVFVKUSVLVMVPURVIUDVRVPUTVMVCVKUSUEURVIVPVNUFVRUPUTUPUQVIVQ
|
|
UIBUJUGVLVMVPURVIUHURVHVCVQUKVBVKUSULUMUNQRSTRSUOT $.
|
|
$}
|
|
|
|
$( Relationship between finite T operator and finite Nc operation in a finite
|
|
universe. Corollary of Theorem X.1.31 of [Rosser] p. 529. (Contributed
|
|
by SF, 24-Jan-2015.) $)
|
|
ncfintfin $p |- ( ( _V e. Fin /\ A e. V ) ->
|
|
_T[fin] Nc[fin] A = Nc[fin] ~P1 A ) $=
|
|
( cvv cfin wcel wa cncfin cnnc cpw1 wceq ncfinprop simpld tfincl syl pw1exg
|
|
ctfin sylan2 tfinpw1 simprd nnceleq syl22anc ) CDEZABEZFZAGZPZHEZAIZGZHEZUH
|
|
UFEZUHUIEZUFUIJUDUEHEZUGUDUMAUEEZABKZLUEMNUDUJULUCUBUHCEUJULFABOUHCKQZLUDUM
|
|
UNFUKUOAUERNUDUJULUPSUHUFUITUA $.
|
|
|
|
${
|
|
$d M a b c $. $d N a b c $.
|
|
$( The finite T operation distributes over non-empty cardinal sum. Theorem
|
|
X.1.32 of [Rosser] p. 529. (Contributed by SF, 26-Jan-2015.) $)
|
|
tfindi $p |- ( ( M e. Nn /\ N e. Nn /\ ( M +c N ) =/= (/) ) ->
|
|
_T[fin] ( M +c N ) = ( _T[fin] M +c _T[fin] N ) ) $=
|
|
( va vb vc cnnc wcel cplc c0 ctfin wceq cv cpw1 nncaddccl 3adant3 tfinpw1
|
|
wa tfincl syl2anc cin wne wex n0 w3a syl syl2an simp3 wrex eladdc simplll
|
|
cun simplrl simpllr simplrr pw1eq pw1in pw10 3eqtr3g adantl eladdci pw1un
|
|
syl3anc syl6eq eleq1d syl5ibrcom expimpd rexlimdvva syl5bi 3impia nnceleq
|
|
syl22anc 3expia exlimdv ) AFGZBFGZABHZIUAZVPJZAJZBJZHZKZVQCLZVPGZCUBVNVOQ
|
|
ZWBCVPUCWEWDWBCVNVOWDWBVNVOWDUDZVRFGZWAFGZWCMZVRGZWIWAGZWBVNVOWGWDWEVPFGZ
|
|
WGABNZVPRUEOVNVOWHWDVNVSFGVTFGWHVOARBRVSVTNUFOWFWLWDWJVNVOWLWDWMOVNVOWDUG
|
|
WCVPPSVNVOWDWKWDDLZELZTZIKZWCWNWOUKZKZQZEBUHDAUHWEWKWCABDEUIWEWTWKDEABWEW
|
|
NAGZWOBGZQZQZWQWSWKXDWQQZWKWSWNMZWOMZUKZWAGZXEXFVSGZXGVTGZXFXGTZIKZXIXEVN
|
|
XAXJVNVOXCWQUJWEXAXBWQULWNAPSXEVOXBXKVNVOXCWQUMWEXAXBWQUNWOBPSWQXMXDWQWPM
|
|
IMXLIWPIUOWNWOUPUQURUSXFXGVSVTUTVBWSWIXHWAWSWIWRMXHWCWRUOWNWOVAVCVDVEVFVG
|
|
VHVIWIVRWAVJVKVLVMVHVI $.
|
|
$}
|
|
|
|
$( The finite T operator is fixed at ` 0c ` . (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
tfin0c $p |- _T[fin] 0c = 0c $=
|
|
( c0c ctfin cnnc wcel c0 wceq peano1 tfincl ax-mp cpw1 pw10 nulel0c tfinpw1
|
|
mp2an eqeltrri nnceleq mp4an ) ABZCDZACDZERDEADZRAFTSGAHIGEJZERKTUAUBRDGLEA
|
|
MNOLERAPQ $.
|
|
|
|
${
|
|
$d A a b x $.
|
|
$( The finite T operator over a successor. (Contributed by SF,
|
|
30-Jan-2015.) $)
|
|
tfinsuc $p |- ( ( A e. Nn /\ ( A +c 1c ) =/= (/) ) ->
|
|
_T[fin] ( A +c 1c ) = ( _T[fin] A +c 1c ) ) $=
|
|
( va vb vx cnnc wcel c1c cplc ctfin wceq cv cpw1 peano2 tfincl syl adantr
|
|
wa tfinpw1 csn cun c0 wne wex sylan ccompl wrex elsuc adantrr wel elcompl
|
|
n0 wn vex snelpw1 xchbinxr biimpi ad2antll snex syl2anc pw1eq pw1un pw1sn
|
|
elsuci uneq2i syl6eq eleq1d syl5ibrcom rexlimdvva syl5bi nnceleq syl22anc
|
|
eqtri imp ex exlimdv ) AEFZAGHZUAUBZVQIZAIZGHZJZVRBKZVQFZBUCVPWBBVQUKVPWD
|
|
WBBVPWDWBVPWDQVSEFZWAEFZWCLZVSFZWGWAFZWBVPWEWDVPVQEFZWEAMZVQNOPVPWFWDVPVT
|
|
EFWFANVTMOPVPWJWDWHWKWCVQRUDVPWDWIWDWCCKZDKZSZTZJZDWLUEZUFCAUFVPWIDWCACUG
|
|
VPWPWICDAWQVPWLAFZWMWQFZQQZWIWPWLLZWNSZTZWAFZWTXAVTFZWNXAFZULZXDVPWRXEWSW
|
|
LARUHWSXGVPWRWSXGWSDCUIXFWMWLDUMZUJWMWLUNUOUPUQXAVTWNWMURVCUSWPWGXCWAWPWG
|
|
WOLZXCWCWOUTXIXAWNLZTXCWLWNVAXJXBXAWMXHVBVDVLVEVFVGVHVIVMWGVSWAVJVKVNVOVI
|
|
VM $.
|
|
$}
|
|
|
|
$( The finite T operator is idempotent over ` 1c ` . Theorem X.1.34(a) of
|
|
[Rosser] p. 529. (Contributed by SF, 30-Jan-2015.) $)
|
|
tfin1c $p |- _T[fin] 1c = 1c $=
|
|
( c0c c1c cplc ctfin cnnc wcel wne wceq peano1 addcid2 csn 1cex snel1c ne0i
|
|
c0 ax-mp eqnetri tfinsuc mp2an tfineq tfin0c addceq1i eqtri 3eqtr3i ) ABCZD
|
|
ZADZBCZBDZBAEFUEOGUFUHHIUEBOBJZBKZBFBOGBLMBUKNPQARSUEBHUFUIHUJUEBTPUHUEBUGA
|
|
BUAUBUJUCUD $.
|
|
|
|
${
|
|
$d M x y $. $d N x y $.
|
|
$( Lemma for ~ tfinltfin . Prove the forward direction of the theorem.
|
|
(Contributed by SF, 2-Feb-2015.) $)
|
|
tfinltfinlem1 $p |- ( ( M e. Nn /\ N e. Nn ) ->
|
|
( << M , N >> e. <[fin] ->
|
|
<< _T[fin] M , _T[fin] N >> e. <[fin] ) ) $=
|
|
( vx vy cnnc wcel wa c0 wne cv cplc wceq wrex ctfin copk addceq1d syl2anc
|
|
c1c wi cvv cltfin tfinnnul adantrd adantr addcnul1 addccom eqtr3i addceq2
|
|
ex syl6eq eqeq2d rspcev mpan2 eleq1 tfineq tfinnul eqeq1d rexbidv imbi12d
|
|
mpbiri adantld a1dd w3a simp2r simp3r simp3l eqnetrrd addcnnul syl simpld
|
|
simprd cpw1 tfinprop adantl simp1l nncaddccl tfinsuc eqtrd tfindi syl3anc
|
|
3ad2ant3 3expa exp32 com12 pm2.61ine expr rexlimdv imp3a opkltfing tfinex
|
|
jcad wb mp2an a1i 3imtr4d ) AEFZBEFZGZAHIZBACJZKZRKZLZCEMZGZANZHIZBNZXFDJ
|
|
ZKZRKZLZDEMZGZABOUAFXFXHOUAFZWRXEXGXMWPXEXGSWQWPWSXGXDWPWSXGAUBUIUCUDWRWS
|
|
XDXMWRWSXDXMSWRWSGXCXMCEWRWSWTEFZXCXMSZWRWSXPGZGZXQSBHBHLZXSXMXCXTWRXMXRX
|
|
TWQXMWPXTWQXMSHEFZHXKLZDEMZSYAHHRKZLZYCRHKHYDRUERHUFUGYBYEDHEXIHLZXKYDHYF
|
|
XJHRYFXJXFHKHXIHXFUHXFUEUJPUKULUMXTWQYAXMYCBHEUNXTXLYBDEXTXHHXKXTXHHNHBHU
|
|
OUPUJUQURUSUTVAUCVBXSBHIZXQXSYGXCXMWRXRYGXCGZXMWRXRYHVCZWTNZEFZXHXFYJKZRK
|
|
ZLZXMYIXPWTHIZYKWRWSXPYHVDZYIWSYOYIXAHIZWSYOGYIYQRHIZYIXBHIZYQYRGYIBXBHWR
|
|
XRYGXCVEWRXRYGXCVFVGZXARVHVIVJZAWTVHVIVKXPYOGYKXIVLYJFDWTMWTDVMVJQYIXHXAN
|
|
ZRKZYMYIXHXBNZUUCYHWRXHUUDLZXRXCUUEYGBXBUOVNWAYIXAEFZYSUUDUUCLYIWPXPUUFWP
|
|
WQXRYHVOZYPAWTVPQYTXAVQQVRYIUUBYLRYIWPXPYQUUBYLLUUGYPUUAAWTVSVTPVRXLYNDYJ
|
|
EXIYJLZXKYMXHUUHXJYLRXIYJXFUHPUKULQWBWCWDWEWFWGUIWHWKCABEEWIXOXNWLZWRXFTF
|
|
XHTFUUIAWJBWJDXFXHTTWIWMWNWO $.
|
|
$}
|
|
|
|
${
|
|
$d M x y $. $d N x y $.
|
|
$( Ordering rule for the finite T operation. Corollary to theorem X.1.33
|
|
of [Rosser] p. 529. (Contributed by SF, 1-Feb-2015.) $)
|
|
tfinltfin $p |- ( ( M e. Nn /\ N e. Nn ) ->
|
|
( << M , N >> e. <[fin] <->
|
|
<< _T[fin] M , _T[fin] N >> e. <[fin] ) ) $=
|
|
( vx vy cnnc wcel wa copk cltfin ctfin tfinltfinlem1 wi c0 wceq cplc wrex
|
|
wne wn cvv jca cv c1c tfineq tfinnul syl6eq df-ne con2bii intnanrd tfinex
|
|
sylib wb opkltfing mp2an sylnibr pm2.21d a1d w3a tfinprop simpld ltfinirr
|
|
cpw1 syl 3adant2 opkeq2 eleq1d notbid syl5ibcom con2d nsyl simpl1 syl2anc
|
|
imp simpl3 simpl2 simprr simprl ltfinasym sylc expr imnan ancoms simprbda
|
|
3adant3 adantrl mtod w3o ltfintri adantr ecase23d expcom pm2.61ine impbid
|
|
ex 3expa ) AEFZBEFZGZABHIFZAJZBJZHZIFZABKWQXBWRLZLAMAMNZXCWQXDXBWRXDWSMQZ
|
|
WTWSCUAOUBONCEPZGZXBXDXEXFXDWSMNZXERXDWSMJMAMUCUDUEXEXHWSMUFUGUJUHWSSFWTS
|
|
FXBXGUKAUIBUICWSWTSSULUMUNUOUPWQAMQZXCWOWPXIXCWOWPXIUQZXBWRXJXBGZWRABNZBA
|
|
HIFZXKWSWTNZXLXJXBXNRXJXNXBXJWSWSHZIFZRZXNXBRWOXIXQWPWOXIGZWSEFZXQXRXSDUA
|
|
ZVAZWSFDAPADURUSZWSUTVBVCXNXPXBXNXOXAIWSWTWSVDVEVFVGVHVLABUCVIXKXMBMQZWTW
|
|
SHIFZGZXKYCYDRZLYERXJXBYCYFXJXBYCGZGZXSWTEFZGXBYFYHXSYIYHWOXIXSWOWPXIYGVJ
|
|
WOWPXIYGVMYBVKYHWPYCYIWOWPXIYGVNXJXBYCVOWPYCGYIYAWTFDBPBDURUSVKTXJXBYCVPW
|
|
SWTVQVRVSYCYDVTUJXJXBXMYEXJXBXMGZGZYCYDXJXMYCXBXJXMYCABXTOUBONDEPZWOWPXMY
|
|
CYLGUKZXIWPWOYMDBAEEULWAWCWBWDYKWPWOGXMYDYKWPWOWOWPXIYJVNWOWPXIYJVJTXJXBX
|
|
MVOBAKVRTVSWEXJWRXLXMWFXBABWGWHWIWMWNWJWKWL $.
|
|
$}
|
|
|
|
$( Ordering rule for the finite T operation. Theorem X.1.33 of [Rosser]
|
|
p. 529. (Contributed by SF, 2-Feb-2015.) $)
|
|
tfinlefin $p |- ( ( M e. Nn /\ N e. Nn ) ->
|
|
( << M , N >> e. <_[fin] <->
|
|
<< _T[fin] M , _T[fin] N >> e. <_[fin] ) ) $=
|
|
( cnnc wcel wa copk cltfin wn ctfin clefin tfinltfin ancoms notbid lenltfin
|
|
wb tfincl syl2an 3bitr4d ) ACDZBCDZEZBAFGDZHBIZAIZFGDZHZABFJDUDUCFJDZUAUBUE
|
|
TSUBUEOBAKLMABNSUDCDUCCDUGUFOTAPBPUDUCNQR $.
|
|
|
|
${
|
|
$d a b $. $d a c $. $d a n $. $d a t $. $d a x $. $d b c $. $d b n $.
|
|
$d b t $. $d b x $. $d c n $. $d c t $. $d c x $. $d n x $. $d n t $.
|
|
$d t x $.
|
|
|
|
$( The set of all even naturals exists. (Contributed by SF,
|
|
20-Jan-2015.) $)
|
|
evenfinex $p |- Even[fin] e. _V $=
|
|
( vx vn va vt vc vb cssetk csik csn wrex wa wcel copk wn exbii snex bitri
|
|
wex 3bitri opksnelsik cevenfin cins2k cvv cxpk cin cins3k c1c cpw1 ccompl
|
|
cimak cun csymdif cnnc c0 cdif cv cplc wceq wne df-evenfin eldifsn elimak
|
|
cab vex wb opkex elpw121c anbi1i 19.41v bitr4i df-rex excom opkeq1 eleq1d
|
|
3bitr4i ceqsexv elsymdif otkelins2k elssetk otkelins3k r2ex elpw141c elin
|
|
eladdc opkelxpk mpbiran anbi12i ndisjrelk notbii elcompl con2bii elpw171c
|
|
wel df-ne wo orbi12i elun bibi12i wal dfcleq rexbii abbi2i eqtr4i ssetkex
|
|
alex ins2kex vvex xpkex inex ins3kex 1cex pw1ex imakex complex sikex unex
|
|
symdifex nncex difex eqeltri ) UAGUBZYAUBZUCYAUDZUEZGUFZYAUEZUGUHZUHZUJZU
|
|
IZHZHZHZUFZGHZUFZUBZUBZYOHZHZHZHZUFZYTUFZUBZUKZULZYHUHZUHZUHZUHZUHZUJZUIZ
|
|
UEZUEZUUIUJZYHUJZUFZULZYHUJZUIZUMUJZUNIZUOZUCUAAUPZBUPZUVGUQZURZBUMJZUVFU
|
|
NUSZKZAVCUVEABUTUVLAUVEUVFUVELUVFUVCLZUVKKUVLUVFUVCUNVAUVMUVJUVKUVMUVGUVF
|
|
MZUVBLZBUMJUVJBUVBUMUVFAVDZVBUVOUVIBUMUVNUVALZNCAWMZCUPZUVHLZVEZNZCRZNZUV
|
|
OUVIUVQUWCUVQDUPZUVNMZUUTLZDYHJZUWEUVSIZIZIZURZUWGKZDRZCRZUWCDUUTYHUVNUVG
|
|
UVFVFZVBUWEYHLZUWGKZDRUWMCRZDRUWHUWOUWRUWSDUWRUWLCRZUWGKUWSUWQUWTUWGCUWEV
|
|
GVHUWLUWGCVIVJOUWGDYHVKUWMCDVLVOUWNUWBCUWNUWKUVNMZUUTLZUXAYALZUXAUUSLZVEZ
|
|
NUWBUWGUXBDUWKUWJPUWLUWFUXAUUTUWEUWKUVNVMVNVPUXAYAUUSVQUXEUWAUXCUVRUXDUVT
|
|
UXCUWIUVFMGLUVRUWIUVGUVFGUVSPZBVDZUVPVRUVSUVFCVDZUVPVSQUWIUVGMZUURLZUWEEU
|
|
PZIZIZIZURZUWEUXIMZUUQLZKZDRZERZUXDUVTUXJUXQDYHJUWQUXQKZDRZUXTDUUQYHUXIUW
|
|
IUVGVFZVBUXQDYHVKUYBUXRERZDRUXTUYAUYDDUYAUXOERZUXQKUYDUWQUYEUXQEUWEVGVHUX
|
|
OUXQEVIVJOUXREDVLVJSUWIUVGUVFUURUXFUXGUVPVTUVTFUPZUXKUEZUNURZUVSUYFUXKUKZ
|
|
URZKZEUVGJFUVGJFBWMZEBWMZKZUYKKZERFRZUXTUVSUVGUVGFEWDUYKFEUVGUVGWAUYPUYOF
|
|
RZERUXTUYOFEVLUXSUYQEUXSUXNUXIMZUUQLZUWEUYFIZIZIZIZIZURZUWEUYRMZUUPLZKZDR
|
|
ZFRZUYQUXQUYSDUXNUXMPZUXOUXPUYRUUQUWEUXNUXIVMVNVPUYSVUGDUUIJUWEUUILZVUGKZ
|
|
DRZVUJDUUPUUIUYRUXNUXIVFZVBVUGDUUIVKVUNVUHFRZDRVUJVUMVUPDVUMVUEFRZVUGKVUP
|
|
VULVUQVUGFUWEWBVHVUEVUGFVIVJOVUHFDVLVJSVUIUYOFVUIVUDUYRMZUUPLZVURYDLZVURU
|
|
UOLZKUYOVUGVUSDVUDVUCPZVUEVUFVURUUPUWEVUDUYRVMVNVPVURYDUUOWCVUTUYNVVAUYKV
|
|
UTVURYBLZVURYCLZKUYNVURYBYCWCVVCUYLVVDUYMVVCVUBUXIMYALUYTUVGMGLUYLVUBUXNU
|
|
XIYAVUAPZVUKUYCVRUYTUWIUVGGUYFPZUXFUXGVRUYFUVGFVDZUXGVSSVVDUYRYALZUXLUVGM
|
|
GLUYMVVDVUDUCLVVHVVBVUDUYRUCYAVVBVUOWEWFUXLUWIUVGGUXKPZUXFUXGVRUXKUVGEVDZ
|
|
UXGVSSWGQVVAVURYNLZVURUUNLZKUYKVURYNUUNWCVVKUYHVVLUYJVVKVUBUXNMYMLVUAUXMM
|
|
YLLZUYHVUBUXNUXIYMVVEVUKUYCVTVUAUXMYLUYTPZUXLPZTVVMUYTUXLMYKLUYFUXKMZYJLZ
|
|
UYHUYTUXLYKVVFVVITUYFUXKYJVVGVVJTVVPYILZNUYGUNUSZNVVQUYHVVRVVSUYFUXKVVGVV
|
|
JWHWIVVPYIUYFUXKVFWJVVSUYHUYGUNWNWKVOSSVURUUMLZNACWMZUVFUYILZVEZNZARZNZVV
|
|
LUYJVVTVWEVVTUWEVURMZUUGLZDUULJZUWEUVFIZIZIZIZIZIZIZIZURZVWHKZDRZARZVWEDU
|
|
UGUULVURVUDUYRVFZVBUWEUULLZVWHKZDRVWSARZDRVWIVXAVXDVXEDVXDVWRARZVWHKVXEVX
|
|
CVXFVWHAUWEWLVHVWRVWHAVIVJOVWHDUULVKVWSADVLVOVWTVWDAVWTVWQVURMZUUGLZVXGYR
|
|
LZVXGUUFLZVEZNVWDVWHVXHDVWQVWPPVWRVWGVXGUUGUWEVWQVURVMVNVPVXGYRUUFVQVXKVW
|
|
CVXIVWAVXJVWBVXIVWOUYRMZYQLVWMUXIMYPLZVWAVWOVUDUYRYQVWNPZVVBVUOVRVWMUXNUX
|
|
IYPVWLPZVUKUYCVRVXMVWKUWIMYOLVWJUVSMGLVWAVWKUWIUVGYOVWJPZUXFUXGVTVWJUVSGU
|
|
VFPZUXHTUVFUVSUVPUXHVSSSVXGUUCLZVXGUUELZWOAFWMZAEWMZWOVXJVWBVXRVXTVXSVYAV
|
|
XRVWOVUDMUUBLVWNVUCMUUALZVXTVWOVUDUYRUUBVXNVVBVUOVTVWNVUCUUAVWMPVUBPTVYBV
|
|
WMVUBMYTLVWLVUAMYSLZVXTVWMVUBYTVXOVVETVWLVUAYSVWKPZVVNTVYCVWKUYTMYOLVWJUY
|
|
FMGLVXTVWKUYTYOVXPVVFTVWJUYFGVXQVVGTUVFUYFUVPVVGVSSSSVXSVXLUUDLVWMUXNMYTL
|
|
ZVYAVWOVUDUYRUUDVXNVVBVUOVRVWMUXNUXIYTVXOVUKUYCVTVYEVWLUXMMYSLVWKUXLMYOLZ
|
|
VYAVWLUXMYSVYDVVOTVWKUXLYOVXPVVITVYFVWJUXKMGLVYAVWJUXKGVXQVVJTUVFUXKUVPVV
|
|
JVSQSSWPVXGUUCUUEWQUVFUYFUXKWQVOWRWISOSWIVURUUMVXBWJUYJVWCAWSVWFAUVSUYIWT
|
|
VWCAXEQVOWGQWGSOSOVJSVOWRWISOSWIUVNUVAUWPWJUVIUWACWSUWDCUVFUVHWTUWACXEQVO
|
|
XAQVHQXBXCUVCUVDUVBUMUVAUUTYHYAUUSGXDXFZUURUUQYHUUPUUIYDUUOYBYCYAVYGXFUCY
|
|
AXGVYGXHXIYNUUNYMYLYKYJYIYFYHYEYAGXDXJVYGXIYGUGXKXLXLZXMXNXOXOXOXJUUMUUGU
|
|
ULYRUUFYQYPYOGXDXOZXJXFXFUUCUUEUUBUUAYTYSYOVYIXOXOZXOXOXJUUDYTVYJXJXFXPXQ
|
|
UUKUUJUUIUUHYHVYHXLXLZXLXLXLXMXNXIXIVYKXMVYHXMXJXQVYHXMXNXRXMUNPXSXT $.
|
|
|
|
$d a y $. $d b y $. $d n y $. $d t y $. $d x y $.
|
|
|
|
$( The set of all odd naturals exists. (Contributed by SF,
|
|
20-Jan-2015.) $)
|
|
oddfinex $p |- Odd[fin] e. _V $=
|
|
( vx va vt vb vc vy cssetk csn wceq wrex wa wcel copk wn wex 3bitr4i snex
|
|
exbii 3bitri opksnelsik coddfin cins2k cvv cxpk cin cins3k c1c cpw1 cimak
|
|
vn ccompl csik cun csymdif ccnvk cidk cnnc c0 cdif cplc wne cab df-oddfin
|
|
cv eldifsn vex elimak wb opkex elpw121c anbi1i 19.41v bitr4i df-rex excom
|
|
opkeq1 eleq1d ceqsexv elsymdif otkelins2k elssetk bitri elpw131c elpw141c
|
|
elin opkelxpk mpbiran anbi12i otkelins3k ndisjrelk notbii elcompl con2bii
|
|
wel df-ne elpw171c wo orbi12i elun bibi12i wal dfcleq alex r2ex opkelcnvk
|
|
eladdc weq sneqb opkelidkg mp2an elsnc elsuc rexbii abbi2i eqtr4i ssetkex
|
|
ins2kex vvex xpkex inex ins3kex 1cex pw1ex imakex complex symdifex cnvkex
|
|
sikex unex idkex nncex difex eqeltri ) UAGUBZYNUBZUCYNUDZUEZGUFZYNUEZUGUH
|
|
ZUHZUIZUKZULZULZULZUFZGULZUFZUBZUBZUUHULZULZULZULZUFZUUMUFZUBZUMZUNZUUAUH
|
|
ZUHZUHZUHZUHZUIZUKZUEZUEZUVBUIZUUAUIZUBZUBZGUKZULZULZULZUOZUFZUEZUUKUUPUP
|
|
UFZUBZUMZUNZUVEUIZUKZUEZUVBUIZUVAUIZUFZUNZUUAUIZUKZUQUIZURHZUSZUCUAAVDZUJ
|
|
VDZUWRUTZUGUTZIZUJUQJZUWQURVAZKZAVBUWPAUJVCUXDAUWPUWQUWPLUWQUWNLZUXCKUXDU
|
|
WQUWNURVEUXEUXBUXCUXEUWRUWQMZUWMLZUJUQJUXBUJUWMUQUWQAVFZVGUXGUXAUJUQUXFUW
|
|
LLZNBAWNZBVDZUWTLZVHZNZBOZNZUXGUXAUXIUXOUXICVDZUXFMZUWKLZCUUAJZUXQUXKHZHZ
|
|
HZIZUXSKZCOZBOZUXOCUWKUUAUXFUWRUWQVIZVGUXQUUALZUXSKZCOUYEBOZCOUXTUYGUYJUY
|
|
KCUYJUYDBOZUXSKUYKUYIUYLUXSBUXQVJVKUYDUXSBVLVMRUXSCUUAVNUYEBCVOPUYFUXNBUY
|
|
FUYCUXFMZUWKLZUYMYNLZUYMUWJLZVHZNUXNUXSUYNCUYCUYBQZUYDUXRUYMUWKUXQUYCUXFV
|
|
PVQVRUYMYNUWJVSUYQUXMUYOUXJUYPUXLUYOUYAUWQMGLUXJUYAUWRUWQGUXKQZUJVFZUXHVT
|
|
UXKUWQBVFZUXHWAWBUYAUWRMZUWILZDVDZUWSLZUWQVUDUKZLZKZUXKVUDUWQHZUMZIZKZDOZ
|
|
AOZUYPUXLVUCUXQVUBMZUWHLZCUVAJZUXQVUIHZHZHZIZVUPKZCOZAOZVUNCUWHUVAVUBUYAU
|
|
WRVIZVGVUQUXQUVALZVUPKZCOVVBAOZCOVVDVUPCUVAVNVVGVVHCVVGVVAAOZVUPKVVHVVFVV
|
|
IVUPAUXQWCVKVVAVUPAVLVMRVVBCAVOSVVCVUMAVVCVUTVUBMZUWHLZUXQVUDHZHZHZHZHZIZ
|
|
UXQVVJMZUWGLZKZCOZDOZVUMVUPVVKCVUTVUSQZVVAVUOVVJUWHUXQVUTVUBVPVQVRVVKVVSC
|
|
UVBJUXQUVBLZVVSKZCOZVWBCUWGUVBVVJVUTVUBVIZVGVVSCUVBVNVWFVVTDOZCOVWBVWEVWH
|
|
CVWEVVQDOZVVSKVWHVWDVWIVVSDUXQWDVKVVQVVSDVLVMRVVTDCVOVMSVWAVULDVWAVVPVVJM
|
|
ZUWGLZVWJUVTLZVWJUWFLZKVULVVSVWKCVVPVVOQZVVQVVRVWJUWGUXQVVPVVJVPVQVRVWJUV
|
|
TUWFWEVWLVUHVWMVUKVWLVWJUVMLZVWJUVSLZKVUHVWJUVMUVSWEVWOVUEVWPVUGVWOVVNVUB
|
|
MUVLLVVLUWRMZUVKLZVUEVVNVUTVUBUVLVVMQZVWCVVEVTVVLUYAUWRUVKVUDQZUYSUYTVTVW
|
|
RBUJWNZEUJWNZKZUXKEVDZUEZURIZVUDUXKVXDUMZIZKZKZBOZEOZVUEVWRUXQVWQMZUVJLZC
|
|
UUAJZUXQVXDHZHZHZIZVXNKZCOZEOZVXLCUVJUUAVWQVVLUWRVIZVGUYIVXNKZCOVXTEOZCOV
|
|
XOVYBVYDVYECVYDVXSEOZVXNKVYEUYIVYFVXNEUXQVJVKVXSVXNEVLVMRVXNCUUAVNVXTECVO
|
|
PVYAVXKEVYAVXRVWQMZUVJLZUXQUYCHZHZIZUXQVYGMZUVILZKZCOZBOZVXKVXNVYHCVXRVXQ
|
|
QZVXSVXMVYGUVJUXQVXRVWQVPVQVRVYHVYMCUVBJVWDVYMKZCOZVYPCUVIUVBVYGVXRVWQVIZ
|
|
VGVYMCUVBVNVYSVYNBOZCOVYPVYRWUACVYRVYKBOZVYMKWUAVWDWUBVYMBUXQWDVKVYKVYMBV
|
|
LVMRVYNBCVOVMSVYOVXJBVYOVYJVYGMZUVILZWUCYQLZWUCUVHLZKVXJVYMWUDCVYJVYIQZVY
|
|
KVYLWUCUVIUXQVYJVYGVPVQVRWUCYQUVHWEWUEVXCWUFVXIWUEWUCYOLZWUCYPLZKVXCWUCYO
|
|
YPWEWUHVXAWUIVXBWUHUYCVWQMYNLVUBGLVXAUYCVXRVWQYNUYRVYQVYCVTUYAVVLUWRGUYSV
|
|
WTUYTVTUXKUWRVUAUYTWASWUIVYGYNLZVXPUWRMGLVXBWUIVYJUCLWUJWUGVYJVYGUCYNWUGV
|
|
YTWFWGVXPVVLUWRGVXDQZVWTUYTVTVXDUWREVFZUYTWASWHWBWUFWUCUUGLZWUCUVGLZKVXIW
|
|
UCUUGUVGWEWUMVXFWUNVXHWUMUYCVXRMUUFLUYBVXQMUUELZVXFUYCVXRVWQUUFUYRVYQVYCW
|
|
IUYBVXQUUEUYAQZVXPQZTWUOUYAVXPMUUDLUXKVXDMZUUCLZVXFUYAVXPUUDUYSWUKTUXKVXD
|
|
UUCVUAWULTWURUUBLZNVXEURVAZNWUSVXFWUTWVAUXKVXDVUAWULWJWKWURUUBUXKVXDVIWLW
|
|
VAVXFVXEURWOWMPSSWUCUVFLZNADWNZUWQVXGLZVHZNZAOZNZWUNVXHWVBWVGWVBUXQWUCMZU
|
|
UTLZCUVEJZUXQVUTHZHZHZHZIZWVJKZCOZAOZWVGCUUTUVEWUCVYJVYGVIZVGUXQUVELZWVJK
|
|
ZCOWVQAOZCOWVKWVSWWBWWCCWWBWVPAOZWVJKWWCWWAWWDWVJAUXQWPVKWVPWVJAVLVMRWVJC
|
|
UVEVNWVQACVOPWVRWVFAWVRWVOWUCMZUUTLZWWEUUKLZWWEUUSLZVHZNWVFWVJWWFCWVOWVNQ
|
|
WVPWVIWWEUUTUXQWVOWUCVPVQVRWWEUUKUUSVSWWIWVEWWGWVCWWHWVDWWGWVMVYGMZUUJLVU
|
|
TVWQMUUILZWVCWVMVYJVYGUUJWVLQZWUGVYTVTVUTVXRVWQUUIVWCVYQVYCVTWWKVURVVLMZU
|
|
UHLVUIVUDMZGLZWVCVURVVLUWRUUHVUIQZVWTUYTWIVUIVUDGUWQQZDVFZTUWQVUDUXHWWRWA
|
|
ZSSWWEUUPLZWWEUURLZWQABWNZAEWNZWQWWHWVDWWTWXBWXAWXCWWTWVMVYJMUUOLWVLVYIMU
|
|
UNLZWXBWVMVYJVYGUUOWWLWUGVYTWIWVLVYIUUNVUTQUYCQTWXDVUTUYCMUUMLVUSUYBMUULL
|
|
ZWXBVUTUYCUUMVWCUYRTVUSUYBUULVURQZWUPTWXEVURUYAMUUHLVUIUXKMGLWXBVURUYAUUH
|
|
WWPUYSTVUIUXKGWWQVUATUWQUXKUXHVUAWASSSWXAWWJUUQLVUTVXRMUUMLZWXCWVMVYJVYGU
|
|
UQWWLWUGVYTVTVUTVXRVWQUUMVWCVYQVYCWIWXGVUSVXQMUULLVURVXPMUUHLZWXCVUSVXQUU
|
|
LWXFWUQTVURVXPUUHWWPWUKTWXHVUIVXDMGLWXCVUIVXDGWWQWULTUWQVXDUXHWULWAWBSSWR
|
|
WWEUUPUURWSUWQUXKVXDWSPWTWKSRSWKWUCUVFWVTWLVXHWVEAXAWVHAVUDVXGXBWVEAXCWBP
|
|
WHWBWHSRSRSVUEVXIEUWRJBUWRJVXJEOBOVXLVUDUWRUWRBEXFVXIBEUWRUWRXDVXJBEVOSVM
|
|
SVWPVVNVUTMUVRLVUTVVNMUVQLZVUGVVNVUTVUBUVRVWSVWCVVEWIVVNVUTUVQVWSVWCXEWXI
|
|
VUSVVMMUVPLWWMUVOLZVUGVUSVVMUVPWXFVVLQZTVURVVLUVOWWPVWTTWXJWWNUVNLZVUGVUI
|
|
VUDUVNWWQWWRTWWONWVCNWXLVUGWWOWVCWWSWKWWNGVUIVUDVIWLUWQVUDUXHWLPWBSSWHWBV
|
|
WJUWELZNFBWNZFVDZVUJLZVHZNZFOZNZVWMVUKWXMWXSWXMUXQVWJMZUWDLZCUVEJZUXQWXOH
|
|
ZHZHZHZHZHZHZHZIZWYBKZCOZFOZWXSCUWDUVEVWJVVPVVJVIZVGWWAWYBKZCOWYMFOZCOWYC
|
|
WYOWYQWYRCWYQWYLFOZWYBKWYRWWAWYSWYBFUXQWPVKWYLWYBFVLVMRWYBCUVEVNWYMFCVOPW
|
|
YNWXRFWYNWYKVWJMZUWDLZWYTUUKLZWYTUWCLZVHZNWXRWYBXUACWYKWYJQWYLWYAWYTUWDUX
|
|
QWYKVWJVPVQVRWYTUUKUWCVSXUDWXQXUBWXNXUCWXPXUBWYIVVJMZUUJLWYGVUBMUUILZWXNW
|
|
YIVVPVVJUUJWYHQZVWNVWGVTWYGVUTVUBUUIWYFQZVWCVVEVTXUFWYEUYAMUUHLWYDUXKMGLW
|
|
XNWYEUYAUWRUUHWYDQZUYSUYTWIWYDUXKGWXOQZVUATWXOUXKFVFZVUAWASSWYTUUPLZWYTUW
|
|
BLZWQFDWNZWXOVUILZWQXUCWXPXULXUNXUMXUOXULWYIVVPMUUOLWYHVVOMUUNLZXUNWYIVVP
|
|
VVJUUOXUGVWNVWGWIWYHVVOUUNWYGQVVNQTXUPWYGVVNMUUMLWYFVVMMUULLZXUNWYGVVNUUM
|
|
XUHVWSTWYFVVMUULWYEQZWXKTXUQWYEVVLMUUHLWYDVUDMGLXUNWYEVVLUUHXUIVWTTWYDVUD
|
|
GXUJWWRTWXOVUDXUKWWRWASSSXUMXUEUWALWYGVUTMUPLZXUOWYIVVPVVJUWAXUGVWNVWGVTW
|
|
YGVUTVUBUPXUHVWCVVEWIWYGVUTIZFAXGZXUSXUOXUTWYFVUSIWYEVURIZXVAWYFVUSXURXHW
|
|
YEVURXUIXHXVBWYDVUIIXVAWYDVUIXUJXHWXOUWQXUKXHWBSWYGUCLVUTUCLXUSXUTVHXUHVW
|
|
CWYGVUTUCUCXIXJWXOUWQXUKXKPSWRWYTUUPUWBWSWXOVUDVUIWSPWTWKSRSWKVWJUWEWYPWL
|
|
VUKWXQFXAWXTFUXKVUJXBWXQFXCWBPWHSRSRSUYAUWRUWQUWIUYSUYTUXHWIUXLVUKAVUFJDU
|
|
WSJVULAODOVUNAUXKUWSDXLVUKDAUWSVUFXDVULDAVOSPWTWKSRSWKUXFUWLUYHWLUXAUXMBX
|
|
AUXPBUWQUWTXBUXMBXCWBPXMWBVKWBXNXOUWNUWOUWMUQUWLUWKUUAYNUWJGXPXQZUWIUWHUV
|
|
AUWGUVBUVTUWFUVMUVSUVLUVKUVJUUAUVIUVBYQUVHYOYPYNXVCXQUCYNXRXVCXSXTUUGUVGU
|
|
UFUUEUUDUUCUUBYSUUAYRYNGXPYAXVCXTYTUGYBYCYCZYDYEYHYHYHYAUVFUUTUVEUUKUUSUU
|
|
JUUIUUHGXPYHZYAXQXQZUUPUURUUOUUNUUMUULUUHXVEYHYHZYHYHYAZUUQUUMXVGYAXQYIYF
|
|
UVDUVCUVBUVAUUAXVDYCZYCZYCYCYCZYDYEXTXTXVJYDXVDYDXQXQUVRUVQUVPUVOUVNGXPYE
|
|
YHYHYHYGYAXTUWEUWDUVEUUKUWCXVFUUPUWBXVHUWAUPYJYAXQYIYFXVKYDYEXTXVJYDXVIYD
|
|
YAYFXVDYDYEYKYDURQYLYM $.
|
|
$}
|
|
|
|
${
|
|
$d n x $.
|
|
$( Cardinal zero is even. (Contributed by SF, 20-Jan-2015.) $)
|
|
0ceven $p |- 0c e. Even[fin] $=
|
|
( vn vx c0c cevenfin wcel cv cplc wceq cnnc wrex c0 peano1 addcid2 eqcomi
|
|
wne addceq12 anidms eqeq2d rspcev wa mp2an snid df-0c eleqtrri ne0i ax-mp
|
|
csn 0ex 0cex eqeq1 rexbidv neeq1 anbi12d df-evenfin elab2 mpbir2an ) CDEC
|
|
AFZUQGZHZAIJZCKOZCIECCCGZHZUTLVBCCMNUSVCACIUQCHZURVBCVDURVBHUQUQCCPQRSUAK
|
|
CEVAKKUGCKUHUBUCUDCKUEUFBFZURHZAIJZVEKOZTUTVATBCDUIVECHZVGUTVHVAVIVFUSAIV
|
|
ECURUJUKVECKULUMBAUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A n x $.
|
|
$( An even finite cardinal is a finite cardinal. (Contributed by SF,
|
|
20-Jan-2015.) $)
|
|
evennn $p |- ( A e. Even[fin] -> A e. Nn ) $=
|
|
( vn vx cevenfin wcel cv cplc wceq cnnc wrex c0 wne eqeq1 rexbidv anbi12d
|
|
wa neeq1 df-evenfin elab2g syl wi nncaddccl anidms eleq1a rexlimiv adantr
|
|
ibi ) ADEZABFZUIGZHZBIJZAKLZPZAIEZUHUNCFZUJHZBIJZUPKLZPUNCADDUPAHZURULUSU
|
|
MUTUQUKBIUPAUJMNUPAKQOCBRSUGULUOUMUKUOBIUIIEZUJIEZUKUOUAVAVBUIUIUBUCUJIAU
|
|
DTUEUFT $.
|
|
$}
|
|
|
|
${
|
|
$d A n x $.
|
|
$( An odd finite cardinal is a finite cardinal. (Contributed by SF,
|
|
20-Jan-2015.) $)
|
|
oddnn $p |- ( A e. Odd[fin] -> A e. Nn ) $=
|
|
( vn vx coddfin wcel cv cplc c1c wceq cnnc wrex c0 wa eqeq1 rexbidv neeq1
|
|
wne anbi12d df-oddfin elab2g ibi wi nncaddccl anidms peano2 3syl rexlimiv
|
|
eleq1a adantr syl ) ADEZABFZULGZHGZIZBJKZALQZMZAJEZUKURCFZUNIZBJKZUTLQZMU
|
|
RCADDUTAIZVBUPVCUQVDVAUOBJUTAUNNOUTALPRCBSTUAUPUSUQUOUSBJULJEZUMJEZUNJEUO
|
|
USUBVEVFULULUCUDUMUEUNJAUHUFUGUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A n x $.
|
|
$( An even number is non-empty. (Contributed by SF, 22-Jan-2015.) $)
|
|
evennnul $p |- ( A e. Even[fin] -> A =/= (/) ) $=
|
|
( vn vx cevenfin wcel cv cplc wceq cnnc wrex c0 wne eqeq1 rexbidv anbi12d
|
|
wa neeq1 df-evenfin elab2g ibi simprd ) ADEZABFZUCGZHZBIJZAKLZUBUFUGPZCFZ
|
|
UDHZBIJZUIKLZPUHCADDUIAHZUKUFULUGUMUJUEBIUIAUDMNUIAKQOCBRSTUA $.
|
|
|
|
$( An odd number is non-empty. (Contributed by SF, 22-Jan-2015.) $)
|
|
oddnnul $p |- ( A e. Odd[fin] -> A =/= (/) ) $=
|
|
( vn vx coddfin wcel cv cplc c1c wceq cnnc wrex c0 wa eqeq1 rexbidv neeq1
|
|
wne anbi12d df-oddfin elab2g ibi simprd ) ADEZABFZUDGHGZIZBJKZALQZUCUGUHM
|
|
ZCFZUEIZBJKZUJLQZMUICADDUJAIZULUGUMUHUNUKUFBJUJAUENOUJALPRCBSTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A m x $.
|
|
$( The successor of an even natural is odd. (Contributed by SF,
|
|
20-Jan-2015.) $)
|
|
sucevenodd $p |- ( ( A e. Even[fin] /\ ( A +c 1c ) =/= (/) ) ->
|
|
( A +c 1c ) e. Odd[fin] ) $=
|
|
( vm vx cevenfin wcel c1c cplc c0 wa coddfin wceq cnnc wrex eqeq1 rexbidv
|
|
wne cv neeq1 anbi12d cvv df-evenfin elab2g ibi addceq1 reximi adantr 1cex
|
|
syl anim1i wb addcexg mpan2 df-oddfin mpbird ) ADEZAFGZHPZIUPJEZUPBQZUSGZ
|
|
FGZKZBLMZUQIZUOVCUQUOAUTKZBLMZAHPZIZVCUOVHCQZUTKZBLMZVIHPZIVHCADDVIAKZVKV
|
|
FVLVGVMVJVEBLVIAUTNOVIAHRSCBUAUBUCVFVCVGVEVBBLAUTFUDUEUFUHUIUOURVDUJZUQUO
|
|
UPTEZVNUOFTEVOUGAFDTUKULVIVAKZBLMZVLIVDCUPJTVIUPKZVQVCVLUQVRVPVBBLVIUPVAN
|
|
OVIUPHRSCBUMUBUHUFUN $.
|
|
$}
|
|
|
|
${
|
|
$d A m n x $.
|
|
$( The successor of an odd natural is even. (Contributed by SF,
|
|
22-Jan-2015.) $)
|
|
sucoddeven $p |- ( ( A e. Odd[fin] /\ ( A +c 1c ) =/= (/) ) ->
|
|
( A +c 1c ) e. Even[fin] ) $=
|
|
( vm vn vx coddfin wcel c1c cplc c0 wa cevenfin cv wceq cnnc wrex rexbidv
|
|
wne eqeq1 syl cvv neeq1 anbi12d df-oddfin elab2g wi peano2 addc32 addcass
|
|
addceq1i eqtri addceq12 anidms eqeq2d rspcev mpan2 addceq1 eqeq1d biimprd
|
|
ibi com12 rexlimiv adantr anim1i wb 1cex addcexg df-evenfin mpbird ) AEFZ
|
|
AGHZIQZJVJKFZVJBLZVMHZMZBNOZVKJZVIVPVKVIACLZVRHGHZMZCNOZAIQZJZVPVIWCDLZVS
|
|
MZCNOZWDIQZJWCDAEEWDAMZWFWAWGWBWHWEVTCNWDAVSRPWDAIUAUBDCUCUDUSWAVPWBVTVPC
|
|
NVRNFZVSGHZVNMZBNOZVTVPUEWIVRGHZNFZWLVRUFWNWJWMWMHZMZWLWJWMVRHZGHWOVSWQGV
|
|
RVRGUGUIWMVRGUHUJWKWPBWMNVMWMMZVNWOWJWRVNWOMVMVMWMWMUKULUMUNUOSVTWLVPVTVP
|
|
WLVTVOWKBNVTVJWJVNAVSGUPUQPURUTSVAVBSVCVIVLVQVDZVKVIVJTFZWSVIGTFWTVEAGETV
|
|
FUOWDVNMZBNOZWGJVQDVJKTWDVJMZXBVPWGVKXCXAVOBNWDVJVNRPWDVJIUAUBDBVGUDSVBVH
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x n $.
|
|
$( Alternate definition of even number. (Contributed by SF,
|
|
25-Jan-2015.) $)
|
|
dfevenfin2 $p |- Even[fin] =
|
|
{ x | E. n e. Nn ( x = ( n +c n ) /\ ( n +c n ) =/= (/) ) } $=
|
|
( cevenfin cv cplc wceq cnnc wrex c0 wne df-evenfin r19.41v neeq1 pm5.32i
|
|
wa cab rexbii bitr3i abbii eqtri ) CADZBDZUBEZFZBGHUAIJZOZAPUDUCIJZOZBGHZ
|
|
APABKUFUIAUFUDUEOZBGHUIUDUEBGLUJUHBGUDUEUGUAUCIMNQRST $.
|
|
|
|
$( Alternate definition of odd number. (Contributed by SF,
|
|
25-Jan-2015.) $)
|
|
dfoddfin2 $p |- Odd[fin] =
|
|
{ x | E. n e. Nn ( x = ( ( n +c n ) +c 1c ) /\
|
|
( ( n +c n ) +c 1c ) =/= (/) ) } $=
|
|
( coddfin cv cplc c1c wceq cnnc wrex c0 wne cab df-oddfin r19.41v pm5.32i
|
|
wa neeq1 rexbii bitr3i abbii eqtri ) CADZBDZUCEFEZGZBHIUBJKZPZALUEUDJKZPZ
|
|
BHIZALABMUGUJAUGUEUFPZBHIUJUEUFBHNUKUIBHUEUFUHUBUDJQORSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x k n m $.
|
|
$( Every non-empty finite cardinal is either even or odd. Theorem X.1.35
|
|
of [Rosser] p. 529. (Contributed by SF, 20-Jan-2015.) $)
|
|
evenoddnnnul $p |- ( Even[fin] u. Odd[fin] ) = ( Nn \ { (/) } ) $=
|
|
( vx vn vm vk cevenfin coddfin cnnc c0 wss cv wcel wne ssriv wi c0c neeq1
|
|
c1c wo eleq1 imbi12d cun cdif wa evennn evennnul eldifsn sylanbrc oddnnul
|
|
csn oddnn pm3.2i unss mpbi cplc cab cvv wn wceq elsnc df-ne con2bii bitri
|
|
vex orbi1i elun imor 3bitr4i abbi2i snex evenfinex oddfinex unex eqeltrri
|
|
weq ssun1 0ceven sselii a1i addcnnul simpld sucevenodd sucoddeven orim12d
|
|
expcom orcom 3imtr4g embantd com12 finds imp sylbi eqssi ) EFUAZGHUIZUBZE
|
|
WOIZFWOIZUCWMWOIWPWQAEWOAJZEKWRGKZWRHLZWRWOKZWRUDWRUEWRGHUFZUGMAFWOWRFKWS
|
|
WTXAWRUJWRUHXBUGMUKEFWOULUMBWOWMBJZWOKXCGKZXCHLZUCXCWMKZXCGHUFXDXEXFCJZHL
|
|
ZXGWMKZNZOHLZOWMKZNDJZHLZXMWMKZNZXMQUNZHLZXQWMKZNZXEXFNCDXCWNWMUAZXJCUOUP
|
|
XJCYAXGWNKZXIRXHUQZXIRXGYAKXJYBYCXIYBXGHURZYCXGHCVCUSXHYDXGHUTVAVBVDXGWNW
|
|
MVEXHXIVFVGVHWNWMHVIEFVJVKVLVLVMXGOURXHXKXIXLXGOHPXGOWMSTCDVNXHXNXIXOXGXM
|
|
HPXGXMWMSTXGXQURXHXRXIXSXGXQHPXGXQWMSTCBVNXHXEXIXFXGXCHPXGXCWMSTXLXKEWMOE
|
|
FVOVPVQVRXPXTNXMGKXRXPXSXRXNXOXSXRXNQHLXMQVSVTXRXMEKZXMFKZRXQFKZXQEKZRZXO
|
|
XSXRYEYGYFYHYEXRYGXMWAWDYFXRYHXMWBWDWCXMEFVEXSYHYGRYIXQEFVEYHYGWEVBWFWGWH
|
|
VRWIWJWKMWL $.
|
|
$}
|
|
|
|
${
|
|
$d j n x a b y t $.
|
|
$( Lemma for ~ evenodddisj . Establish stratification for induction.
|
|
(Contributed by SF, 25-Jan-2015.) $)
|
|
evenodddisjlem1 $p |- { j | ( ( j +c j ) =/= (/) -> A.
|
|
n e. Nn ( ( ( n +c n ) +c 1c ) =/= (/) -> ( j +c j ) =/= ( ( n +c n ) +c 1c ) )
|
|
) } e. _V $=
|
|
( vx vy va vb vt cssetk csn wceq wn wcel copk wex wrex exbii 3bitr4i snex
|
|
wa 3bitri cins3k cins2k cin c1c cpw1 cimak ccompl csik csymdif ccnvk cdif
|
|
cun c0 cimagek cidk cvv cxpk cnnc cv cplc wne wi cab wo df-or vex elimakv
|
|
wral elin wel wal opkex elimak elpw121c anbi1i 19.41v bitr4i df-rex excom
|
|
wb opkeq1 ceqsexv elsymdif otkelins3k elssetk bitri otkelins2k opksnelsik
|
|
eleq1d elpw141c ndisjrelk notbii elcompl con2bii elpw171c orbi12i bibi12i
|
|
df-ne elun dfcleq alex anbi12i rexcom df-addc eqeq2i abeq2 opkelxpk elsnc
|
|
weq elpw11c opkelcnvk opkelimagek dfaddc2 addcex addceq1 eqeq2d opkelidkg
|
|
mpbiran2 eldif mp2an equcom 1cex eqeq2 eqeq1 notbid anbi12d annim ssetkex
|
|
rexbii ins3kex ins2kex inex pw1ex imakex complex sikex unex symdifex vvex
|
|
xpkex imbi12i con34b ralbii dfral2 neeq1 imbi2d ralbidv orbi12d addcexlem
|
|
imbi1i abbi2i cnvkex imagekex idkex difex nncex eqeltrri ) HUAZHUBZUUSUBZ
|
|
UURUUSUCZUDUEZUEZUFZUGZUHZUHZUHZUAZHUHZUAZUBZUBZUVJUHZUHZUHZUHZUAZUVOUAZU
|
|
BZULZUIZUVCUEZUEZUEZUEZUEZUFZUGZUCZUCZUWDUFZUCZUVCUFZUBZUIZUVCUFZUGZUMIZU
|
|
URUUSUUTUVDUHZUHZUHZUJZUAZUGZUVMUVTUVRULZUIZUWGUFZUGZUCZUCZUWDUFZUCZUVCUF
|
|
ZUBZUIZUVCUFZUGZUBZUVEUAUUTUURUBUVNUAULUIUWDUFUKZUVCUFZUNZUAZUCZUVBUFZUAZ
|
|
UOUWSUPUQZUKZUBZUCZUVBUFZURUFZUGZULZUPUQZUCZUPUFZAUSZUYRUTZUMVAZBUSZVUAUT
|
|
ZUDUTZUMVAZUYSVUCVAZVBZBURVHZVBZAVCUPVUHAUYQUYSUMJZVUGVDZVUIKZVUGVBUYRUYQ
|
|
LZVUHVUIVUGVEVULCUSZUYRMZUYPLZCNVUMUYSJZVUMUMJZVUDVUMVUCVAZVBZBURVHZVDZSZ
|
|
CNVUJCUYPUYRAVFZVGVUOVVBCVUOVUNUWRLZVUNUYOLZSVVBVUNUWRUYOVIVVDVUPVVEVVAVV
|
|
DDCVJZEUSZFUSZUCZUMJZDUSZVVGVVHULZJZSZFUYROEUYROZVTZDVKZVUPVUNUWQLZKVVPKZ
|
|
DNZKVVDVVQVVRVVTVVRGUSZVUNMZUWPLZGUVCOZVWAVVKIZIZIZJZVWCSZGNZDNZVVTGUWPUV
|
|
CVUNVUMUYRVLZVMVWAUVCLZVWCSZGNVWIDNZGNVWDVWKVWNVWOGVWNVWHDNZVWCSVWOVWMVWP
|
|
VWCDVWAVNZVOVWHVWCDVPVQPVWCGUVCVRVWIDGVSQVWJVVSDVWJVWGVUNMZUWPLZVWRUURLZV
|
|
WRUWOLZVTZKVVSVWCVWSGVWGVWFRZVWHVWBVWRUWPVWAVWGVUNWAWIWBVWRUURUWOWCVXBVVP
|
|
VWTVVFVXAVVOVWTVWEVUMMHLZVVFVWEVUMUYRHVVKRZCVFZVVCWDVVKVUMDVFZVXFWEZWFVXA
|
|
VWEUYRMZUWNLZVVNEUYROZFUYROZVVOVWEVUMUYRUWNVXEVXFVVCWGVXJFAVJZVXKSZFNZVXL
|
|
VXJVWAVXIMZUWMLZGUVCOZVWAVVHIZIZIZJZVXQSZGNZFNZVXOGUWMUVCVXIVWEUYRVLZVMVW
|
|
MVXQSZGNVYCFNZGNVXRVYEVYGVYHGVYGVYBFNZVXQSVYHVWMVYIVXQFVWAVNVOVYBVXQFVPVQ
|
|
PVXQGUVCVRVYCFGVSQVYDVXNFVYDVYAVXIMZUWMLZVYJUUSLZVYJUWLLZSVXNVXQVYKGVYAVX
|
|
TRZVYBVXPVYJUWMVWAVYAVXIWAWIWBVYJUUSUWLVIVYLVXMVYMVXKVYLVXSUYRMHLVXMVXSVW
|
|
EUYRHVVHRZVXEVVCWGVVHUYRFVFZVVCWEWFVYMEAVJZVVNSZENZVXKVYMVWAVYJMZUWKLZGUW
|
|
DOZVWAVVGIZIZIZIZIZJZWUASZGNZENZVYSGUWKUWDVYJVYAVXIVLZVMVWAUWDLZWUASZGNWU
|
|
IENZGNWUBWUKWUNWUOGWUNWUHENZWUASWUOWUMWUPWUAEVWAWJVOWUHWUAEVPVQPWUAGUWDVR
|
|
WUIEGVSQWUJVYREWUJWUGVYJMZUWKLZWUQUUTLZWUQUWJLZSVYRWUAWURGWUGWUFRZWUHVYTW
|
|
UQUWKVWAWUGVYJWAWIWBWUQUUTUWJVIWUSVYQWUTVVNWUSWUEVXIMUUSLWUCUYRMHLVYQWUEV
|
|
YAVXIUUSWUDRZVYNVYFWGWUCVWEUYRHVVGRZVXEVVCWGVVGUYREVFZVVCWETWUTWUQUVILZWU
|
|
QUWILZSVVNWUQUVIUWIVIWVEVVJWVFVVMWVEWUEVYAMZUVHLWUDVXTMZUVGLZVVJWUEVYAVXI
|
|
UVHWVBVYNVYFWDWUDVXTUVGWUCRZVXSRZWHWVIWUCVXSMZUVFLVVGVVHMZUVELZVVJWUCVXSU
|
|
VFWVCVYOWHVVGVVHUVEWVDVYPWHWVMUVDLZKVVIUMVAZKZWVNVVJWVOWVPVVGVVHWVDVYPWKZ
|
|
WLWVMUVDVVGVVHVLWMWVPVVJVVIUMWRWNZQTTWUQUWHLZKCDVJZVUMVVLLZVTZKZCNZKZWVFV
|
|
VMWVTWWEWVTVWAWUQMZUWBLZGUWGOZVWAVUMIZIZIZIZIZIZIZIZJZWWHSZGNZCNZWWEGUWBU
|
|
WGWUQWUGVYJVLZVMVWAUWGLZWWHSZGNWWSCNZGNWWIWXAWXDWXEGWXDWWRCNZWWHSWXEWXCWX
|
|
FWWHCVWAWOZVOWWRWWHCVPVQPWWHGUWGVRWWSCGVSQWWTWWDCWWTWWQWUQMZUWBLZWXHUVMLZ
|
|
WXHUWALZVTZKWWDWWHWXIGWWQWWPRZWWRWWGWXHUWBVWAWWQWUQWAWIWBWXHUVMUWAWCWXLWW
|
|
CWXJWWAWXKWWBWXJWWOVYJMZUVLLWWMVXIMUVKLZWWAWWOWUGVYJUVLWWNRZWVAWULWGWWMVY
|
|
AVXIUVKWWLRZVYNVYFWGWXOWWKVWEMUVJLZWWJVVKMHLZWWAWWKVWEUYRUVJWWJRZVXEVVCWD
|
|
WWJVVKHVUMRZVXGWHZVUMVVKVXFVXGWEZTTWXHUVRLZWXHUVTLZVDCEVJZCFVJZVDZWXKWWBW
|
|
YDWYFWYEWYGWYDWWOWUGMUVQLWWNWUFMUVPLZWYFWWOWUGVYJUVQWXPWVAWULWDWWNWUFUVPW
|
|
WMRZWUERWHWYIWWMWUEMUVOLZWWLWUDMUVNLZWYFWWMWUEUVOWXQWVBWHWWLWUDUVNWWKRZWV
|
|
JWHZWYLWWKWUCMUVJLZWWJVVGMHLZWYFWWKWUCUVJWXTWVCWHZWWJVVGHWYAWVDWHZVUMVVGV
|
|
XFWVDWEZTTTWYEWXNUVSLWWMVYAMUVOLZWYGWWOWUGVYJUVSWXPWVAWULWGWWMVYAVXIUVOWX
|
|
QVYNVYFWDWYTWWLVXTMUVNLZWWKVXSMUVJLZWYGWWLVXTUVNWYMWVKWHZWWKVXSUVJWXTVYOW
|
|
HZXUBWWJVVHMHLZWYGWWJVVHHWYAVYPWHZVUMVVHVXFVYPWEZWFTTWPWXHUVRUVTWSVUMVVGV
|
|
VHWSZQWQWLTPTWLWUQUWHWXBWMVVMWWCCVKWWFCVVKVVLWTWWCCXAWFZQXBWFXBTPTVVNEUYR
|
|
VRVQXBTPTVXKFUYRVRVQVVNFEUYRUYRXCTWQWLTPTWLVUNUWQVWLWMVVPDXAQVUPVUMVVODVC
|
|
ZJVVQUYSXUJVUMDEFUYRUYRXDXEVVODVUMXFWFVQVVEVUMUYNLZVUMUWSLZVUMUYMLZVDVVAV
|
|
VEXUKUYRUPLVVCVUMUYRUYNUPVXFVVCXGXRVUMUWSUYMWSXULVUQXUMVUTVUMUMVXFXHVUMUY
|
|
LLZKVUMVUCJZVUCUMJZVBZKZBUROZKZXUMVUTXUNXUSXUNVUAVUMMZUYKLZBUROXUSBUYKURV
|
|
UMVXFVMXVBXURBURXVBVVKVUCJZCDXIZVVKUMJZKZSZSZDNZXUOXUPKZSZXURXVBVWAXVAMZU
|
|
YJLZGUVBOZVWAVWFJZXVMSZGNZDNZXVIGUYJUVBXVAVUAVUMVLVMVWAUVBLZXVMSZGNXVPDNZ
|
|
GNXVNXVRXVTXWAGXVTXVODNZXVMSXWAXVSXWBXVMDVWAXJVOXVOXVMDVPVQPXVMGUVBVRXVPD
|
|
GVSQXVQXVHDXVQVWFXVAMZUYJLZXWCUYFLZXWCUYILZSXVHXVMXWDGVWFVWERXVOXVLXWCUYJ
|
|
VWAVWFXVAWAWIWBXWCUYFUYIVIXWEXVCXWFXVGXWEVVKVUAMZUYELZVUMVUBJZVVKVUMUDUTZ
|
|
JZSZCNZXVCVVKVUAVUMUYEVXGBVFZVXFWDXWHVWAXWGMZUYDLZGUVBOZVWAWWKJZXWPSZGNZC
|
|
NZXWMGUYDUVBXWGVVKVUAVLVMXVSXWPSZGNXWSCNZGNXWQXXAXXBXXCGXXBXWRCNZXWPSXXCX
|
|
VSXXDXWPCVWAXJVOXWRXWPCVPVQPXWPGUVBVRXWSCGVSQXWTXWLCXWTWWKXWGMZUYDLZXXEUX
|
|
SLZXXEUYCLZSXWLXWPXXFGWWKWXTXWRXWOXXEUYDVWAWWKXWGWAWIWBXXEUXSUYCVIXXGXWIX
|
|
XHXWKVUMVUAMZUXRLZVVFVVNFVUAOZEVUAOZVTZDVKZXXGXWIXXIUXQLZKXXMKZDNZKXXJXXN
|
|
XXOXXQXXOVWAXXIMZUXPLZGUVCOZVWHXXSSZGNZDNZXXQGUXPUVCXXIVUMVUAVLZVMVWMXXSS
|
|
ZGNXYADNZGNXXTXYCXYEXYFGXYEVWPXXSSXYFVWMVWPXXSVWQVOVWHXXSDVPVQPXXSGUVCVRX
|
|
YADGVSQXYBXXPDXYBVWGXXIMZUXPLZXYGUURLZXYGUXOLZVTZKXXPXXSXYHGVWGVXCVWHXXRX
|
|
YGUXPVWAVWGXXIWAWIWBXYGUURUXOWCXYKXXMXYIVVFXYJXXLXYIVXDVVFVWEVUMVUAHVXEVX
|
|
FXWNWDVXHWFVWEVUAMZUXNLZEBVJZXXKSZENZXYJXXLXYMVWAXYLMZUXMLZGUVCOZVWAWUEJZ
|
|
XYRSZGNZENZXYPGUXMUVCXYLVWEVUAVLZVMVWMXYRSZGNYUAENZGNXYSYUCYUEYUFGYUEXYTE
|
|
NZXYRSYUFVWMYUGXYREVWAVNVOXYTXYREVPVQPXYRGUVCVRYUAEGVSQYUBXYOEYUBWUEXYLMZ
|
|
UXMLZYUHUUSLZYUHUXLLZSXYOXYRYUIGWUEWVBXYTXYQYUHUXMVWAWUEXYLWAWIWBYUHUUSUX
|
|
LVIYUJXYNYUKXXKYUJWUCVUAMHLXYNWUCVWEVUAHWVCVXEXWNWGVVGVUAWVDXWNWEWFYUKFBV
|
|
JZVVNSZFNZXXKYUKVWAYUHMZUXKLZGUWDOZVWAVYAIZIZJZYUPSZGNZFNZYUNGUXKUWDYUHWU
|
|
EXYLVLZVMWUMYUPSZGNYVAFNZGNYUQYVCYVEYVFGYVEYUTFNZYUPSYVFWUMYVGYUPFVWAWJVO
|
|
YUTYUPFVPVQPYUPGUWDVRYVAFGVSQYVBYUMFYVBYUSYUHMZUXKLZYVHUUTLZYVHUXJLZSYUMY
|
|
UPYVIGYUSYURRZYUTYUOYVHUXKVWAYUSYUHWAWIWBYVHUUTUXJVIYVJYULYVKVVNYVJVYAXYL
|
|
MUUSLVXSVUAMHLYULVYAWUEXYLUUSVYNWVBYUDWGVXSVWEVUAHVYOVXEXWNWGVVHVUAVYPXWN
|
|
WETYVKYVHUXELZYVHUXILZSVVNYVHUXEUXIVIYVMVVJYVNVVMYVHUXDLZKWVQYVMVVJYVOWVP
|
|
YVOVYAWUEMUXCLWVGUXBLZWVPVYAWUEXYLUXCVYNWVBYUDWDVYAWUEUXBVYNWVBXKYVPWVHUX
|
|
ALWVLUWTLZWVPWUDVXTUXAWVJWVKWHWUCVXSUWTWVCVYOWHYVQWVOWVPVVGVVHUVDWVDVYPWH
|
|
WVRWFTTWLYVHUXDYUSYUHVLZWMWVSQYVHUXHLZKWWFYVNVVMYVSWWEYVSVWAYVHMZUXGLZGUW
|
|
GOZWWRYWASZGNZCNZWWEGUXGUWGYVHYVRVMWXCYWASZGNYWCCNZGNYWBYWEYWFYWGGYWFWXFY
|
|
WASYWGWXCWXFYWAWXGVOWWRYWACVPVQPYWAGUWGVRYWCCGVSQYWDWWDCYWDWWQYVHMZUXGLZY
|
|
WHUVMLZYWHUXFLZVTZKWWDYWAYWIGWWQWXMWWRYVTYWHUXGVWAWWQYVHWAWIWBYWHUVMUXFWC
|
|
YWLWWCYWJWWAYWKWWBYWJWWOYUHMZUVLLWWMXYLMUVKLZWWAWWOYUSYUHUVLWXPYVLYVDWGWW
|
|
MWUEXYLUVKWXQWVBYUDWGYWNWXRWXSWWAWWKVWEVUAUVJWXTVXEXWNWDWYBWYCTTYWHUVTLZY
|
|
WHUVRLZVDWYHYWKWWBYWOWYFYWPWYGYWOYWMUVSLWYKWYFWWOYUSYUHUVSWXPYVLYVDWGWWMW
|
|
UEXYLUVOWXQWVBYUDWDWYKWYLWYOWYFWYNWYQWYOWYPWYFWYRWYSWFTTYWPWWOYUSMUVQLWWN
|
|
YURMUVPLZWYGWWOYUSYUHUVQWXPYVLYVDWDWWNYURUVPWYJVYARWHYWQWYTXUAWYGWWMVYAUV
|
|
OWXQVYNWHXUCXUAXUBXUEWYGXUDXUFXUGTTTWPYWHUVTUVRWSXUHQWQWLTPTWLYVHUXHYVRWM
|
|
XUIQXBWFXBTPTVVNFVUAVRVQXBTPTVWEVUMVUAUXNVXEVXFXWNWGXXKEVUAVRQWQWLTPTWLXX
|
|
IUXQXYDWMXXMDXAQVUMVVKVUAUXRVXFVXGXWNWGXWIVUMXXLDVCZJXXNVUBYWRVUMDEFVUAVU
|
|
AXDXEXXLDVUMXFWFQVUMVVKMUYBLVVKUYAVUMUFZJXXHXWKVUMVVKUYAVXFVXGXLVUMVVKVUA
|
|
UYBVXFVXGXWNWDXWJYWSVVKVUMUDXMXEQXBTPTXWKXVCCVUBVUAVUAXWNXWNXNZXWIXWJVUCV
|
|
VKVUMVUBUDXOXPWBTXWFVVKVUMMZUYHLYXAUOLZYXAUYGLZKZSXVGVVKVUAVUMUYHVXGXWNVX
|
|
FWGYXAUOUYGXSYXBXVDYXDXVFYXBDCXIZXVDVVKUPLVUMUPLZYXBYXEVTVXGVXFVVKVUMUPUP
|
|
XQXTDCYAWFYXCXVEYXCVVKUWSLZXVEYXCYXGYXFVXFVVKVUMUWSUPVXGVXFXGXRVVKUMVXGXH
|
|
WFWLXBTXBTPTXVGXVKDVUCVUBUDYWTYBXNXVCXVDXUOXVFXVJVVKVUCVUMYCXVCXVEXUPVVKV
|
|
UCUMYDYEYFWBXUOXUPYGTYIWFWLVUMUYLVXFWMVUTXUQBURVHXUTVUSXUQBURVUSXVJXUOKZV
|
|
BXUQVUDXVJVURYXHVUCUMWRVUMVUCWRUUAXUOXUPUUBVQUUCXUQBURUUDWFQWPTXBWFPVVAVU
|
|
JCUYSUYRUYRVVCVVCXNVUPVUQVUIVUTVUGVUMUYSUMYDVUPVUSVUFBURVUPVURVUEVUDVUMUY
|
|
SVUCUUEUUFUUGUUHWBTUYTVUKVUGUYSUMWRUUJQUUKUYPUPUWRUYOUWQUWPUVCUURUWOHYHYJ
|
|
ZUWNUWMUVCUUSUWLHYHYKZUWKUWDUUTUWJUUSYXJYKZUVIUWIUVHUVGUVFUVEUVDUVAUVCUUR
|
|
UUSYXIYXJYLUVBUDYBYMZYMZYNZYOYPYPYPYJUWHUWBUWGUVMUWAUVLUVKUVJHYHYPZYJYKYK
|
|
ZUVRUVTUVQUVPUVOUVNUVJYXOYPYPZYPYPYJZUVSUVOYXQYJYKZYQYRUWFUWEUWDUWCUVCYXM
|
|
YMYMZYMYMYMZYNYOYLYLYXTYNYLYXMYNYKYRYXMYNYOUYNUPUWSUYMUMRZUYLUYKURUYJUVBU
|
|
YFUYIUYEUYDUVBUXSUYCUXRUXQUXPUVCUURUXOYXIUXNUXMUVCUUSUXLYXJUXKUWDUUTUXJYX
|
|
KUXEUXIUXDUXCUXBUXAUWTUVDYXNYPYPYPUULYJYOUXHUXGUWGUVMUXFYXPUVTUVRYXSYXRYQ
|
|
YRYYAYNYOYLYLYXTYNYLYXMYNYKYRYXMYNYOYKUYBUYAUXTUVCUUIYXMYNUUMYJYLYXLYNYJU
|
|
YHUOUYGUUNUWSUPYYBYSYTUUOYKYLYXLYNUUPYNYOYQYSYTYLYSYNUUQ $.
|
|
$}
|
|
|
|
${
|
|
$d m n $. $d m x $. $d n x $. $d k n $. $d k m $. $d k p $. $d k q $.
|
|
$d n p $. $d n q $. $d p q $. $d j k $. $d j m $. $d j n $. $d j p $.
|
|
$d k x $. $d m p $. $d m q $.
|
|
|
|
$( The even finite cardinals and the odd ones are disjoint. Theorem X.1.36
|
|
of [Rosser] p. 529. (Contributed by SF, 22-Jan-2015.) $)
|
|
evenodddisj $p |- ( Even[fin] i^i Odd[fin] ) = (/) $=
|
|
( vx vn vp cv cplc wceq c0 wne wa cnnc c1c wn wi wcel c0c addceq12 anidms
|
|
wral neeq1d imbi12d vk vj vm vq cevenfin coddfin cin dfevenfin2 dfoddfin2
|
|
wrex cab ineq12i inab eqtri evenodddisjlem1 addcid2 syl6eq imbi2d ralbidv
|
|
addceq1d neeq2d cbvralv syl6bb addc32 addceq1i eqtr3i 0cnsuc necomi rgenw
|
|
weq addcass a1i neeq1i addcnnul simpld sylbi adantl simprl nnc0suc mpbiri
|
|
wo sylib simpr rspcv addc4 adantr eqeq2i simplll nncaddccl simplrr peano2
|
|
syl 3syl simpllr prepeano4 syl22anc ex syl5bir necon3d embantd syld com23
|
|
expr imp31 com12 anbi2d syl5ibrcom rexlimiv adantrl mpd simplrl ralrimiva
|
|
finds df-ne imbi2i con2b bitri ralbii ralnex syl6ib anbi1d rexbidv notbid
|
|
jaod imnan eqeq1 imp3a mpbi abf ) UEUFUGZADZUADZYLEZFZYMGHZIZUAJUJZYKBDZY
|
|
REZKEZFZYTGHZIZBJUJZIZAUKZGYJYQAUKZUUDAUKZUGUUFUEUUGUFUUHAUAUHABUIULYQUUD
|
|
AUMUNUUEAYQUUDLZMUUELYPUUIUAJYLJNZYNYOUUIUUJYOUUIMYNYOYMYTFZUUBIZBJUJZLZM
|
|
UUJYOUUBYMYTHZMZBJRZUUNUBDZUUREZGHZUUBUUSYTHZMZBJRZMOGHZUUBOYTHZMZBJRZMUC
|
|
DZUVHEZGHZCDZUVKEZKEZGHZUVIUVMHZMZCJRZMZUVIKEZKEZGHZUUBUVTYTHZMZBJRZMYOUU
|
|
QMUBUCYLUBBUOUUROFZUUTUVDUVCUVGUWEUUSOGUWEUUSOOEZOUWEUUSUWFFUURUUROOPQOUP
|
|
ZUQZSUWEUVBUVFBJUWEUVAUVEUUBUWEUUSOYTUWHSURUSTUBUCVJZUUTUVJUVCUVQUWIUUSUV
|
|
IGUWIUUSUVIFUURUURUVHUVHPQZSUWIUVCUUBUVIYTHZMZBJRUVQUWIUVBUWLBJUWIUVAUWKU
|
|
UBUWIUUSUVIYTUWJSURUSUWLUVPBCJBCVJZUUBUVNUWKUVOUWMYTUVMGUWMYSUVLKUWMYSUVL
|
|
FYRYRUVKUVKPQUTZSUWMYTUVMUVIUWNVATVBVCTUURUVHKEZFZUUTUWAUVCUWDUWPUUSUVTGU
|
|
WPUUSUWOUWOEZUVTUWPUUSUWQFUURUURUWOUWOPQUWOUVHEZKEUWQUVTUWOUVHKVKUWRUVSKU
|
|
VHKUVHVDVEVFUQZSUWPUVBUWCBJUWPUVAUWBUUBUWPUUSUVTYTUWSSURUSTUBUAVJZUUTYOUV
|
|
CUUQUWTUUSYMGUWTUUSYMFUURUURYLYLPQZSUWTUVBUUPBJUWTUVAUUOUUBUWTUUSYMYTUXAS
|
|
URUSTUVGUVDUVFBJUVEUUBYTOYSVGVHVLVIVLUVHJNZUWAUVRUWDUXBUWAUVRUWDMUXBUWAIZ
|
|
UVJUVQUWDUWAUVJUXBUWAUVIKKEZEZGHZUVJUVTUXEGUVIKKVKVMUXFUVJUXDGHZUVIUXDVNV
|
|
OVPVQUXCUVQUWDUXCUVQIZUWCBJUXHYRJNZUUBUWBUXHUXIUUBIZIZUVSYSHZUWBUXKYROFZY
|
|
RUDDZKEZFZUDJUJZWAZUXLUXKUXIUXRUXHUXIUUBVRUDYRVSWBUXKUXMUXLUXQUXMUXLMUXKU
|
|
XMUXLUVSOHUVIVGUXMYSOUVSUXMYSUWFOUXMYSUWFFYRYROOPQUWGUQVAVTVLUXHUUBUXQUXL
|
|
MUXIUXQUXHUUBIZUXLUXPUXSUXLMZUDJUXNJNZUXTUXPUXHUXOUXOEZKEZGHZIZUVSUYBHZMU
|
|
YEUYAUYFUXCUVQUYDUYAUYFMZUXCUYDUVQUYGUXCUYDUVQUYGMUXCUYDIUYAUVQUYFUXCUYDU
|
|
YAUVQUYFMUXCUYDUYAIZIZUVQUXNUXNEZKEZGHZUVIUYKHZMZUYFUYIUYAUVQUYNMUYHUYAUX
|
|
CUYDUYAWCVQUVPUYNCUXNJCUDVJZUVNUYLUVOUYMUYOUVMUYKGUYOUVLUYJKUYOUVLUYJFUVK
|
|
UVKUXNUXNPQUTZSUYOUVMUYKUVIUYPVATWDWLUYIUYLUYMUYFUYHUYLUXCUYDUYLUYAUYDUYK
|
|
UXDEZGHZUYLUYCUYQGUYCUYJUXDEZKEUYQUYBUYSKUXNKUXNKWEVEUYJUXDKVDUNVMUYRUYLU
|
|
XGUYKUXDVNVOVPWFVQUYIUVSUYBUVIUYKUVSUYBFUVSUYKKEZFZUYIUVIUYKFZUYTUYBUVSUY
|
|
TUXOUXNEZKEUYBUYKVUCKUXNUXNKVDVEUXOUXNKVKUNWGUYIVUAVUBUYIVUAIZUVIJNZUYKJN
|
|
ZVUAUVSGHZVUBVUDUXBVUEUXBUWAUYHVUAWHUXBVUEUVHUVHWIQZWLVUDUYAUYJJNZVUFUXCU
|
|
YDUYAVUAWJUYAVUIUXNUXNWIQUYJWKWMUYIVUAWCVUDUWAVUGUXBUWAUYHVUAWNUWAVUGKGHU
|
|
VSKVNVOWLUVIUYKWOWPWQWRWSWTXAXCXBWQXBXDXEUXPUXSUYEUXLUYFUXPUUBUYDUXHUXPYT
|
|
UYCGUXPYSUYBKUXPYSUYBFYRYRUXOUXOPQZUTSXFUXPYSUYBUVSVUJVATXGXHXEXIYDXJUXKU
|
|
VTYTUVSYSUXKUVTYTFZUVSYSFZUXKVUKIZUVSJNZYSJNZVUKUWAVULVUMUXBVUEVUNUXKUXBV
|
|
UKUXBUWAUVQUXJWHWFVUHUVIWKWMVUMUXIVUOUXHUXIUUBVUKXKUXIVUOYRYRWIQWLUXKVUKW
|
|
CUXKUWAVUKUXBUWAUVQUXJWNWFUVSYSWOWPWQWSXJXCXLWQWTWQXBXMUUQUULLZBJRUUNUUPV
|
|
UPBJUUPUUKUUBLMZVUPUUPUUBUUKLZMVUQUUOVURUUBYMYTXNXOUUBUUKXPXQUUKUUBYEXQXR
|
|
UULBJXSXQXTYNUUIUUNYOYNUUDUUMYNUUCUULBJYNUUAUUKUUBYKYMYTYFYAYBYCURXGYGXHY
|
|
QUUDYEYHYIUN $.
|
|
$}
|
|
|
|
${
|
|
$d M m n x $.
|
|
$( If ` M ` is even , then so is ` _T[fin] M ` . Theorem X.1.37 of
|
|
[Rosser] p. 530. (Contributed by SF, 26-Jan-2015.) $)
|
|
eventfin $p |- ( M e. Even[fin] -> _T[fin] M e. Even[fin] ) $=
|
|
( vn vx vm cevenfin wcel cv cplc wceq cnnc wrex c0 wa ctfin eqeq1 rexbidv
|
|
wne neeq1 anbi12d wi df-evenfin elab2g ibi addceq2 addcnul1 cpw1 tfinprop
|
|
syl6eq necon3i simpld sylan2 tfindi 3anidm12 anidms eqeq2d rspcev syl2anc
|
|
addceq12 nncaddccl tfinnnul sylan jca tfinex sylibr tfineq eleq1d imbi12d
|
|
elab2 ex biimprd com12 syl rexlimiv imp ) AEFZABGZVPHZIZBJKZALQZMZANZEFZV
|
|
OWACGZVQIZBJKZWDLQZMWACAEEWDAIZWFVSWGVTWHWEVRBJWDAVQOPWDALRSCBUAUBUCVSVTW
|
|
CVRVTWCTZBJVPJFZVQLQZVQNZEFZTZVRWITWJWKWMWJWKMZWLDGZWPHZIZDJKZWLLQZMZWMWO
|
|
WSWTWOVPNZJFZWLXBXBHZIZWSWKWJVPLQZXCVPLVQLVPLIVQVPLHLVPLVPUDVPUEUHUIWJXFM
|
|
XCWDUFXBFCVPKVPCUGUJUKWJWKXEVPVPULUMWRXEDXBJWPXBIZWQXDWLXGWQXDIWPWPXBXBUR
|
|
UNUOUPUQWJVQJFZWKWTWJXHVPVPUSUNVQUTVAVBWDWQIZDJKZWGMXACWLEVQVCWDWLIZXJWSW
|
|
GWTXKXIWRDJWDWLWQOPWDWLLRSCDUAVHVDVIVRWNWIVRWIWNVRVTWKWCWMAVQLRVRWBWLEAVQ
|
|
VEVFVGVJVKVLVMVNVL $.
|
|
$}
|
|
|
|
${
|
|
$d M n m x $.
|
|
$( If ` M ` is odd , then so is ` _T[fin] M ` . Theorem X.1.38 of [Rosser]
|
|
p. 530. (Contributed by SF, 26-Jan-2015.) $)
|
|
oddtfin $p |- ( M e. Odd[fin] -> _T[fin] M e. Odd[fin] ) $=
|
|
( vn vx vm coddfin wcel cv cplc c1c wceq cnnc c0 wne wa ctfin eqeq1 neeq1
|
|
wrex wi syl rexbidv anbi12d df-oddfin elab2g ibi addceq2 addcnul1 addceq1
|
|
syl6eq addccom eqtri necon3i cpw1 tfinprop simpld sylan2 nncaddccl anidms
|
|
1cnnc tfindi mp3an2 sylan addcnnul 3anidm12 tfin1c addceq12 eqeq2d rspcev
|
|
mpan2 eqtrd syl2anc peano2 tfinnnul jca tfinex elab2 sylibr tfineq eleq1d
|
|
ex imbi12d biimprd com12 rexlimiv imp ) AEFZABGZWGHZIHZJZBKRZALMZNZAOZEFZ
|
|
WFWMCGZWIJZBKRZWPLMZNWMCAEEWPAJZWRWKWSWLWTWQWJBKWPAWIPUAWPALQUBCBUCUDUEWK
|
|
WLWOWJWLWOSZBKWGKFZWILMZWIOZEFZSZWJXASXBXCXEXBXCNZXDDGZXHHZIHZJZDKRZXDLMZ
|
|
NZXEXGXLXMXGWGOZKFZXDXOXOHZIHZJZXLXCXBWGLMZXPWGLWILWGLJZWILIHZLYAWHLJWIYB
|
|
JYAWHWGLHLWGLWGUFWGUGUIWHLIUHTYBILHLLIUJIUGUKUIULXBXTNXPWPUMXOFCWGRWGCUNU
|
|
OUPXGXDWHOZIOZHZXRXBWHKFZXCXDYEJZXBYFWGWGUQURZYFIKFXCYGUSWHIUTVAVBXGYCXQJ
|
|
ZYEXRJZXCXBWHLMZYIXCYKILMWHIVCUOXBYKYIWGWGUTVDUPYIYDIJYJVEYCYDXQIVFVITVJX
|
|
KXSDXOKXHXOJZXJXRXDYLXIXQJZXJXRJYLYMXHXHXOXOVFURXIXQIUHTVGVHVKXBWIKFZXCXM
|
|
XBYFYNYHWHVLTWIVMVBVNWPXJJZDKRZWSNXNCXDEWIVOWPXDJZYPXLWSXMYQYOXKDKWPXDXJP
|
|
UAWPXDLQUBCDUCVPVQVTWJXFXAWJXAXFWJWLXCWOXEAWILQWJWNXDEAWIVRVSWAWBWCTWDWET
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d n l y x b $. $d b t $. $d b z $. $d l t $. $d l z $. $d n t $.
|
|
$d n z $. $d t x $. $d t y $. $d t z $. $d x z $. $d y z $.
|
|
|
|
$( Lemma for ~ nnadjoin . Establish stratification. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
nnadjoinlem1 $p |- { n | A. l e. n ( y e. ~ U. l -> { x |
|
|
E. b e. l x = ( b u. { y } ) } e. n ) } e. _V $=
|
|
( vz vt cssetk wel cvv c1c csn wcel wn copk wa wex exbii 3bitr4i 3bitri
|
|
cab cpw1 cxpk cin ccompl cins3k cins2k cv cun csymdif csik cdif cuni wceq
|
|
cimak wrex wral snex opkeq1 eleq1d ceqsexv elin vex elssetk eldif elcompl
|
|
wi elimak el1c anbi1i 19.41v bitr4i df-rex excom opkelxpk mpbiran2 elequ2
|
|
snelpw1 elab anbi12i eluni xchbinx wb wal abeq2 opkex elpw121c otkelins3k
|
|
df-clel opksnelsik elsymdif bitri otkelins2k alex dfcleq wo elsnc orbi12i
|
|
sneqb elun bibi12i 3bitr4ri notbii annim dfral2 abbi2i ssetkex setswithex
|
|
weq pw1ex vvex xpkex inex 1cex imakex complex ins3kex unex symdifex sikex
|
|
ins2kex difex eqeltrri ) HBFIZFUAZUBZJUCZHUDZKUOZUBZUEZJUCZHUFZHUGZYNHBUH
|
|
ZLZLZJUCZUIZUFZUJZKUBZUBZUOZUEZUKZUFZUDZUUCUOZUGZUJZUUCUOZUEZUKZUFZYNUDZU
|
|
UCUOZULZUDZKUOZUEZYOEUHZUMZUEMZAUHZDUHZYPUIZUNZDUVBUPZAUAZCUHZMZVGZEUVKUQ
|
|
ZCUAJUVNCUVAUVKUUTMZNUVMNZEUVKUPZNUVKUVAMUVNUVOUVQGUHZUVBLZUNZUVRUVKOZUUS
|
|
MZPZGQZEQZECIZUVPPZEQUVOUVQUWDUWGEUWDUVSUVKOZUUSMZUWHHMZUWHUURMZPUWGUWBUW
|
|
IGUVSUVBURZUVTUWAUWHUUSUVRUVSUVKUSUTVAUWHHUURVBUWJUWFUWKUVPUVBUVKEVCZCVCZ
|
|
VDUWKUWHYLMZUWHUUQMZNZPUVDUVLNZPUVPUWHYLUUQVEUWOUVDUWQUWRUVSYKMZYOUVCMZNU
|
|
WOUVDUWSUVSYJMZUWTUVSYJUWLVFUVBYIMZBAIZAEIZPZAQZUXAUWTUXBUVRUVBOZYHMZGKUP
|
|
ZUVRUVELZUNZUXHPZGQZAQZUXFGYHKUVBUWMVHUVRKMZUXHPZGQUXLAQZGQUXIUXNUXPUXQGU
|
|
XPUXKAQZUXHPUXQUXOUXRUXHAUVRVIVJUXKUXHAVKVLRUXHGKVMUXLAGVNSUXMUXEAUXMUXJU
|
|
VBOZYHMZUXSYGMZUXSHMZPUXEUXHUXTGUXJUVEURZUXKUXGUXSYHUVRUXJUVBUSUTVAUXSYGH
|
|
VBUYAUXCUYBUXDUYAUXJYFMZUVEYEMUXCUYAUYDUVBJMUWMUXJUVBYFJUYCUWMVOVPUVEYEVR
|
|
YDUXCFUVEAVCZFABVQVSTUVEUVBUYEUWMVDVTTRTUVBYIVRAYOUVBWASWBUWOUWSUVKJMUWNU
|
|
VSUVKYKJUWLUWNVOVPYOUVCBVCVFSUWPUVLFUHZUVJUNZFCIZPZFQAFIZUVIWCZAWDZUYHPZF
|
|
QZUVLUWPUYIUYMFUYGUYLUYHUVIAUYFWEVJRFUVJUVKWIUWPUVRUWHOZUUPMZGUUCUPZUVRUY
|
|
FLZLZLZUNZUYPPZGQZFQZUYNGUUPUUCUWHUVSUVKWFVHUVRUUCMZUYPPZGQVUBFQZGQUYQVUD
|
|
VUFVUGGVUFVUAFQZUYPPVUGVUEVUHUYPFUVRWGZVJVUAUYPFVKVLRUYPGUUCVMVUBFGVNSVUC
|
|
UYMFVUCUYTUWHOZUUPMZVUJUUOMZVUJYNMZPUYMUYPVUKGUYTUYSURZVUAUYOVUJUUPUVRUYT
|
|
UWHUSUTVAVUJUUOYNVBVULUYLVUMUYHVULUYRUVSOUUNMUYFUVBOZUUMMZUYLUYRUVSUVKUUN
|
|
UYFURZUWLUWNWHUYFUVBUUMFVCZUWMWJVUOUULMZNUYKNZAQZNVUPUYLVUSVVAVUSUVRVUOOZ
|
|
UUKMZGUUCUPZUVRUXJLZLZUNZVVCPZGQZAQZVVAGUUKUUCVUOUYFUVBWFZVHVUEVVCPZGQVVH
|
|
AQZGQVVDVVJVVLVVMGVVLVVGAQZVVCPVVMVUEVVNVVCAUVRWGVJVVGVVCAVKVLRVVCGUUCVMV
|
|
VHAGVNSVVIVUTAVVIVVFVUOOZUUKMZVUTVVCVVPGVVFVVEURVVGVVBVVOUUKUVRVVFVUOUSUT
|
|
VAVVPVVOYMMZVVOUUJMZWCUYKVVOYMUUJWKVVQUYJVVRUVIVVQUXJUYFOHMUYJUXJUYFUVBHU
|
|
YCVURUWMWHUVEUYFUYEVURVDWLUXSUUIMZDEIZUVHPZDQZVVRUVIVVSUVRUXSOZUUHMZGUUCU
|
|
PZUVRUVFLZLZLZUNZVWDPZGQZDQZVWBGUUHUUCUXSUXJUVBWFVHVUEVWDPZGQVWJDQZGQVWEV
|
|
WLVWMVWNGVWMVWIDQZVWDPVWNVUEVWOVWDDUVRWGVJVWIVWDDVKVLRVWDGUUCVMVWJDGVNSVW
|
|
KVWADVWKVWHUXSOZUUHMZVWPYNMZVWPUUGMZPVWAVWDVWQGVWHVWGURVWIVWCVWPUUHUVRVWH
|
|
UXSUSUTVAVWPYNUUGVBVWRVVTVWSUVHVWRVWFUVBOHMVVTVWFUXJUVBHUVFURZUYCUWMWMUVF
|
|
UVBDVCZUWMVDWLVWSVWFUXJOUUFMUVFUVEOZUUEMZUVHVWFUXJUVBUUFVWTUYCUWMWHUVFUVE
|
|
UUEVXAUYEWJFAIZUYFUVGMZWCZFWDVXFNZFQZNUVHVXCVXFFWNFUVEUVGWOVXCVXBUUDMZVXH
|
|
VXBUUDUVFUVEWFZVFVXIUVRVXBOZUUAMZGUUCUPZVUAVXLPZGQZFQZVXHGUUAUUCVXBVXJVHV
|
|
UEVXLPZGQVXNFQZGQVXMVXPVXQVXRGVXQVUHVXLPVXRVUEVUHVXLVUIVJVUAVXLFVKVLRVXLG
|
|
UUCVMVXNFGVNSVXOVXGFVXOUYTVXBOZUUAMZVXGVXLVXTGUYTVUNVUAVXKVXSUUAUVRUYTVXB
|
|
USUTVAVXTVXSYNMZVXSYTMZWCVXFVXSYNYTWKVYAVXDVYBVXEVYAUYRUVEOHMVXDUYRUVFUVE
|
|
HVUQVXAUYEWMUYFUVEVURUYEVDWLVYBUYRUVFOZYSMZVXEUYRUVFUVEYSVUQVXAUYEWHVYCHM
|
|
ZVYCYRMZWPFDIZUYFYPMZWPVYDVXEVYEVYGVYFVYHUYFUVFVURVXAVDUYRYQMZFBXIZVYFVYH
|
|
VYIUYRYPUNVYJUYRYPVUQWQUYFYOVURWSWLVYFVYIUVFJMVXAUYRUVFYQJVUQVXAVOVPUYFYO
|
|
VURWQSWRVYCHYRWTUYFUVFYPWTSWLXAWBWLRTWBXBTVTTRTUXJUYFUVBUUIUYCVURUWMWMUVH
|
|
DUVBVMSXAWBWLRTXCVUOUULVVKVFUYKAWNSTVUMUYRUVKOHMUYHUYRUVSUVKHVUQUWLUWNWMU
|
|
YFUVKVURUWNVDWLVTTRTXBXCVTUVDUVLXDTVTTRUVOUWBGKUPZUWEGUUSKUVKUWNVHUXOUWBP
|
|
ZGQUWCEQZGQVYKUWEVYLVYMGVYLUVTEQZUWBPVYMUXOVYNUWBEUVRVIVJUVTUWBEVKVLRUWBG
|
|
KVMUWCEGVNSWLUVPEUVKVMSXCUVKUUTUWNVFUVMEUVKXESXFUUTUUSKHUURXGYLUUQYKJYJYI
|
|
YHKYGHYFJYEFYOXHXJXKXLXGXMXNXOXJXPXKXLUUPUUCUUOYNUUNUUMUULUUKUUCYMUUJHXGX
|
|
QUUIUUHUUCYNUUGHXGYAZUUFUUEUUDUUAUUCYNYTVYOYSHYRXGYQJYPURXKXLXRXQXSUUBKXN
|
|
XJXJZXOXPXTXQXMVYPXOYAXSVYPXOXPXTXQVYOXMVYPXOYBXMXNXOXPYC $.
|
|
$}
|
|
|
|
${
|
|
$d b l $. $d b y $. $d L b $. $d L l $. $d l n $. $d l x $. $d L x $.
|
|
$d l y $. $d L y $. $d N l $. $d N y $. $d X b $. $d X x $. $d x y $.
|
|
$d X y $. $d a b $. $d a c $. $d a k $. $d a l $. $d a n $. $d a x $.
|
|
$d a y $. $d a z $. $d b c $. $d b k $. $d b n $. $d b x $. $d b z $.
|
|
$d c k $. $d c l $. $d c x $. $d c y $. $d c z $. $d k l $. $d k n $.
|
|
$d k x $. $d k y $. $d k z $. $d l z $. $d N n $. $d n x $. $d n y $.
|
|
$d x z $. $d y z $.
|
|
|
|
$( Adjoining a new element to every member of ` L ` does not change its
|
|
size. Theorem X.1.39 of [Rosser] p. 530. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
nnadjoin $p |- ( ( N e. Nn /\ L e. N /\ X e. ~ U. L ) ->
|
|
{ x | E. b e. L x = ( b u. { X } ) } e. N ) $=
|
|
( vl vc vz wcel cuni ccompl cv cun wceq wrex cab wi c0 wn wa vy vn vk csn
|
|
va cnnc sneq uneq2d eqeq2d rexbidv abbidv eleq1d imbi2d wral wal c1c cplc
|
|
nnadjoinlem1 c0c eleq2 el0c ab0 bitri syl6bb raleqbi1dv df-ral imbi1i 0ex
|
|
albii unieq compleqd eleq2d notbid albidv imbi12d ceqsalv 3bitrri syl6bbr
|
|
rexeq weq cbvralv rex0 ax-gen a1i elsuc wel cin adantr adantl elin simp3l
|
|
rspcv w3a unisn compleqi eleq2i anbi2i simpr simpl2r elcompl sylib eleq1a
|
|
vex mtod wb simpl3r elunii expcom con3d mpan9 adj11 syl2anc mtbird nrexdv
|
|
eqeq1 elabg nsyl snex unex elsuci ex syl3an3b embantd 3expia syl5bi com23
|
|
ibi syld imp an32s uniun iunin eqtri syl6eq unab df-sn uneq2i rexun uneq1
|
|
wo rexsn orbi2i abbii 3eqtr4ri syl5ibrcom rexlimdvva ralrimiv finds com3r
|
|
rspccv syl vtoclga com3l 3imp ) CUFIZBCIZDBJZKZIZALZELZDUDZMZNZEBOZAPZCIZ
|
|
UUSUUOUUPUVGUUOUUPUUTUVAUALZUDZMZNZEBOZAPZCIZQZQUUOUUPUVGQZQUADUURUVHDNZU
|
|
VOUVPUUOUVQUVNUVGUUPUVQUVMUVFCUVQUVLUVEAUVQUVKUVDEBUVQUVJUVCUUTUVQUVIUVBU
|
|
VAUVHDUGUHUIUJUKULUMUMUUOUUPUVHUURIZUVNUUOUVHFLZJZKZIZUVKEUVSOZAPZCIZQZFC
|
|
UNZUUPUVRUVNQZQUWBUWDUBLZIZQZFUWIUNZUVHRJZKZIZUVKEROZSZAUOZQZUWBUWDUCLZIZ
|
|
QZFUWTUNZUVHUELZJZKZIZUVKEUXDOZAPZUWTUPUQZIZQZUEUXJUNZUWGUBUCCAUAUBEFURUW
|
|
IUSNZUWLUWBUWCSZAUOZQZFUSUNZUWSUWKUXQFUWIUSUXNUWJUXPUWBUXNUWJUWDUSIZUXPUW
|
|
IUSUWDUTUXSUWDRNUXPUWDVAUWCAVBVCVDUMVEUXRUVSUSIZUXQQZFUOUVSRNZUXQQZFUOUWS
|
|
UXQFUSVFUYAUYCFUXTUYBUXQUVSVAVGVIUXQUWSFRVHUYBUWBUWOUXPUWRUYBUWAUWNUVHUYB
|
|
UVTUWMUVSRVJVKVLUYBUXOUWQAUYBUWCUWPUVKEUVSRVSVMVNVOVPVQVRUWKUXBFUWIUWTUBU
|
|
CVTUWJUXAUWBUWIUWTUWDUTUMVEUWIUXJNZUWLUWBUWDUXJIZQZFUXJUNUXMUWKUYFFUWIUXJ
|
|
UYDUWJUYEUWBUWIUXJUWDUTUMVEUYFUXLFUEUXJFUEVTZUWBUXGUYEUXKUYGUWAUXFUVHUYGU
|
|
VTUXEUVSUXDVJVKVLUYGUWDUXIUXJUYGUWCUXHAUVKEUVSUXDVSUKULVOWAVDUWKUWFFUWICU
|
|
WICNUWJUWEUWBUWICUWDUTUMVEUWRUWOUWQAUVKEWBWCWDUWTUFIZUXCUXMUYHUXCTZUXLUEU
|
|
XJUXDUXJIUXDGLZHLZUDZMZNZHUYJKZOGUWTOUYIUXLHUXDUWTGWEUYIUYNUXLGHUWTUYOUYI
|
|
GUCWFZUYKUYOIZTZTUXLUYNUVHUYJJZKZUYLJZKZWGZIZUVKEUYJOZAPZUYKUVIMZUDZMZUXJ
|
|
IZQZUYHUYRUXCVUKUYHUYRTZUXCVUKVULUXCUVHUYTIZVUFUWTIZQZVUKUYRUXCVUOQZUYHUY
|
|
PVUPUYQUXBVUOFUYJUWTFGVTZUWBVUMUXAVUNVUQUWAUYTUVHVUQUVTUYSUVSUYJVJVKVLVUQ
|
|
UWDVUFUWTVUQUWCVUEAUVKEUVSUYJVSUKULVOWLWHWIVULVUDVUOVUJVUDVUMUVHVUBIZTZVU
|
|
LVUOVUJQZUVHUYTVUBWJUYHUYRVUSVUTUYHUYRVUSWMVUMVUNVUJUYHUYRVUMVURWKVUSUYHU
|
|
YRVUMUVHUYKKZIZTZVUNVUJQVURVVBVUMVUBVVAUVHVUAUYKUYKHXCZWNWOWPWQUYHUYRVVCW
|
|
MZVUNVUJVVEVUNTVUNVUGVUFIZSZVUJVVEVUNWRVVEVVGVUNVVEVUGUVJNZEUYJOZVVFVVEVV
|
|
HEUYJVVEEGWFZTZVVHHEVTZVVKVVLHGWFZVVKUYQVVMSUYPUYQUYHVVCVVJWSUYKUYJVVDWTX
|
|
AVVJVVLVVMQVVEUVAUYJUYKXBWIXDVVKUAHWFSZUAEWFZSZVVHVVLXEVVKVVBVVNVUMVVBUYH
|
|
UYRVVJXFUVHUYKUAXCZWTXAVVEUVHUYSIZSZVVJVVPVVEVUMVVSUYHUYRVUMVVBWKUVHUYSVV
|
|
QWTXAVVJVVOVVRVVOVVJVVRUVHUVAUYJXGXHXIXJUYKUVAUVHXKXLXMXNVVFVVIVUEVVIAVUG
|
|
VUFUUTVUGNZUVKVVHEUYJUUTVUGUVJXOUJXPYGXQWHVUFUWTVUGUYKUVIVVDUVHXRXSXTXLYA
|
|
YBYCYDYEYFYHYIYJUYNUXGVUDUXKVUJUYNUXFVUCUVHUYNUXFUYMJZKZVUCUYNUXEVWAUXDUY
|
|
MVJVKVWBUYSVUAMZKVUCVWAVWCUYJUYLYKWOUYSVUAYLYMYNVLUYNUXIVUIUXJUYNUXIUVKEU
|
|
YMOZAPZVUIUYNUXHVWDAUVKEUXDUYMVSUKVUFVVTAPZMVUEVVTYTZAPVUIVWEVUEVVTAYOVUH
|
|
VWFVUFAVUGYPYQVWDVWGAVWDVUEUVKEUYLOZYTVWGUVKEUYJUYLYRVWHVVTVUEUVKVVTEUYKV
|
|
VDEHVTUVJVUGUUTUVAUYKUVIYSUIUUAUUBVCUUCUUDYNULVOUUEUUFYEUUGYAUUHUWFUWHFBC
|
|
UVSBNZUWBUVRUWEUVNVWIUWAUURUVHVWIUVTUUQUVSBVJVKVLVWIUWDUVMCVWIUWCUVLAUVKE
|
|
UVSBVSUKULVOUUJUUKUUIUULUUMUUN $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a t $. $d A t $. $d b t $. $d M b $.
|
|
$d N b $. $d X a $. $d X b $. $d X t $.
|
|
|
|
$( Adjoining an element to a power class. Theorem X.1.40 of [Rosser]
|
|
p. 530. (Contributed by SF, 27-Jan-2015.) $)
|
|
nnadjoinpw $p |- ( ( ( M e. Nn /\ N e. Nn ) /\ ( A e. M /\ X e. ~ A ) /\
|
|
~P A e. N ) -> ~P ( A u. { X } ) e. ( N +c N ) ) $=
|
|
( va vb vt cnnc wcel wa ccompl cpw cun cv wceq wrex wss wn wral wal simp3
|
|
w3a csn cab cplc pwadjoin cin cuni simp1r simp2r unipw compleqi syl6eleqr
|
|
c0 nnadjoin syl3anc elcomplg ibi syl wb snssg mtbid intnand ralrimivw weq
|
|
disjr eqeq1 rexbidv ralab ralcom4 vex snex unex eleq1 notbid ceqsalv elpw
|
|
unss bitr4i xchbinx ralbii r19.23v albii 3bitr3ri 3bitri eladdci syl5eqel
|
|
wi sylibr ) BHIZCHIZJZABIZDAKZIZJZALZCIZUBZADUCZMLWQENZFNZWTMZOZFWQPZEUDZ
|
|
MZCCUEZADEFUFWSWRXFCIZWQXFUGUNOZXGXHIWLWPWRUAZWSWKWRDWQUHZKZIXIWJWKWPWRUI
|
|
XKWSDWNXMWLWMWOWRUJZXLAAUKULUMEWQCDFUOUPWSXBAQZWTAQZJZRZFWQSZXJWSXRFWQWSX
|
|
PXOWSDAIZXPWSWOXTRZXNWOYADAWNUQURUSWSWOXTXPUTXNDAWNVAUSVBVCVDXJGNZWQIZRZG
|
|
XFSYBXCOZFWQPZYDWHZGTZXSGWQXFVFXEYFYDGEEGVEXDYEFWQXAYBXCVGVHVIYEYDWHZGTZF
|
|
WQSYIFWQSZGTXSYHYIFGWQVJYJXRFWQYJXCWQIZXQYDYLRGXCXBWTFVKDVLVMZYEYCYLYBXCW
|
|
QVNVOVPYLXCAQXQXCAYMVQXBWTAVRVSVTWAYKYGGYEYDFWQWBWCWDWEWIWQXFCCWFUPWG $.
|
|
$}
|
|
|
|
${
|
|
$d m a b n t x $.
|
|
$( Lemma for ~ nnpweq . Establish stratification for induction.
|
|
(Contributed by SF, 26-Jan-2015.) $)
|
|
nnpweqlem1 $p |- { m | A. a e. m A. b e. m E. n e. Nn ( ~P a
|
|
e. n /\ ~P b e. n ) } e. _V $=
|
|
( vt vx cssetk cnnc wcel wa wrex csn copk snex bitri exbii df-rex 3bitr4i
|
|
wex 3bitri csik cins2k cins3k csymdif c1c cpw1 cimak ccnvk cin cdif cuni1
|
|
ccompl cv cpw wral cab cvv vex eluni1 wceq wel opkeq1 eleq1d ceqsexv elin
|
|
opksnelsik elssetk opkex elimak elpw131c anbi1i 19.41v bitr4i excom eldif
|
|
wn otkelins2k otkelins3k elpw12 r19.41v rexcom4 elpw121c eqpwrelk anbi12i
|
|
df-clel rexbii notbii rexnal 3bitr2i elpw11c elcompl dfral2 ssetkex sikex
|
|
opkelcnvk abbi2i ins2kex ins3kex symdifex 1cex imakex complex cnvkex inex
|
|
pw1ex nncex difex uni1ex eqeltrri ) GUAZXJUBZGUBZXJUCZUDZUEUFZUFZUGZULZUH
|
|
ZUAZUBZGUCZUIZXPUGZUBZXRUAZUHZUBZYBUIZXPUGZUCZUIZHUFZUFZUGZUCZUJZXPUFZUGZ
|
|
UIZXOUGZULZUKZCUMZUNZBUMZIZDUMZUNZUUFIZJZBHKZDAUMZUOZCUUMUOZAUPUQUUOAUUCU
|
|
UMUUCIUUMLZUUBIZUUOUUMUUBAURZUSUUPUUAIZVPUUNVPZCUUMKZVPUUQUUOUUSUVAEUMZUU
|
|
DLZLZUTZUVBUUPMZYTIZJZESZCSZCAVAZUUTJZCSUUSUVAUVIUVLCUVIUVDUUPMZYTIZUVMXJ
|
|
IZUVMYSIZJUVLUVGUVNEUVDUVCNZUVEUVFUVMYTUVBUVDUUPVBVCVDUVMXJYSVEUVOUVKUVPU
|
|
UTUVOUVCUUMMGIUVKUVCUUMGUUDNZUURVFUUDUUMCURZUURVGOUVPDAVAZUULVPZJZDSZUWAD
|
|
UUMKUUTUVPUVBUVMMZYQIZEYRKZUVBUUHLZLZLZLZUTZUWEJZESZDSZUWCEYQYRUVMUVDUUPV
|
|
HVIUVBYRIZUWEJZESUWLDSZESUWFUWNUWPUWQEUWPUWKDSZUWEJUWQUWOUWRUWEDUVBVJVKUW
|
|
KUWEDVLVMPUWEEYRQUWLDEVNRUWMUWBDUWMUWJUVMMZYQIZUWSXKIZUWSYPIZVPZJUWBUWEUW
|
|
TEUWJUWINUWKUWDUWSYQUVBUWJUVMVBVCVDUWSXKYPVOUXAUVTUXCUWAUXAUWHUUPMXJIUWGU
|
|
UMMGIUVTUWHUVDUUPXJUWGNZUVQUUMNZVQUWGUUMGUUHNZUURVFUUHUUMDURZUURVGTUXBUUL
|
|
UXBUWHUVDMZYOIZUVBUUFLZLZUTZUVBUXHMZYLIZJZESZBHKZUULUWHUVDUUPYOUXDUVQUXEV
|
|
RUXIUXNEYNKZUXQEYLYNUXHUWHUVDVHVIUVBYNIZUXNJZESUXOBHKZESUXRUXQUXTUYAEUXTU
|
|
XLBHKZUXNJUYAUXSUYBUXNBUVBHVSVKUXLUXNBHVTVMPUXNEYNQUXOBEHWAROUXPUUKBHUXPU
|
|
XKUXHMZYLIZUYCYEIZUYCYKIZJUUKUXNUYDEUXKUXJNUXLUXMUYCYLUVBUXKUXHVBVCVDUYCY
|
|
EYKVEUYEUUGUYFUUJUUFUVDMZYDIZFUMZUUEUTZFBVAZJZFSZUYEUUGUYHUVBUYGMZYCIZEXP
|
|
KZUVBUYILZLZLZUTZUYOJZESZFSZUYMEYCXPUYGUUFUVDVHVIUVBXPIZUYOJZESVUAFSZESUY
|
|
PVUCVUEVUFEVUEUYTFSZUYOJVUFVUDVUGUYOFUVBWBZVKUYTUYOFVLVMPUYOEXPQVUAFEVNRV
|
|
UBUYLFVUBUYSUYGMZYCIZVUIYAIZVUIYBIZJUYLUYOVUJEUYSUYRNZUYTUYNVUIYCUVBUYSUY
|
|
GVBVCVDVUIYAYBVEVUKUYJVULUYKVUKUYQUVDMXTIUYIUVCMXSIZUYJUYQUUFUVDXTUYINZBU
|
|
RZUVQVQUYIUVCXSFURZUVRVFVUNUVCUYIMXRIUYJUYIUVCXRVUQUVRWOUUDUYIUVSVUQWCOTV
|
|
ULUYQUUFMGIZUYKUYQUUFUVDGVUOVUPUVQVRUYIUUFVUQVUPVGZOWDTPTUUFUWHUVDYDVUPUX
|
|
DUVQVQFUUEUUFWERUUFUWHMZYJIZUYIUUIUTZUYKJZFSZUYFUUJVVAUVBVUTMZYIIZEXPKZUY
|
|
TVVFJZESZFSZVVDEYIXPVUTUUFUWHVHVIVUDVVFJZESVVHFSZESVVGVVJVVKVVLEVVKVUGVVF
|
|
JVVLVUDVUGVVFVUHVKUYTVVFFVLVMPVVFEXPQVVHFEVNRVVIVVCFVVIUYSVUTMZYIIZVVMYHI
|
|
ZVVMYBIZJVVCVVFVVNEUYSVUMUYTVVEVVMYIUVBUYSVUTVBVCVDVVMYHYBVEVVOVVBVVPUYKV
|
|
VOUYQUWHMYGIUWHUYQMYFIZVVBUYQUUFUWHYGVUOVUPUXDVQUYQUWHYFVUOUXDWOVVQUWGUYI
|
|
MXRIVVBUWGUYIXRUXFVUQVFUUHUYIUXGVUQWCOTVVPVURUYKUYQUUFUWHGVUOVUPUXDVRVUSO
|
|
WDTPTUUFUWHUVDYJVUPUXDUVQVRFUUIUUFWERWDTWFTWGWDTPTUWADUUMQUULDUUMWHWIWDTP
|
|
UUSUVGEXOKZUVJEYTXOUUPUXEVIUVBXOIZUVGJZESUVHCSZESVVRUVJVVTVWAEVVTUVECSZUV
|
|
GJVWAVVSVWBUVGCUVBWJVKUVEUVGCVLVMPUVGEXOQUVHCEVNROUUTCUUMQRWGUUPUUAUXEWKU
|
|
UNCUUMWLROWPUUBUUAYTXOXJYSGWMWNZYQYRXKYPXJVWCWQYOYLYNYEYKYDYCXPYAYBXTXSXR
|
|
XQXNXPXLXMGWMWQXJVWCWRWSXOUEWTXEZXEZXAXBZXCWNWQGWMWRZXDVWEXAWQYJYIXPYHYBY
|
|
GYFXRVWFWNXCWQVWGXDVWEXAWRXDYMHXFXEXEXAWRXGXPVWEXEXAXDVWDXAXBXHXI $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c d e f x y m n j k $. $d B a b c d e f x y m n j k $.
|
|
$d M a b c d e f x y m n j k $.
|
|
$( If two sets are the same finite size, then so are their power classes.
|
|
Theorem X.1.41 of [Rosser] p. 530. (Contributed by SF, 26-Jan-2015.) $)
|
|
nnpweq $p |- ( ( M e. Nn /\ A e. M /\ B e. M ) ->
|
|
E. n e. Nn ( ~P A e. n /\ ~P B e. n ) ) $=
|
|
( va vb ve vf cnnc wcel cpw cv wa wrex wral c0 c0c wceq pweq eleq1d vm vk
|
|
vc vj vd vx vy csn c1c cplc nnpweqlem1 raleq raleqbi1dv wal df-ral imbi1i
|
|
wi albii 0ex pw0 syl6eq anbi1d rexbidv ralbidv ceqsalv bitri anbi2d anidm
|
|
el0c syl6bb 3bitri weq cbvral2v eleq2 anbi12d cbvrexv 1cnnc snel1c rspcev
|
|
2ralbii mp2an cun ccompl reeanv 2rexbii elsuc anbi12i 3bitr4ri wel rspc2v
|
|
adantl w3a nncaddccl anidms 3ad2ant1 simp1l simp1r simp2ll simp3l simp2rl
|
|
nnadjoinpw syl221anc simp2lr simp3r simp2rr syl12anc syl5ibrcom rexlimdvv
|
|
bi2anan9 3expia expr an32s rexlimdva syld imp rexlimdvva syl5bi ralrimivv
|
|
ex finds syl5com 3impib ) DIJZADJZBDJZAKZCLZJZBKZYGJZMZCINZYCELZKZYGJZFLZ
|
|
KZYGJZMZCINZFDOZEDOZYDYEMYLYTFUALZOZEUUCOZPUHZYGJZCINZYTFUBLZOZEUUIOZUCLZ
|
|
KZUDLZJZUELZKZUUNJZMZUDINZUEUUIUIUJZOUCUVAOZUUBUAUBDUACEFUKUUCQRUUEYTFQOZ
|
|
EQOZUUHUUDUVCEUUCQYTFUUCQULUMUVDYMQJZUVCUQZEUNYMPRZUVCUQZEUNZUUHUVCEQUOUV
|
|
FUVHEUVEUVGUVCYMVIUPURUVIUUGYRMZCINZFQOZYPPRZUVKUQZFUNZUUHUVCUVLEPUSUVGYT
|
|
UVKFQUVGYSUVJCIUVGYOUUGYRUVGYNUUFYGUVGYNPKZUUFYMPSUTVATVBVCVDVEUVLYPQJZUV
|
|
KUQZFUNUVOUVKFQUOUVRUVNFUVQUVMUVKYPVIUPURVFUVKUUHFPUSUVMUVJUUGCIUVMUVJUUG
|
|
UUGMUUGUVMYRUUGUUGUVMYQUUFYGUVMYQUVPUUFYPPSUTVATVGUUGVHVJVCVEVKVKVJUUDUUJ
|
|
EUUCUUIYTFUUCUUIULUMUUCUVARUUEYTFUVAOZEUVAOZUVBUUDUVSEUUCUVAYTFUUCUVAULUM
|
|
UVTUUMYGJZUUQYGJZMZCINZUEUVAOUCUVAOUVBYTUWDUWAYRMZCINEFUCUEUVAUVAEUCVLZYS
|
|
UWECIUWFYOUWAYRUWFYNUUMYGYMUULSTVBVCFUEVLZUWEUWCCIUWGYRUWBUWAUWGYQUUQYGYP
|
|
UUPSTVGVCVMUWDUUTUCUEUVAUVAUWCUUSCUDICUDVLUWAUUOUWBUURYGUUNUUMVNYGUUNUUQV
|
|
NVOVPVTVFVJUUDUUAEUUCDYTFUUCDULUMUIIJUUFUIJZUUHVQPUSVRUUGUWHCUIIYGUIUUFVN
|
|
VSWAUUIIJZUUKUVBUWIUUKMZUUTUCUEUVAUVAUULUVAJZUUPUVAJZMZUULGLZUFLZUHWBZRZU
|
|
UPHLZUGLZUHWBZRZMZUGUWRWCZNUFUWNWCZNZHUUINGUUINZUWJUUTUWQUFUXDNZUXAUGUXCN
|
|
ZMZHUUINGUUINUXGGUUINZUXHHUUINZMUXFUWMUXGUXHGHUUIUUIWDUXEUXIGHUUIUUIUWQUX
|
|
AUFUGUXDUXCWDWEUWKUXJUWLUXKUFUULUUIGWFUGUUPUUIHWFWGWHUWJUXEUUTGHUUIUUIUWI
|
|
GUBWIZHUBWIZMZUUKUXEUUTUQZUWIUXNMZUUKUXOUXPUUKUWNKZYGJZUWRKZYGJZMZCINZUXO
|
|
UXNUUKUYBUQUWIYTUYBUXRYRMZCINEFUWNUWRUUIUUIEGVLZYSUYCCIUYDYOUXRYRUYDYNUXQ
|
|
YGYMUWNSTVBVCFHVLZUYCUYACIUYEYRUXTUXRUYEYQUXSYGYPUWRSTVGVCWJWKUXPUYAUXOCI
|
|
UWIYGIJZUXNUYAUXOUQUWIUYFMZUXNUYAUXOUYGUXNUYAMZMUXBUUTUFUGUXDUXCUYGUYHUWO
|
|
UXDJZUWSUXCJZMZUXBUUTUQUYGUYHUYKWLZUUTUXBUWPKZUUNJZUWTKZUUNJZMZUDINZUYLYG
|
|
YGUJZIJZUYMUYSJZUYOUYSJZUYRUYGUYHUYTUYKUYFUYTUWIUYFUYTYGYGWMWNWKWOUYLUWIU
|
|
YFUXLUYIUXRVUAUWIUYFUYHUYKWPZUWIUYFUYHUYKWQZUXLUXMUYAUYGUYKWRUYGUYHUYIUYJ
|
|
WSUXRUXTUXNUYGUYKWTUWNUUIYGUWOXAXBUYLUWIUYFUXMUYJUXTVUBVUCVUDUXLUXMUYAUYG
|
|
UYKXCUYGUYHUYIUYJXDUXRUXTUXNUYGUYKXEUWRUUIYGUWSXAXBUYQVUAVUBMUDUYSIUUNUYS
|
|
RUYNVUAUYPVUBUUNUYSUYMVNUUNUYSUYOVNVOVSXFUXBUUSUYQUDIUWQUUOUYNUXAUURUYPUW
|
|
QUUMUYMUUNUULUWPSTUXAUUQUYOUUNUUPUWTSTXIVCXGXJXHXKXLXMXNXOXLXPXQXRXSXTYTY
|
|
LYHYRMZCINEFABDDYMARZYSVUECIVUFYOYHYRVUFYNYFYGYMASTVBVCYPBRZVUEYKCIVUGYRY
|
|
JYHVUGYQYIYGYPBSTVGVCWJYAYB $.
|
|
$}
|
|
|
|
${
|
|
$d A t $. $d A x $. $d A y $. $d B t $. $d B x $. $d B y $. $d t x $.
|
|
$d t y $. $d t z $. $d x y $. $d x z $. $d y z $.
|
|
srelk.1 $e |- A e. _V $.
|
|
srelk.2 $e |- B e. _V $.
|
|
$( Binary relationship form of the ` _S[fin] ` relationship. (Contributed
|
|
by SF, 23-Jan-2015.) $)
|
|
srelk $p |- ( << A , B >> e. ( ( Nn X._k Nn ) i^i ( (
|
|
Ins3_k ( ( Ins3_k SI_k ( ( ~P 1c X._k _V ) \ ( ( Ins3_k _S_k (+) Ins2_k SI_k
|
|
_S_k ) "_k ~P1 ~P1 ~P1 1c ) ) i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c )
|
|
i^i Ins2_k ( ( Ins3_k SI_k ~ ( ( Ins3_k _S_k (+) Ins2_k SI_k _S_k ) "_k
|
|
~P1 ~P1 1c ) i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) ) "_k ~P1 ~P1
|
|
~P1 1c ) ) <->
|
|
_S[fin] ( A , B ) ) $=
|
|
( vx vt vy vz copk cnnc wcel cssetk wa wex wceq exbii 3bitr4i snex 3bitri
|
|
csn cxpk c1c cpw cvv cins3k csik cins2k csymdif cpw1 cimak cdif ccompl cv
|
|
cin wsfin opkelxpk wrex opkex elimak elpw131c anbi1i 19.41v bitr4i df-rex
|
|
excom opkeq1 eleq1d ceqsexv elin elpw121c otkelins3k opksnelsik eqpw1relk
|
|
w3a vex otkelins2k elssetk bitri anbi12i df-clel wel elsymdif opkelssetkg
|
|
wss wb wal wn mp2an bibi12i notbii elcompl alex df-pw eqeq2i abeq2 df-3an
|
|
cab df-sfin ) ABIZJJUAZKZWSUBUCUDUALUEZLUFZUGZUHZUBUIUIZUIZUJUKZUFZUEZLUG
|
|
ZUNZXFUJZUEZXEXFUJZULZUFZUEZXKUNZXFUJZUGZUNZXGUJZKZMZAJKZBJKZEUMZUIZAKZYH
|
|
UCZBKZMZENZVNZWSWTYCUNKABUOYEYFYGMZYNMYOXAYPYDYNABJJCDUPYDFUMZWSIZYBKZFXG
|
|
UQZYQYHTZTZTZTZOZYSMZFNZENZYNFYBXGWSABURUSYQXGKZYSMZFNUUFENZFNYTUUHUUJUUK
|
|
FUUJUUEENZYSMUUKUUIUULYSEYQUTVAUUEYSEVBVCPYSFXGVDUUFEFVEQUUGYMEUUGUUDWSIZ
|
|
YBKZUUMXNKZUUMYAKZMYMYSUUNFUUDUUCRUUEYRUUMYBYQUUDWSVFVGVHUUMXNYAVIUUOYJUU
|
|
PYLUUBAIZXMKZGUMZYIOZUUSAKZMZGNZUUOYJUURYQUUQIZXLKZFXFUQZYQUUSTZTZTZOZUVE
|
|
MZFNZGNZUVCFXLXFUUQUUBAURUSYQXFKZUVEMZFNUVKGNZFNUVFUVMUVOUVPFUVOUVJGNZUVE
|
|
MUVPUVNUVQUVEGYQVJZVAUVJUVEGVBVCPUVEFXFVDUVKGFVEQUVLUVBGUVLUVIUUQIZXLKZUV
|
|
SXJKZUVSXKKZMUVBUVEUVTFUVIUVHRZUVJUVDUVSXLYQUVIUUQVFVGVHUVSXJXKVIUWAUUTUW
|
|
BUVAUWAUVGUUBIZXIKUUSUUAIZXHKUUTUVGUUBAXIUUSRZUUARZCVKUUSUUAXHGVOZYHRZVLU
|
|
USYHUWHEVOZVMSUWBUVGAILKUVAUVGUUBALUWFUWGCVPUUSAUWHCVQVRVSSPSUUBABXMUWGCD
|
|
VKGYIAVTQUUBBIZXTKZUUSYKOZUUSBKZMZGNZUUPYLUWLYQUWKIZXSKZFXFUQZUVJUWRMZFNZ
|
|
GNZUWPFXSXFUWKUUBBURUSUVNUWRMZFNUWTGNZFNUWSUXBUXCUXDFUXCUVQUWRMUXDUVNUVQU
|
|
WRUVRVAUVJUWRGVBVCPUWRFXFVDUWTGFVEQUXAUWOGUXAUVIUWKIZXSKZUXEXRKZUXEXKKZMU
|
|
WOUWRUXFFUVIUWCUVJUWQUXEXSYQUVIUWKVFVGVHUXEXRXKVIUXGUWMUXHUWNUXGUWDXQKUWE
|
|
XPKZUWMUVGUUBBXQUWFUWGDVKUUSUUAXPUWHUWIVLUXIHGWAZHUMZYHWDZWEZHWFZUWMUWEXO
|
|
KZWGUXMWGZHNZWGUXIUXNUXOUXQUXOYQUWEIZXEKZFXFUQZYQUXKTZTZTZOZUXSMZFNZHNZUX
|
|
QFXEXFUWEUUSUUAURZUSUVNUXSMZFNUYEHNZFNUXTUYGUYIUYJFUYIUYDHNZUXSMUYJUVNUYK
|
|
UXSHYQVJVAUYDUXSHVBVCPUXSFXFVDUYEHFVEQUYFUXPHUYFUYCUWEIZXEKZUYLXBKZUYLXDK
|
|
ZWEZWGUXPUXSUYMFUYCUYBRUYDUXRUYLXEYQUYCUWEVFVGVHUYLXBXDWBUYPUXMUYNUXJUYOU
|
|
XLUYNUYAUUSILKUXJUYAUUSUUALUXKRZUWHUWIVKUXKUUSHVOZUWHVQVRUYOUYAUUAIXCKUXK
|
|
YHILKZUXLUYAUUSUUAXCUYQUWHUWIVPUXKYHLUYRUWJVLUXKUDKYHUDKUYSUXLWEUYRUWJUXK
|
|
YHUDUDWCWHSWIWJSPSWJUWEXOUYHWKUXMHWLQUWMUUSUXLHWQZOUXNYKUYTUUSHYHWMWNUXLH
|
|
UUSWOVRVCSUXHUVGBILKUWNUVGUUBBLUWFUWGDVPUUSBUWHDVQVRVSSPSUUBABXTUWGCDVPGY
|
|
KBVTQVSSPSVSYFYGYNWPVCWSWTYCVIABEWRQ $.
|
|
$}
|
|
|
|
$( The expression at the core of ~ srelk exists. (Contributed by SF,
|
|
30-Jan-2015.) $)
|
|
srelkex $p |- ( ( Nn X._k Nn ) i^i ( (
|
|
Ins3_k ( ( Ins3_k SI_k ( ( ~P 1c X._k _V ) \ ( ( Ins3_k _S_k (+) Ins2_k SI_k
|
|
_S_k ) "_k ~P1 ~P1 ~P1 1c ) ) i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c )
|
|
i^i Ins2_k ( ( Ins3_k SI_k ~ ( ( Ins3_k _S_k (+) Ins2_k SI_k _S_k ) "_k
|
|
~P1 ~P1 1c ) i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) ) "_k ~P1 ~P1
|
|
~P1 1c ) ) e. _V $=
|
|
( cnnc cxpk c1c cvv cssetk cins3k csik cins2k cimak cin nncex xpkex ssetkex
|
|
cpw1 ins3kex sikex ins2kex pw1ex imakex inex cpw cdif ccompl 1cex pwex vvex
|
|
csymdif symdifex difex complex ) AABCUAZDBZEFZEGZHZUGZCNZNZNZIZUBZGZFZEHZJZ
|
|
URIZFZUPURIZUCZGZFZVDJZURIZHZJZUSIAAKKLVOUSVGVNVFVEURVCVDVBVAULUTUKDCUDUEUF
|
|
LUPUSUMUOEMOUNEMPQUHZURUQCUDRRZRZSUIPOEMQZTVQSOVMVLURVKVDVJVIVHUPURVPVQSUJP
|
|
OVSTVQSQTVRST $.
|
|
|
|
${
|
|
$d A y $. $d B y $. $d C y $.
|
|
$( Equality theorem for the finite S relationship. (Contributed by SF,
|
|
27-Jan-2015.) $)
|
|
sfineq1 $p |- ( A = B -> ( _S[fin] ( A , C ) <-> _S[fin] ( B , C ) ) ) $=
|
|
( vy wceq cnnc wcel cv cpw1 cpw wex w3a wsfin eleq1 eleq2 anbi1d 3anbi13d
|
|
wa exbidv df-sfin 3bitr4g ) ABEZAFGZCFGZDHZIZAGZUEJCGZRZDKZLBFGZUDUFBGZUH
|
|
RZDKZLACMBCMUBUCUKUJUNUDABFNUBUIUMDUBUGULUHABUFOPSQACDTBCDTUA $.
|
|
|
|
$( Equality theorem for the finite S relationship. (Contributed by SF,
|
|
27-Jan-2015.) $)
|
|
sfineq2 $p |- ( A = B -> ( _S[fin] ( C , A ) <-> _S[fin] ( C , B ) ) ) $=
|
|
( vy wceq cnnc wcel cv cpw1 cpw wex w3a wsfin eleq1 eleq2 anbi2d 3anbi23d
|
|
wa exbidv df-sfin 3bitr4g ) ABEZCFGZAFGZDHZICGZUEJZAGZRZDKZLUCBFGZUFUGBGZ
|
|
RZDKZLCAMCBMUBUDUKUJUNUCABFNUBUIUMDUBUHULUFABUGOPSQCADTCBDTUA $.
|
|
$}
|
|
|
|
$( Zero and one satisfy ` _S[fin] ` . Theorem X.1.42 of [Rosser] p. 530.
|
|
(Contributed by SF, 30-Jan-2015.) $)
|
|
sfin01 $p |- _S[fin] ( 0c , 1c ) $=
|
|
( va c0c c1c wsfin cnnc wcel cv cpw1 cpw wex peano1 1cnnc wceq csn pw10 0ex
|
|
wa c0 snel1c el0c pw1eq eqeq1d syl5bb pw0 syl6eq eleq1d anbi12d spcev mp2an
|
|
pweq df-sfin mpbir3an ) BCDBEFCEFAGZHZBFZUMIZCFZQZAJZKLRHZRMZRNZCFZUSORPSUR
|
|
VAVCQARPUMRMZUOVAUQVCUOUNRMVDVAUNTVDUNUTRUMRUAUBUCVDUPVBCVDUPRIVBUMRUJUDUEU
|
|
FUGUHUIBCAUKUL $.
|
|
|
|
${
|
|
$d M x y n k $. $d N x y n k $. $d P x y n k $.
|
|
$( Equality law for the finite S operator. Theorem X.1.43 of [Rosser]
|
|
p. 530. (Contributed by SF, 27-Jan-2015.) $)
|
|
sfin112 $p |- ( ( _S[fin] ( M , N ) /\ _S[fin] ( M , P ) ) -> N = P ) $=
|
|
( vx vy vn vk wsfin wa cnnc wcel cpw1 cpw wex w3a wceq df-sfin wel wrex
|
|
cv 3an6 eeanv 3anbi3i anbi12i 3bitr4ri simpllr simprll simprrl ncfinlower
|
|
syl3anc nnpweq 3expb simp1rl simp3l simp2lr simp3rl nnceleq simp1rr 3expa
|
|
syl22anc simp2rr simp3rr eqtr4d expr rexlimdva syl5 exp3a rexlimdv mpd ex
|
|
exlimdvv 3impia sylbi ) BCHZBAHZIZBJKZVQIZCJKZAJKZIZDTZLBKZWBMZCKZIZETZLB
|
|
KZWGMZAKZIZIZENDNZOZCAPZVRWAWFDNZWKENZIZOVQVSWPOZVQVTWQOZIWNVPVQVQVSVTWPW
|
|
QUAWMWRVRWAWFWKDEUBUCVNWSVOWTBCDQBAEQUDUEVRWAWMWOVRWAIZWLWODEXAWLWOXAWLIZ
|
|
DFRZEFRZIZFJSZWOXBVQWCWHXFVQVQWAWLUFXAWCWEWKUGXAWFWHWJUHWBWGFBUIUJXBXEWOF
|
|
JXBFTZJKZXEWOXHXEIWDGTZKZWIXIKZIZGJSZXBWOXHXCXDXMWBWGGXGUKULXBXLWOGJXBXIJ
|
|
KZXLWOXAWLXNXLIZWOXAWLXOOZCXIAXPVSXNWEXJCXIPVSVTVRWLXOUMXAWLXNXLUNZWCWEWK
|
|
XAXOUOXJXKXNXAWLUPWDCXIUQUTXPVTXNWJXKAXIPVSVTVRWLXOURXQWHWJWFXAXOVAXJXKXN
|
|
XAWLVBWIAXIUQUTVCUSVDVEVFVGVHVIVJVKVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d A b $. $d a n $. $d A x $. $d a y $. $d A y $. $d a z $. $d A z $.
|
|
$d b n $. $d b x $. $d b y $. $d b z $. $d k n $. $d k y $. $d k z $.
|
|
$d M a $. $d M b $. $d M k $. $d M n $. $d M x $. $d M y $. $d M z $.
|
|
$d n x $. $d n y $. $d n z $. $d x y $. $d x z $. $d y z $.
|
|
$( If the unit power set of a set is in the successor of a finite cardinal,
|
|
then there is a natural that is smaller than the finite cardinal and
|
|
whose double is smaller than the successor of the cardinal. Theorem
|
|
X.1.44 of [Rosser] p. 531. (Contributed by SF, 30-Jan-2015.) $)
|
|
sfindbl $p |- ( ( M e. Nn /\ ~P1 A e. ( M +c 1c ) ) ->
|
|
E. n e. Nn ( _S[fin] ( M , n ) /\
|
|
_S[fin] ( ( M +c 1c ) , ( n +c n ) ) ) ) $=
|
|
( vb vx vy vz vk va cpw1 wcel cnnc cv cun wceq wrex wa wex vex eleq1d c1c
|
|
cplc csn ccompl wsfin elsuc w3a pw1eqadj wi wel wn wb eleq1 adantr eleq12
|
|
compleq snex elcompl snelpw1 xchbinx syl6bb sylan2 ancoms anbi2d 3anidm23
|
|
anbi12d ncfinlower adantrr cpw simp3l simp3rr syl3anc simpl2l simprrr weq
|
|
nnpweq simpl1 simprl pw1eq pweq spcev syl2anc df-sfin syl3anbrc nncaddccl
|
|
peano2 syl anidms pw1un pw1sn uneq2i eqtri simpl2r sylnibr elsuci simpl3l
|
|
syl5eqel sylibr nnadjoinpw syl221anc unex jca expr reximdva mpd rexlimdva
|
|
3expa syl6bi 3adant1 com12 exlimdvv syl5bi rexlimdvva imp sylan2b ) AJZCU
|
|
AUBZKCLKZXPDMZEMZUCNOZEXSUDZPDCPZCBMZUEZXQYDYDUBZUEZQZBLPZEXPCDUFXRYCYIXR
|
|
YAYIDECYBYAAFMZGMZUCZNZOZXSYJJZOZXTYLOZUGZGRFRXRXSCKZXTYBKZQZQZYIFGXSXTAD
|
|
SESUHUUBYRYIFGYRUUBYIYPYQUUBYIUIYNYPYQQZUUBXRYOCKZGFUJZUKZQZQZYIUUCUUAUUG
|
|
XRUUCYSUUDYTUUFYPYSUUDULYQXSYOCUMUNYQYPYTUUFULZYPYQYBYOUDZOZUUIXSYOUPYQUU
|
|
KQYTYLUUJKZUUFXTYLYBUUJUOUULYLYOKZUUEYLYOYKUQZURYKYJUSZUTVAVBVCVFVDUUHFHU
|
|
JZUUPQZHLPZYIXRUUDUURUUFXRUUDUURYJYJHCVGVEVHUUHUUQYIHLUUHHMZLKZUUQYIXRUUG
|
|
UUTUUQQZYIXRUUGUVAUGZYJVIZYDKZUVDQZBLPZYIUVBUUTUUPUUPUVFXRUUGUUTUUQVJUUPU
|
|
UPUUTXRUUGVKZUVGYJYJBUUSVPVLUVBUVEYHBLUVBYDLKZUVEYHUVBUVHUVEQZQZYEYGUVJXR
|
|
UVHIMZJZCKZUVKVIZYDKZQZIRZYEXRUUGUVAUVIVQZUVBUVHUVEVRZUVJUUDUVDUVQUUDUUFX
|
|
RUVAUVIVMZUVBUVHUVDUVDVNZUVPUUDUVDQIYJFSZIFVOZUVMUUDUVOUVDUWCUVLYOCUVKYJV
|
|
STUWCUVNUVCYDUVKYJVTTVFWAWBCYDIWCWDUVJXQLKZYFLKZUVLXQKZUVNYFKZQZIRZYGUVJX
|
|
RUWDUVRCWFWGUVJUVHUWEUVSUVHUWEYDYDWEWHWGUVJYMJZXQKZYMVIZYFKZUWIUVJUWJYOYL
|
|
UCZNZXQUWJYOYLJZNUWOYJYLWIUWPUWNYOYKGSZWJWKWLUVJUUDUUMUKUWOXQKUVTUVJUUEUU
|
|
MUUDUUFXRUVAUVIWMZUUOWNYOCYLUUNWOWBWQUVJUUTUVHUUPYKYJUDKZUVDUWMUUTUUQXRUU
|
|
GUVIWPUVSUVBUUPUVIUVGUNUVJUUFUWSUWRYKYJUWQURWRUWAYJUUSYDYKWSWTUWHUWKUWMQI
|
|
YMYJYLUWBUUNXAUVKYMOZUWFUWKUWGUWMUWTUVLUWJXQUVKYMVSTUWTUVNUWLYFUVKYMVTTVF
|
|
WAWBXQYFIWCWDXBXCXDXEXGXCXFXEXHXIXJXKXLXMXNXO $.
|
|
$}
|
|
|
|
${
|
|
$d m n t x y $.
|
|
$( Lemma for ~ sfintfin . Set up induction stratification. (Contributed
|
|
by SF, 31-Jan-2015.) $)
|
|
sfintfinlem1 $p |- { m | A. n ( _S[fin] ( m , n ) ->
|
|
_S[fin] ( _T[fin] m , _T[fin] n ) ) } e. _V $=
|
|
( vt vy vx cxpk c1c cins3k cins2k cimak cin csn cv wcel wn wex copk exbii
|
|
wa 3bitri cnnc cpw cvv cssetk csik csymdif cpw1 cdif ccompl c0 ccnvk cidk
|
|
cun cuni1 wsfin ctfin wi wal cab eluni1 wrex wceq snex elimak el1c anbi1i
|
|
vex 19.41v bitr4i df-rex excom 3bitr4i opkeq1 eleq1d opkelcnvk opksnelsik
|
|
ceqsexv eldif srelk elpw11c elin otkelins2k eqtfinrelk otkelins3k anbi12i
|
|
bitri opkex tfinex sfineq1 sfineq2 notbii annim elcompl alex abbi2i sikex
|
|
srelkex tfinrelkex cnvkex ins2kex ins3kex inex pw1ex imakex difex complex
|
|
1cex uni1ex eqeltrri ) UAUAFGUBUCFUDHUDUEIZUFZGUGZUGZUGZJUHUEHUDIZKXMJHZX
|
|
KXMJUIUEHXOKXMJIKXNJKZUEZUJLZLZXSFXOUDUKHUAUCFXJXPKXNJKIULHUFXLJIUHXLJHUF
|
|
XMJUIXTUCFUHUMZUKZIZYCXQHZKZXLJZHZKZXLJZUHZUKZGJZUIZUNZAMZBMZUOZYOUPZYPUP
|
|
ZUOZUQZBURZAUSUCUUBAYNYOYNNYOLZYMNZUUBYOYMAVGZUTUUCYLNZOUUAOZBPZOUUDUUBUU
|
|
FUUHUUFCMZUUCQZYKNZCGVAZUUIYPLZVBZUUKSZCPZBPZUUHCYKGUUCYOVCZVDUUIGNZUUKSZ
|
|
CPUUOBPZCPUULUUQUUTUVACUUTUUNBPZUUKSUVAUUSUVBUUKBUUIVEVFUUNUUKBVHVIRUUKCG
|
|
VJUUOBCVKVLUUPUUGBUUPUUMUUCQZYKNZUUCUUMQZYJNZUUGUUKUVDCUUMYPVCZUUNUUJUVCY
|
|
KUUIUUMUUCVMVNVQUUMUUCYJUVGUURVOUVFUVEXRNZUVEYINZOZSYQYTOZSUUGUVEXRYIVRUV
|
|
HYQUVJUVKUVHYOYPQXQNYQYOYPXQUUEBVGZVPYOYPUUEUVLVSWFUVIYTUVIUUIDMZLZLZVBZU
|
|
UIUVEQZYHNZSZCPZDPZUVMYSVBZYRUVMUOZSZDPYTUVIUVRCXLVAUUIXLNZUVRSZCPZUWACYH
|
|
XLUVEUUCUUMWGVDUVRCXLVJUWGUVSDPZCPUWAUWFUWHCUWFUVPDPZUVRSUWHUWEUWIUVRDUUI
|
|
VTVFUVPUVRDVHVIRUVSDCVKVITUVTUWDDUVTUVOUVEQZYHNZUWJYCNZUWJYGNZSUWDUVRUWKC
|
|
UVOUVNVCUVPUVQUWJYHUUIUVOUVEVMVNVQUWJYCYGWAUWLUWBUWMUWCUWLUVMUUMQYBNUUMUV
|
|
MQYANUWBUVMUUCUUMYBDVGZUURUVGWBUVMUUMYAUWNUVGVOYPUVMUVLUWNWCTUWMUVMUUCQZY
|
|
FNZEMZYRVBZUWQUVMUOZSZEPZUWCUVMUUCUUMYFUWNUURUVGWDUWPUUIUWOQZYENZCXLVAZUU
|
|
IUWQLZLZVBZUXCSZCPZEPZUXACYEXLUWOUVMUUCWGVDUWEUXCSZCPUXHEPZCPUXDUXJUXKUXL
|
|
CUXKUXGEPZUXCSUXLUWEUXMUXCEUUIVTVFUXGUXCEVHVIRUXCCXLVJUXHECVKVLUXIUWTEUXI
|
|
UXFUWOQZYENZUXNYCNZUXNYDNZSUWTUXCUXOCUXFUXEVCUXGUXBUXNYEUUIUXFUWOVMVNVQUX
|
|
NYCYDWAUXPUWRUXQUWSUXPUWQUUCQYBNUUCUWQQYANUWRUWQUVMUUCYBEVGZUWNUURWBUWQUU
|
|
CYAUXRUURVOYOUWQUUEUXRWCTUXQUWQUVMQXQNUWSUWQUVMUUCXQUXRUWNUURWDUWQUVMUXRU
|
|
WNVSWFWETRTUWSUWCEYRYOWHUWQYRUVMWIVQTWETRUWCYTDYSYPWHUVMYSYRWJVQTWKWEYQYT
|
|
WLTTRTWKUUCYLUURWMUUABWNVLWFWOYMYLYKGYJXRYIXQWQWPYHXLYCYGYBYAWRWSWTZYFYEX
|
|
LYCYDUXSXQWQXAXBGXGXCZXDXAXBUXTXDXEWSXGXDXFXHXI $.
|
|
$}
|
|
|
|
${
|
|
$d M a k m n p q $. $d N a k n $.
|
|
$( If two numbers obey ` _S[fin] ` , then do their T raisings. Theorem
|
|
X.1.45 of [Rosser] p. 532. (Contributed by SF, 30-Jan-2015.) $)
|
|
sfintfin $p |- ( _S[fin] ( M , N ) ->
|
|
_S[fin] ( _T[fin] M , _T[fin] N ) ) $=
|
|
( va vn vk cnnc wcel wa wsfin ctfin wi c0c c1c wceq sfineq1 wb tfineq syl
|
|
imbi12d sfineq2 vm vp vq cv cpw wex w3a df-sfin 3simpa sylbi sfintfinlem1
|
|
cpw1 wal cplc tfin0c syl6eq albidv weq cbvalv syl6bb sfin01 sfin112 mpan2
|
|
tfin1c mpbiri simp3bi 3ad2ant3 wrex sfindbl 3ad2antl1 spv simprrl simplrl
|
|
ax-gen adantl simprrr ad2antlr syl2anc simp2 simp1bi simp1l simp3 tfinpw1
|
|
peano2 c0 wne ne0i tfinsuc eleqtrd simp3l addceq12 anidms biimprcd 3expia
|
|
mpd rexlimdvw 3adant3 adantrd exlimdv simpll adantr exlimiv 3syl sylan9bb
|
|
tfindi syl3anc mpbird ex embantd exp32 com34 com23 exp3a rexlimdv alrimiv
|
|
syl5 3imp finds spcgv mpan9 mpcom ) AFGZBFGZHZABIZAJZBJZIZYEYBYCCUDZULZAG
|
|
YIUEZBGHCUFZUGYDABCUHYBYCYLUIUJYBADUDZIZYFYMJZIZKZDUMZYCYEYHKZEUDZYMIZYTJ
|
|
ZYOIZKZDUMZLYMIZLYOIZKZDUMUAUDZUBUDZIZUUIJZUUJJZIZKZUBUMZUUIMUNZYMIZUUQJZ
|
|
YOIZKZDUMZYREUAAEDUKYTLNZUUDUUHDUVCUUAUUFUUCUUGYTLYMOUVCUUBLNUUCUUGPUVCUU
|
|
BLJLYTLQUOUPUUBLYOORSUQEUAURZUUEUUIYMIZUULYOIZKZDUMUUPUVDUUDUVGDUVDUUAUVE
|
|
UUCUVFYTUUIYMOUVDUUBUULNUUCUVFPYTUUIQUUBUULYOORSUQUVGUUODUBDUBURZUVEUUKUV
|
|
FUUNYMUUJUUITUVHYOUUMNUVFUUNPYMUUJQYOUUMUULTRSUSUTYTUUQNZUUDUVADUVIUUAUUR
|
|
UUCUUTYTUUQYMOUVIUUBUUSNUUCUUTPYTUUQQUUBUUSYOORSUQYTANZUUDYQDUVJUUAYNUUCY
|
|
PYTAYMOUVJUUBYFNUUCYPPYTAQUUBYFYOORSUQUUHDUUFYMMNZUUGUUFLMIZUVKVAMLYMVBVC
|
|
UVKUUGUVLVAUVKYOMNUUGUVLPUVKYOMJMYMMQVDUPYOMLTRVERVNUUIFGZUUPUVBUVMUUPHUV
|
|
ADUVMUUPUURUUTUVMUUPUURUGZYJUUQGZYKYMGZHZCUFZUUTUURUVMUVRUUPUURUUQFGZYMFG
|
|
UVRUUQYMCUHVFVGUVNUVQUUTCUVNUVOUUTUVPUVNUVOUUTUVNUVOHUUIUCUDZIZUUQUVTUVTU
|
|
NZIZHZUCFVHZUUTUVMUUPUVOUWEUURYIUCUUIVIVJUVNUWEUUTKUVOUVNUWDUUTUCFUVNUVTF
|
|
GZUWDUUTUVMUUPUURUWFUWDHZUUTKZUVMUURUUPUWHUVMUURUWGUUPUUTUVMUURUWGUUPUUTK
|
|
UUPUWAUULUVTJZIZKZUVMUURUWGHZHZUUTUUOUWKUBUCUBUCURZUUKUWAUUNUWJUUJUVTUUIT
|
|
UWNUUMUWINUUNUWJPUUJUVTQUUMUWIUULTRSVKUWMUWAUWJUUTUWLUWAUVMUURUWFUWAUWCVL
|
|
VOUWMUWJUUTUWMUWJHZYMUWBNZUUTUWOUURUWCUWPUVMUURUWGUWJVMUWLUWCUVMUWJUURUWF
|
|
UWAUWCVPVQZUWBUUQYMVBVRUWOUUSUWBJZIZUWPUUTKUWOUWSUULMUNZUWIUWIUNZIZUWOUVO
|
|
YKUWBGZHZCUFZUXBUWOUWCUXEUWQUWCUVSUWBFGUXEUUQUWBCUHVFZRUWOUXDUXBCUWOUVOUX
|
|
BUXCUWMUWJUVOUXBUWMUWJUVOUGZUULYTIZUWTYTYTUNZIZHZEFVHZUXBUXGUULFGZYJULZUW
|
|
TGUXLUXGUWJUXMUWMUWJUVOVSUWJUXMUWIFGYJUULGYKUWIGHCUFUULUWICUHVTRUXGUXNUUS
|
|
UWTUXGUVSUVOUXNUUSGUXGUVMUVSUVMUWLUWJUVOWAZUUIWDRUWMUWJUVOWBYJUUQWCVRUXGU
|
|
VMUUQWEWFZUUSUWTNZUXOUVOUWMUXPUWJUUQYJWGZVGUUIWHZVRWIYJEUULVIVRUWMUWJUXLU
|
|
XBKUVOUWOUXKUXBEFUWMUWJUXKUXBUWMUWJUXKUGZUWIYTNZUXBUXTUWJUXHUYAUWMUWJUXKV
|
|
SUWMUWJUXHUXJWJYTUULUWIVBVRUXKUWMUYAUXBKZUWJUXJUYBUXHUYAUXBUXJUYAUXAUXINZ
|
|
UXBUXJPUYAUYCUWIUWIYTYTWKWLUXAUXIUWTTRWMVOVGWOWNWPWQWOWNWRWSWOUWOUXQUWRUX
|
|
ANZUWSUXBPUWOUVMUXPUXQUVMUWLUWJWTUWOUWCUXEUXPUWQUXFUXDUXPCUVOUXPUXCUXRXAX
|
|
BXCUXSVRUWOUWFUWFUWBWEWFZUYDUWMUWFUWJUVMUURUWFUWDVLXAZUYFUWOUWCUXEUYEUWQU
|
|
XFUXDUYECUXCUYEUVOUWBYKWGVOXBXCUVTUVTXEXFUXQUWSUWTUWRIUYDUXBUUSUWTUWROUWR
|
|
UXAUWTTXDVRXGUWPUUTUWSUWPYOUWRNUUTUWSPYMUWBQYOUWRUUSTRWMRWOXHXIXPXJXKXLXQ
|
|
XMXNXAWOXHWRWSWOWNXOXHXRYQYSDBFYMBNZYNYEYPYHYMBATUYGYOYGNYPYHPYMBQYOYGYFT
|
|
RSXSXTYA $.
|
|
$}
|
|
|
|
${
|
|
$d n y a x $. $d a t $. $d a w $. $d a z $. $d n t $. $d n z $.
|
|
$d t w $. $d t x $. $d t y $. $d t z $. $d w x $. $d w y $. $d w z $.
|
|
$d x z $. $d y z $.
|
|
|
|
$( Lemma for ~ tfinnn . Establish stratification. (Contributed by SF,
|
|
30-Jan-2015.) $)
|
|
tfinnnlem1 $p |- { n | A. y e. n ( y C_ Nn -> { a | E. x e.
|
|
y a = _T[fin] x } e. _T[fin] n ) } e. _V $=
|
|
( vt vz vw cssetk csn cimak wcel wn copk wa wex snex bitri 3bitri 3bitr4i
|
|
exbii csik cnnc cpw cpw1 cvv cxpk c0 cins2k ccnvk cins3k c1c csymdif cdif
|
|
cin cidk ccompl cun cuni1 cv wss ctfin wceq wrex cab wi vex eluni1 opkeq1
|
|
wral wel eleq1d ceqsexv elin opksnelsik elssetk opkelxpk mpbiran2 snelpw1
|
|
eldif otkelins2k opkelcnvk eqtfinrelk opkex elimak elpw121c anbi1i 19.41v
|
|
bitr4i df-rex excom wb elsymdif otkelins3k anbi12i bibi12i notbii elcompl
|
|
elpw abeq2 df-clel elpw11c tfinex clel3 annim dfral2 abbi2i ssetkex sikex
|
|
wal alex nncex pwex pw1ex vvex xpkex tfinrelkex ins2kex ins3kex inex 1cex
|
|
cnvkex imakex symdifex complex difex uni1ex eqeltrri ) HUAZUBUCZUDZUEUFZU
|
|
AZUGIZIZYMUFHUHZHUIUJUBUEUFYHUHZUKUCUEUFHUJZYPULUKUDZUDZUDZJUMUAUJYOUNYSJ
|
|
UJUNYTJUNUHUOUJULYRJUHUMYRJUJULYSJUPYNUEUFUMUQZUIZUHZYQYOUUAUJZUNZYSJZUAZ
|
|
UHZULZYSJZUPZUAZUHZYQUNZYSJZUJZUNZYRJZUMZUNZYRJZUPZURZBUSZUBUTZDUSZAUSZVA
|
|
VBZAUVDVCZDVDZCUSZVAZKZVEZBUVKVIZCVDUEUVOCUVCUVKUVCKUVKIZUVBKZUVOUVKUVBCV
|
|
FZVGUVPUVAKZLUVNLZBUVKVCZLUVQUVOUVSUWAEUSZUVDIZIZVBZUWBUVPMZUUTKZNZEOZBOZ
|
|
BCVJZUVTNZBOUVSUWAUWIUWLBUWIUWDUVPMZUUTKZUWMYHKZUWMUUSKZNUWLUWGUWNEUWDUWC
|
|
PZUWEUWFUWMUUTUWBUWDUVPVHVKVLUWMYHUUSVMUWOUWKUWPUVTUWOUWCUVKMZHKUWKUWCUVK
|
|
HUVDPZUVRVNUVDUVKBVFZUVRVOQUWPUWMYLKZUWMUURKZLZNUVEUVMLZNUVTUWMYLUURVSUXA
|
|
UVEUXCUXDUXAUWRYKKZUVDYIKZUVEUWCUVKYKUWSUVRVNUXEUWCYJKZUXFUXEUXGUVKUEKUVR
|
|
UWCUVKYJUEUWSUVRVPVQUVDYIVRQUVDUBUWTWRRUXBUVMUWBFUSZIZIZVBZUWBUWMMZUUQKZN
|
|
ZEOZFOZUXHUVLVBZUVJUXHKZNZFOUXBUVMUXOUXSFUXOUXJUWMMZUUQKZUXTUUCKZUXTUUPKZ
|
|
NUXSUXMUYAEUXJUXIPUXKUXLUXTUUQUWBUXJUWMVHVKVLUXTUUCUUPVMUYBUXQUYCUXRUYBUX
|
|
HUVPMUUBKUVPUXHMUUAKUXQUXHUWDUVPUUBFVFZUWQUVKPZVTUXHUVPUUAUYDUYEWAUVKUXHU
|
|
VRUYDWBRUXHUWDMZUUOKZGUSZUVJVBZGFVJZNZGOZUYCUXRUYGUWBUYFMZUUNKZEYSVCZUWBU
|
|
YHIZIZIZVBZUYNNZEOZGOZUYLEUUNYSUYFUXHUWDWCWDUWBYSKZUYNNZEOUYTGOZEOUYOVUBV
|
|
UDVUEEVUDUYSGOZUYNNVUEVUCVUFUYNGUWBWEWFUYSUYNGWGWHTUYNEYSWIUYTGEWJSVUAUYK
|
|
GVUAUYRUYFMZUUNKZVUGUUMKZVUGYQKZNUYKUYNVUHEUYRUYQPUYSUYMVUGUUNUWBUYRUYFVH
|
|
VKVLVUGUUMYQVMVUIUYIVUJUYJVUIUYPUWDMUULKUYHUWCMZUUKKZUYIUYPUXHUWDUULUYHPZ
|
|
UYDUWQVTUYHUWCUUKGVFZUWSVNVUKUUJKZLDGVJZUVIWKZLZDOZLZVULUYIVUOVUSVUOUWBVU
|
|
KMZUUIKZEYSVCZUWBUVFIZIZIZVBZVVBNZEOZDOZVUSEUUIYSVUKUYHUWCWCZWDVUCVVBNZEO
|
|
VVHDOZEOVVCVVJVVLVVMEVVLVVGDOZVVBNVVMVUCVVNVVBDUWBWEWFVVGVVBDWGWHTVVBEYSW
|
|
IVVHDEWJSVVIVURDVVIVVFVUKMZUUIKZVVOYQKZVVOUUHKZWKZLVURVVBVVPEVVFVVEPVVGVV
|
|
AVVOUUIUWBVVFVUKVHVKVLVVOYQUUHWLVVSVUQVVQVUPVVRUVIVVQVVDUYHMHKVUPVVDUYHUW
|
|
CHUVFPZVUNUWSWMUVFUYHDVFZVUNVOQVVRVVDUWCMUUGKZUVIVVDUYHUWCUUGVVTVUNUWSVTU
|
|
VFUVDMZUUFKZABVJZUVHNZAOZVWBUVIVWDUWBVWCMZUUEKZEYSVCZUWBUVGIZIZIZVBZVWINZ
|
|
EOZAOZVWGEUUEYSVWCUVFUVDWCWDVUCVWINZEOVWOAOZEOVWJVWQVWRVWSEVWRVWNAOZVWINV
|
|
WSVUCVWTVWIAUWBWEWFVWNVWIAWGWHTVWIEYSWIVWOAEWJSVWPVWFAVWPVWMVWCMZUUEKZVXA
|
|
YOKZVXAUUDKZNVWFVWIVXBEVWMVWLPVWNVWHVXAUUEUWBVWMVWCVHVKVLVXAYOUUDVMVXCVWE
|
|
VXDUVHVXCVWKUVDMHKVWEVWKUVFUVDHUVGPZVWAUWTVTUVGUVDAVFZUWTVOQVXDVWKUVFMUUA
|
|
KUVHVWKUVFUVDUUAVXEVWAUWTWMUVGUVFVXFVWAWBQWNRTRUVFUVDUUFVWAUWTVNUVHAUVDWI
|
|
SQWOWPRTRWPVUKUUJVVKWQUYIVUQDXIVUTUVIDUYHWSVUQDXJQSRVUJUYPUXHMHKUYJUYPUXH
|
|
UWDHVUMUYDUWQWMUYHUXHVUNUYDVOQWNRTRUXHUWDUVPUUOUYDUWQUYEWMGUVJUXHWTSWNRTU
|
|
XBUXMEYRVCZUXPEUUQYRUWMUWDUVPWCWDUWBYRKZUXMNZEOUXNFOZEOVXGUXPVXIVXJEVXIUX
|
|
KFOZUXMNVXJVXHVXKUXMFUWBXAWFUXKUXMFWGWHTUXMEYRWIUXNFEWJSQFUVJUVLUVKXBXCSW
|
|
PWNUVEUVMXDRWNRTUVSUWGEYRVCZUWJEUUTYRUVPUYEWDVXHUWGNZEOUWHBOZEOVXLUWJVXMV
|
|
XNEVXMUWEBOZUWGNVXNVXHVXOUWGBUWBXAWFUWEUWGBWGWHTUWGEYRWIUWHBEWJSQUVTBUVKW
|
|
ISWPUVPUVAUYEWQUVNBUVKXESQXFUVBUVAUUTYRYHUUSHXGXHYLUURYKYJUEYIUBXKXLXMXNX
|
|
OXHUUQYRUUCUUPUUBUUAXPYAXQUUOUUNYSUUMYQUULUUKUUJUUIYSYQUUHHXGXRZUUGUUFUUE
|
|
YSYOUUDHXGXQUUAXPXRXSYRUKXTXMZXMZYBXHXQYCVXRYBYDXHXQVXPXSVXRYBXRXSVXQYBYE
|
|
XSVXQYBYDYFYG $.
|
|
$}
|
|
|
|
${
|
|
$d N a x $. $d A a x $. $d a y $. $d A y $. $d n y $. $d N y $.
|
|
$d x y $. $d a b $. $d a k $. $d a n $. $d a w $. $d a z $. $d b k $.
|
|
$d b w $. $d b x $. $d b y $. $d b z $. $d k n $. $d k w $. $d k x $.
|
|
$d k y $. $d k z $. $d N n $. $d n x $. $d n z $. $d w x $. $d w y $.
|
|
$d w z $. $d x z $. $d y z $.
|
|
|
|
$( T-raising of a set of naturals. Theorem X.1.46 of [Rosser] p. 532.
|
|
(Contributed by SF, 30-Jan-2015.) $)
|
|
tfinnn $p |- ( ( N e. Nn /\ A C_ Nn /\ A e. N ) ->
|
|
{ a | E. x e. A a = _T[fin] x } e. _T[fin] N ) $=
|
|
( vy vb vw cnnc wcel wss cv ctfin wceq wrex cab wi c0 wn c0c wa vn vk wal
|
|
vz wral c1c cplc tfinnnlem1 tfineq tfin0c syl6eq eleq2d imbi2d raleqbi1dv
|
|
df-ral el0c ab0 bitri imbi2i imbi12i albii 0ex sseq1 rexeq notbid imbi12d
|
|
albidv ceqsalv 3bitri syl6bb weq abbidv eleq1d cbvralv ax-gen a1i csn cun
|
|
rex0 ccompl elsuc wel rspcv ad2antrl simprl w3a simp3 simplrr vex elcompl
|
|
sylib elequ1 syl5ibcom con2d imp simpll simprr simplr sseldd simpr tfin11
|
|
syl syl3anc mtand nrexdv 3adant3 tfinex eqeq1 rexbidv elab sylnibr elsuci
|
|
syl2anc 3expia embantd ex com23 snss anbi2i unss bitr2i rexun df-sn rexsn
|
|
wo eqeq2d abbii eqtr4i uneq2i unab eqtr2i biimprcd syl6 syld an32s syl5bi
|
|
rexlimdvva imp32 wne ne0i tfinsuc eleqtrrd expr ralrimiva finds rspccv
|
|
3imp ) CHIZBHJZBCIZDKZAKZLZMZABNZDOZCLZIZUUHUUJUUIUURUUHEKZHJZUUNAUUSNZDO
|
|
ZUUQIZPZECUEZUUJUUIUURPZPUUTUVBUAKZLZIZPZEUVGUEZQHJZUUNAQNZRZDUCZPZUUTUVB
|
|
UBKZLZIZPZEUVQUEZUDKZHJZUUNAUWBNZDOZUVQUFUGZLZIZPZUDUWFUEZUVEUAUBCAEUADUH
|
|
UVGSMZUVKUUTUVBSIZPZESUEZUVPUVJUWMEUVGSUWKUVIUWLUUTUWKUVHSUVBUWKUVHSLSUVG
|
|
SUIUJUKULUMUNUWNUUSSIZUWMPZEUCUUSQMZUUTUVARZDUCZPZPZEUCUVPUWMESUOUWPUXAEU
|
|
WOUWQUWMUWTUUSUPUWLUWSUUTUWLUVBQMUWSUVBUPUVADUQURUSUTVAUWTUVPEQVBUWQUUTUV
|
|
LUWSUVOUUSQHVCUWQUWRUVNDUWQUVAUVMUUNAUUSQVDVEVGVFVHVIVJUVJUVTEUVGUVQUAUBV
|
|
KZUVIUVSUUTUXBUVHUVRUVBUVGUVQUIULUMUNUVGUWFMZUVKUUTUVBUWGIZPZEUWFUEUWJUVJ
|
|
UXEEUVGUWFUXCUVIUXDUUTUXCUVHUWGUVBUVGUWFUIULUMUNUXEUWIEUDUWFEUDVKZUUTUWCU
|
|
XDUWHUUSUWBHVCUXFUVBUWEUWGUXFUVAUWDDUUNAUUSUWBVDVLVMVFVNVJUVJUVDEUVGCUVGC
|
|
MZUVIUVCUUTUXGUVHUUQUVBUVGCUIULUMUNUVOUVLUVNDUUNAVSVOVPUVQHIZUWAUWJUXHUWA
|
|
TZUWIUDUWFUXIUWBUWFIZUWCUWHUXIUXJUWCTZTZUWEUVRUFUGZUWGUXIUXJUWCUWEUXMIZUX
|
|
JUWBFKZGKZVQZVRZMZGUXOVTZNFUVQNUXIUWCUXNPZGUWBUVQFWAUXIUXSUYAFGUVQUXTUXHF
|
|
UBWBZUXPUXTIZTZUWAUXSUYAPZUXHUYDTZUWAUYEUYFUWAUXOHJZUUNAUXONZDOZUVRIZPZUY
|
|
EUYBUWAUYKPUXHUYCUVTUYKEUXOUVQEFVKZUUTUYGUVSUYJUUSUXOHVCUYLUVBUYIUVRUYLUV
|
|
AUYHDUUNAUUSUXOVDVLVMVFWCWDUYFUYKUYGUXPHIZTZUYIUXPLZVQZVRZUXMIZPZUYEUYFUY
|
|
NUYKUYRUYFUYNUYKUYRPUYFUYNTZUYGUYJUYRUYFUYGUYMWEZUYFUYNUYJUYRUYFUYNUYJWFZ
|
|
UYJUYOUYIIZRUYRUYFUYNUYJWGVUBUYOUUMMZAUXONZVUCUYFUYNVUERUYJUYTVUDAUXOUYTA
|
|
FWBZTZVUDGAVKZUYTVUFVUHRUYTVUHVUFUYTGFWBZRZVUHVUFRUYTUYCVUJUXHUYBUYCUYNWH
|
|
UXPUXOGWIZWJWKVUHVUIVUFGAFWLVEWMWNWOVUGVUDTZUYMUULHIVUDVUHVULUYTUYMUYTVUF
|
|
VUDWPZUYFUYGUYMWQXBVULUXOHUULVULUYTUYGVUMVUAXBUYTVUFVUDWRWSVUGVUDWTUXPUUL
|
|
XAXCXDXEXFUYHVUEDUYOUXPXGZUUKUYOMZUUNVUDAUXOUUKUYOUUMXHXIXJXKUYIUVRUYOVUN
|
|
XLXMXNXOXPXQUXSUYAUYSUXSUWCUYNUXNUYRUXSUWCUXRHJZUYNUWBUXRHVCUYNUYGUXQHJZT
|
|
VUPUYMVUQUYGUXPHVUKXRXSUXOUXQHXTYAVJUXSUWEUYQUXMUXSUWEUYHUUNAUXQNZYEZDOZU
|
|
YQUXSUWDVUSDUXSUWDUUNAUXRNVUSUUNAUWBUXRVDUUNAUXOUXQYBVJVLUYQUYIVURDOZVRVU
|
|
TUYPVVAUYIUYPVUODOVVADUYOYCVURVUODUUNVUOAUXPVUKAGVKUUMUYOUUKUULUXPUIYFYDY
|
|
GYHYIUYHVURDYJYKUKVMVFYLYMYNWOYOYQYPYRUXLUXHUWFQYSZUWGUXMMUXHUWAUXKWPUXJV
|
|
VBUXIUWCUWFUWBYTWDUVQUUAXMUUBUUCUUDXPUUEUVDUVFEBCUUSBMZUUTUUIUVCUURUUSBHV
|
|
CVVCUVBUUPUUQVVCUVAUUODUUNAUUSBVDVLVMVFUUFXBXQUUG $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a r $. $d a s $. $d b r $. $d b s $. $d M a $. $d M b $.
|
|
$d M r $. $d M s $. $d N a $. $d N b $. $d N r $. $d N s $. $d P a $.
|
|
$d P b $. $d P r $. $d P s $. $d Q a $. $d Q b $. $d Q r $. $d Q s $.
|
|
$d r s $. $d a d $. $d a g $. $d a t $. $d a x $. $d b d $. $d b g $.
|
|
$d b t $. $d b x $. $d d g $. $d d r $. $d d s $. $d d t $. $d d x $.
|
|
$d g r $. $d g s $. $d g t $. $d g x $. $d M d $. $d M g $. $d M t $.
|
|
$d M x $. $d N d $. $d N g $. $d N t $. $d N x $. $d P d $. $d P g $.
|
|
$d P t $. $d P x $. $d Q d $. $d Q g $. $d Q t $. $d Q x $. $d r t $.
|
|
$d r x $. $d s t $. $d s x $. $d t x $. $d a n $. $d a u $. $d b n $.
|
|
$d b u $. $d d n $. $d d u $. $d g n $. $d g u $. $d m n $. $d M n $.
|
|
$d m u $. $d M u $. $d N n $. $d n r $. $d n s $. $d n t $. $d n u $.
|
|
$d N u $. $d P n $. $d P u $. $d Q n $. $d Q u $. $d r u $. $d s u $.
|
|
$d t u $.
|
|
$( Ordering law for finite smaller than. Theorem X.1.47 of [Rosser]
|
|
p. 532. (Contributed by SF, 30-Jan-2015.) $)
|
|
sfinltfin $p |- ( ( ( _S[fin] ( M , N ) /\ _S[fin] ( P , Q ) ) /\
|
|
<< M , P >> e. <[fin] ) -> << N , Q >> e. <[fin] ) $=
|
|
( va vb vr vx wa cltfin wcel cnnc cv wrex wceq c0 wn syl2anc syl cplc cpw
|
|
vs vt vg vd vu vn vm wsfin copk cpw1 wex w3a wi df-sfin 3an6 eeanv simp1l
|
|
wel simp3ll ncfinlower syl3anc simp1r simp3rl reeanv simpl anim12i cin wo
|
|
ctfin simprll simprrl tfinpw1 elin sylanbrc n0i simpl1l wne ne0i tfinprop
|
|
adantr simpld nndisjeq sylc simprlr simprrr simpl1r wb tfinltfin ad2antrl
|
|
orel1 c1c opkltfing simp2rr simp3r eleqtrd addcass syl6eleq 0nelsuc eleq1
|
|
cun eladdc syl5ibcom mtoi df-ne sylibr wss ssun2 sseq2 mpbiri sseld disjr
|
|
n0 rsp sylbi anim12ii ancoms ad2antll vex snelpw sylib eleq2 imp 3ad2ant1
|
|
wral ad2antrr cfin cvv pwex c0c syl5bir eleq1d syl5ibrcom mpd 3expa exp32
|
|
rexlimdv syl5bi rexlimdvv mp2and csn notbii anbi12i ssun1 adantl biimprcd
|
|
sspwb anim2i dfpss2 simp2ll simp2rl nnpweq simpr2l simp3lr simpr1 simp12l
|
|
wpss con3d cdif simp3rr simp12r elunii df-fin syl6eleqr difex difss ssfin
|
|
mp3an13 elfin undif1 eqtr3i simp23 pssssd ssequn2 syl5eqr simp22r disjdif
|
|
uncom a1i eladdci eqeltrrd simp21 simp3l nncaddccl df-pss ssdif0 simplbi2
|
|
cuni necon3d biimpcd syl6ib necon3ad nnc0suc addceq2 syl6eqr reximi mp2an
|
|
eqss el0c addcex opkeq2 opkeq1 3exp2 mpand exlimdv adantld sylbid sylbird
|
|
syl5 syld opkeq12 imbi1d syl7 3expia exlimdvv 3impia sylbir syl2anb ) CDU
|
|
IZABUIZICAUJZJKZDBUJZJKZUXSCLKZDLKZEMZUKZCKZUYGUAZDKZIZEULZUMZALKZBLKZFMZ
|
|
UKZAKZUYQUAZBKZIZFULZUMZUYBUYDUNZUXTCDEUOABFUOUYNVUDIUYEUYOIZUYFUYPIZUYMV
|
|
UCIZUMVUEUYEUYOUYFUYPUYMVUCUPVUFVUGVUHVUEVUHUYLVUBIZFULEULVUFVUGIZVUEUYLV
|
|
UBEFUQVUJVUIVUEEFVUFVUGVUIVUEVUFVUGVUIUMZEGUSZVULIZGLNZFUBUSZVUOIZUBLNZVU
|
|
EVUKUYEUYIUYIVUNUYEUYOVUGVUIURUYIUYKVUBVUFVUGUTZVURUYGUYGGCVAVBVUKUYOUYSU
|
|
YSVUQUYEUYOVUGVUIVCUYSVUAUYLVUFVUGVDZVUSUYQUYQUBAVAVBVUNVUQIVUMVUPIZUBLNG
|
|
LNVUKVUEVUMVUPGUBLLVEVUKVUTVUEGUBLLVUTVULVUOIZVUKGMZLKZUBMZLKZIZVUEVUMVUL
|
|
VUPVUOVULVULVFVUOVUOVFVGVUKVVFVVAVUEVUKVVFVVAIZIZCVVBVJZOZAVVDVJZOZVUEVVH
|
|
CVVIVHZPOZQZVVNVVJVIZVVJVVHUYHVVMKZVVOVVHUYIUYHVVIKZVVQVUKUYIVVGVURWAVVHV
|
|
VCVULVVRVUKVVCVVEVVAVKZVUKVVFVULVUOVLZUYGVVBVMRUYHCVVIVNVOVVMUYHVPSVVHUYE
|
|
VVILKZVVPUYEUYOVUGVUIVVGVQVVHVVCVVBPVRZVWAVVSVVHVULVWBVVTVVBUYGVSSVVCVWBI
|
|
VWAVVREVVBNVVBEVTWBRCVVIWCRVVNVVJWKWDVVHAVVKVHZPOZQZVWDVVLVIZVVLVVHUYRVWC
|
|
KZVWEVVHUYSUYRVVKKZVWGVUKUYSVVGVUSWAVVHVVEVUOVWHVUKVVCVVEVVAWEZVUKVVFVULV
|
|
UOWFZUYQVVDVMRUYRAVVKVNVOVWCUYRVPSVVHUYOVVKLKZVWFUYEUYOVUGVUIVVGWGVVHVVEV
|
|
VDPVRZVWKVWIVVHVUOVWLVWJVVDUYQVSSVVEVWLIVWKUYHVVKKEVVDNVVDEVTWBRAVVKWCRVW
|
|
DVVLWKWDVVHVUEVVJVVLIZVVIVVKUJZJKZUYDUNVVHVWOVVBVVDUJJKZUYDVVFVWPVWOWHVUK
|
|
VVAVVBVVDWIWJVVHVWPVWBVVDVVBUCMZTWLTZOZUCLNZIZUYDVVFVWPVXAWHVUKVVAUCVVBVV
|
|
DLLWMWJVVHVWTUYDVWBVVHVWSUYDUCLVVHVWQLKZVWSUYDVUKVVGVXBVWSIZUYDVUKVVGVXCU
|
|
MZUYQVVBVWQWLTZTZKZUYDVXDUYQVWRVXFVXDUYQVVDVWRVULVUOVVFVUKVXCWNVUKVVGVXBV
|
|
WSWOWPVVBVWQWLWQWRVXGUDMZUEMZVHPOZUYQVXHVXIXAZOZIZUEVXENUDVVBNVXDUYDUYQVV
|
|
BVXEUDUEXBVXDVXMUYDUDUEVVBVXEVXDUDGUSZVXIVXEKZIZVXMUYDVXDVXPVXMIZIZVXIPVR
|
|
ZUYDVXRVXIPOZQVXSVXRVXTPVXEKZVWQWSVXRVXOVXTVYAVXDVXNVXOVXMWEVXIPVXEWTXCXD
|
|
VXIPXEXFVXSHUEUSZHULVXRUYDHVXIXMVXRVYBUYDHVXRVYBHFUSZHUDUSZQZIZUYDVXMVYBV
|
|
YFUNZVXDVXPVXLVXJVYGVXLVYBVYCVXJVYEVXLVXIUYQHMZVXLVXIUYQXGVXIVXKXGVXIVXHX
|
|
HUYQVXKVXIXIXJXKVXJVYEHVXIYEVYBVYEUNHVXHVXIXLVYEHVXIXNXOXPXQXRVYFVYHUUAZU
|
|
YTKZVYIVXHUAZKZQZIZVXRUYDVYJVYCVYMVYEVYHUYQHXSZXTVYLVYDVYHVXHVYOXTUUBUUCV
|
|
XRVYKUYTXGZVYNUYDVXMVYPVXDVXPVXLVYPVXJVXLVXHUYQXGZVYPVXLVYQVXHVXKXGVXHVXI
|
|
UUDUYQVXKVXHXIXJVXHUYQUUGYAUUEXRVYPVYNIZVYKUYTUUQZVXRUYDVYRVYPVYKUYTOZQZI
|
|
VYSVYNWUAVYPVYJVYMWUAVYJVYTVYLVYTVYLVYJVYKUYTVYIYBUUFUURYCUUHVYKUYTUUIXFV
|
|
XRUYJUFMZKZVYKWUBKZIZUFLNZVYSUYDUNZVXRVVCVULVXNWUFVXDVVCVXQVVCVVEVVAVUKVX
|
|
CUUJWAVXDVULVXQVULVUOVVFVUKVXCUUKWAVXDVXNVXOVXMVKUYGVXHUFVVBUULVBVXRWUEWU
|
|
GUFLVXRWUBLKZWUEVYSUYDVXRWUHWUEVYSUMZIZWUBDOZUYDWUJWUBDVHZPOZQZWUMWUKVIZW
|
|
UKWUJUYJWULKZWUNWUJWUCUYKWUPWUCWUDWUHVYSVXRUUMVXDUYKVXQWUIVUKVVGUYKVXCUYI
|
|
UYKVUBVUFVUGUUNYDYFUYJWUBDVNVOWULUYJVPSWUJWUHUYFWUOVXRWUHWUEVYSUUOVXDUYFV
|
|
XQWUIUYFUYPVUFVUIVVGVXCUUPYFWUBDWCRWUMWUKWKWDWUJWUBBUJZJKZWUKUYDWUJUYTVYK
|
|
UUSZYGKZWURWUJUYTYGKZWUTWUJUYTLUWHZYGWUJVUAUYPUYTWVBKVXDVUAVXQWUIVUKVVGVU
|
|
AVXCUYSVUAUYLVUFVUGUUTYDZYFVXDUYPVXQWUIUYFUYPVUFVUIVVGVXCUVAZYFUYTBLUVBRU
|
|
VCUVDWUSYHKWVAWUSUYTXGWUTUYTVYKUYQFXSYIVXHUDXSYIUVEUYTVYKUVFWUSUYTYHUVGUV
|
|
HSWUTWUSUGMZKZUGLNWUJWURUGWUSUVIWUJWVFWURUGLWUJWVELKZWVFWURVXRWUIWVGWVFIZ
|
|
WURVXRWUIWVHUMZBWUBWVETZOZWURWVIBWVJVHZPOZQZWVMWVKVIZWVKWVIUYTWVLKZWVNWVI
|
|
VUAUYTWVJKWVPVXRWUIVUAWVHVXDVUAVXQWVCWAYDWVIVYKWUSXAZUYTWVJWVIWVQUYTVYKXA
|
|
ZUYTWUSVYKXAWVRWVQUYTVYKUVJWUSVYKUVRUVKWVIVYPWVRUYTOWVIVYKUYTVXRWUHWUEVYS
|
|
WVHUVLZUVMVYKUYTUVNYAUVOWVIWUDWVFVYKWUSVHPOZWVQWVJKWUCWUDWUHVYSVXRWVHUVPZ
|
|
VXRWUIWVGWVFWOZWVTWVIVYKUYTUVQUVSVYKWUSWUBWVEUVTVBUWAUYTBWVJVNVOWVLUYTVPS
|
|
WVIUYPWVJLKZWVOVXRWUIUYPWVHVXDUYPVXQWVDWAYDWVIWUHWVGWWCVXRWUHWUEVYSWVHUWB
|
|
VXRWUIWVGWVFUWCZWUBWVEUWDRBWVJWCRWVMWVKWKWDWVIWURWVKWUBWVJUJZJKZWVIWUBPVR
|
|
ZWVJWUBUHMZTWLTZOZUHLNZWWFWVIWUDWWGWWAWUBVYKVSSWVIWVEWWHWLTZOZUHLNZWWKWVI
|
|
WVEYJOZQZWWOWWNVIZWWNWVIWVFWUSPVRZWWPWWBWVIVYSWWRWVSVYSVYPVYKUYTVRZIWWRVY
|
|
KUYTUWEVYPWWSWWRVYPWUSPVYKUYTWUSPOZUYTVYKXGZVYPVYTUYTVYKUWFVYTVYPWXAVYKUY
|
|
TUWRUWGYKUWIYCXOSWVFWWOWUSPWVFWWOWUSYJKZWWTWWOWVFWXBWVEYJWUSYBUWJWUSUWSUW
|
|
KUWLWDWVIWVGWWQWWDUHWVEUWMYAWWOWWNWKWDWWMWWJUHLWWMWVJWUBWWLTWWIWVEWWLWUBU
|
|
WNWUBWWHWLWQUWOUWPSWUBYHKWVJYHKWWFWWGWWKIWHUFXSZWUBWVEWXCUGXSUWTUHWUBWVJY
|
|
HYHWMUWQVOWVKWUQWWEJBWVJWUBUXAYLYMYNYOYPYQYRYNWUKWUQUYCJWUBDBUXBYLXCYNUXC
|
|
YQYNUXIUXDYKUXJUXEYRYNYPYSYRYNYOYPYQUXFUXGUXHVWMUYBVWOUYDVWMUYAVWNJCAVVIV
|
|
VKUXKYLUXLYMYTYPUXMYSYKYTUXNUXOYKUXPUXQUXRYC $.
|
|
$}
|
|
|
|
${
|
|
$d M a $. $d N a $. $d P a $.
|
|
$( The finite smaller relationship is one-to-one in its first argument.
|
|
Theorem X.1.48 of [Rosser] p. 533. (Contributed by SF, 29-Jan-2015.) $)
|
|
sfin111 $p |- ( ( _S[fin] ( M , P ) /\ _S[fin] ( N , P ) ) ->
|
|
M = N ) $=
|
|
( va wsfin wa copk cltfin wcel wo wn wceq cnnc wex df-sfin adantl simp1bi
|
|
sfinltfin adantr sylib cv cpw1 cpw simp2bi ltfinirr syl mtand ancoms mtod
|
|
wi ex ioran sylanbrc w3o wne ne0i exlimiv 3ad2ant3 sylbi ltfintri syl3anc
|
|
c0 w3a df-3or or32 orel1 sylc ) BAEZCAEZFZBCGHIZCBGHIZJZKZVMBCLZJZVOVJVKK
|
|
VLKVNVJVKAAGHIZVJAMIZVQKVIVRVHVICMIZVRDUAZUBZCIVTUCAIZFDNZCADOZUDPAUEUFZC
|
|
ABARUGVJVLVQWEVIVHVLVQUJVIVHFVLVQBACARUKUHUIVKVLULUMVJVKVOJVLJZVPVJVKVOVL
|
|
UNZWFVJBMIZVSBVBUOZWGVHWHVIVHWHVRWABIZWBFZDNZBADOZQSVIVSVHVIVSVRWCWDQPVHW
|
|
IVIVHWHVRWLVCWIWMWLWHWIVRWKWIDWJWIWBBWAUPSUQURUSSBCUTVAVKVOVLVDTVKVOVLVET
|
|
VMVOVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d a x $. $d a z $. $d t x $. $d x z $. $d a t $. $d t z $.
|
|
$( ` Sp[fin] ` is a set. (Contributed by SF, 20-Jan-2015.) $)
|
|
spfinex $p |- Sp[fin] e. _V $=
|
|
( va vz vx vt cv wcel cssetk c1c cins3k csik cimak cin wa wn wex copk csn
|
|
bitr4i exbii 3bitri cspfin cvv cncfin cab cnnc cxpk cpw csymdif cpw1 cdif
|
|
cins2k ccompl cint wsfin wel wi wal wral df-spfin wrex wceq elimak anbi1i
|
|
el1c 19.41v df-rex excom 3bitr4i snex opkeq1 eleq1d ceqsexv elssetk opkex
|
|
elin elpw121c eldif otkelins3k opksnelsik srelk otkelins2k notbii anbi12i
|
|
bitri exanali elcompl dfral2 abbi2i ineq2i eqtri inteqi eqtr4i setswithex
|
|
vex inab ssetkex srelkex sikex ins3kex ins2kex difex pw1ex imakex complex
|
|
1cex inex intex eqeltri ) UAUBUCZAEZFZAUDZGUEUEUFHUGUBUFGIGJUKUHZHUIZUIZU
|
|
IZKUJJIGUKZLXOKIXMXOKULJIXQLXOKUKLXPKLZJZIZXQUJZXOKZLZHKZULZLZUMZUBUAXKBE
|
|
ZCEZUNZBAUOZUPBUQZCXJURZMAUDZUMYGCBAUSYFYNYFXLYMAUDZLYNYEYOXLYMAYEXJYDFZN
|
|
YLNZCXJUTZNXJYEFYMYPYRYPCAUOZYQMZCOZYRYPDEZXJPZYCFZDHUTZUUBYIQZVAZUUDMZDO
|
|
ZCOZUUADYCHXJAWNZVBUUBHFZUUDMZDOUUHCOZDOUUEUUJUUMUUNDUUMUUGCOZUUDMUUNUULU
|
|
UOUUDCUUBVDVCUUGUUDCVERSUUDDHVFUUHCDVGVHUUIYTCUUIUUFXJPZYCFZUUPGFZUUPYBFZ
|
|
MYTUUDUUQDUUFYIVIZUUGUUCUUPYCUUBUUFXJVJVKVLUUPGYBVOUURYSUUSYQYIXJCWNZUUKV
|
|
MUUSUUBYHQZQZQZVAZUUBUUPPZYAFZMZDOZBOZYJYKNZMZBOYQUUSUVGDXOUTUUBXOFZUVGMZ
|
|
DOZUVJDYAXOUUPUUFXJVNVBUVGDXOVFUVOUVHBOZDOUVJUVNUVPDUVNUVEBOZUVGMUVPUVMUV
|
|
QUVGBUUBVPVCUVEUVGBVERSUVHBDVGRTUVIUVLBUVIUVDUUPPZYAFZUVRXTFZUVRXQFZNZMUV
|
|
LUVGUVSDUVDUVCVIUVEUVFUVRYAUUBUVDUUPVJVKVLUVRXTXQVQUVTYJUWBUVKUVTUVBUUFPX
|
|
SFYHYIPXRFYJUVBUUFXJXSYHVIZUUTUUKVRYHYIXRBWNZUVAVSYHYIUWDUVAVTTUWAYKUWAUV
|
|
BXJPGFYKUVBUUFXJGUWCUUTUUKWAYHXJUWDUUKVMWDWBWCTSYJYKBWETWCTSTYQCXJVFRWBXJ
|
|
YDUUKWFYLCXJWGVHWHWIXKYMAWOWJWKWLYFXLYEAXIWMYDYCHGYBWPYAXOXTXQXSXRWQWRWSG
|
|
WPWTXAXNHXEXBXBXCXFXEXCXDXFXGXH $.
|
|
$}
|
|
|
|
${
|
|
$d a x z $.
|
|
$( The cardinality of the universe is in the finite Sp set. Theorem X.1.49
|
|
of [Rosser] p. 534. (Contributed by SF, 27-Jan-2015.) $)
|
|
ncvspfin $p |- Nc[fin] _V e. Sp[fin] $=
|
|
( va vz vx cvv cncfin cv wcel wsfin wel wi wal wral wa cab cspfin ncfinex
|
|
cint elintab simpl mpgbir df-spfin eleqtrri ) DEZUCAFZGZBFCFHBAIJBKCUDLZM
|
|
ZANQZOUCUHGUGUEJAUGAUCDPRUEUFSTCBAUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d a p $. $d a q $. $d a x $. $d a z $. $d p q $. $d p x $. $d q x $.
|
|
$d q z $. $d X x $. $d X y $. $d x z $. $d Z x $. $d Z y $. $d Z z $.
|
|
|
|
$( If ` X ` is in ` Sp[fin] ` and ` Z ` is smaller than ` X ` , then ` Z `
|
|
is also in ` Sp[fin] ` . Theorem X.1.50 of [Rosser] p. 534.
|
|
(Contributed by SF, 27-Jan-2015.) $)
|
|
spfinsfincl $p |- ( ( X e. Sp[fin] /\ _S[fin] ( Z , X ) ) ->
|
|
Z e. Sp[fin] ) $=
|
|
( vy vz vx va vq vp wsfin cspfin wcel wi cnnc cv wa eleq1 imbi12d wel wal
|
|
wceq cpw1 cpw wex w3a df-sfin sfineq1 imbi2d sfineq2 imbi1d cncfin albidv
|
|
cvv wral weq rspcv spv com12 syl9r com23 adantld a2d alimdv cint df-spfin
|
|
cab eleq2i vex elintab bitri 3imtr4g vtocl2g 3adant3 sylbi pm2.43i impcom
|
|
) BAIZAJKZBJKZVPVQVRLZVPBMKZAMKZCNZUABKWBUBAKOCUCZUDVPVSLZBACUEVTWAWDWCDN
|
|
ZENZIZWFJKZWEJKZLZLBWFIZWHVRLZLWDDEBAMMWEBTZWGWKWJWLWEBWFUFWMWIVRWHWEBJPU
|
|
GQWFATZWKVPWLVSWFABUHWNWHVQVRWFAJPUIQWGULUJFNZKZGNZHNZIZGFRZLZGSZHWOUMZOZ
|
|
EFRZLZFSZXDDFRZLZFSZWHWIWGXFXIFWGXDXEXHWGXCXEXHLWPWGXEXCXHXEXCWQWFIZWTLZG
|
|
SZWGXHXBXMHWFWOHEUNZXAXLGXNWSXKWTWRWFWQUHUIUKUOXMWGXHXLWGXHLGDGDUNXKWGWTX
|
|
HWQWEWFUFWQWEWOPQUPUQURUSUTVAVBWHWFXDFVEVCZKXGJXOWFHGFVDZVFXDFWFEVGVHVIWI
|
|
WEXOKXJJXOWEXPVFXDFWEDVGVHVIVJVKVLVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
$d a x $. $d a z $. $d B a $. $d B x $. $d B z $. $d x z $.
|
|
|
|
$( Inductive principle for ` Sp[fin] ` . Theorem X.1.51 of [Rosser]
|
|
p. 534. (Contributed by SF, 27-Jan-2015.) $)
|
|
spfininduct $p |- ( ( B e. V /\ Nc[fin] _V e. B /\
|
|
A. x e. Sp[fin] A. z ( ( x e. B /\ _S[fin] ( z , x ) ) -> z e. B ) ) ->
|
|
Sp[fin] C_ B ) $=
|
|
( va wcel cspfin cvv cv wa wi wal wral wss mpan elin df-ral 19.21v bitr4i
|
|
albii cin cncfin wsfin spfinex inexg ncvspfin biimpri spfinsfincl adantrl
|
|
a1d ancrd syl6ibr ex a2d exp4a a2i syl5bi 2alimi 3imtr4i w3a wel cab cint
|
|
imp3a df-spfin wceq eleq2 imbi2d albidv raleqbi1dv anbi12d biimprd 3impib
|
|
elabg intss1 syl syl5eqss inss2 syl6ss syl3an ) CDFZGCUAZHFZHUBZCFZWDWBFZ
|
|
AIZCFZBIZWGUCZJZWICFZKZBLZAGMZWJWIWBFZKZBLZAWBMZGCNGHFWAWCUDGCHDUEOWDGFZW
|
|
EWFUFWFWTWEJWDGCPUGOWGGFZWMKZBLZALZWGWBFZWQKZBLZALZWOWSXBXFABXEXAWHJXBWQW
|
|
GGCPXBXAWHWQXAWMWHWQKXAWMWHWJWPXAWKWLWPXAWKWLWPKXAWKJZWLWIGFZWLJWPXIWLXJX
|
|
IXJWLXAWJXJWHWGWIUHUIUJUKWIGCPULUMUNUOUPVDUQURWOXAWNKZALXDWNAGQXCXKAXAWMB
|
|
RTSWSXEWRKZALXHWRAWBQXGXLAXEWQBRTSUSWCWFWSUTZGWBCXMGWDEIZFZWJBEVAZKZBLZAX
|
|
NMZJZEVBZVCZWBABEVEXMWBYAFZYBWBNWCWFWSYCWCYCWFWSJZXTYDEWBHXNWBVFZXOWFXSWS
|
|
XNWBWDVGXRWRAXNWBYEXQWQBYEXPWPWJXNWBWIVGVHVIVJVKVNVLVMWBYAVOVPVQGCVRVSVT
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( If the universe is finite, then ` Sp[fin] ` is a subset of the non-empty
|
|
naturals. Theorem X.1.53 of [Rosser] p. 534. (Contributed by SF,
|
|
27-Jan-2015.) $)
|
|
vfinspnn $p |- ( _V e. Fin -> Sp[fin] C_ ( Nn \ { (/) } ) ) $=
|
|
( vx vz vy cvv cfin wcel cncfin cnnc c0 csn cv wsfin wa wi wal cspfin wne
|
|
cdif ne0i eldifsn wral wss vvex ncfinprop mpan2 anim2i syl sylibr cpw wex
|
|
cpw1 w3a df-sfin adantr exlimiv biimpri sylan2 3adant2 sylbi adantl rgenw
|
|
ax-gen nncex snex difex spfininduct mp3an1 sylancl ) DEFZDGZHIJZRZFZAKZVL
|
|
FZBKZVNLZMVPVLFZNZBOZAPUAZPVLUBZVIVJHFZVJIQZMZVMVIWCDVJFZMZWEVIDDFWGUCDDU
|
|
DUEWFWDWCVJDSUFUGVJHITUHVTAPVSBVQVRVOVQVPHFZVNHFZCKZUKZVPFZWJUIVNFZMZCUJZ
|
|
ULVRVPVNCUMWHWOVRWIWOWHVPIQZVRWNWPCWLWPWMVPWKSUNUOVRWHWPMVPHITUPUQURUSUTV
|
|
BVAVLDFVMWAWBHVKVCIVDVEABVLDVFVGVH $.
|
|
$}
|
|
|
|
$( If the universe is finite, then ` Nc[fin] 1c ` is the base two log of
|
|
` Nc[fin] _V ` . Theorem X.1.54 of [Rosser] p. 534. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
1cvsfin $p |- ( _V e. Fin -> _S[fin] ( Nc[fin] 1c , Nc[fin] _V ) ) $=
|
|
( va cvv cfin wcel c1c cncfin cnnc cv cpw1 cpw wa wex 1cex ncfinprop simpld
|
|
w3a mpan2 vvex simprd eleq1d wsfin wceq pw1eq df1c2 syl6eqr pweq pwv syl6eq
|
|
anbi12d spcev syl2anc 3jca df-sfin sylibr ) BCDZEFZGDZBFZGDZAHZIZUPDZUTJZUR
|
|
DZKZALZPUPURUAUOUQUSVFUOEBDZUQMUOVGKUQEUPDZEBNZOQUOBBDZUSRUOVJKUSBURDZBBNZO
|
|
QUOVHVKVFUOUQVHUOVGUQVHKMVIQSUOUSVKUOVJUSVKKRVLQSVEVHVKKABRUTBUBZVBVHVDVKVM
|
|
VAEUPVMVABIEUTBUCUDUETVMVCBURVMVCBJBUTBUFUGUHTUIUJUKULUPURAUMUN $.
|
|
|
|
$( If the universe is finite, then the size of ` 1c ` is in ` Sp[fin] ` .
|
|
Corollary of Theorem X.1.54 of [Rosser] p. 534. (Contributed by SF,
|
|
29-Jan-2015.) $)
|
|
1cspfin $p |- ( _V e. Fin -> Nc[fin] 1c e. Sp[fin] ) $=
|
|
( cvv cfin wcel cncfin cspfin wsfin ncvspfin 1cvsfin spfinsfincl sylancr
|
|
c1c ) ABCADZECKDZLFMECGHLMIJ $.
|
|
|
|
$( If the universe is finite, then the T-raising of the size of the universe
|
|
is equal to the size of ` 1c ` . Theorem X.1.55 of [Rosser] p. 534.
|
|
(Contributed by SF, 29-Jan-2015.) $)
|
|
tncveqnc1fin $p |- ( _V e. Fin -> _T[fin] Nc[fin] _V = Nc[fin] 1c ) $=
|
|
( cvv cfin wcel cncfin ctfin cpw1 c1c wceq vvex ncfintfin mpan2 df1c2 ax-mp
|
|
ncfineq syl6eqr ) ABCZADEZAFZDZGDZPAACQSHIAAJKGRHTSHLGRNMO $.
|
|
|
|
$( If the universe is finite, then the T-raising of the size of ` 1c ` is
|
|
smaller than the size itself. Corollary of theorem X.1.56 of [Rosser]
|
|
p. 534. (Contributed by SF, 29-Jan-2015.) $)
|
|
t1csfin1c $p |- ( _V e. Fin ->
|
|
_S[fin] ( _T[fin] Nc[fin] 1c , Nc[fin] 1c ) ) $=
|
|
( cvv cfin wcel c1c cncfin ctfin wsfin 1cvsfin sfintfin syl wb tncveqnc1fin
|
|
wceq sfineq2 mpbid ) ABCZDEZFZAEZFZGZRQGZPQSGUAHQSIJPTQMUAUBKLTQRNJO $.
|
|
|
|
${
|
|
$d N a $.
|
|
$( If the universe is finite, then the T-raising of all non-empty naturals
|
|
are no greater than the size of ` 1c ` . Theorem X.1.56 of [Rosser]
|
|
p. 534. (Contributed by SF, 30-Jan-2015.) $)
|
|
vfintle $p |- ( ( _V e. Fin /\ N e. Nn /\ N =/= (/) ) ->
|
|
<< _T[fin] N , Nc[fin] 1c >> e. <_[fin] ) $=
|
|
( va cvv cfin wcel cnnc c0 w3a cncfin copk clefin ctfin wa wceq ncfinprop
|
|
simp2 simpld mpan2 adantr 3ad2ant1 wne c1c cv wex n0 3adant2 simp3 simprd
|
|
nnceleq syl22anc 3expia wi ccompl simpr cun uncompl ncfineq ax-mp vex cin
|
|
complex incompl ncfindi mp3an23 syl5eqr ncfinex lefinaddc sylancr eqeltrd
|
|
cplc ex syld exlimdv syl5bi 3impia wb vvex tfinlefin syl2anc tncveqnc1fin
|
|
opkeq12d opkeq2d eleq1d bitrd mpbid ) CDEZAFEZAGUAZHZACIZJZKEZALZUBIZJZKE
|
|
ZWFWGWHWLWHBUCZAEZBUDWFWGMZWLBAUEWSWRWLBWSWRAWQIZNZWLWFWGWRXAWFWGWRHZWGWT
|
|
FEZWRWQWTEZXAWFWGWRPXBXCXDWFWRXCXDMWGWQAOUFZQWFWGWRUGXBXCXDXEUHWQAWTUIUJU
|
|
KWFXAWLULWGWFXAWLWFXAMZWKWTWTWQUMZIZVJZJZKXFAWTWJXIWFXAUNWFWJXINXAWFWJWQX
|
|
GUOZIZXIXKCNXLWJNWQUPXKCUQURWFWQCEZXLXINZBUSZWFXMMXGCEZWQXGUTGNXNWQXOVAZW
|
|
QVBWQXGCCVCVDRVESWAWFXJKEZXAWFWTCEXHFEZXRWQVFWFXSXGXHEZWFXPXSXTMXQXGCORQW
|
|
TXHCVGVHSVIVKSVLVMVNVOWIWLWMWJLZJZKEZWPWIWGWJFEZWLYCVPWFWGWHPWIYDCWJEZWFW
|
|
GYDYEMZWHWFCCEYFVQCCORTQAWJVRVSWIYBWOKWIYAWNWMWFWGYAWNNWHVTTWBWCWDWE $.
|
|
$}
|
|
|
|
$( If the universe is finite, then ` 1c ` is strictly smaller than the
|
|
universe. Theorem X.1.57 of [Rosser] p. 534. (Contributed by SF,
|
|
30-Jan-2015.) $)
|
|
vfin1cltv $p |- ( _V e. Fin -> << Nc[fin] 1c , Nc[fin] _V >> e. <[fin] ) $=
|
|
( cvv wcel c1c cncfin cplc copk cltfin wceq ax-mp 1cex wa c0 mpan2 c0c cnnc
|
|
wn ncfinprop simprd wne wb cfin cun uncompl ncfineq complex incompl ncfindi
|
|
ccompl cin mp3an23 syl5reqr opkeq2d 0nel1c 0ex elcompl mpbir n0i syl5ibrcom
|
|
eleq2 el0c syl6ib mtoi addcid1 eqeq1i simpld peano1 a1i eqnetrd preaddccan2
|
|
ne0i syl syl31anc syl5bbr mtbird clefin wo ncfinex lefinaddc sylancr addcex
|
|
lefinlteq mp3an12 mpbid orcomd ord mpd eqeltrrd ) AUABZCDZWICUHZDZEZFZWIADZ
|
|
FGWHWLWNWIWHWNCWJUBZDZWLWOAHWPWNHCUCWOAUDIWHCABZWPWLHZJWHWQKWJABZCWJUILHWRC
|
|
JUEZCUFCWJAAUGUJMUKULWHWIWLHZPWMGBZWHXANWKHZWHXCWJLHZLWJBZXDPXELCBPUMLCUNUO
|
|
UPWJLUQIWHXCWJNBZXDWHXFXCWJWKBZWHWKOBZXGWHWSXHXGKWTWJAQMZRNWKWJUSURWJUTVAVB
|
|
XAWINEZWLHZWHXCXJWIWLWIVCZVDWHWIOBZNOBZXHXJLSXKXCTWHXMCWIBZWHWQXMXOKJCAQMZV
|
|
EXNWHVFVGWHXHXGXIVEZWHXJWILXJWIHWHXLVGWHXOWILSZWHXMXOXPRWICVJVKZVHWKWINVIVL
|
|
VMVNWHXAXBWHXBXAWHWMVOBZXBXAVPZWHWIABZXHXTCVQZXQWIWKAVRVSWHXRXTYATZXSYBWLAB
|
|
XRYDYCWIWKYCWJVQVTWIWLAAWAWBVKWCWDWEWFWG $.
|
|
|
|
${
|
|
$d N a $.
|
|
$( If the universe is finite, then the size of the universe is not the
|
|
T-raising of a natural. Theorem X.1.58 of [Rosser] p. 534.
|
|
(Contributed by SF, 29-Jan-2015.) $)
|
|
vfinncvntnn $p |- ( ( _V e. Fin /\ N e. Nn ) ->
|
|
_T[fin] N =/= Nc[fin] _V ) $=
|
|
( va cvv wcel cnnc wa ctfin cncfin wi c0 wceq vvex ncfinprop mpan2 cltfin
|
|
wne copk simpld 3ad2ant1 c1c cfin simprd syl necomd tfineq tfinnul syl6eq
|
|
ne0i neeq1d syl5ibr adantrd w3a wn ltfinirr clefin vfintle vfin1cltv cpw1
|
|
cv wrex tfinprop 3adant1 leltfintr syl3anc mp2and opkeq1 eleq1d syl5ibcom
|
|
1cex mtod df-ne sylibr 3expa expcom pm2.61ine ) CUADZAEDZFZAGZCHZPZIAJAJK
|
|
ZVPWAVQVPWAWBJVTPVPVTJVPCVTDZVTJPVPVTEDZWCVPCCDZWDWCFLCCMZNUBVTCUHUCUDWBV
|
|
SJVTWBVSJGJAJUEUFUGUIUJUKVRAJPZWAVPVQWGWAVPVQWGULZVSVTKZUMWAWHWIVTVTQZODZ
|
|
VPVQWKUMZWGVPWDWLVPWEWDLVPWEFWDWCWFRNZVTUNUCSWHVSVTQZODZWIWKWHVSTHZQUODZW
|
|
PVTQODZWOAUPVPVQWRWGUQSWHVSEDZWPEDZWDWQWRFWOIVQWGWSVPVQWGFWSBUSURVSDBAUTA
|
|
BVARVBVPVQWTWGVPTCDZWTVIVPXAFWTTWPDTCMRNSVPVQWDWGWMSVSWPVTVCVDVEWIWNWJOVS
|
|
VTVTVFVGVHVJVSVTVKVLVMVNVO $.
|
|
$}
|
|
|
|
${
|
|
$d a x $.
|
|
$( If the universe is finite, then its size is not a T raising of an
|
|
element of ` Sp[fin] ` . Corollary of theorem X.1.58 of [Rosser]
|
|
p. 534. (Contributed by SF, 27-Jan-2015.) $)
|
|
vfinncvntsp $p |- ( _V e. Fin ->
|
|
-. Nc[fin] _V e. { a | E. x e. Sp[fin] a = _T[fin] x } ) $=
|
|
( cvv cfin wcel cncfin cv ctfin wceq cspfin wrex cab wa wne cnnc csn cdif
|
|
wn c0 vfinspnn difss syl6ss sselda vfinncvntnn syldan necomd df-ne nrexdv
|
|
sylib ncfinex eqeq1 rexbidv elab sylnibr ) CDEZCFZAGZHZIZAJKZUPBGZURIZAJK
|
|
ZBLEUOUSAJUOUQJEZMZUPURNUSRVEURUPUOVDUQOEURUPNUOJOUQUOJOSPZQOTOVFUAUBUCUQ
|
|
UDUEUFUPURUGUIUHVCUTBUPCUJVAUPIVBUSAJVAUPURUKULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d n z x a b p g d $.
|
|
$( Lemma for ~ vfinspss . Establish part of the inductive step.
|
|
(Contributed by SF, 3-Feb-2015.) $)
|
|
vfinspsslem1 $p |- (
|
|
( ( _V e. Fin /\ _T[fin] n e. Sp[fin] ) /\
|
|
( n e. Sp[fin] /\ _S[fin] ( z , _T[fin] n ) ) ) ->
|
|
E. x e. Sp[fin] z = _T[fin] x ) $=
|
|
( vp va vb vd vg cvv wcel cv cspfin wa c1c wceq cnnc wrex syl cpw1 wex c0
|
|
cfin ctfin wsfin cncfin cplc copk clefin simpl cdif vfinspnn difss syl6ss
|
|
wne csn sselda eldifsn simprbi vfintle syl3anc cltfin wn t1csfin1c adantr
|
|
ad2ant2r wi simpr sfinltfin ex syl2an con3d wb tfincl 1cex mpan2 ad2antrr
|
|
ncfinprop simpld lenltfin syl2anc df-sfin simp1bi ad2antll 3imtr4d tfinex
|
|
cpw mpd vex opklefing mp2an sylib df1c2 pw1eq tfinpw1 syl5eqelr syl5ibcom
|
|
ax-mp eleq2 cin cun eladdc wel wss ssun1 sseq2 mpbiri ssv biantrur bitr4i
|
|
sspw1 exbii anbi1i 19.41v excom pw1ex eqeq2d ceqsexv bitri 3bitri biimpac
|
|
eleq1 nnceleq syl22anc pwex simprr simprd weq eleq1d pweq spcev syl3anbrc
|
|
anbi12d sfintfin sfin112 tfin11 simprl eqeltrrd anbi2d syld syl5 sfineq1
|
|
spfinsfincl risset tfineq eqcomd reximi sylbi eqeq1 rexbidv imbi12d com12
|
|
expdimp exlimdv adantld adantrr rexlimdvva syl5bi rexlimdvw ) IUBJZCKZUCZ
|
|
LJZMZUUTLJZBKZUVAUDZMZMZNUEZUCZUVEDKZUFZOZDPQZUVEAKZUCZOZALQZUVHUVEUVJUGU
|
|
HJZUVNUVHUVAUVIUGUHJZUVSUUSUVDUVTUVBUVFUUSUVDMZUUSUUTPJZUUTUAUNZUVTUUSUVD
|
|
UIUUSLPUUTUUSLPUAUOZUJZPUKPUWDULUMUPZUWAUUTUWEJZUWCUUSLUWEUUTUKUPUWGUWBUW
|
|
CUUTPUAUQURRUUTUSUTVEUVHUVIUVAUGVAJZVBZUVJUVEUGVAJZVBZUVTUVSUVHUWJUWHUVCU
|
|
VJUVIUDZUVFUWJUWHVFUVGUUSUWLUVBVCVDUVDUVFVGUWLUVFMUWJUWHUVEUVAUVJUVIVHVIV
|
|
JVKUVHUVAPJZUVIPJZUVTUWIVLUVHUWBUWMUUSUVDUWBUVBUVFUWFVEUUTVMRUVHUWNNUVIJZ
|
|
UUSUWNUWOMZUVBUVGUUSNIJUWPVNNIVQVOVPZVRZUVAUVIVSVTUVHUVEPJZUVJPJZUVSUWKVL
|
|
UVFUWSUVCUVDUVFUWSUWMEKZSZUVEJUXAWFZUVAJMETUVEUVAEWAWBWCZUVHUWNUWTUWRUVIV
|
|
MRUVEUVJVSVTWDWGUVEIJUVJIJUVSUVNVLBWHUVIWEDUVEUVJIIWIWJWKUVHUVMUVRDPUVHUV
|
|
MISZSZUVLJZUVRUVHUXFUVJJUVMUXGUVHUXFNSZUVJNUXEOUXHUXFOWLNUXEWMWQUVHUWPUXH
|
|
UVJJUWQNUVIWNRWOUVJUVLUXFWRWPUXGUXAFKZWSUAOZUXFUXAUXIWTZOZMZFUVKQEUVEQUVH
|
|
UVRUXFUVEUVKEFXAUVHUXMUVREFUVEUVKUVHEBXBZUXMUVRVFFDXBUVHUXNMZUXLUVRUXJUXL
|
|
UXAGKZSZSZOZGTZUXOUVRUXLUXAUXFXCZUXTUXLUYAUXAUXKXCUXAUXIXDUXFUXKUXAXEXFUY
|
|
AHKZUXEXCZUXAUYBSZOZMZHTUYBUXQOZUYEMZGTZHTZUXTHUXAUXEEWHXJUYFUYIHUYFUYGGT
|
|
ZUYEMUYIUYCUYKUYEUYCUXPIXCZUYGMZGTUYKGUYBIHWHXJUYGUYMGUYLUYGUXPXGXHXKXIXL
|
|
UYGUYEGXMXIXKUYJUYHHTZGTUXTUYHHGXNUYNUXSGUYEUXSHUXQUXPGWHZXOZUYGUYDUXRUXA
|
|
UYBUXQWMXPXQXKXRXSWKUXOUXSUVRGUVHUXNUXSUVRUXNUXSMUXRUVEJZUVHUVRUXSUXNUYQU
|
|
XAUXRUVEYAXTUVHUYQUVEUXQUEZUCZOZUVRUVHUYQUYTUVHUYQMUWSUYSPJZUYQUXRUYSJZUY
|
|
TUVHUWSUYQUXDVDUVHVUAUYQUVHUYRPJZVUAUVHVUCUXQUYRJZUUSVUCVUDMZUVBUVGUUSUXQ
|
|
IJVUEUYPUXQIVQVOZVPZVRUYRVMRVDUVHUYQVGUVHVUBUYQUVHVUEVUBVUGUXQUYRWNRVDUXR
|
|
UVEUYSYBYCVIUYTUVHUVRUYTUVHUVRVFUVCUVDUYSUVAUDZMZMZUYSUVPOZALQZVFVUJUYRLJ
|
|
ZVULVUJUXPWFZUEZLJUYRVUOUDZVUMVUJUUTVUOLVUJUWBVUOPJZUVAVUOUCZOZUUTVUOOUUS
|
|
UVDUWBUVBVUHUWFVEUUSVUQUVBVUIUUSVUQVUNVUOJZUUSVUNIJVUQVUTMUXPUYOYDVUNIVQV
|
|
OZVRZVPVUJVUHUYSVURUDZVUSUVCUVDVUHYEVUJVUPVVCUUSVUPUVBVUIUUSVUCVUQUXBUYRJ
|
|
ZUXCVUOJZMZETZVUPUUSVUCVUDVUFVRVVBUUSVUDVUTVVGUUSVUCVUDVUFYFUUSVUQVUTVVAY
|
|
FVVFVUDVUTMEUXPUYOEGYGZVVDVUDVVEVUTVVHUXBUXQUYRUXAUXPWMYHVVHUXCVUNVUOUXAU
|
|
XPYIYHYLYJVTUYRVUOEWAYKVPZUYRVUOYMRVURUYSUVAYNVTUUTVUOYOUTUVCUVDVUHYPYQVV
|
|
IVUOUYRUUBVTVUMUVOUYROZALQVULAUYRLUUCVVJVUKALVVJUVPUYSUVOUYRUUDUUEUUFUUGR
|
|
UYTUVHVUJUVRVULUYTUVGVUIUVCUYTUVFVUHUVDUVEUYSUVAUUAYRYRUYTUVQVUKALUVEUYSU
|
|
VPUUHUUIUUJXFUUKYSYTUULUUMYTUUNUUOUUPUUQYSUURWG $.
|
|
$}
|
|
|
|
${
|
|
$d a t $. $d a w $. $d a x $. $d a z $. $d t x $. $d w x $. $d w z $.
|
|
$d x z $. $d n w $. $d n x $. $d n z $.
|
|
$( If the universe is finite, then ` Sp[fin] ` is a subset of its ` T `
|
|
raisings and the cardinality of the universe. Theorem X.1.59 of
|
|
[Rosser] p. 534. (Contributed by SF, 29-Jan-2015.) $)
|
|
vfinspss $p |- ( _V e. Fin ->
|
|
Sp[fin] C_
|
|
( { a | E. x e. Sp[fin] a = _T[fin] x } u. { Nc[fin] _V } ) ) $=
|
|
( vw vz vn vt cvv wcel cv wceq cspfin wrex wsfin wa wi bitri cins3k cimak
|
|
csn vex cfin ctfin cab cncfin cun wal wral wss wo weq tfineq vfinspsslem1
|
|
eqeq2d cbvrexv expr anbi2d anbi1d sfineq2 imbi1d imbi12d mpbiri rexlimdva
|
|
eleq1 com12 syl5bi biimpa c1c 1cvsfin tncveqnc1fin eqcomd ncvspfin rspcev
|
|
sfin111 mpan eqeq1 rexbidv biimpd syl5 mpand exp3a adantr jaod imp3a elun
|
|
elab elsnc orbi12i anbi1i 3imtr4g ssun1 sseli syl6 alrimiv ralrimiva cxpk
|
|
syl c0 cssetk cins2k cnnc csik cpw csymdif cpw1 cdif cin cidk ccompl copk
|
|
ccnvk elimak df-rex elpw1 r19.41v bitr4i exbii rexcom4 snex opkeq1 eleq1d
|
|
ceqsexv eqtfinrelk rexbii abbi2i tfinrelkex spfinex pw1ex imakex eqeltrri
|
|
wex unex ssun2 ncfinex snid sselii spfininduct mp3an12 ) GUAHZCIZBIZAIZUB
|
|
ZJZAKLZBUCZGUDZSZUEZHZDIZYSMZNZUUJUUHHZOZDUFZCKUGZKUUHUHZYRUUOCKYRYSKHZNZ
|
|
UUNDUUSUULUUJUUEHZUUMUUSYSUUBJZAKLZYSUUFJZUIZUUKNUUJUUBJZAKLZUULUUTUUSUVD
|
|
UUKUVFUUSUVBUUKUVFOZUVCUVBYSEIZUBZJZEKLUUSUVGUVAUVJAEKAEUJUUBUVIYSUUAUVHU
|
|
KUMUNUUSUVJUVGEKUVJUUSUVHKHZNZUVGUVJUVLUVGOYRUVIKHZNZUVKNZUUJUVIMZUVFOZOU
|
|
VNUVKUVPUVFADEULUOUVJUVLUVOUVGUVQUVJUUSUVNUVKUVJUURUVMYRYSUVIKVCUPUQUVJUU
|
|
KUVPUVFYSUVIUUJURUSUTVAVDVBVEYRUVCUVGOUURYRUVCUUKUVFUVCUUKNUUJUUFMZYRUVFU
|
|
VCUUKUVRYSUUFUUJURVFYRVGUDZUUFMZUVRUVFVHUVTUVRNUVSUUJJZYRUVFUUFUVSUUJVMYR
|
|
UVSUUBJZAKLZUWAUVFOYRUVSUUFUBZJZUWCYRUWDUVSVIVJUUFKHUWEUWCVKUWBUWEAUUFKUU
|
|
AUUFJUUBUWDUVSUUAUUFUKUMVLVNWPUWAUWCUVFUWAUWCUVFUWAUWBUVEAKUVSUUJUUBVOVPV
|
|
QVDWPVRVSVRVTWAWBWCUUIUVDUUKUUIYSUUEHZYSUUGHZUIUVDYSUUEUUGWDUWFUVBUWGUVCU
|
|
UDUVBBYSCTZBCUJUUCUVAAKYTYSUUBVOVPWEYSUUFUWHWFWGPWHUUDUVFBUUJDTBDUJUUCUVE
|
|
AKYTUUJUUBVOVPWEWIUUEUUHUUJUUEUUGWJWKWLWMWNUUHGHUUFUUHHUUPUUQUUEUUGWQSZSZ
|
|
UWIWOWRWSZWRXJQWTGWOWRXAWSZVGXBGWOWRQUWLXCVGXDZXDZXDZRXEXAQUWKXFUWNRQXFUW
|
|
ORXFWSXGQXCUWMRWSXEUWMRQXCUWNRXHUWJGWOXEUEZKXDZRZUUEGUUDBUWRYTUWRHZFIZUUA
|
|
SZJZUWTYTXIZUWPHZNZFYJZAKLZUUDUWSUXDFUWQLZUXGFUWPUWQYTBTZXKUXHUWTUWQHZUXD
|
|
NZFYJZUXGUXDFUWQXLUXLUXEAKLZFYJUXGUXKUXMFUXKUXBAKLZUXDNUXMUXJUXNUXDAUWTKX
|
|
MWHUXBUXDAKXNXOXPUXEAFKXQXOPPUXFUUCAKUXFUXAYTXIZUWPHZUUCUXDUXPFUXAUUAXRUX
|
|
BUXCUXOUWPUWTUXAYTXSXTYAUUAYTATUXIYBPYCPYDUWPUWQYEKYFYGYHYIUUFXRYKUUGUUHU
|
|
UFUUGUUEYLUUFGYMYNYOCDUUHGYPYQWP $.
|
|
$}
|
|
|
|
${
|
|
$d X x y z $.
|
|
$( If the universe is finite, then ` Sp[fin] ` is closed under T-raising.
|
|
Theorem X.1.60 of [Rosser] p. 536. (Contributed by SF, 30-Jan-2015.) $)
|
|
vfinspclt $p |- ( ( _V e. Fin /\ X e. Sp[fin] ) ->
|
|
_T[fin] X e. Sp[fin] ) $=
|
|
( vx vy vz cvv wcel cspfin wa cv ctfin c1c wceq tfineq eleq1d elab cssetk
|
|
cxpk cins2k cins3k cimak cfin cab cncfin wi wal wral tncveqnc1fin 1cspfin
|
|
wsfin eqeltrd ncfinex sylibr simprl sfintfin ad2antll spfinsfincl syl2anc
|
|
wss ex vex weq anbi1i 3imtr4g alrimiv ralrimiva c0 csn ccnvk cnnc csymdif
|
|
csik cpw cpw1 cdif cin cidk ccompl cuni1 wrex copk snex elimak eqtfinrelk
|
|
cun opkelcnvk bitri rexbii eluni1 risset 3bitr4i abbi2i tfinrelkex cnvkex
|
|
spfinex imakex uni1ex eqeltrri spfininduct mp3an1 sselda wb elabg adantl
|
|
mpbid ) EUAFZAGFZHABIZJZGFZBUBZFZAJZGFZXEGXJAXEEUCZXJFZCIZXJFZDIZXPUIZHZX
|
|
RXJFZUDZDUEZCGUFZGXJURZXEXNJZGFZXOXEYFKUCGUGUHUJXIYGBXNEUKXGXNLXHYFGXGXNM
|
|
NOULXEYCCGXEXPGFHZYBDYHXPJZGFZXSHZXRJZGFZXTYAYHYKYMYHYKHYJYLYIUIZYMYHYJXS
|
|
UMXSYNYHYJXRXPUNUOYIYLUPUQUSXQYJXSXIYJBXPCUTZBCVAXHYIGXGXPMNOVBXIYMBXRDUT
|
|
BDVAXHYLGXGXRMNOVCVDVEXJEFXOYDYEVFVGZVGZYPQPRZPVHSVIEQPVKRZKVLEQPSYSVJKVM
|
|
ZVMZVMZTVNVKSYRVOUUATSVOUUBTVORVPSVJYTTRVNYTTSVJUUATVQYQEQVNWDZVHZGTZVRZX
|
|
JEXIBUUFXGVGZUUEFZXPXHLZCGVSZXGUUFFXIUUHXPUUGVTUUDFZCGVSUUJCUUDGUUGXGWAZW
|
|
BUUKUUICGUUKUUGXPVTUUCFUUIXPUUGUUCYOUULWEXGXPBUTZYOWCWFWGWFXGUUEUUMWHCXHG
|
|
WIWJWKUUEUUDGUUCWLWMWNWOWPWQCDXJEWRWSUQWTXFXKXMXAXEXIXMBAGXGALXHXLGXGAMNX
|
|
BXCXD $.
|
|
$}
|
|
|
|
${
|
|
$d a x $.
|
|
$( If the universe is finite, then ` Sp[fin] ` is equal to its T raisings
|
|
and the cardinality of the universe. Theorem X.1.61 of [Rosser]
|
|
p. 536. (Contributed by SF, 29-Jan-2015.) $)
|
|
vfinspeqtncv $p |- ( _V e. Fin ->
|
|
Sp[fin] = ( { a | E. x e. Sp[fin] a = _T[fin] x } u.
|
|
{ Nc[fin] _V } ) ) $=
|
|
( cvv cfin wcel cspfin cv ctfin wceq wrex cab cncfin csn cun vfinspss wss
|
|
wa wi vfinspclt eleq1 biimprd com12 syl rexlimdva abssdv ncvspfin ncfinex
|
|
snss mpbi jctir unss sylib eqssd ) CDEZFBGZAGZHZIZAFJZBKZCLZMZNZABOUNUTFP
|
|
ZVBFPZQVCFPUNVDVEUNUSBFUNURUOFEZAFUNUPFEQUQFEZURVFRUPSURVGVFURVFVGUOUQFTU
|
|
AUBUCUDUEVAFEVEUFVAFCUGUHUIUJUTVBFUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d a x t $.
|
|
$( If the universe is finite, then the size of ` Sp[fin] ` is equal to the
|
|
successor of its T-raising. Theorem X.1.62 of [Rosser] p. 536.
|
|
(Contributed by SF, 20-Jan-2015.) $)
|
|
vfinncsp $p |- ( _V e. Fin ->
|
|
Nc[fin] Sp[fin] = ( _T[fin] Nc[fin] Sp[fin] +c 1c ) ) $=
|
|
( va vx cvv wcel cspfin cncfin wceq wrex csn c1c cin cssetk cins2k cins3k
|
|
vt cxpk cnnc cpw1 cimak wa cfin cv ctfin cab cun vfinspeqtncv ncfineq syl
|
|
cplc c0 vfinncvntsp disjsn sylibr ccnvk csik cpw csymdif cdif cidk ccompl
|
|
wn copk wex elimak df-rex elpw1 anbi1i r19.41v bitr4i exbii rexcom4 bitri
|
|
vex snex opkeq1 eleq1d ceqsexv eqtfinrelk rexbii tfinrelkex spfinex pw1ex
|
|
abbi2i imakex eqeltrri ncfindi mp3an2 mpanl2 mpdan ncfinprop mpan2 simpld
|
|
tfincl simprd wss vfinspnn syl6ss tfinnn syl3anc nnceleq syl22anc ncfinex
|
|
difss ncfinsn addceq12d 3eqtrd ) CUADZEFZAUBZBUBZUCGZBEHZAUDZCFZIZUEZFZXM
|
|
FZXOFZUIZXHUCZJUIXGEXPGXHXQGBAUFEXPUGUHXGXMXOKUJGZXQXTGZXGXNXMDVAYBBAUKXM
|
|
XNULUMXGXMCDZYBYCUJIZIZYEPLMZLUNNQCPLUOMZJUPCPLNYHUQJRZRZRZSURUONYGKYJSNK
|
|
YKSKMUSNUQYISMURYISNUQYJSUTYFCPURUEZERZSZXMCXLAYNXIYNDZOUBZXJIZGZYPXIVBZY
|
|
LDZTZOVCZBEHZXLYOYTOYMHZUUCOYLYMXIAVMZVDUUDYPYMDZYTTZOVCZUUCYTOYMVEUUHUUA
|
|
BEHZOVCUUCUUGUUIOUUGYRBEHZYTTUUIUUFUUJYTBYPEVFVGYRYTBEVHVIVJUUABOEVKVIVLV
|
|
LUUBXKBEUUBYQXIVBZYLDZXKYTUULOYQXJVNYRYSUUKYLYPYQXIVOVPVQXJXIBVMUUEVRVLVS
|
|
VLWCYLYMVTEWAWBWDWEZXGYDTXOCDYBYCXNVNXMXOCCWFWGWHWIXGXRYAXSJXGXRQDZYAQDZX
|
|
MXRDZXMYADZXRYAGXGUUNUUPXGYDUUNUUPTUUMXMCWJWKZWLXGXHQDZUUOXGUUSEXHDZXGECD
|
|
UUSUUTTWAECWJWKZWLZXHWMUHXGUUNUUPUURWNXGUUSEQWOUUTUUQUVBXGEQYEURQWPQYEXCW
|
|
QXGUUSUUTUVAWNBEXHAWRWSXMXRYAWTXAXGXNCDXSJGCXBXNCXDWKXEXF $.
|
|
$}
|
|
|
|
$( The universe is infinite. Theorem X.1.63 of [Rosser] p. 536.
|
|
(Contributed by SF, 20-Jan-2015.) $)
|
|
vinf $p |- -. _V e. Fin $=
|
|
( cvv cfin wcel cspfin cncfin c0 noel cevenfin coddfin cin cnnc wa vfinncsp
|
|
wne adantr adantl eqnetrrd syl2anc eqeltrd ex cun cdif ncfinprop mpan2 ne0i
|
|
csn spfinex anim2i syl eldifsn sylibr evenoddnnnul syl6eleqr ctfin c1c cplc
|
|
wo wceq eventfin evennnul sucevenodd ancld oddtfin oddnnul sucoddeven ancrd
|
|
jaod elun elin 3imtr4g mpd evenodddisj syl6eleq mto ) ABCZDEZFCVPGVOVPHIJZF
|
|
VOVPHIUAZCZVPVQCZVOVPKFUFUBZVRVOVPKCZVPFNZLZVPWACVOWBDVPCZLZWDVODACWFUGDAUC
|
|
UDWEWCWBVPDUEUHUIVPKFUJUKULUMVOVPHCZVPICZUQWGWHLZVSVTVOWGWIWHVOWGWHVOWGWHVO
|
|
WGLZVPVPUNZUOUPZIVOVPWLURZWGMOZWJWKHCZWLFNZWLICWGWOVOVPUSPWJVPWLFWNWGWCVOVP
|
|
UTPQWKVARSTVBVOWHWGVOWHWGVOWHLZVPWLHVOWMWHMOZWQWKICZWPWLHCWHWSVOVPVCPWQVPWL
|
|
FWRWHWCVOVPVDPQWKVERSTVFVGVPHIVHVPHIVIVJVKVLVMVN $.
|
|
|
|
${
|
|
$d n m x a $.
|
|
$( The empty class is not a natural. Theorem X.1.65 of [Rosser] p. 536.
|
|
(Contributed by SF, 20-Jan-2015.) $)
|
|
nulnnn $p |- -. (/) e. Nn $=
|
|
( vx vn vm va c0 cnnc wcel cv wceq wne c0c cab csn ccompl cvv wn neeq1 wa
|
|
ne0i ex wrex c1c cplc complab df-sn compleqi df-ne abbii 3eqtr4ri complex
|
|
snex eqeltri nulel0c ax-mp wel wex n0 cfin vinf cuni elunii ancoms df-fin
|
|
syl6eleqr mtoi eleq1 notbid syl5ibrcom necon2ad compleqb necon3bii complV
|
|
imp sylib neeq2i vex elcompl cun elsuci syl sylan2b adantl exlimdv syl5bi
|
|
wi mpd finds neneqd nrex risset mtbir ) EFGAHZEIZAFUAWMAFWLFGWLEBHZEJZKEJ
|
|
ZCHZEJZWQUBUCZEJZWLEJBCWLWOBLZEMZNZOWNEIZBLZNXDPZBLXCXAXDBUDXBXEBEUEUFWOX
|
|
FBWNEUGUHUIXBEUKUJULWNKEQWNWQEQWNWSEQWNWLEQEKGWPUMKESUNWRDCUOZDUPWQFGZWTD
|
|
WQUQXHXGWTDXHXGWTXHXGRZDHZNZEJZWTXIXKONZJZXLXIXJOJZXNXHXGXOXHXGXJOXHXGPXJ
|
|
OIZOWQGZPXHXQOURGZUSXHXQXRXHXQROFUTZURXQXHOXSGOWQFVAVBVCVDTVEXPXGXQXJOWQV
|
|
FVGVHVIVMXJOXKXMXJOVJVKVNXMEXKVLVOVNXLWLXKGZAUPXIWTAXKUQXIXTWTAXGXTWTWEXH
|
|
XGXTWTXTXGADUOPZWTWLXJAVPZVQXGYARXJWLMVRZWSGWTXJWQWLYBVSWSYCSVTWATWBWCWDW
|
|
FTWCWDWGWHWIAEFWJWK $.
|
|
$}
|
|
|
|
$( The successor operation is one-to-one over the finite cardinals. Theorem
|
|
X.1.66 of [Rosser] p. 537. (Contributed by SF, 20-Jan-2015.) $)
|
|
peano4 $p |- ( ( M e. Nn /\ N e. Nn /\ ( M +c 1c ) = ( N +c 1c ) ) ->
|
|
M = N ) $=
|
|
( cnnc wcel c1c cplc wceq w3a wa c0 3simpa simp3 peano2 nulnnn eleq1 mtbiri
|
|
wne necon2ai syl 3ad2ant1 prepeano4 syl12anc ) ACDZBCDZAEFZBEFGZHUCUDIUFUEJ
|
|
QZABGUCUDUFKUCUDUFLUCUDUGUFUCUECDZUGAMUHUEJUEJGUHJCDNUEJCOPRSTABUAUB $.
|
|
|
|
$( Successor cancellation law for finite cardinals. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
suc11nnc $p |- ( ( M e. Nn /\ N e. Nn ) ->
|
|
( ( M +c 1c ) = ( N +c 1c ) <-> M = N ) ) $=
|
|
( cnnc wcel wa c1c cplc wceq peano4 3expia addceq1 impbid1 ) ACDZBCDZEAFGBF
|
|
GHZABHZMNOPABIJABFKL $.
|
|
|
|
$( Cancellation law for natural addition. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
addccan2 $p |- ( ( M e. Nn /\ N e. Nn /\ P e. Nn ) ->
|
|
( ( M +c N ) = ( M +c P ) <-> N = P ) ) $=
|
|
( cnnc wcel w3a cplc c0 wne wceq wb wa nncaddccl nulnnn mtbiri necon2ai syl
|
|
eleq1 3adant3 preaddccan2 mpdan ) BDEZCDEZADEZFBCGZHIZUEBAGJCAJKUBUCUFUDUBU
|
|
CLUEDEZUFBCMUGUEHUEHJUGHDENUEHDROPQSABCTUA $.
|
|
|
|
$( Cancellation law for natural addition. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
addccan1 $p |- ( ( M e. Nn /\ N e. Nn /\ P e. Nn ) ->
|
|
( ( M +c P ) = ( N +c P ) <-> M = N ) ) $=
|
|
( cplc wceq cnnc wcel w3a addccom eqeq12i wb addccan2 3coml syl5bb ) BADZCA
|
|
DZEABDZACDZEZBFGZCFGZAFGZHBCEZOQPRBAICAIJUBTUASUCKCABLMN $.
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Ordered Pairs, Relationships, and Functions
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordered Pairs
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare the constant symbols for ordered pairs. $)
|
|
$c <. >. Phi Proj1 Proj2 $.
|
|
|
|
$( Declare the syntax for an ordered pair. $)
|
|
cop $a class <. A , B >. $.
|
|
|
|
$( Declare the syntax for the Phi operation. $)
|
|
cphi $a class Phi A $.
|
|
|
|
$( Declare the syntax for the first projection operation. $)
|
|
cproj1 $a class Proj1 A $.
|
|
|
|
$( Declare the syntax for the second projection operation. $)
|
|
cproj2 $a class Proj2 A $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
|
|
$( Define the phi operator. This operation increments all the naturals in
|
|
` A ` and leaves all its other members the same. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
df-phi $a |- Phi A = { y |
|
|
E. x e. A y = if ( x e. Nn , ( x +c 1c ) , x ) } $.
|
|
|
|
$( Define the type-level ordered pair. Definition from [Rosser] p. 281.
|
|
(Contributed by SF, 3-Feb-2015.) $)
|
|
df-op $a |- <. A , B >. =
|
|
( { x | E. y e. A x = Phi y } u.
|
|
{ x | E. y e. B x = ( Phi y u. { 0c } ) } ) $.
|
|
|
|
$( Define the first projection operation. This operation recovers the
|
|
first element of an ordered pair. Definition from [Rosser] p. 281.
|
|
(Contributed by SF, 3-Feb-2015.) $)
|
|
df-proj1 $a |- Proj1 A = { x | Phi x e. A } $.
|
|
|
|
$( Define the second projection operation. This operation recovers the
|
|
second element of an ordered pair. Definition from [Rosser] p. 281.
|
|
(Contributed by SF, 3-Feb-2015.) $)
|
|
df-proj2 $a |- Proj2 A = { x | ( Phi x u. { 0c } ) e. A } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Express the phi operator in terms of the Kuratowski set construction
|
|
functions. (Contributed by SF, 3-Feb-2015.) $)
|
|
dfphi2 $p |- Phi A = ( ( (
|
|
Image_k
|
|
( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) "_k A ) $=
|
|
( vx vy vz cssetk cins3k cins2k cin c1c cpw1 cimak cnnc cidk cv wcel wceq
|
|
cvv wa wo wb cphi ccompl csik cun csymdif cdif cimagek cxpk cplc cif wrex
|
|
copk weq wn iftrue eqeq2d iba simpr con2i biorf syl syl6bb 3bitrd iffalse
|
|
orcom con3i pm2.61i equcom elcompl anbi12i orbi2i bitr4i elun opkelimagek
|
|
vex dfaddc2 eqeq2i opkelxpk mpbiran2 bitri opkelidkg mp2an orbi12i rexbii
|
|
elin eqeq1 rexbidv df-phi elab2 elimak 3bitr4i eqriv ) BAUAZEFZEGZHIJJZKU
|
|
BFWOGWNGEUCUCFUDUEWPJJKUFWPKZUGZLQUHZHZMLUBZQUHZHZUDZAKZBNZCNZLOZXGIUIZXG
|
|
UJZPZCAUKZXGXFULZXDOZCAUKXFWMOXFXEOXKXNCAXKXFXIPZXHRZCBUMZXGXAOZRZSZXNXKX
|
|
PBCUMZXHUNZRZSZXTXHXKYDTXHXKXOXPYDXHXJXIXFXHXIXGUOUPXHXOUQXHXPYCXPSZYDXHY
|
|
CUNXPYETYCXHYAYBURUSYCXPUTVAYCXPVEVBVCYBXKYAYCYDYBXJXGXFXHXIXGVDUPYBYAUQY
|
|
BXPUNYCYDTXPXHXOXHURVFXPYCUTVAVCVGXSYCXPXQYAXRYBCBVHXGLCVOZVIVJVKVLXNXMWT
|
|
OZXMXCOZSXTXMWTXCVMYGXPYHXSYGXMWROZXMWSOZRXPXMWRWSWEYIXOYJXHYIXFWQXGKZPXO
|
|
XGXFWQYFBVOZVNXIYKXFXGIVPVQVLYJXHXFQOZYLXGXFLQYFYLVRVSVJVTYHXMMOZXMXBOZRX
|
|
SXMMXBWEYNXQYOXRXGQOYMYNXQTYFYLXGXFQQWAWBYOXRYMYLXGXFXAQYFYLVRVSVJVTWCVTV
|
|
LWDDNZXJPZCAUKXLDXFWMYLDBUMYQXKCAYPXFXJWFWGCDAWHWICXDAXFYLWJWKWL $.
|
|
$}
|
|
|
|
$( Equality law for the Phi operation. (Contributed by SF, 3-Feb-2015.) $)
|
|
phieq $p |- ( A = B -> Phi A = Phi B ) $=
|
|
( wceq cssetk cins3k cins2k cin c1c cpw1 cimak ccompl csik cun csymdif cdif
|
|
cnnc cvv cxpk cphi dfphi2 cimagek cidk imakeq2 3eqtr4g ) ABCDEZDFZGHIIZJKEU
|
|
FFUEFDLLEMNUGIIJOUGJUAPQRGUBPKQRGMZAJUHBJASBSABUHUCATBTUD $.
|
|
|
|
$( The phi operator preserves sethood. (Contributed by SF, 3-Feb-2015.) $)
|
|
phiexg $p |- ( A e. V -> Phi A e. _V ) $=
|
|
( wcel cssetk cins3k cins2k cin c1c cpw1 ccompl csik cun cnnc cvv cxpk cidk
|
|
cimak pw1ex nncex vvex cphi csymdif cdif dfphi2 addcexlem 1cex imakex xpkex
|
|
cimagek imagekex inex idkex complex unex imakexg mpan syl5eqel ) ABCZAUADEZ
|
|
DFZGHIZIZQJEUTFUSFDKKELUBVBIIQUCZVBQZUIZMNOZGZPMJZNOZGZLZAQZNAUDVKNCURVLNCV
|
|
GVJVEVFVDVCVBUEVAHUFRRUGUJMNSTUHUKPVIULVHNMSUMTUHUKUNVKANBUOUPUQ $.
|
|
|
|
${
|
|
phiex.1 $e |- A e. _V $.
|
|
$( The phi operator preserves sethood. (Contributed by SF, 3-Feb-2015.) $)
|
|
phiex $p |- Phi A e. _V $=
|
|
( cvv wcel cphi phiexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z t A $. $d x y z t B $.
|
|
$( Lemma for ~ dfop2 and ~ dfproj22 . (Contributed by SF, 2-Jan-2015.) $)
|
|
dfop2lem1 $p |- ( << x , y >> e. ~ ( ( Ins2_k _S_k (+) Ins3_k ( ( `'_k
|
|
Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k )
|
|
"_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) )
|
|
) "_k
|
|
~P1 ~P1 1c ) <-> y = ( Phi x u. { 0c } ) ) $=
|
|
( vz vt cv copk cssetk cins3k cpw1 cimak cun cvv c0c csn wcel wn wex wceq
|
|
wa 3bitr4i cins2k cin c1c ccompl csik csymdif cdif cimagek cnnc cxpk cidk
|
|
ccnvk ccomk cphi wb wrex opkex elimak elpw121c anbi1i 19.41v bitr4i exbii
|
|
wel df-rex excom snex opkeq1 eleq1d ceqsexv elsymdif vex otkelins2k bitri
|
|
elssetk otkelins3k elun ancom opkelimagek opkelcnvk dfphi2 eqeq2i anbi12i
|
|
opkelcok phiex clel3 opkelxpk mpbiran2 sneqb elsnc orbi12i 3bitri bibi12i
|
|
wo 3bitr4ri notbii elcompl wal dfcleq alex ) AEZBEZFZGUAZGHZXDUBUCIIZJUDH
|
|
XDUAXEUAGUEUEHKUFXFIIJUGXFJUHUILUJUBUKUIUDLUJUBKZUHZULZGUMZMNZNZLUJZKZHZU
|
|
FZXFJZOZPCBVDZCEZXAUNZXKKZOZUOZPZCQZPZXCXQUDOXBYBRZXRYFXRDEZXCFZXPOZDXFUP
|
|
ZYIXTNZNZNZRZYKSZDQZCQZYFDXPXFXCXAXBUQZURYIXFOZYKSZDQYQCQZDQYLYSUUBUUCDUU
|
|
BYPCQZYKSUUCUUAUUDYKCYIUSUTYPYKCVAVBVCYKDXFVEYQCDVFTYRYECYRYOXCFZXPOZUUEX
|
|
DOZUUEXOOZUOZPYEYKUUFDYOYNVGYPYJUUEXPYIYOXCVHVIVJUUEXDXOVKUUIYDUUGXSUUHYC
|
|
UUGYMXBFGOZXSYMXAXBGXTVGZAVLZBVLZVMXTXBCVLZUUMVOZVNUUHYMXAFZXNOUUPXJOZUUP
|
|
XMOZWNZYCYMXAXBXNUUKUULUUMVPUUPXJXMVQUUSXTYAOZXTXKOZWNYCUUQUUTUURUVAUUJXB
|
|
XAFXIOZSZBQXBYARZXSSZBQUUQUUTUVCUVEBUVCUVBUUJSUVEUUJUVBVRUVBUVDUUJXSXCXHO
|
|
XBXGXAJZRUVBUVDXAXBXGUULUUMVSXBXAXHUUMUULVTYAUVFXBXAWAWBTUUOWCVNVCBYMXAXI
|
|
GUUKUULWDBXTYAXAUULWEWFTUURYMXLOZUVAUURUVGXALOUULYMXAXLLUUKUULWGWHYMXKRXT
|
|
MRUVGUVAXTMUUNWIYMXKUUKWJXTMUUNWJWOVBWKXTYAXKVQVBWLWMWPWLVCWLWPXCXQYTWQYH
|
|
YDCWRYGCXBYBWSYDCWTVNT $.
|
|
|
|
$( Lemma for ~ dfop2 (Contributed by SF, 2-Jan-2015.) $)
|
|
dfop2lem2 $p |- ( ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k
|
|
( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u. ( _I_k i^i ( ~ Nn X._k _V
|
|
) ) ) o._k
|
|
_S_k ) u. ( { { 0c } } X._k _V ) ) ) "_k ~P1 ~P1 1c ) "_k B
|
|
)
|
|
= { x | E. y e. B x = ( Phi y u. { 0c } ) } $=
|
|
( cv csn cun wrex cssetk cins2k cins3k cin cpw1 cimak ccompl csik csymdif
|
|
cimagek cnnc cvv cxpk cphi c0c wceq c1c cdif cidk ccnvk ccomk wcel elimak
|
|
copk vex dfop2lem1 rexbii bitri abbi2i ) ADZBDZUAUBEZFUCZBCGZAHIZHJZVBKUD
|
|
LLZMNJVBIVCIHOOJFPVDLLMUEVDMQRSTKUFRNSTKFQUGHUHUSESTFJPVDMNZCMZUQVFUIURUQ
|
|
UKVEUIZBCGVABVECUQAULUJVGUTBCBAUMUNUOUP $.
|
|
|
|
$( Express the ordered pair via the set construction functors.
|
|
(Contributed by SF, 2-Jan-2015.) $)
|
|
dfop2 $p |- <. A , B >. =
|
|
( ( Image_k ( ( Image_k
|
|
( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u. ( _I_k i^i ( ~ Nn X._k _V
|
|
) ) ) "_k A ) u.
|
|
( ~ ( ( Ins2_k _S_k (+) Ins3_k ( ( `'_k Image_k ( ( Image_k
|
|
( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u.
|
|
( { { 0c } } X._k _V ) ) ) "_k ~P1 ~P1 1c ) "_k B ) ) $=
|
|
( vx vy cv wceq wrex cab csn cun cssetk cins3k cins2k cin cpw1 cimak csik
|
|
ccompl cvv cxpk cop cphi c0c c1c csymdif cdif cnnc cidk ccnvk ccomk df-op
|
|
wcel copk vex elimak dfphi2 eqeq2i opkelimagek bitr4i rexbii bicomi bitri
|
|
cimagek abbi2i dfop2lem2 uneq12i eqtr4i ) ABUACEZDEZUBZFZDAGZCHZVHVJUCIZJ
|
|
FDBGCHZJKLZKMZNUDOOZPRLVQMVPMKQQLJUEVROOPUFVRPVCUGSTNUHUGRSTNJZVCZAPZVQVT
|
|
UIKUJVNISTJLUEVRPRBPZJCDABUKWAVMWBVOVLCWAVHWAULVIVHUMVTULZDAGZVLDVTAVHCUN
|
|
ZUOVLWDVKWCDAVKVHVSVIPZFWCVJWFVHVIUPUQVIVHVSDUNWEURUSUTVAVBVDCDBVEVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Express the first projection operator via the set construction
|
|
functors. (Contributed by SF, 2-Jan-2015.) $)
|
|
dfproj12 $p |- Proj1 A =
|
|
( `'_k Image_k ( ( Image_k
|
|
( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u. ( _I_k i^i ( ~ Nn X._k _V
|
|
) ) ) "_k A ) $=
|
|
( vx vy wcel cssetk cins3k cins2k cin cpw1 cimak ccompl csik cimagek cnnc
|
|
cv cun cvv cxpk wceq wrex cproj1 cphi cab c1c csymdif cdif ccnvk df-proj1
|
|
cidk copk opkelimagek opkelcnvk dfphi2 eqeq2i rexbii risset elimak abbi2i
|
|
vex 3bitr4ri eqtr4i ) AUABOZUBZADZBUCEFZEGZHUDIIZJKFVFGVEGELLFPUEVGIIJUFV
|
|
GJMNQRHUINKQRHPZMZUGZAJZBAUHVDBVKCOZVCSZCATVLVBUJVJDZCATVDVBVKDVMVNCAVBVL
|
|
UJVIDVLVHVBJZSVNVMVBVLVHBUSZCUSZUKVLVBVIVQVPULVCVOVLVBUMUNUTUOCVCAUPCVJAV
|
|
BVPUQUTURVA $.
|
|
|
|
$( Express the second projection operator via the set construction
|
|
functors. (Contributed by SF, 2-Jan-2015.) $)
|
|
dfproj22 $p |- Proj2 A = ( `'_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k
|
|
( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k
|
|
_S_k ) u. ( { { 0c } } X._k _V ) ) ) "_k ~P1 ~P1 1c ) "_k A ) $=
|
|
( vx vy cv csn cun wcel cssetk cins2k cins3k cin cpw1 ccompl csik csymdif
|
|
cimak cimagek cnnc cvv cxpk cproj2 cphi c0c cab c1c cdif cidk ccnvk ccomk
|
|
df-proj2 copk wrex opkelcnvk dfop2lem1 bitri rexbii elimak risset 3bitr4i
|
|
wceq vex abbi2i eqtr4i ) AUABDZUBUCEZFZAGZBUDHIZHJZVHKUELLZPMJVHIVIIHNNJF
|
|
OVJLLPUFVJPQRSTKUGRMSTKFQUHHUIVEESTFJOVJPMZUHZAPZBAUJVGBVMCDZVDUKVLGZCAUL
|
|
VNVFUTZCAULVDVMGVGVOVPCAVOVDVNUKVKGVPVNVDVKCVABVAZUMBCUNUOUPCVLAVDVQUQCVF
|
|
AURUSVBVC $.
|
|
$}
|
|
|
|
$( Equality theorem for ordered pairs. (Contributed by SF, 2-Jan-2015.) $)
|
|
opeq1 $p |- ( A = B -> <. A , C >. = <. B , C >. ) $=
|
|
( cssetk cins3k cins2k cin cpw1 cimak ccompl csik csymdif cimagek cnnc cxpk
|
|
cun cvv csn cop dfop2 wceq c1c cdif cidk ccnvk ccomk imakeq2 uneq1d 3eqtr4g
|
|
c0c ) ABUAZDEZDFZGUBHHZIJEUMFULFDKKEPLUNHHIUCUNIMNQOGUDNJQOGPMZAIZUMUOUEDUF
|
|
UJRRQOPELUNIJCIZPUOBIZUQPACSBCSUKUPURUQABUOUGUHACTBCTUI $.
|
|
|
|
$( Equality theorem for ordered pairs. (Contributed by SF, 2-Jan-2015.) $)
|
|
opeq2 $p |- ( A = B -> <. C , A >. = <. C , B >. ) $=
|
|
( cssetk cins3k cins2k cin cpw1 cimak ccompl csik csymdif cimagek cnnc cxpk
|
|
cun cvv csn cop dfop2 wceq c1c cdif cidk ccnvk ccomk imakeq2 uneq2d 3eqtr4g
|
|
c0c ) ABUAZDEZDFZGUBHHZIJEUMFULFDKKEPLUNHHIUCUNIMNQOGUDNJQOGPMZCIZUMUOUEDUF
|
|
UJRRQOPELUNIJZAIZPUPUQBIZPCASCBSUKURUSUPABUQUGUHCATCBTUI $.
|
|
|
|
$( Equality theorem for ordered pairs. (Contributed by SF, 2-Jan-2015.) $)
|
|
opeq12 $p |- ( ( A = B /\ C = D ) -> <. A , C >. = <. B , D >. ) $=
|
|
( wceq cop opeq1 opeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
opeq1i.1 $e |- A = B $.
|
|
$( Equality inference for ordered pairs. (Contributed by SF,
|
|
16-Dec-2006.) $)
|
|
opeq1i $p |- <. A , C >. = <. B , C >. $=
|
|
( wceq cop opeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for ordered pairs. (Contributed by SF,
|
|
16-Dec-2006.) $)
|
|
opeq2i $p |- <. C , A >. = <. C , B >. $=
|
|
( wceq cop opeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
${
|
|
opeq12i.2 $e |- C = D $.
|
|
$( Equality inference for ordered pairs. (The proof was shortened by
|
|
Eric Schmidt, 4-Apr-2007.) (Contributed by SF, 16-Dec-2006.) $)
|
|
opeq12i $p |- <. A , C >. = <. B , D >. $=
|
|
( cop opeq1i opeq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
opeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for ordered pairs. (Contributed by SF,
|
|
16-Dec-2006.) $)
|
|
opeq1d $p |- ( ph -> <. A , C >. = <. B , C >. ) $=
|
|
( wceq cop opeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for ordered pairs. (Contributed by SF,
|
|
16-Dec-2006.) $)
|
|
opeq2d $p |- ( ph -> <. C , A >. = <. C , B >. ) $=
|
|
( wceq cop opeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
${
|
|
opeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for ordered pairs. (The proof was shortened by
|
|
Andrew Salmon, 29-Jun-2011.) (Contributed by SF, 16-Dec-2006.)
|
|
(Revised by SF, 29-Jun-2011.) $)
|
|
opeq12d $p |- ( ph -> <. A , C >. = <. B , D >. ) $=
|
|
( cop opeq1d opeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
$}
|
|
|
|
$( An ordered pair of two sets is a set. (Contributed by SF, 2-Jan-2015.) $)
|
|
opexg $p |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V ) $=
|
|
( wcel cssetk cins3k cins2k cin c1c cpw1 ccompl csik cun cnnc cvv cxpk vvex
|
|
cimak xpkex wa cop csymdif cdif cimagek ccnvk ccomk c0c csn dfop2 addcexlem
|
|
cidk 1cex pw1ex imagekex nncex inex idkex complex unex imakexg mpan ssetkex
|
|
imakex ins2kex cnvkex cokex snex ins3kex symdifex unexg syl2an syl5eqel ) A
|
|
CEZBDEZUAABUBFGZFHZIJKZKZSLGVQHVPHFMMGNUCVSKKSUDZVSSZUEZOPQZIZULOLZPQZIZNZU
|
|
EZASZVQWIUFZFUGZUHUIZUIZPQZNZGZUCZVSSZLZBSZNZPABUJVNWJPEZXAPEZXBPEVOWIPEVNX
|
|
CWHWDWGWBWCWAVTVSUKVRJUMUNUNZVDUOOPUPRTUQULWFURWEPOUPUSRTUQUTUOZWIAPCVAVBWT
|
|
PEVOXDWSWRVSVQWQFVCVEWPWLWOWKFWIXFVFVCVGWNPWMVHRTUTVIVJXEVDUSWTBPDVAVBWJXAP
|
|
PVKVLVM $.
|
|
|
|
${
|
|
opex.1 $e |- A e. _V $.
|
|
opex.2 $e |- B e. _V $.
|
|
$( An ordered pair of two sets is a set. (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
opex $p |- <. A , B >. e. _V $=
|
|
( cvv wcel cop opexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
$( Equality theorem for first projection operator. (Contributed by SF,
|
|
2-Jan-2015.) $)
|
|
proj1eq $p |- ( A = B -> Proj1 A = Proj1 B ) $=
|
|
( wceq cssetk cins3k cins2k cin c1c cpw1 cimak ccompl csik cun csymdif cnnc
|
|
cimagek cvv cxpk cproj1 dfproj12 cdif cidk ccnvk imakeq2 3eqtr4g ) ABCDEZDF
|
|
ZGHIIZJKEUGFUFFDLLEMNUHIIJUAUHJPOQRGUBOKQRGMPUCZAJUIBJASBSABUIUDATBTUE $.
|
|
|
|
$( Equality theorem for second projection operator. (Contributed by SF,
|
|
2-Jan-2015.) $)
|
|
proj2eq $p |- ( A = B -> Proj2 A = Proj2 B ) $=
|
|
( cssetk cins2k cins3k cin cpw1 cimak ccompl csik csymdif cimagek cnnc cxpk
|
|
cun cvv ccnvk csn cproj2 dfproj22 wceq c1c cdif cidk ccomk imakeq2 3eqtr4g
|
|
c0c ) ABUACDZCEZUIFUBGGZHIEUIDUJDCJJEOKUKGGHUCUKHLMPNFUDMIPNFOLQCUEUHRRPNOE
|
|
KUKHIQZAHULBHASBSABULUFATBTUG $.
|
|
|
|
$( The first projection of a set is a set. (Contributed by SF,
|
|
2-Jan-2015.) $)
|
|
proj1exg $p |- ( A e. V -> Proj1 A e. _V ) $=
|
|
( wcel cssetk cins3k cins2k cin c1c cpw1 cimak ccompl csik cun cimagek cnnc
|
|
cvv cxpk cidk pw1ex imagekex cproj1 csymdif ccnvk dfproj12 addcexlem imakex
|
|
cdif 1cex nncex vvex xpkex inex idkex complex unex cnvkex imakexg syl5eqel
|
|
mpan ) ABCZAUADEZDFZGHIZIZJKEVBFVAFDLLEMUBVDIIJUGZVDJZNZOPQZGZROKZPQZGZMZNZ
|
|
UCZAJZPAUDVOPCUTVPPCVNVMVIVLVGVHVFVEVDUEVCHUHSSUFTOPUIUJUKULRVKUMVJPOUIUNUJ
|
|
UKULUOTUPVOAPBUQUSUR $.
|
|
|
|
$( The second projection of a set is a set. (Contributed by SF,
|
|
2-Jan-2015.) $)
|
|
proj2exg $p |- ( A e. V -> Proj2 A e. _V ) $=
|
|
( wcel cssetk cins2k cins3k cin c1c cpw1 cimak ccompl csik cun csymdif cnnc
|
|
cimagek cvv cxpk vvex xpkex cproj2 cdif cidk ccnvk c0c csn dfproj22 ssetkex
|
|
ccomk ins2kex addcexlem 1cex pw1ex imakex imagekex nncex inex idkex complex
|
|
unex cnvkex cokex snex ins3kex symdifex imakexg mpan syl5eqel ) ABCZAUADEZD
|
|
FZVJGHIZIZJKFVJEVKEDLLFMNVMIIJUBZVMJZPZOQRZGZUCOKZQRZGZMZPZUDZDUIZUEUFZUFZQ
|
|
RZMZFZNZVMJZKZUDZAJZQAUGWNQCVIWOQCWMWLWKVMVJWJDUHUJWIWEWHWDDWCWBVRWAVPVQVOV
|
|
NVMUKVLHULUMUMZUNUOOQUPSTUQUCVTURVSQOUPUSSTUQUTUOVAUHVBWGQWFVCSTUTVDVEWPUNU
|
|
SVAWNAQBVFVGVH $.
|
|
|
|
${
|
|
projex.1 $e |- A e. _V $.
|
|
|
|
$( The first projection of a set is a set. (Contributed by Scott Fenton,
|
|
16-Apr-2021.) $)
|
|
proj1ex $p |- Proj1 A e. _V $=
|
|
( cvv wcel cproj1 proj1exg ax-mp ) ACDAECDBACFG $.
|
|
|
|
$( The second projection of a set is a set. (Contributed by Scott Fenton,
|
|
16-Apr-2021.) $)
|
|
proj2ex $p |- Proj2 A e. _V $=
|
|
( cvv wcel cproj2 proj2exg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Lemma for ~ phi11 . (Contributed by SF, 3-Feb-2015.) $)
|
|
phi11lem1 $p |- ( Phi A = Phi B -> A C_ B ) $=
|
|
( vz vy vx cphi wceq cv wcel cnnc wi wa wrex eqeq2d rexbidv elab2 biimpac
|
|
c1c syl5 syl cplc cif iftrue eqcomd weq eleq1 addceq1 id ifbieq12d rspcev
|
|
sylan2 ancoms vex 1cex addcex eqeq1 df-phi sylibr eleq2 wn iffalse peano2
|
|
syl5ibcom expdimp pm2.18d simpl simpr eqtr2d peano4 syl3anc 3adant2 simp2
|
|
w3a eqeltrrd 3expia rexlimdva syl5bi exp3a adantr mpd eleq1a sylbid com12
|
|
con3d impcom adantlr simplr ex pm2.61ian ssrdv ) AFZBFZGZCABCHZAIZWMWNBIZ
|
|
WNJIZWOWMWPKZWQWOLZWNRUAZWKIZWRWSWTDHZJIZXBRUAZXBUBZGZDAMZXAWOWQXGWQWOWTW
|
|
QWTWNUBZGZXGWQXHWTWQWTWNUCUDXFXIDWNADCUEZXEXHWTXJXCWQXDXBWTWNXBWNJUFXBWNR
|
|
UGXJUHUIZNUJUKULEHZXEGZDAMZXGEWTWKWNRCUMZUNUOZXLWTGZXMXFDAXLWTXEUPZODEAUQ
|
|
ZPURWQXAWRKWOWQXAWMWPXAWMLWTWLIZWQWPWMXAXTWKWLWTUSQXTXFDBMZWQWPXMDBMZYAEW
|
|
TWLXPXQXMXFDBXRODEBUQZPWQXFWPDBWQXBBIZXFWPWQYDXFVMXBWNBWQXFXJYDWQXFLZXCWQ
|
|
XDWTGXJYEXCWQXFXCUTZXCXFYFLWTXBGZWQXCYFXFYGYFXEXBWTXCXDXBVAZNQWQWTJIYGXCW
|
|
NVBWTXBJUFVCSVDVEZWQXFVFYEWTXEXDWQXFVGYEXCXEXDGYIXCXDXBUCZTVHXBWNVIVJVKWQ
|
|
YDXFVLVNVOVPVQSVRVSVTWQUTZWOLZWNWKIZWRYLWNXEGZDAMZYMWOYKYOYKWOWNXHGZYOYKX
|
|
HWNWQWTWNVAUDYNYPDWNAXJXEXHWNXKNUJUKULXNYOEWNWKXOECUEZXMYNDAXLWNXEUPZOXSP
|
|
URYKYMWRKWOYKYMWMWPYMWMLWNWLIZYKWPWMYMYSWKWLWNUSQYSYNDBMZYKWPYBYTEWNWLXOY
|
|
QXMYNDBYROYCPYKYNWPDBYKYDLZYNWPUUAYNLXBWNBYKYNXJYDYKYNLZWNXEXBYKYNVGUUBYF
|
|
XEXBGYNYKYFYNXCWQXCYNWQXCYNWNXDGZWQXCXEXDWNYJNXCXDJIUUCWQKXBVBXDJWNWATWBW
|
|
CWDWEYHTVHWFYKYDYNWGVNWHVPVQSVRVSVTWIWCWJ $.
|
|
$}
|
|
|
|
$( The phi operator is one-to-one. Theorem X.2.2 of [Rosser] p. 282.
|
|
(Contributed by SF, 3-Feb-2015.) $)
|
|
phi11 $p |- ( A = B <-> Phi A = Phi B ) $=
|
|
( wceq cphi phieq phi11lem1 wss eqcoms eqssd impbii ) ABCADZBDZCZABEMABABFB
|
|
AGLKBAFHIJ $.
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Cardinal zero is not a member of a phi operation. Theorem X.2.3 of
|
|
[Rosser] p. 282. (Contributed by SF, 3-Feb-2015.) $)
|
|
0cnelphi $p |- -. 0c e. Phi A $=
|
|
( vy vx c0c cphi wcel cv cnnc c1c cplc cif wceq wrex wn 0cnsuc df-ne mpbi
|
|
wne wa eqeq2d iffalse biimpac peano1 syl6eqelr iftrue eqcom syl6bb biimpd
|
|
ex pm2.18d mpcom mto a1i nrex 0cex eqeq1 rexbidv df-phi elab2 mtbir ) DAE
|
|
ZFDBGZHFZVBIJZVBKZLZBAMZVFBAVFNVBAFVFVDDLZVDDRVHNVBOVDDPQVCVFVHVFVCVFVCNZ
|
|
VCVFVISVBDHVIVFDVBLVIVEVBDVCVDVBUATUBUCUDUIUJVCVFVHVCVFDVDLVHVCVEVDDVCVDV
|
|
BUETDVDUFUGUHUKULUMUNCGZVELZBAMVGCDVAUOVJDLVKVFBAVJDVEUPUQBCAURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d A z $. $d B z $.
|
|
$( Lemma for ~ phi011 . (Contributed by SF, 3-Feb-2015.) $)
|
|
phi011lem1 $p |- ( ( Phi A u. { 0c } ) = ( Phi B u. { 0c } ) ->
|
|
Phi A C_ Phi B ) $=
|
|
( vz cphi c0c csn wceq cv wcel wn ssun1 sseli eleq2 syl5ib 0cnelphi eleq1
|
|
cun wi mtbiri wo con2i a1i elun df-sn abeq2i orbi2i bitri biimpi ord ee22
|
|
orcomd ssrdv ) ADZEFZQZBDZUNQZGZCUMUPURCHZUMIZUSUQIZUSEGZJZUSUPIZUTUSUOIU
|
|
RVAUMUOUSUMUNKLUOUQUSMNUTVCRURVBUTVBUTEUMIAOUSEUMPSUAUBVAVBVDVAVDVBVAVDVB
|
|
TZVAVDUSUNIZTVEUSUPUNUCVFVBVDVBCUNCEUDUEUFUGUHUKUIUJUL $.
|
|
$}
|
|
|
|
$( ` ( Phi A u. { 0c } ) ` is one-to-one. Theorem X.2.4 of [Rosser] p. 282.
|
|
(Contributed by SF, 3-Feb-2015.) $)
|
|
phi011 $p |- ( A = B <-> ( Phi A u. { 0c } ) = ( Phi B u. { 0c } ) ) $=
|
|
( wceq cphi c0c csn cun phi11 uneq1 phi011lem1 eqcoms eqssd impbii bitri
|
|
wss ) ABCADZBDZCZPEFZGZQSGZCZABHRUBPQSIUBPQABJQPOUATBAJKLMN $.
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( The first projection operator applied to an ordered pair yields its
|
|
first member. Theorem X.2.7 of [Rosser] p. 282. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
proj1op $p |- Proj1 <. A , B >. = A $=
|
|
( vz vy vx cop cproj1 cv cphi wcel c0c cun wceq wrex wo cab eqeq1 rexbidv
|
|
weq elab csn df-op eleq2i phiex phi11 equcom bitr3i syl6bb risset syl6bbr
|
|
elun vex orbi12i 3bitri phieq eleq1d df-proj1 elab2 wn 0cnelphi 0cex snid
|
|
ssun2 sselii eleq2 mpbiri mto a1i nrex biorfi 3bitr4i eqriv ) CABFZGZACHZ
|
|
IZVMJZVOAJZVPDHZIZKUAZLZMZDBNZOZVOVNJVRVQVPEHZVTMZDANZEPZWFWBMZDBNZEPZLZJ
|
|
VPWIJZVPWLJZOWEVMWMVPEDABUBUCVPWIWLUKWNVRWOWDWHVREVPVOCULZUDZWFVPMZWHDCSZ
|
|
DANVRWRWGWSDAWRWGVPVTMZWSWFVPVTQWTCDSWSVOVSUECDUFUGUHRDVOAUIUJTWKWDEVPWQW
|
|
RWJWCDBWFVPWBQRTUMUNWFIZVMJVQEVOVNWPECSXAVPVMWFVOUOUPEVMUQURWDVRWCDBWCUSV
|
|
SBJWCKVPJZVOUTWCXBKWBJWAWBKWAVTVCKVAVBVDVPWBKVEVFVGVHVIVJVKVL $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( The second projection operator applied to an ordered pair yields its
|
|
second member. Theorem X.2.8 of [Rosser] p. 283. (Contributed by SF,
|
|
3-Feb-2015.) $)
|
|
proj2op $p |- Proj2 <. A , B >. = B $=
|
|
( vz vy vx cv cphi c0c cun wcel wceq wrex wo cab rexbidv elab weq 3bitr4i
|
|
eqeq1 bitri cop cproj2 csn df-op eleq2i elun vex phiex snex phi011 equcom
|
|
unex bitr3i rexbii risset orbi12i phieq uneq1d df-proj2 elab2 wn 0cnelphi
|
|
eleq1d ssun2 0cex snid sselii eleq2 mpbii mto a1i nrex biorfi orcom eqriv
|
|
) CABUAZUBZBCFZGZHUCZIZVPJZWADFZGZKZDALZVRBJZMZVRVQJWGWBWAEFZWDKZDALZENZW
|
|
IWDVTIZKZDBLZENZIZJZWHVPWQWAEDABUDUEWRWAWLJZWAWPJZMWHWAWLWPUFWSWFWTWGWKWF
|
|
EWAVSVTVRCUGZUHHUIULZWIWAKZWJWEDAWIWAWDSOPWAWMKZDBLZDCQZDBLWTWGXDXFDBXDCD
|
|
QXFVRWCUJCDUKUMUNWOXEEWAXBXCWNXDDBWIWAWMSOPDVRBUORUPTTWIGZVTIZVPJWBEVRVQX
|
|
AECQZXHWAVPXIXGVSVTWIVRUQURVCEVPUSUTWGWGWFMWHWFWGWEDAWEVAWCAJWEHWDJZWCVBW
|
|
EHWAJXJVTWAHVTVSVDHVEVFVGWAWDHVHVIVJVKVLVMWGWFVNTRVO $.
|
|
$}
|
|
|
|
$( The ordered pair theorem. Two ordered pairs are equal iff their
|
|
components are equal. (Contributed by SF, 2-Jan-2015.) $)
|
|
opth $p |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) $=
|
|
( cop wceq cproj1 proj1eq proj1op 3eqtr3g cproj2 proj2eq proj2op jca opeq12
|
|
wa impbii ) ABEZCDEZFZACFZBDFZPTUAUBTRGSGACRSHABICDIJTRKSKBDRSLABMCDMJNACBD
|
|
OQ $.
|
|
|
|
$( An ordered pair is a set iff its components are sets. (Contributed by SF,
|
|
2-Jan-2015.) $)
|
|
opexb $p |- ( <. A , B >. e. _V <-> ( A e. _V /\ B e. _V ) ) $=
|
|
( cop cvv wcel wa cproj1 proj1op proj1exg syl5eqelr cproj2 proj2op proj2exg
|
|
jca opexg impbii ) ABCZDEZADEZBDEZFRSTRAQGDABHQDIJRBQKDABLQDMJNABDDOP $.
|
|
|
|
${
|
|
$d z w A $. $d z w B $. $d x z w $.
|
|
nfop.1 $e |- F/_ x A $.
|
|
nfop.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for ordered pairs. (Contributed by
|
|
SF, 2-Jan-2015.) $)
|
|
nfop $p |- F/_ x <. A , B >. $=
|
|
( vz vw cop cv cphi wceq wrex cab c0c csn cun df-op nfv nfrex nfab nfcxfr
|
|
nfun ) ABCHFIZGIJZKZGBLZFMZUCUDNOPKZGCLZFMZPFGBCQAUGUJUFAFUEAGBDUEARSTUIA
|
|
FUHAGCEUHARSTUBUA $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d A z $. $d B z $. $d ph z $.
|
|
nfopd.1 $e |- ( ph -> F/_ x A ) $.
|
|
nfopd.2 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nfop .
|
|
(Contributed by SF, 2-Jan-2015.) $)
|
|
nfopd $p |- ( ph -> F/_ x <. A , B >. ) $=
|
|
( vz cv wcel wal cab cop wnfc nfaba1 nfop wb wa nfnfc1 wceq abidnf adantr
|
|
nfan adantl opeq12d nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUHDIZBJGKZLZMZBC
|
|
DLZMZBUJULUIBGNUKBGNOABCMZBDMZUNUPPEFUQURQZBUMUOUQURBBCRBDRUBUSUJCULDUQUJ
|
|
CSURBGCTUAURULDSUQBGDTUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
eqvinop.1 $e |- B e. _V $.
|
|
eqvinop.2 $e |- C e. _V $.
|
|
$( A variable introduction law for ordered pairs. Analog of Lemma 15 of
|
|
[Monk2] p. 109. (Contributed by NM, 28-May-1995.) $)
|
|
eqvinop $p |- ( A = <. B , C >. <-> E. x E. y ( A = <. x , y >. /\
|
|
<. x , y >. = <. B , C >. ) ) $=
|
|
( cv cop wceq wa wex opth ancom bitri anbi2i an13 exbii eqeq2d ceqsexv
|
|
19.42v opeq2 3bitri opeq1 bitr2i ) CAHZBHZIZJZUHDEIZJZKZBLZALUFDJZCUFEIZJ
|
|
ZKZALCUJJZUMUQAUMUNUGEJZUIKZKZBLUNUTBLZKUQULVABULUIUSUNKZKVAUKVCUIUKUNUSK
|
|
VCUFUGDEMUNUSNOPUIUSUNQORUNUTBUAVBUPUNUIUPBEGUSUHUOCUGEUFUBSTPUCRUPURADFU
|
|
NUOUJCUFDEUDSTUE $.
|
|
$}
|
|
|
|
${
|
|
$d x z w A $. $d y z w A $. $d z w ph $.
|
|
$( Substitution of class ` A ` for ordered pair ` <. x , y >. ` .
|
|
(Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon,
|
|
25-Jul-2011.) $)
|
|
copsexg $p |- ( A = <. x , y >. ->
|
|
( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) $=
|
|
( vz vw cv cop wceq wa wex wb wi vex 19.8a weq syl5bi syl5 weu euequ1 syl
|
|
eqvinop ex opth anbi1i 2exbii nfe1 wal anass a1i anim2d eximd biidd drex1
|
|
nfae sylibd exbii 19.40 nfnae dveeq2 nfd 19.9d anim1d syl6 pm2.61i exlimi
|
|
wn equcom eubii mpbi eupick mpan com12 sylan9 sylbi impbid anbi1d 2exbidv
|
|
eqeq1 bibi2d imbi12d mpbiri adantr exlimivv pm2.43i ) DBGZCGZHZIZAWIAJZCK
|
|
BKZLZWIDEGZFGZHZIZWOWHIZJZFKEKWIWLMZEFDWFWGBNCNUBWRWSEFWPWSWQWPWSWQAWQAJZ
|
|
CKZBKZLZMWQAXBWQAXBWTXAXBWTCOXABOUAUCWQEBPZFCPZJZXBAMWMWNWFWGUDZXBXFAJZCK
|
|
ZBKZXFAWTXHBCWQXFAXGUEUFXJXDXEAJZCKZJZBKZXFAXIXNBXMBUGCBPCUHZXIXNMXOXIXMC
|
|
KXNXOXHXMCCBCUOXHXDXKJZXOXMXDXEAUIZXOXKXLXDXKXLMXOXKCOUJUKQULXMXMCBXOXMUM
|
|
UNUPXOVGZXIXMXNXIXPCKZXRXMXHXPCXQUQXSXDCKZXLJXRXMXDXKCURXRXTXDXLXDXRCXRXD
|
|
CCBCUSCBEUTVAVBVCRQXMBOVDVEVFXDXNXLXEAXNXDXLXDBSZXNXDXLMBEPZBSYABETYBXDBB
|
|
EVHVIVJXDXLBVKVLVMXLXEAXECSZXLXEAMCFPZCSYCCFTYDXECCFVHVIVJXEACVKVLVMVNRQV
|
|
OVPWPWIWQWLXCDWOWHVSZWPWKXBAWPWJWTBCWPWIWQAYEVQVRVTWAWBWCWDVOWE $.
|
|
$}
|
|
|
|
${
|
|
$d x y ps $. $d x y A $. $d x y B $.
|
|
$( Closed theorem form of ~ copsex2g . (Contributed by NM,
|
|
17-Feb-2013.) $)
|
|
copsex2t $p |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
|
|
/\ ( A e. V /\ B e. W ) ) ->
|
|
( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) $=
|
|
( wcel wa cv wceq wb wal wex cop elisset nfe1 nfv nfbi anim12i eeanv nfa1
|
|
wi sylibr nfa2 nfex opeq12 copsexg eqcoms syl adantl sp 19.21bi bitr3d ex
|
|
imp exlimd sylan2 ) EGIZFHIZJZCKZELZDKZFLZJZABMZUDZDNZCNZVGDOZCOZEFPZVCVE
|
|
PZLAJZDOZCOZBMZVBVDCOZVFDOZJVMUTVTVAWACEGQDFHQUAVDVFCDUBUEVKVMVSVKVLVSCVJ
|
|
CUCVRBCVQCRBCSTVKVGVSDVIDCUFVRBDVQDCVPDRUGBDSTVKVGVSVKVGJAVRBVGAVRMZVKVGV
|
|
OVNLWBVCEVEFUHWBVNVOACDVNUIUJUKULVKVGVHVKVIDVJCUMUNUQUOUPURURUQUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y ps $. $d x y A $. $d x y B $.
|
|
copsex2g.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( Implicit substitution inference for ordered pairs. (Contributed by NM,
|
|
28-May-1995.) $)
|
|
copsex2g $p |- ( ( A e. V /\ B e. W ) ->
|
|
( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) $=
|
|
( wcel cv wceq wex cop wa wb elisset nfe1 nfv nfbi eeanv nfex copsexg syl
|
|
opeq12 eqcoms bitr3d exlimi sylbir syl2an ) EGJCKZELZCMZDKZFLZDMZEFNZUKUN
|
|
NZLAOZDMZCMZBPZFHJCEGQDFHQUMUPOULUOOZDMZCMVBULUOCDUAVDVBCVABCUTCRBCSTVCVB
|
|
DVABDUTDCUSDRUBBDSTVCAVABVCURUQLAVAPZUKEUNFUEVEUQURACDUQUCUFUDIUGUHUHUIUJ
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d x y z w D $.
|
|
$d x y z w ps $. $d x y z w R $. $d x y z w S $.
|
|
copsex4g.1 $e |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) ->
|
|
( ph <-> ps ) ) $.
|
|
$( An implicit substitution inference for 2 ordered pairs. (Contributed by
|
|
NM, 5-Aug-1995.) $)
|
|
copsex4g $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) ->
|
|
( E. x E. y E. z E. w ( ( <. A , B >. = <. x , y >. /\
|
|
<. C , D >. = <. z , w >. ) /\ ph ) <-> ps ) ) $=
|
|
( cop cv wceq wa wex wcel eqcom opth anbi12i anbi1i 2exbii cgsex4g syl5bb
|
|
bitri id ) GHNZCOZDOZNZPZIJNZEOZFOZNZPZQZAQZFRERZDRCRUJGPUKHPQZUOIPUPJPQZ
|
|
QZAQZFRERZDRCRGKSHLSQIKSJLSQQBVAVFCDUTVEEFUSVDAUMVBURVCUMULUIPVBUIULTUJUK
|
|
GHUAUGURUQUNPVCUNUQTUOUPIJUAUGUBUCUDUDABVDCDEFGHIJKLVDUHMUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d A z t $. $d B z t $. $d C t z $.
|
|
$( Express equality to an ordered pair. (Contributed by SF,
|
|
6-Jan-2015.) $)
|
|
eqop $p |- ( A = <. B , C >. <->
|
|
A. z ( z e. A <->
|
|
( E. t e. B z = Phi t \/ E. t e. C z = ( Phi t u. { 0c } ) ) ) ) $=
|
|
( cop wceq cv wcel wb wal cphi wrex c0c csn cun wo cab bitri abid orbi12i
|
|
dfcleq df-op eleq2i elun bibi2i albii ) CDEFZGAHZCIZUIUHIZJZAKUJUIBHLZGBD
|
|
MZUIUMNOPGBEMZQZJZAKACUHUBULUQAUKUPUJUKUIUNARZIZUIUOARZIZQZUPUKUIURUTPZIV
|
|
BUHVCUIABDEUCUDUIURUTUESUSUNVAUOUNATUOATUASUFUGS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( "At most one" remains true inside ordered pair quantification.
|
|
(Contributed by NM, 28-Aug-2007.) $)
|
|
mosubopt $p |- ( A. y A. z E* x ph ->
|
|
E* x E. y E. z ( A = <. y , z >. /\ ph ) ) $=
|
|
( wmo wal cv cop wceq wex wa nfa1 nfe1 nfmo wi nfex sps exlimd wn copsexg
|
|
mobidv biimpcd simpl 2eximi exlimiv con3i exmo ori syl pm2.61d1 ) ABFZDGZ
|
|
CGZECHDHIJZDKZCKZUOALZDKZCKZBFZUNUPVACUMCMUTCBUSCNOUMUPVAPCUMUOVADULDMUTD
|
|
BUSDCURDNQOULUOVAPDUOULVAUOAUTBACDEUAUBUCRSRSUQTUTBKZTVAVBUQUTUQBURUOCDUO
|
|
AUDUEUFUGVBVAUTBUHUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
mosubop.1 $e |- E* x ph $.
|
|
$( "At most one" remains true inside ordered pair quantification.
|
|
(Contributed by NM, 28-May-1995.) $)
|
|
mosubop $p |- E* x E. y E. z ( A = <. y , z >. /\ ph ) $=
|
|
( wmo wal cv cop wceq wa wex gen2 mosubopt ax-mp ) ABGZDHCHECIDIJKALDMCMB
|
|
GQCDFNABCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The phi operation distributes over union. (Contributed by SF,
|
|
20-Feb-2015.) $)
|
|
phiun $p |- Phi ( A u. B ) = ( Phi A u. Phi B ) $=
|
|
( vx vy cv cnnc wcel c1c cplc cif wceq cun wrex cab wo rexun abbii df-phi
|
|
cphi uneq12i unab eqtri 3eqtr4i ) CEDEZFGUDHIUDJKZDABLZMZCNUEDAMZUEDBMZOZ
|
|
CNZUFSASZBSZLZUGUJCUEDABPQDCUFRUNUHCNZUICNZLUKULUOUMUPDCARDCBRTUHUICUAUBU
|
|
C $.
|
|
|
|
$( The phi operation applied to a set disjoint from the naturals has no
|
|
effect. (Contributed by SF, 20-Feb-2015.) $)
|
|
phidisjnn $p |- ( ( A i^i Nn ) = (/) -> Phi A = A ) $=
|
|
( vx vy cnnc cin c0 wceq cv wcel c1c cplc cif wrex wb wal cphi wa syl6bbr
|
|
weq wn wral disj biimpi r19.21bi iffalse syl eqeq2d equcom risset alrimiv
|
|
rexbidva cab df-phi eqeq1i abeq1 bitri sylibr ) ADEFGZBHZCHZDIZUTJKZUTLZG
|
|
ZCAMZUSAIZNZBOZAPZAGZURVGBURVECBSZCAMVFURVDVKCAURUTAIQZVDBCSVKVLVCUTUSVLV
|
|
ATZVCUTGURVMCAURVMCAUACADUBUCUDVAVBUTUEUFUGCBUHRUKCUSAUIRUJVJVEBULZAGVHVI
|
|
VNACBAUMUNVEBAUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $.
|
|
phiall.1 $e |- A e. _V $.
|
|
|
|
$( Lemma for ~ phiall . Any set of numbers without zero is the
|
|
Phi of a set. (Contributed by Scott Fenton, 14-Apr-2021.) $)
|
|
phialllem1 $p |- ( ( A C_ Nn /\ -. 0c e. A ) -> E. x A = Phi x ) $=
|
|
( vz vy vw cnnc c0c wcel wa cv c1c wceq wrex cab eqeq2d cssetk cpw1 cimak
|
|
cins3k wss wn cplc crab cphi wex weq wo eleq1 biimpcd con3d adantll ssel2
|
|
impcom adantlr nnc0suc sylib orel1 anidm anbi2d syl5bbr rexbidv syl5ibcom
|
|
sylc eqeq1 eqtr3 rexlimivw impbid1 rexbidva risset rexcom 3bitr4g abbi2dv
|
|
cif df-phi addceq1 rexrab r19.41v syl6bbr rexbiia bitri abbii syl6eqr cin
|
|
iftrue eqtri cvv dfrab2 cins2k ccompl csik cun csymdif cdif cimagek ccnvk
|
|
copk elimak opkelimagek opkelcnvk dfaddc2 eqeq2i 3bitr4i rexbii addcexlem
|
|
vex abbi2i 1cex pw1ex imakex imagekex cnvkex eqeltrri nncex eqeltri phieq
|
|
inex spcev syl ) BGUAZHBIZUBZJZBDKZEKZLUCZMZDBNZEGUDZUEZMZBAKZUEZMZAUFYCB
|
|
YDYLLUCZMZFKZYOMZJZDBNZAGNZFOZYJYCUUAFBYCDFUGZDBNYSAGNZDBNYQBIUUAYCUUCUUD
|
|
DBYCYDBIZJZUUCUUDUUFYPAGNZUUCUUDUUFYDHMZUBZUUHUUGUHZUUGYBUUEUUIXTUUEYBUUI
|
|
UUEUUHYAUUHUUEYAYDHBUIUJUKUNULUUFYDGIZUUJXTUUEUUKYBBGYDUMUOAYDUPUQUUHUUGU
|
|
RVDUUCYPYSAGYPYPYPJUUCYSYPUSUUCYPYRYPYDYQYOVEUTVAVBVCYSUUCAGYDYQYOVFVGVHV
|
|
IDYQBVJYSADGBVKVLVMYJYQYLGIZYOYLVNZMZAYINZFOUUBAFYIVOUUOUUAFUUOYPDBNZUUNJ
|
|
ZAGNUUAYHUUPUUNAEGEAUGZYGYPDBUURYFYOYDYEYLLVPPVBVQUUQYTAGUULUUQUUPYRJYTUU
|
|
LUUNYRUUPUULUUMYOYQUULYOYLWEPUTYPYRDBVRVSVTWAWBWFWCYNYKAYIYIYHEOZGWDWGYHE
|
|
GWHUUSGQTZQWIZWDLRZRZSWJTUVAWIUUTWIQWKWKTWLWMUVCRRSWNZUVCSZWOZWPZBSZUUSWG
|
|
YHEUVHYEUVHIYDYEWQUVGIZDBNYHDUVGBYEEXFZWRUVIYGDBYEYDWQUVFIYDUVEYESZMUVIYG
|
|
YEYDUVEUVJDXFZWSYDYEUVFUVLUVJWTYFUVKYDYELXAXBXCXDWAXGUVGBUVFUVEUVDUVCXEUV
|
|
BLXHXIXIXJXKXLCXJXMXNXQXOYLYIMYMYJBYLYIXPPXRXS $.
|
|
|
|
|
|
$( Lemma for ~ phiall . Any set without ` 0c ` is equal to the ` Phi `
|
|
of a set. (Contributed by Scott Fenton, 8-Apr-2021.) $)
|
|
phialllem2 $p |- ( -. 0c e. A -> E. x A = Phi x ) $=
|
|
( vy c0c wcel wn cnnc cin cv cphi wceq wex wss inss2 nncex cun eqtri syl
|
|
c0 inss1 sseli con3i phialllem1 sylancr uncom inundif uneq2 syl5eqr phiun
|
|
inex cdif incom disjdif phidisjnn ax-mp uneq1i syl6eqr difex phieq eqeq2d
|
|
vex unex spcev exlimiv ) EBFZGZBHIZDJZKZLZDMZBAJZKZLZAMZVGVHHNEVHFZGVLBHO
|
|
VQVFVHBEBHUAUBUCDVHBHCPUKUDUEVKVPDVKBBHULZVIQZKZLZVPVKBVRVJQZVTVKBVRVHQZW
|
|
BWCVHVRQBVRVHUFBHUGRVHVJVRUHUIVTVRKZVJQWBVRVIUJWDVRVJVRHIZTLWDVRLWEHVRITV
|
|
RHUMHBUNRVRUOUPUQRURVOWAAVSVRVIBHCPUSDVBVCVMVSLVNVTBVMVSUTVAVDSVES $.
|
|
|
|
$( Any set is equal to either the ` Phi ` of another set or to a ` Phi `
|
|
with ` 0c ` adjoined. (Contributed by Scott Fenton, 8-Apr-2021.) $)
|
|
phiall $p |- E. x ( A = Phi x \/ A = ( Phi x u. { 0c } ) ) $=
|
|
( c0c wcel cv cphi csn cun wo wex wn phialllem2 cin c0 disjsn mpbir eximi
|
|
wceq syl cdif neldifsn snex difex ax-mp wa 0cnelphi eqtr4i biantru unineq
|
|
bitri difsnid eqeq1d syl5bbr exbidv mpbii olc orc pm2.61i ) DBEZBAFZGZSZB
|
|
VBDHZIZSZJZAKZUTVFAKZVHUTBVDUAZVBSZAKZVIDVJELZVLDBUBZAVJBVDCDUCUDMUEUTVKV
|
|
FAVKVJVDIZVESZUTVFVPVPVJVDNZVBVDNZSZUFVKVSVPVQOVRVQOSVMVNVJDPQVROSDVBELVA
|
|
UGVBDPQUHUIVJVBVDUJUKUTVOBVEBDULUMUNUOUPVFVGAVFVCUQRTUTLVCAKVHABCMVCVGAVC
|
|
VFURRTUS $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Any class is equal to an ordered pair. (Contributed by Scott Fenton,
|
|
8-Apr-2021.) $)
|
|
opeq $p |- A = <. Proj1 A , Proj2 A >. $=
|
|
( vx vy vz cv cphi wceq wrex cab cun wex crab wcel wa rexeqi rexab ancom
|
|
eleq1d 3bitri eqtr4i cproj1 cop c0c csn df-op df-proj1 weq phieq pm5.32i
|
|
cproj2 eleq1 bitr4i exbii 19.41v abbii df-rab df-proj2 uneq1d uneq12i wo
|
|
unrab rabid2 vex phiall 19.43 mpbi a1i mprgbir 3eqtrri ) AUAZAUJZUBBEZCE
|
|
ZFZGZCVJHZBIZVLVNUCUDZJZGZCVKHZBIZJVOCKZBALZVTCKZBALZJZABCVJVKUEVQWDWBWF
|
|
VQVLAMZWCNZBIWDVPWIBVPVOCDEZFZAMZDIZHVNAMZVONZCKZWIVOCVJWMDAUFOWLWNVOCDD
|
|
CUGZWKVNAWJVMUHZRPWPVOWHNZCKWCWHNWIWOWSCWOVOWNNWSWNVOQVOWHWNVLVNAUKUIULU
|
|
MVOWHCUNWCWHQSSUOWCBAUPTWBWHWENZBIWFWAWTBWAVTCWKVRJZAMZDIZHVSAMZVTNZCKZW
|
|
TVTCVKXCDAUQOXBXDVTCDWQXAVSAWQWKVNVRWRURRPXFVTWHNZCKWEWHNWTXEXGCXEVTXDNX
|
|
GXDVTQVTWHXDVLVSAUKUIULUMVTWHCUNWEWHQSSUOWEBAUPTUSWGWCWEUTZBALZAWCWEBAVA
|
|
AXIGXHBAXHBAVBXHWHVOVTUTCKXHCVLBVCVDVOVTCVEVFVGVHTVI $.
|
|
|
|
$( A class is a set iff it is equal to an ordered pair. (Contributed by
|
|
Scott Fenton, 19-Apr-2021.) $)
|
|
opeqexb $p |- ( A e. _V <-> E. x E. y A = <. x , y >. ) $=
|
|
( cproj1 cproj2 cop cvv wcel wa cv wceq wex opexb opeq eleq1i eeanv eqcom
|
|
eqeq1i isset 3bitr4i opth 3bitri 2exbii anbi12i ) CDZCEZFZGHUEGHZUFGHZIZC
|
|
GHCAJZBJZFZKZBLALZUEUFMCUGGCNZOUKUEKZULUFKZIZBLALUQALZURBLZIUOUJUQURABPUN
|
|
USABUNUGUMKUMUGKUSCUGUMUPRUGUMQUKULUEUFUAUBUCUHUTUIVAAUESBUFSUDTT $.
|
|
|
|
$( Any set is equal to some ordered pair. (Contributed by Scott Fenton,
|
|
16-Apr-2021.) $)
|
|
opeqex $p |- ( A e. V -> E. x E. y A = <. x , y >. ) $=
|
|
( wcel cvv cv cop wceq wex elex opeqexb sylib ) CDECFECAGBGHIBJAJCDKABCLM
|
|
$.
|
|
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordered-pair class abstractions (class builders)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend class notation to include ordered-pair class abstraction (class
|
|
builder). $)
|
|
copab $a class { <. x , y >. | ph } $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( Define the class abstraction of a collection of ordered pairs.
|
|
Definition 3.3 of [Monk1] p. 34. Usually ` x ` and ` y ` are distinct,
|
|
although the definition doesn't strictly require it (see ~ dfid2 for a
|
|
case where they are not distinct). The brace notation is called "class
|
|
abstraction" by Quine; it is also (more commonly) called a "class
|
|
builder" in the literature. (Contributed by SF, 12-Jan-2015.) $)
|
|
df-opab $a |- { <. x , y >. | ph } =
|
|
{ z | E. x E. y ( z = <. x , y >. /\ ph ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $. $d z ps $. $d z ch $.
|
|
opabbid.1 $e |- F/ x ph $.
|
|
opabbid.2 $e |- F/ y ph $.
|
|
opabbid.3 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal ordered-pair class abstractions (deduction
|
|
rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew
|
|
Salmon, 9-Jul-2011.) $)
|
|
opabbid $p |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } ) $=
|
|
( vz cv cop wceq wa wex cab copab anbi2d exbid abbidv df-opab 3eqtr4g ) A
|
|
IJDJEJKLZBMZENZDNZIOUBCMZENZDNZIOBDEPCDEPAUEUHIAUDUGDFAUCUFEGABCUBHQRRSBD
|
|
EITCDEITUA $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
opabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal ordered-pair class abstractions (deduction
|
|
rule). (Contributed by NM, 15-May-1995.) $)
|
|
opabbidv $p |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } ) $=
|
|
( nfv opabbid ) ABCDEADGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $. $d z ps $.
|
|
opabbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Equivalent wff's yield equal class abstractions. (Contributed by NM,
|
|
15-May-1995.) $)
|
|
opabbii $p |- { <. x , y >. | ph } = { <. x , y >. | ps } $=
|
|
( vz cv wceq copab eqid wb a1i opabbidv ax-mp ) FGZOHZACDIBCDIHOJPABCDABK
|
|
PELMN $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d ph w $.
|
|
nfopab.1 $e |- F/ z ph $.
|
|
$( Bound-variable hypothesis builder for class abstraction. (Contributed
|
|
by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were
|
|
removed by Andrew Salmon, 11-Jul-2011.) $)
|
|
nfopab $p |- F/_ z { <. x , y >. | ph } $=
|
|
( vw copab cv cop wceq wa wex cab df-opab nfv nfan nfex nfab nfcxfr ) DAB
|
|
CGFHBHCHIJZAKZCLZBLZFMABCFNUCDFUBDBUADCTADTDOEPQQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z ph $.
|
|
$( The first abstraction variable in an ordered-pair class abstraction
|
|
(class builder) is effectively not free. (Contributed by NM,
|
|
16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
nfopab1 $p |- F/_ x { <. x , y >. | ph } $=
|
|
( vz copab cv cop wceq wa wex cab df-opab nfe1 nfab nfcxfr ) BABCEDFBFCFG
|
|
HAICJZBJZDKABCDLQBDPBMNO $.
|
|
|
|
$( The second abstraction variable in an ordered-pair class abstraction
|
|
(class builder) is effectively not free. (Contributed by NM,
|
|
16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
nfopab2 $p |- F/_ y { <. x , y >. | ph } $=
|
|
( vz copab cv cop wceq wa wex cab df-opab nfe1 nfex nfab nfcxfr ) CABCEDF
|
|
BFCFGHAIZCJZBJZDKABCDLSCDRCBQCMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v $. $d v ph $. $d v ps $.
|
|
cbvopab.1 $e |- F/ z ph $.
|
|
cbvopab.2 $e |- F/ w ph $.
|
|
cbvopab.3 $e |- F/ x ps $.
|
|
cbvopab.4 $e |- F/ y ps $.
|
|
cbvopab.5 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables in an ordered-pair class
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
14-Sep-2003.) $)
|
|
cbvopab $p |- { <. x , y >. | ph } = { <. z , w >. | ps } $=
|
|
( vv cv cop wceq wa wex cab nfv nfan copab weq opeq12 eqeq2d cbvex2 abbii
|
|
anbi12d df-opab 3eqtr4i ) LMZCMZDMZNZOZAPZDQCQZLRUJEMZFMZNZOZBPZFQEQZLRAC
|
|
DUABEFUAUPVBLUOVACDEFUNAEUNESGTUNAFUNFSHTUTBCUTCSITUTBDUTDSJTCEUBDFUBPZUN
|
|
UTABVCUMUSUJUKUQULURUCUDKUGUEUFACDLUHBEFLUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $. $d x y ps $.
|
|
cbvopabv.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change bound variables in an ordered-pair class
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
15-Oct-1996.) $)
|
|
cbvopabv $p |- { <. x , y >. | ph } = { <. z , w >. | ps } $=
|
|
( nfv cbvopab ) ABCDEFAEHAFHBCHBDHGI $.
|
|
$}
|
|
|
|
${
|
|
$d v w x y $. $d v w y z $. $d v w ph $. $d v w ps $.
|
|
cbvopab1.1 $e |- F/ z ph $.
|
|
cbvopab1.2 $e |- F/ x ps $.
|
|
cbvopab1.3 $e |- ( x = z -> ( ph <-> ps ) ) $.
|
|
$( Change first bound variable in an ordered-pair class abstraction, using
|
|
explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by
|
|
Mario Carneiro, 14-Oct-2016.) $)
|
|
cbvopab1 $p |- { <. x , y >. | ph } = { <. z , y >. | ps } $=
|
|
( vw vv cv cop wceq wa wex cab copab wsb nfv nfan nfs1v nfex opeq1 eqeq2d
|
|
sbequ12 anbi12d exbidv cbvex nfsb sbequ sbie syl6bb bitri df-opab 3eqtr4i
|
|
abbii ) IKZCKZDKZLZMZANZDOZCOZIPUQEKZUSLZMZBNZDOZEOZIPACDQBEDQVDVJIVDUQJK
|
|
ZUSLZMZACJRZNZDOZJOVJVCVPCJVCJSVOCDVMVNCVMCSACJUATUBURVKMZVBVODVQVAVMAVNV
|
|
QUTVLUQURVKUSUCUDACJUEUFUGUHVPVIJEVOEDVMVNEVMESACJEFUITUBVIJSVKVEMZVOVHDV
|
|
RVMVGVNBVRVLVFUQVKVEUSUCUDVRVNACERBAJECUJABCEGHUKULUFUGUHUMUPACDIUNBEDIUN
|
|
UO $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z $. $d w ph $. $d w ps $.
|
|
cbvopab2.1 $e |- F/ z ph $.
|
|
cbvopab2.2 $e |- F/ y ps $.
|
|
cbvopab2.3 $e |- ( y = z -> ( ph <-> ps ) ) $.
|
|
$( Change second bound variable in an ordered-pair class abstraction, using
|
|
explicit substitution. (Contributed by NM, 22-Aug-2013.) $)
|
|
cbvopab2 $p |- { <. x , y >. | ph } = { <. x , z >. | ps } $=
|
|
( vw cv cop wceq wa wex cab copab nfv nfan opeq2 df-opab anbi12d 3eqtr4i
|
|
eqeq2d cbvex exbii abbii ) IJZCJZDJZKZLZAMZDNZCNZIOUGUHEJZKZLZBMZENZCNZIO
|
|
ACDPBCEPUNUTIUMUSCULURDEUKAEUKEQFRUQBDUQDQGRUIUOLZUKUQABVAUJUPUGUIUOUHSUC
|
|
HUAUDUEUFACDITBCEITUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $.
|
|
$( Change first bound variable in an ordered-pair class abstraction, using
|
|
explicit substitution. (Contributed by NM, 31-Jul-2003.) $)
|
|
cbvopab1s $p |- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph } $=
|
|
( vw cv cop wceq wa wex cab wsb copab nfv nfs1v nfan opeq1 eqeq2d df-opab
|
|
nfex sbequ12 anbi12d exbidv cbvex abbii 3eqtr4i ) EFZBFZCFZGZHZAIZCJZBJZE
|
|
KUGDFZUIGZHZABDLZIZCJZDJZEKABCMURDCMUNVAEUMUTBDUMDNUSBCUQURBUQBNABDOPTUHU
|
|
OHZULUSCVBUKUQAURVBUJUPUGUHUOUIQRABDUAUBUCUDUEABCESURDCESUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z $. $d z ph $. $d x ps $.
|
|
cbvopab1v.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change the first bound variable in an ordered pair
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) $)
|
|
cbvopab1v $p |- { <. x , y >. | ph } = { <. z , y >. | ps } $=
|
|
( nfv cbvopab1 ) ABCDEAEGBCGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $. $d y w ps $.
|
|
cbvopab2v.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change the second bound variable in an ordered pair
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
2-Sep-1999.) $)
|
|
cbvopab2v $p |- { <. x , y >. | ph } = { <. x , z >. | ps } $=
|
|
( vw cv cop wceq wex cab copab opeq2 eqeq2d anbi12d cbvexv exbii df-opab
|
|
wa abbii 3eqtr4i ) GHZCHZDHZIZJZATZDKZCKZGLUCUDEHZIZJZBTZEKZCKZGLACDMBCEM
|
|
UJUPGUIUOCUHUNDEUEUKJZUGUMABUQUFULUCUEUKUDNOFPQRUAACDGSBCEGSUB $.
|
|
$}
|
|
|
|
${
|
|
$d w y z A $. $d w ph $. $d w x y z $.
|
|
$( Move substitution into a class abstraction. (Contributed by NM,
|
|
6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) $)
|
|
csbopabg $p |- ( A e. V -> [_ A / x ]_ { <. y , z >. | ph } =
|
|
{ <. y , z >. | [. A / x ]. ph } ) $=
|
|
( vw cv copab csb wsb wceq wsbc csbeq1 dfsbcq2 opabbidv eqeq12d vex nfs1v
|
|
nfopab sbequ12 csbief vtoclg ) BGHZACDIZJZABGKZCDIZLBEUEJZABEMZCDIZLGEFUD
|
|
ELZUFUIUHUKBUDEUENULUGUJCDABGEOPQBUDUEUHGRUGCDBABGSTBHUDLAUGCDABGUAPUBUC
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d ph z $. $d ps z $.
|
|
$( Union of two ordered pair class abstractions. (Contributed by NM,
|
|
30-Sep-2002.) $)
|
|
unopab $p |- ( { <. x , y >. | ph } u. { <. x , y >. | ps } ) =
|
|
{ <. x , y >. | ( ph \/ ps ) } $=
|
|
( vz cv cop wceq wa wex cab wo copab unab 19.43 andi exbii bitr2i df-opab
|
|
cun bitr3i abbii eqtri uneq12i 3eqtr4i ) EFCFDFGHZAIZDJZCJZEKZUFBIZDJZCJZ
|
|
EKZTZUFABLZIZDJZCJZEKZACDMZBCDMZTUPCDMUOUIUMLZEKUTUIUMENVCUSEVCUHULLZCJUS
|
|
UHULCOVDURCURUGUKLZDJVDUQVEDUFABPQUGUKDORQUAUBUCVAUJVBUNACDESBCDESUDUPCDE
|
|
SUE $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Binary relations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend wff notation to include the general binary relation predicate.
|
|
Note that the syntax is simply three class symbols in a row. Since binary
|
|
relations are the only possible wff expressions consisting of three class
|
|
expressions in a row, the syntax is unambiguous. $)
|
|
wbr $a wff A R B $.
|
|
|
|
$( Define a general binary relation. Note that the syntax is simply three
|
|
class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29
|
|
generalized to arbitrary classes. Class ` R ` normally denotes a relation
|
|
that compares two classes ` A ` and ` B ` . This definition is
|
|
well-defined, although not very meaningful, when classes ` A ` and/or
|
|
` B ` are proper classes (i.e. are not sets). On the other hand, we often
|
|
find uses for this definition when ` R ` is a proper class. (Contributed
|
|
by NM, 4-Jun-1995.) $)
|
|
df-br $a |- ( A R B <-> <. A , B >. e. R ) $.
|
|
|
|
$( Equality theorem for binary relations. (Contributed by NM,
|
|
4-Jun-1995.) $)
|
|
breq $p |- ( R = S -> ( A R B <-> A S B ) ) $=
|
|
( wceq cop wcel wbr eleq2 df-br 3bitr4g ) CDEABFZCGLDGABCHABDHCDLIABCJABDJK
|
|
$.
|
|
|
|
$( Equality theorem for a binary relation. (Contributed by NM,
|
|
31-Dec-1993.) $)
|
|
breq1 $p |- ( A = B -> ( A R C <-> B R C ) ) $=
|
|
( wceq cop wcel wbr opeq1 eleq1d df-br 3bitr4g ) ABEZACFZDGBCFZDGACDHBCDHMN
|
|
ODABCIJACDKBCDKL $.
|
|
|
|
$( Equality theorem for a binary relation. (Contributed by NM,
|
|
31-Dec-1993.) $)
|
|
breq2 $p |- ( A = B -> ( C R A <-> C R B ) ) $=
|
|
( wceq cop wcel wbr opeq2 eleq1d df-br 3bitr4g ) ABEZCAFZDGCBFZDGCADHCBDHMN
|
|
ODABCIJCADKCBDKL $.
|
|
|
|
$( Equality theorem for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq12 $p |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) ) $=
|
|
( wceq wbr breq1 breq2 sylan9bb ) ABFACEGBCEGCDFBDEGABCEHCDBEIJ $.
|
|
|
|
${
|
|
breqi.1 $e |- R = S $.
|
|
$( Equality inference for binary relations. (Contributed by NM,
|
|
19-Feb-2005.) $)
|
|
breqi $p |- ( A R B <-> A S B ) $=
|
|
( wceq wbr wb breq ax-mp ) CDFABCGABDGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
breq1i.1 $e |- A = B $.
|
|
$( Equality inference for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq1i $p |- ( A R C <-> B R C ) $=
|
|
( wceq wbr wb breq1 ax-mp ) ABFACDGBCDGHEABCDIJ $.
|
|
|
|
$( Equality inference for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq2i $p |- ( C R A <-> C R B ) $=
|
|
( wceq wbr wb breq2 ax-mp ) ABFCADGCBDGHEABCDIJ $.
|
|
|
|
${
|
|
breq12i.2 $e |- C = D $.
|
|
$( Equality inference for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) (Revised by Eric Schmidt, 4-Apr-2007.) $)
|
|
breq12i $p |- ( A R C <-> B R D ) $=
|
|
( wceq wbr wb breq12 mp2an ) ABHCDHACEIBDEIJFGABCDEKL $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
breq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq1d $p |- ( ph -> ( A R C <-> B R C ) ) $=
|
|
( wceq wbr wb breq1 syl ) ABCGBDEHCDEHIFBCDEJK $.
|
|
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
29-Oct-2011.) $)
|
|
breqd $p |- ( ph -> ( C A D <-> C B D ) ) $=
|
|
( wceq wbr wb breq syl ) ABCGDEBHDECHIFDEBCJK $.
|
|
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breq2d $p |- ( ph -> ( C R A <-> C R B ) ) $=
|
|
( wceq wbr wb breq2 syl ) ABCGDBEHDCEHIFBCDEJK $.
|
|
|
|
${
|
|
breq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for a binary relation. (The proof was shortened by
|
|
Andrew Salmon, 9-Jul-2011.) (Contributed by NM, 8-Feb-1996.)
|
|
(Revised by set.mm contributors, 9-Jul-2011.) $)
|
|
breq12d $p |- ( ph -> ( A R C <-> B R D ) ) $=
|
|
( wceq wbr wb breq12 syl2anc ) ABCIDEIBDFJCEFJKGHBCDEFLM $.
|
|
$}
|
|
|
|
${
|
|
breq123d.2 $e |- ( ph -> R = S ) $.
|
|
breq123d.3 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
29-Oct-2011.) $)
|
|
breq123d $p |- ( ph -> ( A R C <-> B S D ) ) $=
|
|
( wbr breq12d breqd bitrd ) ABDFKCEFKCEGKABCDEFHJLAFGCEIMN $.
|
|
$}
|
|
|
|
${
|
|
breqan12i.2 $e |- ( ps -> C = D ) $.
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breqan12d $p |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) ) $=
|
|
( wceq wbr wb breq12 syl2an ) ACDJEFJCEGKDFGKLBHICDEFGMN $.
|
|
|
|
$( Equality deduction for a binary relation. (Contributed by NM,
|
|
8-Feb-1996.) $)
|
|
breqan12rd $p |- ( ( ps /\ ph ) -> ( A R C <-> B R D ) ) $=
|
|
( wbr wb breqan12d ancoms ) ABCEGJDFGJKABCDEFGHILM $.
|
|
$}
|
|
$}
|
|
|
|
$( Two classes are different if they don't have the same relationship to a
|
|
third class. (Contributed by NM, 3-Jun-2012.) $)
|
|
nbrne1 $p |- ( ( A R B /\ -. A R C ) -> B =/= C ) $=
|
|
( wbr wn wne wceq breq2 biimpcd necon3bd imp ) ABDEZACDEZFBCGMNBCBCHMNBCADI
|
|
JKL $.
|
|
|
|
$( Two classes are different if they don't have the same relationship to a
|
|
third class. (Contributed by NM, 3-Jun-2012.) $)
|
|
nbrne2 $p |- ( ( A R C /\ -. B R C ) -> A =/= B ) $=
|
|
( wbr wn wne wceq breq1 biimpcd necon3bd imp ) ACDEZBCDEZFABGMNABABHMNABCDI
|
|
JKL $.
|
|
|
|
${
|
|
eqbrtr.1 $e |- A = B $.
|
|
eqbrtr.2 $e |- B R C $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqbrtri $p |- A R C $=
|
|
( wbr breq1i mpbir ) ACDGBCDGFABCDEHI $.
|
|
$}
|
|
|
|
${
|
|
eqbrtrd.1 $e |- ( ph -> A = B ) $.
|
|
eqbrtrd.2 $e |- ( ph -> B R C ) $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 8-Oct-1999.) $)
|
|
eqbrtrd $p |- ( ph -> A R C ) $=
|
|
( wbr breq1d mpbird ) ABDEHCDEHGABCDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
eqbrtrr.1 $e |- A = B $.
|
|
eqbrtrr.2 $e |- A R C $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
eqbrtrri $p |- B R C $=
|
|
( eqcomi eqbrtri ) BACDABEGFH $.
|
|
$}
|
|
|
|
${
|
|
eqbrtrrd.1 $e |- ( ph -> A = B ) $.
|
|
eqbrtrrd.2 $e |- ( ph -> A R C ) $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 24-Oct-1999.) $)
|
|
eqbrtrrd $p |- ( ph -> B R C ) $=
|
|
( eqcomd eqbrtrd ) ACBDEABCFHGI $.
|
|
$}
|
|
|
|
${
|
|
breqtr.1 $e |- A R B $.
|
|
breqtr.2 $e |- B = C $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
breqtri $p |- A R C $=
|
|
( wbr breq2i mpbi ) ABDGACDGEBCADFHI $.
|
|
$}
|
|
|
|
${
|
|
breqtrd.1 $e |- ( ph -> A R B ) $.
|
|
breqtrd.2 $e |- ( ph -> B = C ) $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 24-Oct-1999.) $)
|
|
breqtrd $p |- ( ph -> A R C ) $=
|
|
( wbr breq2d mpbid ) ABCEHBDEHFACDBEGIJ $.
|
|
$}
|
|
|
|
${
|
|
breqtrr.1 $e |- A R B $.
|
|
breqtrr.2 $e |- C = B $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 5-Aug-1993.) $)
|
|
breqtrri $p |- A R C $=
|
|
( eqcomi breqtri ) ABCDECBFGH $.
|
|
$}
|
|
|
|
${
|
|
breqtrrd.1 $e |- ( ph -> A R B ) $.
|
|
breqtrrd.2 $e |- ( ph -> C = B ) $.
|
|
$( Substitution of equal classes into a binary relation. (Contributed by
|
|
NM, 24-Oct-1999.) $)
|
|
breqtrrd $p |- ( ph -> A R C ) $=
|
|
( eqcomd breqtrd ) ABCDEFADCGHI $.
|
|
$}
|
|
|
|
${
|
|
3brtr3.1 $e |- A R B $.
|
|
3brtr3.2 $e |- A = C $.
|
|
3brtr3.3 $e |- B = D $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 11-Aug-1999.) $)
|
|
3brtr3i $p |- C R D $=
|
|
( eqbrtrri breqtri ) CBDEACBEGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3brtr4.1 $e |- A R B $.
|
|
3brtr4.2 $e |- C = A $.
|
|
3brtr4.3 $e |- D = B $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 11-Aug-1999.) $)
|
|
3brtr4i $p |- C R D $=
|
|
( eqbrtri breqtrri ) CBDECABEGFIHJ $.
|
|
$}
|
|
|
|
${
|
|
3brtr3d.1 $e |- ( ph -> A R B ) $.
|
|
3brtr3d.2 $e |- ( ph -> A = C ) $.
|
|
3brtr3d.3 $e |- ( ph -> B = D ) $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 18-Oct-1999.) $)
|
|
3brtr3d $p |- ( ph -> C R D ) $=
|
|
( wbr breq12d mpbid ) ABCFJDEFJGABDCEFHIKL $.
|
|
$}
|
|
|
|
${
|
|
3brtr4d.1 $e |- ( ph -> A R B ) $.
|
|
3brtr4d.2 $e |- ( ph -> C = A ) $.
|
|
3brtr4d.3 $e |- ( ph -> D = B ) $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 21-Feb-2005.) $)
|
|
3brtr4d $p |- ( ph -> C R D ) $=
|
|
( wbr breq12d mpbird ) ADEFJBCFJGADBECFHIKL $.
|
|
$}
|
|
|
|
${
|
|
3brtr3g.1 $e |- ( ph -> A R B ) $.
|
|
3brtr3g.2 $e |- A = C $.
|
|
3brtr3g.3 $e |- B = D $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 16-Jan-1997.) $)
|
|
3brtr3g $p |- ( ph -> C R D ) $=
|
|
( wbr breq12i sylib ) ABCFJDEFJGBDCEFHIKL $.
|
|
$}
|
|
|
|
${
|
|
3brtr4g.1 $e |- ( ph -> A R B ) $.
|
|
3brtr4g.2 $e |- C = A $.
|
|
3brtr4g.3 $e |- D = B $.
|
|
$( Substitution of equality into both sides of a binary relation.
|
|
(Contributed by NM, 16-Jan-1997.) $)
|
|
3brtr4g $p |- ( ph -> C R D ) $=
|
|
( wbr breq12i sylibr ) ABCFJDEFJGDBECFHIKL $.
|
|
$}
|
|
|
|
${
|
|
syl5eqbr.1 $e |- A = B $.
|
|
syl5eqbr.2 $e |- ( ph -> B R C ) $.
|
|
$( B chained equality inference for a binary relation. (Contributed by NM,
|
|
11-Oct-1999.) $)
|
|
syl5eqbr $p |- ( ph -> A R C ) $=
|
|
( eqid 3brtr4g ) ACDBDEGFDHI $.
|
|
$}
|
|
|
|
${
|
|
syl5eqbrr.1 $e |- B = A $.
|
|
syl5eqbrr.2 $e |- ( ph -> B R C ) $.
|
|
$( B chained equality inference for a binary relation. (Contributed by NM,
|
|
17-Sep-2004.) $)
|
|
syl5eqbrr $p |- ( ph -> A R C ) $=
|
|
( eqid 3brtr3g ) ACDBDEGFDHI $.
|
|
$}
|
|
|
|
${
|
|
syl5breq.1 $e |- A R B $.
|
|
syl5breq.2 $e |- ( ph -> B = C ) $.
|
|
$( B chained equality inference for a binary relation. (Contributed by NM,
|
|
11-Oct-1999.) $)
|
|
syl5breq $p |- ( ph -> A R C ) $=
|
|
( wbr a1i breqtrd ) ABCDEBCEHAFIGJ $.
|
|
$}
|
|
|
|
${
|
|
syl5breqr.1 $e |- A R B $.
|
|
syl5breqr.2 $e |- ( ph -> C = B ) $.
|
|
$( B chained equality inference for a binary relation. (Contributed by NM,
|
|
24-Apr-2005.) $)
|
|
syl5breqr $p |- ( ph -> A R C ) $=
|
|
( eqcomd syl5breq ) ABCDEFADCGHI $.
|
|
$}
|
|
|
|
${
|
|
syl6eqbr.1 $e |- ( ph -> A = B ) $.
|
|
syl6eqbr.2 $e |- B R C $.
|
|
$( A chained equality inference for a binary relation. (Contributed by NM,
|
|
12-Oct-1999.) $)
|
|
syl6eqbr $p |- ( ph -> A R C ) $=
|
|
( wbr breq1d mpbiri ) ABDEHCDEHGABCDEFIJ $.
|
|
$}
|
|
|
|
${
|
|
syl6eqbrr.1 $e |- ( ph -> B = A ) $.
|
|
syl6eqbrr.2 $e |- B R C $.
|
|
$( A chained equality inference for a binary relation. (Contributed by NM,
|
|
4-Jan-2006.) $)
|
|
syl6eqbrr $p |- ( ph -> A R C ) $=
|
|
( eqcomd syl6eqbr ) ABCDEACBFHGI $.
|
|
$}
|
|
|
|
${
|
|
syl6breq.1 $e |- ( ph -> A R B ) $.
|
|
syl6breq.2 $e |- B = C $.
|
|
$( A chained equality inference for a binary relation. (Contributed by NM,
|
|
11-Oct-1999.) $)
|
|
syl6breq $p |- ( ph -> A R C ) $=
|
|
( eqid 3brtr3g ) ABCBDEFBHGI $.
|
|
$}
|
|
|
|
${
|
|
syl6breqr.1 $e |- ( ph -> A R B ) $.
|
|
syl6breqr.2 $e |- C = B $.
|
|
$( A chained equality inference for a binary relation. (Contributed by NM,
|
|
24-Apr-2005.) $)
|
|
syl6breqr $p |- ( ph -> A R C ) $=
|
|
( eqcomi syl6breq ) ABCDEFDCGHI $.
|
|
$}
|
|
|
|
${
|
|
ssbrd.1 $e |- ( ph -> A C_ B ) $.
|
|
$( Deduction from a subclass relationship of binary relations.
|
|
(Contributed by NM, 30-Apr-2004.) $)
|
|
ssbrd $p |- ( ph -> ( C A D -> C B D ) ) $=
|
|
( cop wcel wbr sseld df-br 3imtr4g ) ADEGZBHMCHDEBIDECIABCMFJDEBKDECKL $.
|
|
$}
|
|
|
|
${
|
|
ssbri.1 $e |- A C_ B $.
|
|
$( Inference from a subclass relationship of binary relations. (The proof
|
|
was shortened by Andrew Salmon, 9-Jul-2011.) (Contributed by NM,
|
|
28-Mar-2007.) (Revised by set.mm contributors, 9-Jul-2011.) $)
|
|
ssbri $p |- ( C A D -> C B D ) $=
|
|
( wss wbr wi ssid a1i ssbrd ax-mp ) AAFZCDAGCDBGHAIMABCDABFMEJKL $.
|
|
$}
|
|
|
|
${
|
|
nfbrd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfbrd.3 $e |- ( ph -> F/_ x R ) $.
|
|
nfbrd.4 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nfbr .
|
|
(Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro,
|
|
14-Oct-2016.) $)
|
|
nfbrd $p |- ( ph -> F/ x A R B ) $=
|
|
( wbr cop wcel df-br nfopd nfeld nfxfrd ) CDEICDJZEKABCDELABPEABCDFHMGNO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nfbr.1 $e |- F/_ x A $.
|
|
nfbr.2 $e |- F/_ x R $.
|
|
nfbr.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for binary relation. (Contributed by
|
|
NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.) $)
|
|
nfbr $p |- F/ x A R B $=
|
|
( wbr wnf wtru wnfc a1i nfbrd trud ) BCDHAIJABCDABKJELADKJFLACKJGLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y z A $. $d y z R $.
|
|
$( Relationship between a binary relation and a class abstraction.
|
|
(Contributed by Andrew Salmon, 8-Jul-2011.) $)
|
|
brab1 $p |- ( x R A <-> x e. { z | z R A } ) $=
|
|
( vy cv wbr wsbc cab wcel cvv wb vex breq1 sbcie2g ax-mp df-sbc bitr3i )
|
|
AFZCDGZBFZCDGZBSHZSUBBIJSKJUCTLAMUBEFZCDGTBESKUAUDCDNUDSCDNOPUBBSQR $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d y D $. $d y R $. $d x y $.
|
|
$( Move substitution in and out of a binary relation. (Contributed by NM,
|
|
13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
|
|
sbcbrg $p |- ( A e. D -> ( [. A / x ]. B R C <->
|
|
[_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) ) $=
|
|
( vy wbr wsb cv csb wsbc dfsbcq2 wceq csbeq1 breq123d nfcsb1v weq csbeq1a
|
|
nfbr sbie vtoclbg ) CDFHZAGIAGJZCKZAUDDKZAUDFKZHZUCABLABCKZABDKZABFKZHGBE
|
|
UCAGBMUDBNUEUIUFUJUGUKAUDBCOAUDBFOAUDBDOPUCUHAGAUEUFUGAUDCQAUDFQAUDDQTAGR
|
|
CUEDUFFUGAUDCSAUDFSAUDDSPUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x R $.
|
|
$( Move substitution in and out of a binary relation. (Contributed by NM,
|
|
13-Dec-2005.) $)
|
|
sbcbr12g $p |- ( A e. D ->
|
|
( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) ) $=
|
|
( wcel wbr wsbc csb sbcbrg csbconstg breqd bitrd ) BEGZCDFHABIABCJZABDJZA
|
|
BFJZHPQFHABCDEFKORFPQABFELMN $.
|
|
$}
|
|
|
|
${
|
|
$d x C $. $d x R $.
|
|
$( Move substitution in and out of a binary relation. (Contributed by NM,
|
|
13-Dec-2005.) $)
|
|
sbcbr1g $p |- ( A e. D ->
|
|
( [. A / x ]. B R C <-> [_ A / x ]_ B R C ) ) $=
|
|
( wcel wbr wsbc csb sbcbr12g csbconstg breq2d bitrd ) BEGZCDFHABIABCJZABD
|
|
JZFHPDFHABCDEFKOQDPFABDELMN $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x R $.
|
|
$( Move substitution in and out of a binary relation. (Contributed by NM,
|
|
13-Dec-2005.) $)
|
|
sbcbr2g $p |- ( A e. D ->
|
|
( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) ) $=
|
|
( wcel wbr wsbc csb sbcbr12g csbconstg breq1d bitrd ) BEGZCDFHABIABCJZABD
|
|
JZFHCQFHABCDEFKOPCQFABCELMN $.
|
|
$}
|
|
|
|
$( Binary relationship implies sethood of both parts. (Contributed by SF,
|
|
7-Jan-2015.) $)
|
|
brex $p |- ( A R B -> ( A e. _V /\ B e. _V ) ) $=
|
|
( cop wcel cvv wbr wa elex df-br opexb bicomi 3imtr4i ) ABDZCENFEZABCGAFEBF
|
|
EHZNCIABCJOPABKLM $.
|
|
|
|
$( Binary relationship implies sethood of domain. (Contributed by SF,
|
|
7-Jan-2018.) $)
|
|
brreldmex $p |- ( A R B -> A e. _V ) $=
|
|
( wbr cvv wcel brex simpld ) ABCDAEFBEFABCGH $.
|
|
|
|
$( Binary relationship implies sethood of range. (Contributed by SF,
|
|
7-Jan-2018.) $)
|
|
brrelrnex $p |- ( A R B -> B e. _V ) $=
|
|
( wbr cvv wcel brex simprd ) ABCDAEFBEFABCGH $.
|
|
|
|
$( The union of two binary relations. (Contributed by NM, 21-Dec-2008.) $)
|
|
brun $p |- ( A ( R u. S ) B <-> ( A R B \/ A S B ) ) $=
|
|
( cop cun wcel wo wbr elun df-br orbi12i 3bitr4i ) ABEZCDFZGNCGZNDGZHABOIAB
|
|
CIZABDIZHNCDJABOKRPSQABCKABDKLM $.
|
|
|
|
$( The intersection of two relations. (Contributed by FL, 7-Oct-2008.) $)
|
|
brin $p |- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) ) $=
|
|
( cop cin wcel wa wbr elin df-br anbi12i 3bitr4i ) ABEZCDFZGNCGZNDGZHABOIAB
|
|
CIZABDIZHNCDJABOKRPSQABCKABDKLM $.
|
|
|
|
$( The difference of two binary relations. (Contributed by Scott Fenton,
|
|
11-Apr-2011.) $)
|
|
brdif $p |- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) ) $=
|
|
( cop cdif wcel wn wa wbr eldif df-br notbii anbi12i 3bitr4i ) ABEZCDFZGPCG
|
|
ZPDGZHZIABQJABCJZABDJZHZIPCDKABQLUARUCTABCLUBSABDLMNO $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Ordered-pair class abstractions (cont.)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
$d x z $. $d y z $. $d ph z $.
|
|
$( The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61.
|
|
(The proof was shortened by Andrew Salmon, 25-Jul-2011.) (Contributed
|
|
by NM, 14-Apr-1995.) (Revised by set.mm contributors, 25-Jul-2011.) $)
|
|
opabid $p |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) $=
|
|
( vz cv cop wceq wa wex copab vex opex copsexg bicomd df-opab elab2 ) DEZ
|
|
BEZCEZFZGZAHCIBIZADTABCJRSBKCKLUAAUBABCQMNABCDOP $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d y z A $. $d z ph $.
|
|
$( Membership in a class abstraction of pairs. (Contributed by NM,
|
|
24-Mar-1998.) $)
|
|
elopab $p |- ( A e. { <. x , y >. | ph } <->
|
|
E. x E. y ( A = <. x , y >. /\ ph ) ) $=
|
|
( vz copab wcel cvv cv cop wceq wex elex vex eleq1 mpbiri adantr exlimivv
|
|
wa opex eqeq1 anbi1d 2exbidv df-opab elab2g pm5.21nii ) DABCFZGDHGZDBIZCI
|
|
ZJZKZASZCLBLZDUGMUMUHBCULUHAULUHUKHGUIUJBNCNTDUKHOPQREIZUKKZASZCLBLUNEDUG
|
|
HUODKZUQUMBCURUPULAUODUKUAUBUCABCEUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d w z A $. $d w x B $. $d w z ph $.
|
|
$( The law of concretion in terms of substitutions. (Contributed by NM,
|
|
30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.) $)
|
|
opelopabsb $p |- ( <. A , B >. e. { <. x , y >. | ph }
|
|
<-> [. A / x ]. [. B / y ]. ph ) $=
|
|
( vz vw cop wcel cvv wsbc sbcex cv wsb wceq opeq1 eleq1d dfsbcq2 bibi12d
|
|
wb copab wbr df-br brex sylbir wex spesbc exlimiv syl jca sbcbidv nfopab1
|
|
opeq2 nfel2 nfs1v nfbi weq sbequ12 nfopab2 opabid chvar vtocl2g pm5.21nii
|
|
wa ) DEHZABCUAZIZDJIZEJIZVDZACEKZBDKZVGDEVFUBVJDEVFUCDEVFUDUEVLVHVIVKBDLV
|
|
LVKBUFVIVKBDUGVKVIBACELUHUIUJFMZGMZHZVFIZACGNZBFNZTZDVNHZVFIZVQBDKZTVGVLT
|
|
FGDEJJVMDOZVPWAVRWBWCVOVTVFVMDVNPQVQBFDRSVNEOZWAVGWBVLWDVTVEVFVNEDUMQWDVQ
|
|
VKBDACGERUKSBMZVNHZVFIZVQTZVSBFVPVRBBVOVFABCULUNVQBFUOUPBFUQZWGVPVQVRWIWF
|
|
VOVFWEVMVNPQVQBFURSWECMZHZVFIZATWHCGWGVQCCWFVFABCUSUNACGUOUPCGUQZWLWGAVQW
|
|
MWKWFVFWJVNWEUMQACGURSABCUTVAVAVBVC $.
|
|
|
|
brabsb.1 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion in terms of substitutions. (Contributed by NM,
|
|
17-Mar-2008.) $)
|
|
brabsb $p |- ( A R B <-> [. A / x ]. [. B / y ]. ph ) $=
|
|
( wbr cop wcel copab wsbc df-br eleq2i opelopabsb 3bitri ) DEFHDEIZFJQABC
|
|
KZJACELBDLDEFMFRQGNABCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ch $.
|
|
$( Closed theorem form of ~ opelopab . (Contributed by NM,
|
|
19-Feb-2013.) $)
|
|
opelopabt $p |- ( ( A. x A. y ( x = A -> ( ph <-> ps ) )
|
|
/\ A. x A. y ( y = B -> ( ps <-> ch ) )
|
|
/\ ( A e. V /\ B e. W ) ) ->
|
|
( <. A , B >. e. { <. x , y >. | ph } <-> ch ) ) $=
|
|
( cop copab wcel cv wceq wa wex wb wi wal w3a elopab prth 2alimi copsex2t
|
|
19.26-2 bitr syl6 sylbir sylan 3impa syl5bb ) FGJZADEKLULDMZEMZJNAOEPDPZU
|
|
MFNZABQZRZESDSZUNGNZBCQZRZESDSZFHLGILOZTCADEULUAUSVCVDUOCQZUSVCOZUPUTOZAC
|
|
QZRZESDSZVDVEVFURVBOZESDSVJURVBDEUEVKVIDEVKVGUQVAOVHUPUQUTVAUBABCUFUGUCUH
|
|
ACDEFGHIUDUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
opelopabga.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by
|
|
Mario Carneiro, 19-Dec-2013.) $)
|
|
opelopabga $p |- ( ( A e. V /\ B e. W ) ->
|
|
( <. A , B >. e. { <. x , y >. | ph } <-> ps ) ) $=
|
|
( cop copab wcel cv wceq wa wex elopab copsex2g syl5bb ) EFJZACDKLTCMDMJN
|
|
AODPCPEGLFHLOBACDTQABCDEFGHIRS $.
|
|
|
|
${
|
|
brabga.2 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion for a binary relation. (Contributed by Mario
|
|
Carneiro, 19-Dec-2013.) $)
|
|
brabga $p |- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) ) $=
|
|
( wbr cop copab wcel wa df-br eleq2i bitri opelopabga syl5bb ) EFGLZEFM
|
|
ZACDNZOZEHOFIOPBUBUCGOUEEFGQGUDUCKRSABCDEFHIJTUA $.
|
|
$}
|
|
|
|
$d x y C $. $d x y D $.
|
|
$( Ordered pair membership in an ordered pair class abstraction.
|
|
(Contributed by Mario Carneiro, 19-Dec-2013.) $)
|
|
opelopab2a $p |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e.
|
|
{ <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) ) $=
|
|
( wcel wa cop cv copab wceq eleq1 bi2anan9 anbi12d opelopabga bianabs ) E
|
|
GJZFHJZKZEFLCMZGJZDMZHJZKZAKZCDNJBUIUCBKCDEFGHUDEOZUFFOZKUHUCABUJUEUAUKUG
|
|
UBUDEGPUFFHPQIRST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ps $.
|
|
opelopaba.1 $e |- A e. _V $.
|
|
opelopaba.2 $e |- B e. _V $.
|
|
opelopaba.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by
|
|
Mario Carneiro, 19-Dec-2013.) $)
|
|
opelopaba $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) $=
|
|
( cvv wcel cop copab wb opelopabga mp2an ) EJKFJKEFLACDMKBNGHABCDEFJJIOP
|
|
$.
|
|
|
|
${
|
|
braba.4 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion for a binary relation. (Contributed by NM,
|
|
19-Dec-2013.) $)
|
|
braba $p |- ( A R B <-> ps ) $=
|
|
( cvv wcel wbr wb brabga mp2an ) ELMFLMEFGNBOHIABCDEFGLLJKPQ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ch $.
|
|
opelopabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
opelopabg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by
|
|
NM, 28-May-1995.) (Revised by set.mm contributors, 19-Dec-2013.) $)
|
|
opelopabg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( <. A , B >. e. { <. x , y >. | ph } <-> ch ) ) $=
|
|
( cv wceq sylan9bb opelopabga ) ACDEFGHIDLFMABELGMCJKNO $.
|
|
|
|
${
|
|
brabg.5 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion for a binary relation. (Contributed by NM,
|
|
16-Aug-1999.) (Revised by set.mm contributors, 19-Dec-2013.) $)
|
|
brabg $p |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) ) $=
|
|
( cv wceq sylan9bb brabga ) ACDEFGJHIDNFOABENGOCKLPMQ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y ch $.
|
|
opelopab2.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
opelopab2.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( Ordered pair membership in an ordered pair class abstraction.
|
|
(Contributed by NM, 14-Oct-2007.) (Revised by set.mm contributors,
|
|
19-Dec-2013.) $)
|
|
opelopab2 $p |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e.
|
|
{ <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ch ) ) $=
|
|
( cv wceq sylan9bb opelopab2a ) ACDEFGHIDLFMABELGMCJKNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ch $.
|
|
opelopab.1 $e |- A e. _V $.
|
|
opelopab.2 $e |- B e. _V $.
|
|
opelopab.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
opelopab.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by
|
|
NM, 16-May-1995.) $)
|
|
opelopab $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) $=
|
|
( cvv wcel cop copab wb opelopabg mp2an ) FLMGLMFGNADEOMCPHIABCDEFGLLJKQR
|
|
$.
|
|
|
|
${
|
|
brab.5 $e |- R = { <. x , y >. | ph } $.
|
|
$( The law of concretion for a binary relation. (Contributed by NM,
|
|
16-Aug-1999.) $)
|
|
brab $p |- ( A R B <-> ch ) $=
|
|
( cvv wcel wbr wb brabg mp2an ) FNOGNOFGHPCQIJABCDEFGNNHKLMRS $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
opelopabaf.x $e |- F/ x ps $.
|
|
opelopabaf.y $e |- F/ y ps $.
|
|
opelopabaf.1 $e |- A e. _V $.
|
|
opelopabaf.2 $e |- B e. _V $.
|
|
opelopabaf.3 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of
|
|
~ opelopab uses bound-variable hypotheses in place of distinct variable
|
|
conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof
|
|
shortened by Mario Carneiro, 18-Nov-2016.) $)
|
|
opelopabaf $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) $=
|
|
( cop copab wcel wsbc opelopabsb cvv wb nfv sbc2iegf mp2an bitri ) EFLACD
|
|
MNADFOCEOZBACDEFPEQNFQNZUCBRIJABCDEFQQGHUDCSKTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
opelopabf.x $e |- F/ x ps $.
|
|
opelopabf.y $e |- F/ y ch $.
|
|
opelopabf.1 $e |- A e. _V $.
|
|
opelopabf.2 $e |- B e. _V $.
|
|
opelopabf.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
opelopabf.4 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
$( The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of
|
|
~ opelopab uses bound-variable hypotheses in place of distinct variable
|
|
conditions." (Contributed by NM, 19-Dec-2008.) $)
|
|
opelopabf $p |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) $=
|
|
( cop wcel wsbc cvv wb sbciegf ax-mp copab opelopabsb nfcv cv wceq 3bitri
|
|
nfsbc sbcbidv ) FGNADEUAOAEGPZDFPZBEGPZCADEFGUBFQOUJUKRJUIUKDFQBDEGDGUCHU
|
|
GDUDFUEABEGLUHSTGQOUKCRKBCEGQIMSTUF $.
|
|
$}
|
|
|
|
${
|
|
$d ph z $. $d ps z $. $d x z $. $d y z $.
|
|
$( Equivalence of ordered pair abstraction subclass and implication.
|
|
(Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro,
|
|
19-May-2013.) $)
|
|
ssopab2 $p |- ( A. x A. y ( ph -> ps ) ->
|
|
{ <. x , y >. | ph } C_ { <. x , y >. | ps } ) $=
|
|
( vz wi wal cv cop wceq wa wex cab copab nfa1 sp anim2d eximd sps df-opab
|
|
ss2abdv 3sstr4g ) ABFZDGZCGZEHCHDHIJZAKZDLZCLZEMUFBKZDLZCLZEMACDNBCDNUEUI
|
|
ULEUEUHUKCUDCOUDUHUKFCUDUGUJDUCDOUDABUFUCDPQRSRUAACDETBCDETUB $.
|
|
$}
|
|
|
|
$( Equivalence of ordered pair abstraction subclass and implication.
|
|
(Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro,
|
|
18-Nov-2016.) $)
|
|
ssopab2b $p |- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } <->
|
|
A. x A. y ( ph -> ps ) ) $=
|
|
( copab wss wi wal nfopab1 nfss nfopab2 cop wcel ssel opabid 3imtr3g alrimi
|
|
cv ssopab2 impbii ) ACDEZBCDEZFZABGZDHZCHUCUECCUAUBACDIBCDIJUCUDDDUAUBACDKB
|
|
CDKJUCCRDRLZUAMUFUBMABUAUBUFNACDOBCDOPQQABCDST $.
|
|
|
|
${
|
|
ssopab2i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference of ordered pair abstraction subclass from implication.
|
|
(Contributed by NM, 5-Apr-1995.) $)
|
|
ssopab2i $p |- { <. x , y >. | ph } C_ { <. x , y >. | ps } $=
|
|
( wi wal copab wss ssopab2 ax-gen mpg ) ABFZDGACDHBCDHICABCDJMDEKL $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $. $d y ph $.
|
|
ssopab2dv.1 $e |- ( ph -> ( ps -> ch ) ) $.
|
|
$( Inference of ordered pair abstraction subclass from implication.
|
|
(Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro,
|
|
24-Jun-2014.) $)
|
|
ssopab2dv $p |- ( ph -> { <. x , y >. | ps } C_ { <. x , y >. | ch } ) $=
|
|
( wi wal copab wss alrimivv ssopab2 syl ) ABCGZEHDHBDEICDEIJANDEFKBCDELM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d z ph $. $d z x $. $d z y $.
|
|
$( Non-empty ordered pair class abstraction. (Contributed by NM,
|
|
10-Oct-2007.) $)
|
|
opabn0 $p |- ( { <. x , y >. | ph } =/= (/) <-> E. x E. y ph ) $=
|
|
( vz copab c0 wne cv wcel wex cop wceq wa n0 elopab exbii exrot3 vex opex
|
|
isseti 19.41v mpbiran 2exbii bitri 3bitri ) ABCEZFGDHZUFIZDJUGBHZCHZKZLZA
|
|
MZCJBJZDJZACJBJZDUFNUHUNDABCUGOPUOUMDJZCJBJUPUMDBCQUQABCUQULDJADUKUIUJBRC
|
|
RSTULADUAUBUCUDUE $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Set construction functions
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c 1st $. $( First member of an ordered pair $)
|
|
$c Swap $.
|
|
$c _S $.
|
|
$c o. $.
|
|
$c " $.
|
|
$c SI $.
|
|
|
|
$( Extend the definition of a class to include the first member an ordered
|
|
pair function. $)
|
|
c1st $a class 1st $.
|
|
|
|
$( Extend the definition of a class to include the swap function. $)
|
|
cswap $a class Swap $.
|
|
|
|
$( Extend the definition of a class to include the subset relationship. $)
|
|
csset $a class _S $.
|
|
|
|
$( Extend the definition of a class to include the singleton image. $)
|
|
csi $a class SI A $.
|
|
|
|
$( Extend the definition of a class to include the composition of two
|
|
classes. $)
|
|
ccom $a class ( A o. B ) $.
|
|
|
|
$( Extend the definition of a class to include the image of one class under
|
|
another. $)
|
|
cima $a class ( A " B ) $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Define a function that extracts the first member, or abscissa, of an
|
|
ordered pair. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-1st $a |- 1st = { <. x , y >. | E. z x = <. y , z >. } $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( Define a function that swaps the two elements of an ordered pair.
|
|
(Contributed by SF, 5-Jan-2015.) $)
|
|
df-swap $a |- Swap = { <. x , y >. |
|
|
E. z E. w ( x = <. z , w >. /\ y = <. w , z >. ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define a relationship that holds between subsets. (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
df-sset $a |- _S = { <. x , y >. | x C_ y } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
$( Define the composition of two classes. (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
df-co $a |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Define the image of one class under another. (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
df-ima $a |- ( A " B ) = { x | E. y e. B y A x } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $.
|
|
$( Define the singleton image of a class. (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
df-si $a |- SI A = { <. x , y >. |
|
|
E. z E. w ( x = { z } /\ y = { w } /\ z A w ) } $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Membership in ` 1st ` . (Contributed by SF, 5-Jan-2015.) $)
|
|
el1st $p |- ( A e. 1st <-> E. x E. y A = <. <. x , y >. , x >. ) $=
|
|
( vz c1st wcel cv cop wceq wex wa copab df-1st eleq2i bitri excom exancom
|
|
elopab vex exbii opex opeq1 eqeq2d ceqsexv exdistr 3bitr3ri ) CEFZCDGZAGZ
|
|
HZIZUHUIBGZHZIZBJZKZAJDJZCUMUIHZIZBJZAJZUGCUODALZFUQEVBCDABMNUODACROUQUPD
|
|
JZAJVAUPDAPVCUTAUKUNKZDJZBJVDBJDJUTVCVDBDPVEUSBVEUNUKKDJUSUKUNDQUKUSDUMUI
|
|
ULASBSUAUNUJURCUHUMUIUBUCUDOTUKUNDBUEUFTOO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w $. $d B x y z w $. $d C x y z w $.
|
|
$( The binary relationship over the ` 1st ` function. (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
br1stg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( <. A , B >. 1st C <-> A = C ) ) $=
|
|
( vz vw vx vy cv cop c1st wbr wceq wb breq1d eqeq1 wex wa bitri opeq1 weq
|
|
bibi12d opeq2 bibi1d wcel df-br el1st w3a eqcom opth anbi1i df-3an bitr4i
|
|
2exbii vex biidd ceqsex2v 3bitri vtocl2g ) FJZGJZKZCLMZVACNZOAVBKZCLMZACN
|
|
ZOABKZCLMZVHOFGABDEVAANZVDVGVEVHVKVCVFCLVAAVBUAPVAACQUCVBBNZVGVJVHVLVFVIC
|
|
LVBBAUDPUEVDVCCKZLUFVMHJZIJZKZVNKZNZIRHRZVEVCCLUGHIVMUHVSHFUBZIGUBZVNCNZU
|
|
IZIRHRVEVRWCHIVRVQVMNZWCVMVQUJWDVPVCNZWBSZWCVPVNVCCUKWFVTWASZWBSWCWEWGWBV
|
|
NVOVAVBUKULVTWAWBUMUNTTUOWBVEVEHIVAVBFUPGUPVNVACQWAVEUQURTUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z t $. $d B x y z t $.
|
|
setconslem1.1 $e |- A e. _V $.
|
|
setconslem1.2 $e |- B e. _V $.
|
|
$( Lemma for the set construction theorems. (Contributed by SF,
|
|
6-Jan-2015.) $)
|
|
setconslem1 $p |- ( << { A } , B >> e.
|
|
( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( (
|
|
Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) <->
|
|
E. x e. B A = Phi x ) $=
|
|
( vt vz csn cv copk cssetk cins3k cpw1 cimak cvv wcel wa wex wceq exbii
|
|
cins2k cin c1c ccompl csik csymdif cdif cimagek cnnc cxpk cidk ccnvk cphi
|
|
cun ccomk wrex w3a wb snex opkelsikg mp2an excom 3anass eqcom sneqb bitri
|
|
vex anbi1i opkeq1 eleq1d anbi2d ceqsexv 3bitri 19.41v 3bitr2i opkelimagek
|
|
anass opkelcnvk dfphi2 eqeq2i 3bitr4i elssetk anbi12i opkelcok df-rex
|
|
ancom ) BHZFIZJKLZKUAZUBUCMMZNUDLWJUAWIUAKUEUELUNUFWKMMNUGWKNUHUIOUJUBUKU
|
|
IUDOUJUBUNZUHZULZUEZPZWHCJZKPZQZFRZAIZCPZBXAUMZSZQZARZWGCJKWOUOPXDACUPWTW
|
|
HXAHZSZBXAJZWNPZWRQZQZARZFRXLFRZARXFWSXMFWSXHXJQZARZWRQXOWRQZARXMWPXPWRWP
|
|
WGGIZHZSZXHXRXAJZWNPZUQZARGRZYCGRZARXPWGOPWHOPWPYDURBUSZFVGGAWGWHWNOOUTVA
|
|
YCGAVBYEXOAYEXRBSZXHYBQZQZGRXOYCYIGYCXTYHQYIXTXHYBVCXTYGYHXTXSWGSYGWGXSVD
|
|
XRBGVGVEVFVHVFTYHXOGBDYGYBXJXHYGYAXIWNXRBXAVIVJVKVLVFTVMVHXOWRAVNXQXLAXHX
|
|
JWRVQTVOTXLAFVBXNXEAXNXJXGCJZKPZQZXDXBQXEXKYLFXGXAUSXHWRYKXJXHWQYJKWHXGCV
|
|
IVJVKVLXJXDYKXBXABJWMPBWLXANZSXJXDXABWLAVGZDVPBXAWMDYNVRXCYMBXAVSVTWAXACY
|
|
NEWBWCXDXBWFVMTVOFWGCKWOYFEWDXDACWEWA $.
|
|
|
|
$( Lemma for the set construction theorems. (Contributed by SF,
|
|
6-Jan-2015.) $)
|
|
setconslem2 $p |- ( << { A } , B >> e. ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) <->
|
|
E. x e. B A = ( Phi x u. { 0c } ) ) $=
|
|
( vt vy vz csn copk cssetk cins3k cimak cvv wcel wceq wex 3bitr4i bitri
|
|
wa cv cins2k cin c1c cpw1 ccompl csik csymdif cdif cimagek cnnc cxpk cidk
|
|
cun ccnvk ccomk c0c wrex elpw121c anbi1i 19.41v bitr4i exbii df-rex excom
|
|
cphi snex opkeq1 eleq1d ceqsexv elin otkelins2k vex otkelins3k opksnelsik
|
|
elssetk wb wal wn elimak elsymdif wel opkelimagek opkelcnvk dfphi2 eqeq2i
|
|
opkex wo anbi12i ancom opkelcok phiex clel3 opkelxpk mpbiran2 sneqb elsnc
|
|
orbi12i elun bibi12i xchbinx exnal 3bitri con2bii dfcleq elcompl 3bitr4ri
|
|
) FUAZBIZCJZJZKUBZXLKLZXLUCUDUEUEZMUFLXLUBXMUBKUGUGLUNUHXNUEUEMUIXNMUJUKN
|
|
ULUCUMUKUFNULUCUNZUJZUOZKUPZUQIZIZNULZUNZLZUHZXNMZUFZUGZLZUCZOZFXNURZAUAZ
|
|
COZBYLVFZXSUNZPZTZAQZXJYIXNMOYPACURYKXHYLIZIZIZPZYJTZFQZAQZYRXHXNOZYJTZFQ
|
|
UUCAQZFQYKUUEUUGUUHFUUGUUBAQZYJTUUHUUFUUIYJAXHUSUTUUBYJAVAVBVCYJFXNVDUUCA
|
|
FVERUUDYQAUUDUUAXJJZYIOZUUJXLOZUUJYHOZTYQYJUUKFUUAYTVGUUBXKUUJYIXHUUAXJVH
|
|
VIVJUUJXLYHVKUULYMUUMYPUULYSCJKOYMYSXICKYLVGZBVGZEVLYLCAVMZEVPSUUMYSXIJYG
|
|
OYLBJZYFOZYPYSXICYGUUNUUOEVNYLBYFUUPDVOGUAZBOZUUSYOOZVQZGVRZUUQYEOZVSYPUU
|
|
RUVDUVCUVDXHUUSIZIZIZPZXHUUQJZYDOZTZFQZGQZUVBVSZGQUVCVSUVDUVJFXNURZUVMFYD
|
|
XNUUQYLBWGZVTUUFUVJTZFQUVKGQZFQUVOUVMUVQUVRFUVQUVHGQZUVJTUVRUUFUVSUVJGXHU
|
|
SUTUVHUVJGVAVBVCUVJFXNVDUVKGFVERSUVLUVNGUVLUVGUUQJZYDOZUVNUVJUWAFUVGUVFVG
|
|
UVHUVIUVTYDXHUVGUUQVHVIVJUWAUVTXLOZUVTYCOZVQUVBUVTXLYCWAUWBUUTUWCUVAUWBUV
|
|
EBJKOUUTUVEYLBKUUSVGZUUPDVLUUSBGVMZDVPSUWCUVEYLJZYBOZUVAUVEYLBYBUWDUUPDVN
|
|
UWFXROZUWFYAOZWHUUSYNOZUUSXSOZWHUWGUVAUWHUWJUWIUWKUVEHUAZJKOZUWLYLJXQOZTZ
|
|
HQUWLYNPZGHWBZTZHQUWHUWJUWOUWRHUWOUWQUWPTUWRUWMUWQUWNUWPUUSUWLUWEHVMZVPYL
|
|
UWLJXPOUWLXOYLMZPUWNUWPYLUWLXOUUPUWSWCUWLYLXPUWSUUPWDYNUWTUWLYLWEWFRWIUWQ
|
|
UWPWJSVCHUVEYLXQKUWDUUPWKHUUSYNYLUUPWLWMRUWIUVEXTOZUWKUWIUXAYLNOUUPUVEYLX
|
|
TNUWDUUPWNWOUVEXSPUUSUQPUXAUWKUUSUQUWEWPUVEXSUWDWQUUSUQUWEWQRSWRUWFXRYAWS
|
|
UUSYNXSWSRSWTXASVCUVBGXBXCXDGBYOXEUUQYEUVPXFXGXCWIXCVCSFYIXNXJXICWGVTYPAC
|
|
VDR $.
|
|
$}
|
|
|
|
${
|
|
$d A x y t $. $d B x y t $. $d C x y t $.
|
|
setconslem3.1 $e |- A e. _V $.
|
|
setconslem3.2 $e |- B e. _V $.
|
|
setconslem3.3 $e |- C e. _V $.
|
|
$( Lemma for set construction functions. Set up a mapping between
|
|
Kuratowski and Quine ordered pairs. (Contributed by SF, 7-Jan-2015.) $)
|
|
setconslem3 $p |- ( << { { A } } , << B , C >> >> e.
|
|
~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) <-> A = <. B , C >. ) $=
|
|
( vx vy vt wcel wceq csn cun copk cssetk cins3k cins2k cimak wex snex c0c
|
|
cv cphi wrex wo wb wal csik cin c1c cpw1 ccompl csymdif cdif cimagek cnnc
|
|
cvv cxpk ccnvk ccomk wn cop wa opkex elimak df-rex elpw141c anbi1i 19.41v
|
|
bitr4i exbii excom 3bitri opkeq1 eleq1d ceqsexv otkelins3k opksnelsik vex
|
|
elsymdif elssetk bitri otkelins2k setconslem1 setconslem2 orbi12i bibi12i
|
|
cidk elun xchbinx exnal con2bii eqop elcompl 3bitr4ri ) GUBZAJZWPHUBUCZKH
|
|
BUDZWPWRUALZMKHCUDZUEZUFZGUGZALZLZBCNZNZOUHZUHZPZOOPZOQZUIUJUKUKZRULPXMQX
|
|
LQXKMUMXNUKUKZRUNXNRUOUPUQURUIWHUPULUQURUIMUOUSZUHUTZPZXMXMXPOUTWTLUQURMP
|
|
UMXNRULUHPUIXNRZQZMZQZUMZXORZJZVAABCVBKXHYDULJYEXDYEIUBZWPLZLZLZLZLZKZYFX
|
|
HNZYCJZVCZISZGSZXCVAZGSXDVAYEYNIXOUDYFXOJZYNVCZISZYQIYCXOXHXFXGVDZVEYNIXO
|
|
VFUUAYOGSZISYQYTUUCIYTYLGSZYNVCUUCYSUUDYNGYFVGVHYLYNGVIVJVKYOGIVLVJVMYPYR
|
|
GYPYKXHNZYCJZYRYNUUFIYKYJTYLYMUUEYCYFYKXHVNVOVPUUFUUEXKJZUUEYBJZUFXCUUEXK
|
|
YBVTUUGWQUUHXBUUGYIXFNXJJYHXENXIJZWQYIXFXGXJYHTZXETZBCVDZVQYHXEXIYGTATVRU
|
|
UIYGANOJWQYGAOWPTZDVRWPAGVSZDWAWBVMUUHYIXGNZYAJUUOXRJZUUOXTJZUEXBYIXFXGYA
|
|
UUJUUKUULWCUUOXRXTWIUUPWSUUQXAUUPYGBNXQJWSYGBCXQUUMEFVQHWPBUUNEWDWBUUQYGC
|
|
NXSJXAYGBCXSUUMEFWCHWPCUUNFWEWBWFVMWGWJWBVKXCGWKVMWLGHABCWMXHYDUUBWNWO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z t $.
|
|
$( Lemma for set construction functions. Create a mapping between the two
|
|
types of ordered pair abstractions. (Contributed by SF, 7-Jan-2015.) $)
|
|
setconslem4 $p |- U.1 U.1 ( ( ( ( _V X._k _V ) X._k _V ) i^i
|
|
`'_k ~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k A ) =
|
|
{ <. x , y >. | << x , y >> e. A } $=
|
|
( vt cvv cxpk cssetk csik cins3k cins2k cin cpw1 cimak ccompl cun wcel wa
|
|
csn wex bitri vz c1c csymdif cdif cimagek cnnc cidk ccnvk ccomk c0c cuni1
|
|
copk copab cop wceq wrex snex elimak df-rex elin anbi2i an12 vex opkelxpk
|
|
cv mpbiran2 elvvk anbi1i 19.41vv bitr4i exbii exrot3 opkex opkeq1 anbi12d
|
|
eleq1 eleq1d ceqsexv setconslem3 ancom 2exbii eluni1 elopab 3bitr4i eqriv
|
|
opkelcnvk ) UAEEFZEFZGHHIZGGIZGJZKUBLLZMNIWKJWJJWIOUCWLLLZMUDWLMUEUFEFKUG
|
|
UFNEFKOUEUHZHUIIWKWKWNGUIUJRREFOIUCWLMNHIKWLMJOJUCWMMNZUHZKZCMZUKZUKZAVEZ
|
|
BVEZULZCPZABUMZUAVEZRZRZWRPZXFXAXBUNUOZXDQZBSASZXFWTPZXFXEPXIDVEZXHULZWQP
|
|
ZDCUPZXLDWQCXHXGUQZURXQXNXCUOZXNCPZXOWPPZQZQZDSZBSASZXLXQXTXPQZDSZYEXPDCU
|
|
SYGYCBSASZDSYEYFYHDYFXTXOWHPZYAQZQZYHXPYJXTXOWHWPUTVAYKYIYBQZYHXTYIYAVBYL
|
|
XSBSASZYBQYHYIYMYBYIXNWGPZYMYIYNXHEPXRXNXHWGEDVCXRVDVFABXNVGTVHXSYBABVIVJ
|
|
TTVKYCDABVLTTYDXKABYDXDXCXHULZWPPZQZXKYBYQDXCXAXBVMZXSXTXDYAYPXNXCCVPXSXO
|
|
YOWPXNXCXHVNVQVOVRYQXDXJQXKYPXJXDYPXHXCULWOPXJXCXHWOYRXRWFXFXAXBUAVCZAVCB
|
|
VCVSTVAXDXJVTTTWATTXMXGWSPXIXFWSYSWBXGWRXFUQWBTXDABXFWCWDWE $.
|
|
$}
|
|
|
|
$( Lemma for set construction theorems. The big expression in the middle of
|
|
~ setconslem4 forms a set. (Contributed by SF, 7-Jan-2015.) $)
|
|
setconslem5 $p |- ~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) e. _V $=
|
|
( cssetk csik cins3k cins2k cin cpw1 cimak ccompl cun csymdif cnnc cvv cxpk
|
|
ssetkex sikex ins3kex pw1ex imakex vvex xpkex cdif cimagek cidk ccnvk ccomk
|
|
c1c c0c csn addcexlem 1cex imagekex nncex inex idkex complex cnvkex ins2kex
|
|
unex cokex snex symdifex ) ABZBZCZAACZADZEUFFZFZGHCVFDVEDVDIJVHFZFZGUAZVHGZ
|
|
UBZKLMZEZUCKHZLMZEZIZUBZUDZBZUEZCZVFVFWAAUEZUGUHZUHZLMZIZCZJZVHGZHZBZCZEZVH
|
|
GZDZIZDZJZVJGXAVJVDWTVCVBANOOPWSWDWRWCAWBNWAVTVSVOVRVMVNVLVKVHUIVGUFUJQQZRU
|
|
KKLULSTUMUCVQUNVPLKULUOSTUMURUKUPZOUSPWQWPVHVFWOANUQZWNWMWLWKVHVFWJXDWIWEWH
|
|
WAAXCNUSWGLWFUTSTURPVAXBRUOOPUMXBRUQURUQVAVIVHXBQQRUO $.
|
|
|
|
${
|
|
$d A x y z w t $.
|
|
$( Lemma for the set construction functions. Invert the expression from
|
|
~ setconslem4 . (Contributed by SF, 7-Jan-2015.) $)
|
|
setconslem6 $p |- ( ( ( _V X._k ( _V X._k _V ) ) i^i
|
|
~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k ~P1 ~P1 A ) =
|
|
{ z | E. x E. y ( z = << x , y >> /\ <. x , y >. e. A ) } $=
|
|
( vt vw cv copk wcel wa wex cvv cxpk cssetk cins3k cins2k cpw1 cimak wrex
|
|
cin wceq cop csik c1c ccompl cun csymdif cdif cimagek cnnc cidk ccnvk c0c
|
|
ccomk csn elimak df-rex elpw12 anbi1i r19.41v bitr4i exbii rexcom4 opkeq1
|
|
vex snex eleq1d ceqsexv 3bitr2i bitri exancom opkelxpk mpbiran elvvk elin
|
|
rexbii 19.41vv 3bitr4i exrot3 anass 19.42v opkeq2 setconslem3 syl6bb opex
|
|
anbi1d exbidv eleq1 pm5.32i 3bitri 2exbii abbi2i ) CGZAGZBGZHZUAZWNWOUBZD
|
|
IZJZBKAKZCLLLMZMZNUCUCOZNNOZNPZTUDQQZRUEOXFPXEPXDUFUGXGQQZRUHXGRUIUJLMTUK
|
|
UJUELMTUFUIULZUCUNOXFXFXINUNUMUOUOLMUFOUGXGRUEUCOTXGRPUFPUGXHRUEZTZDQQZRZ
|
|
WMXMIEGZWMHZXKIZEXLSZFGZUOZUOZWMHZXKIZFDSZXAEXKXLWMCVEZUPXQXNXLIZXPJZEKZY
|
|
CXPEXLUQYGXNXTUAZXPJZFDSZEKYIEKZFDSYCYFYJEYFYHFDSZXPJYJYEYLXPFXNDURUSYHXP
|
|
FDUTVAVBYIFEDVCYKYBFDXPYBEXTXSVFZYHXOYAXKXNXTWMVDVGVHVPVIVJYCXRDIZYBJFKYB
|
|
YNJZFKZXAYBFDUQYNYBFVKYPWQYAXJIZJZYNJZFKZBKAKZXAYPYRBKAKZYNJZFKYSBKAKZFKU
|
|
UAYOUUCFYBUUBYNYAXCIZYQJWQBKAKZYQJYBUUBUUEUUFYQUUEWMXBIZUUFUUEXTLIUUGYMXT
|
|
WMLXBYMYDVLVMABWMVNVJUSYAXCXJVOWQYQABVQVRUSVBUUDUUCFYRYNABVQVBYSFABVSVIYT
|
|
WTABYTWQYQYNJZJZFKWQUUHFKZJWTYSUUIFWQYQYNVTVBWQUUHFWAWQUUJWSWQUUJXRWRUAZY
|
|
NJZFKWSWQUUHUULFWQYQUUKYNWQYQXTWPHZXJIUUKWQYAUUMXJWMWPXTWBVGXRWNWOFVEAVEZ
|
|
BVEZWCWDWFWGYNWSFWRWNWOUUNUUOWEXRWRDWHVHWDWIWJWKVJWJWJWL $.
|
|
$}
|
|
|
|
${
|
|
$d A t x y $. $d B t x y $. $d C t x y $.
|
|
setconslem7.1 $e |- A e. _V $.
|
|
setconslem7.2 $e |- B e. _V $.
|
|
setconslem7.3 $e |- C e. _V $.
|
|
$( Lemma for the set construction theorems. Reorganized version of
|
|
~ setconslem3 . (Contributed by SF, 4-Feb-2015.) $)
|
|
setconslem7 $p |- ( << { { C } } , << A , B >> >> e. ~ ( ( Ins2_k Ins3_k
|
|
_S_k (+)
|
|
( Ins2_k Ins2_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k
|
|
~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins3_k SI_k SI_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) <-> A = <. B , C >. ) $=
|
|
( vx vy vt wcel csn cun copk cssetk cins3k cimak csik wex bitri snex cphi
|
|
cv wceq wrex c0c wal cins2k cin c1c cpw1 ccompl csymdif cdif cimagek cnnc
|
|
wo wb cvv cxpk cidk ccnvk ccomk wn wa opkex elimak df-rex elpw141c anbi1i
|
|
cop 19.41v bitr4i exbii excom opkeq1 eleq1d ceqsexv otkelins2k otkelins3k
|
|
elsymdif elssetk setconslem1 opksnelsik setconslem2 orbi12i bibi12i exnal
|
|
vex elun notbii con2bii eqop elcompl 3bitr4ri ) GUBZAJZWOHUBUAZUCHBUDZWOW
|
|
QUEKZLUCHCUDZUPZUQZGUFZCKZKZABMZMZNOZUGZNXHNUGZUHUIUJUJZPUKOXJUGXINQQOLUL
|
|
XKUJUJZPUMXKPUNUOURUSUHUTUOUKURUSUHLUNVAZQVBZUGZUGZXJXJXMNVBWSKURUSLOULXK
|
|
PUKQOUHXKPZQZQZOZLZULZXLPZJZVCABCVJUCXGYCUKJYDXCYDXBVCZGRZXCVCYDIUBZWOKZK
|
|
ZKZKZKZUCZYGXGMZYBJZVDZIRZGRZYFYDYOIXLUDZYRIYBXLXGXEXFVEZVFYSYGXLJZYOVDZI
|
|
RZYRYOIXLVGUUCYPGRZIRYRUUBUUDIUUBYMGRZYOVDUUDUUAUUEYOGYGVHVIYMYOGVKVLVMYP
|
|
GIVNVLSSYQYEGYQYLXGMZYBJZYEYOUUGIYLYKTYMYNUUFYBYGYLXGVOVPVQUUGUUFXIJZUUFY
|
|
AJZUQZVCYEUUFXIYAVTUUJXBUUHWPUUIXAUUHYJXFMZXHJZWPYJXEXFXHYITZXDTZABVEZVRU
|
|
ULYHAMNJWPYHABNWOTZDEVSWOAGWHZDWASSUUIUUFXPJZUUFXTJZUPXAUUFXPXTWIUURWRUUS
|
|
WTUURUUKXOJZWRYJXEXFXOUUMUUNUUOVRUUTYHBMXNJWRYHABXNUUPDEVRHWOBUUQEWBSSUUS
|
|
YJXEMXSJZWTYJXEXFXSUUMUUNUUOVSUVAYIXDMXRJZWTYIXDXRYHTCTWCUVBYHCMXQJWTYHCX
|
|
QUUPFWCHWOCUUQFWDSSSWESWFWJSSVMSXBGWGSWKGHABCWLXGYCYTWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z t $.
|
|
$( Express the ` 1st ` function via the set construction functions.
|
|
(Contributed by SF, 4-Feb-2015.) $)
|
|
df1st2 $p |- 1st = U.1 U.1 ( ( ( ( _V X._k _V ) X._k _V ) i^i
|
|
`'_k ~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k (
|
|
~ ( ( Ins2_k Ins3_k _S_k (+)
|
|
( Ins2_k Ins2_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k
|
|
~ ( ( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins3_k SI_k SI_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) "_k ~P1 1c ) ) $=
|
|
( vx vy vz vt cssetk cins3k cins2k cin cpw1 cimak ccompl csik cun csymdif
|
|
cv cvv cxpk csn wcel wex copk c1c cdif cimagek cnnc ccnvk ccomk c0c copab
|
|
cidk cop wceq cuni1 c1st wa wrex opkex elimak df-rex anbi1i 19.41v bitr4i
|
|
elpw11c exbii excom bitri snex opkeq1 ceqsexv vex setconslem7 setconslem4
|
|
eleq1d opabbii df-1st 3eqtr4ri ) AOZBOZUAZEFZGZEVTEGZHUBIZIZJKFWBGWAELLFZ
|
|
MNWDIIZJUCWDJUDUEPQHUJUEKPQHMUDUFZLUGZGGWBWBWGEUGUHRRPQMFNWDJKLFHWDJZLLFM
|
|
NWFJKZWCJZSZABUIVQVRCOZUKULZCTZABUIPPQPQWEWHFWIGMGNWFJKUFHWKJUMUMUNWLWOAB
|
|
WLDOZWMRZRZULZWPVSUAZWJSZUOZDTZCTZWOWLXADWCUPZXDDWJWCVSVQVRUQURXEWPWCSZXA
|
|
UOZDTZXDXADWCUSXHXBCTZDTXDXGXIDXGWSCTZXAUOXIXFXJXACWPVCUTWSXACVAVBVDXBCDV
|
|
EVBVFVFXCWNCXCWRVSUAZWJSZWNXAXLDWRWQVGWSWTXKWJWPWRVSVHVMVIVQVRWMAVJBVJCVJ
|
|
VKVFVDVFVNABWKVLABCVOVP $.
|
|
$}
|
|
|
|
$( The ` 1st ` function is a set. (Contributed by SF, 6-Jan-2015.) $)
|
|
1stex $p |- 1st e. _V $=
|
|
( cvv cxpk cssetk csik cins3k cins2k cin cpw1 cimak ccompl cun csymdif cnnc
|
|
vvex xpkex inex ssetkex ins3kex ins2kex imakex c1st cdif cimagek cidk ccnvk
|
|
c1c ccomk c0c cuni1 df1st2 setconslem5 cnvkex addcexlem 1cex pw1ex imagekex
|
|
csn nncex idkex complex unex sikex cokex snex symdifex uni1ex eqeltri ) UAA
|
|
ABZABZCDDEZCCEZCFZGUFHZHZIJEVLFVKFZVJKLVNHZHZIUBZVNIZUCZMABZGZUDMJZABZGZKZU
|
|
CZUEZDZUGZEVLVLWHCUGZUHUQZUQZABZKZEZLZVNIZJZDZEZGZVNIZFKFLVQIJZUEZGZVOWJFZF
|
|
ZXCDZDZEZKZLZVQIZJZVMIZIZUIZUIAUJXRXQXFXPVIXEVHAAANNONOXDUKULPXOVMXNXMVQVOX
|
|
LVKCQRSXHXKXGWJCWIQWHWGWFWBWEVTWAVSVRVNUMVMUFUNUOZUOZTUPMAURNOPUDWDUSWCAMUR
|
|
UTNOPVAUPULZVBVCSSXJXIXCXBVNVLXACQSZWTWSWRWQVNVLWPYBWOWKWNWHCYAQVCWMAWLVDNO
|
|
VARVEXTTUTVBRPXTTVBVBRVAVEVPVNXTUOUOTUTXSTTVFVFVG $.
|
|
|
|
${
|
|
$d A x y z w $.
|
|
$( Membership in the ` Swap ` function. (Contributed by SF,
|
|
6-Jan-2015.) $)
|
|
elswap $p |- ( A e. Swap <->
|
|
E. x E. y A = <. <. x , y >. , <. y , x >. >. ) $=
|
|
( vz vw cswap wcel cv cop wceq wex copab df-swap eleq2i bitri 2exbii opex
|
|
wa vex eqeq2d elopab exrot4 19.42vv w3a df-3an ancom opeq1 opeq2 ceqsex2v
|
|
bitr2i 3bitr3i ) CFGZCDHZEHZIZJZUMAHZBHZIZJZUNURUQIZJZRZBKAKZRZEKDKZCUSVA
|
|
IZJZBKAKZULCVDDELZGVFFVJCDEABMNVDDECUAOUPVCRZBKAKZEKDKVKEKDKZBKAKVFVIVKDE
|
|
ABUBVLVEDEUPVCABUCPVMVHABVMUTVBUPUDZEKDKVHVKVNDEVNVCUPRVKUTVBUPUEVCUPUFUJ
|
|
PUPCUSUNIZJVHDEUSVAUQURASZBSZQURUQVQVPQUTUOVOCUMUSUNUGTVBVOVGCUNVAUSUHTUI
|
|
OPUKO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w t u v $.
|
|
$( Express the ` Swap ` function via set construction operators.
|
|
(Contributed by SF, 6-Jan-2015.) $)
|
|
dfswap2 $p |- Swap = ( ( ~ ( ( Ins2_k Ins2_k _S_k (+)
|
|
( ( ( Ins2_k
|
|
( Ins2_k Ins3_k ( _S_k o._k
|
|
SI_k
|
|
`'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u. ( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins3_k SI_k SI_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) i^i
|
|
Ins3_k SI_k SI_k SI_k SI_k SI_k
|
|
Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k
|
|
) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u. ( _I_k i^i ( ~ Nn X._k _V ) ) ) )
|
|
"_k
|
|
~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c ) u.
|
|
( ( Ins2_k
|
|
( Ins3_k SI_k SI_k
|
|
( _S_k o._k
|
|
SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k
|
|
_S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u. ( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k Ins3_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) i^i
|
|
Ins3_k SI_k SI_k SI_k SI_k SI_k
|
|
~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k
|
|
( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i Ins2_k _S_k )
|
|
"_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u. ( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k
|
|
_S_k ) u.
|
|
( { { 0c } } X._k _V ) ) ) "_k
|
|
~P1 ~P1 1c ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 ~P1 ~P1 1c ) ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c )
|
|
"_k ~P1 1c ) "_k _V ) $=
|
|
( vz vt vw vv vu cins3k cimak csik cun csn wceq wex copk wcel wrex bitr4i
|
|
wa 3bitri snex opksnelsik vx vy cssetk cins2k cin c1c cpw1 ccompl csymdif
|
|
cswap cdif cimagek cnnc cvv cxpk cidk ccnvk ccomk c0c cv cop opkex elimak
|
|
df-rex elpw11c anbi1i 19.41v exbii excom opkeq1 eleq1d ceqsexv wel wb wal
|
|
wn elpw141c elsymdif otkelins2k vex elssetk wo cphi elpw161c 3bitr4i elin
|
|
otkelins3k setconslem1 setconslem2 bitri orbi12i df-op eleq2i opkelimagek
|
|
elun cab abid dfphi2 eqeq2i anbi12i dfop2lem1 bibi12i notbii exnal dfcleq
|
|
con2bii elcompl 3bitr4ri bitr2i elswap elimakv eqriv ) UAUJUCUDZUDZUCUCFZ
|
|
XMUEUFUGZUGZGUHFXNXOUDUCHHFIUIXQUGUGZGUKXQGULUMUNUOUEUPUMUHUNUOUEIZULZUQZ
|
|
HURZFZUDZXMXMYAUCURUSJZJUNUOIFUIXQGUHZHZFUEXQGZHZHZFZIZUDZXTHZHZHZHZHZFZU
|
|
EZXRUGUGZGZYBHZHZFZYHFZUDZIZUDZYGHZHZHZHZFZUEZUUAGZIZUIZXRGZUHZXPGZUNGZUA
|
|
UTZUBUTZAUTZVAZUVEUVDVAZVAZKZALZUBLUVDUVCMZUVANZUBLUVCUJNUVCUVBNUVJUVLUBU
|
|
VLBUTZUVEJZJZKZUVMUVKMZUUTNZQZBLZALZUVJUVLUVRBXPOUVMXPNZUVRQZBLZUWABUUTXP
|
|
UVKUVDUVCVBZVCUVRBXPVDUWDUVSALZBLUWAUWCUWFBUWCUVPALZUVRQUWFUWBUWGUVRAUVMV
|
|
EVFUVPUVRAVGPVHUVSABVIPRUVTUVIAUVTUVOUVKMZUUTNZUVIUVRUWIBUVOUVNSZUVPUVQUW
|
|
HUUTUVMUVOUVKVJVKVLCUAVMZCUTZUVHNZVNZCVOZUWHUUSNZVPUVIUWIUWPUWOUWPUVMUWLJ
|
|
ZJZJZJZJZKZUVMUWHMZUURNZQZBLZCLZUWNVPZCLUWOVPUWPUXDBXROUVMXRNZUXDQZBLZUXG
|
|
BUURXRUWHUVOUVKVBZVCUXDBXRVDUXKUXECLZBLUXGUXJUXMBUXJUXBCLZUXDQUXMUXIUXNUX
|
|
DCUVMVQVFUXBUXDCVGPVHUXECBVIPRUXFUXHCUXFUXAUWHMZUURNZUXOXNNZUXOUUQNZVNZVP
|
|
UXHUXDUXPBUXAUWTSZUXBUXCUXOUURUVMUXAUWHVJVKVLUXOXNUUQVRUXSUWNUXQUWKUXRUWM
|
|
UXQUWSUVKMXMNUWQUVCMUCNUWKUWSUVOUVKXMUWRSZUWJUWEVSUWQUVDUVCUCUWLSZUBVTZUA
|
|
VTZVSUWLUVCCVTZUYDWARUXOUUBNZUXOUUPNZWBUWLDUTZWCZKZDUVFOZUWLUYIYEIKZDUVGO
|
|
ZWBZUXRUWMUYFUYKUYGUYMUYFUYHUVFNZUYJQZDLZUYKUYFUVMUXOMZYTNZBUUAOZUVMUYHJZ
|
|
JZJZJZJZJZJZKZUYSQZBLZDLZUYQBYTUUAUXOUXAUWHVBZVCUVMUUANZUYSQZBLVUIDLZBLUY
|
|
TVUKVUNVUOBVUNVUHDLZUYSQVUOVUMVUPUYSDUVMWDZVFVUHUYSDVGPVHUYSBUUAVDVUIDBVI
|
|
WEVUJUYPDVUJVUGUXOMZYTNZVURYMNZVURYSNZQUYPUYSVUSBVUGVUFSZVUHUYRVURYTUVMVU
|
|
GUXOVJZVKVLVURYMYSWFVUTUYOVVAUYJVUEUWHMZYLNZUYHEUTWCZKZEUVDOZUYHVVFYEIKZE
|
|
UVEOZWBZVUTUYOVVEVVDYDNZVVDYKNZWBVVKVVDYDYKWOVVLVVHVVMVVJVVLVUCUVKMZYCNVU
|
|
AUVDMZYBNVVHVUCUVOUVKYCVUBSZUWJUWEVSVUAUVDUVCYBUYHSZUYCUYDWGEUYHUVDDVTZUY
|
|
CWHRVVMVUCUVOMZYJNVUBUVNMZYINZVVJVUCUVOUVKYJVVPUWJUWEWGVUBUVNYIVUASZUVESZ
|
|
TVWAVUAUVEMZYHNVVJVUAUVEYHVVQAVTZTEUYHUVEVVRVWEWIWJRWKWJVUEUXAUWHYLVUDSZU
|
|
XTUXLVSUYOUYHVVHDWPZVVJDWPZIZNUYHVWGNZUYHVWHNZWBVVKUVFVWIUYHDEUVDUVEWLWMU
|
|
YHVWGVWHWOVWJVVHVWKVVJVVHDWQVVJDWQWKRWEVVAVUEUXAMZYRNVUDUWTMZYQNZUYJVUEUX
|
|
AUWHYRVWFUXTUXLWGVUDUWTYQVUCSZUWSSZTVWNVUCUWSMZYPNVUBUWRMZYONZUYJVUCUWSYP
|
|
VVPUYATVUBUWRYOVWBUWQSZTVWSVUAUWQMZYNNUYHUWLMZXTNZUYJVUAUWQYNVVQUYBTUYHUW
|
|
LXTVVRUYETVXCUWLXSUYHGZKUYJUYHUWLXSVVRUYEWNUYIVXDUWLUYHWRWSPRRRWTRVHRUYJD
|
|
UVFVDPVUHUYRUUONZQZBLZDLZUYHUVGNZUYLQZDLUYGUYMVXGVXJDVXGVURUUONZVURUUINZV
|
|
URUUNNZQVXJVXEVXKBVUGVVBVUHUYRVURUUOVVCVKVLVURUUIUUNWFVXLVXIVXMUYLVXLVVGE
|
|
UVEOZVVIEUVDOZWBZVXIVXLVVDUUHNVVDUUENZVVDUUGNZWBVXPVUEUXAUWHUUHVWFUXTUXLV
|
|
SVVDUUEUUGWOVXQVXNVXRVXOVXQVVSUUDNVVTUUCNZVXNVUCUVOUVKUUDVVPUWJUWEWGVUBUV
|
|
NUUCVWBVWCTVXSVWDYBNVXNVUAUVEYBVVQVWETEUYHUVEVVRVWEWHWJRVXRVVNUUFNVVOYHNV
|
|
XOVUCUVOUVKUUFVVPUWJUWEVSVUAUVDUVCYHVVQUYCUYDWGEUYHUVDVVRUYCWIRWKRVXIUYHV
|
|
XNDWPZVXODWPZIZNUYHVXTNZUYHVYANZWBVXPUVGVYBUYHDEUVEUVDWLWMUYHVXTVYAWOVYCV
|
|
XNVYDVXOVXNDWQVXODWQWKRPVXMVWLUUMNVWMUULNZUYLVUEUXAUWHUUMVWFUXTUXLWGVUDUW
|
|
TUULVWOVWPTVYEVWQUUKNVWRUUJNZUYLVUCUWSUUKVVPUYATVUBUWRUUJVWBVWTTVYFVXAYGN
|
|
VXBYFNUYLVUAUWQYGVVQUYBTUYHUWLYFVVRUYETDCXARRRWTRVHUYGVXEBUUAOZVXFDLZBLZV
|
|
XHBUUOUUAUXOVULVCVYGVUMVXEQZBLVYIVXEBUUAVDVYJVYHBVYJVUPVXEQVYHVUMVUPVXEVU
|
|
QVFVUHVXEDVGPVHWJVXFBDVIRUYLDUVGVDWEWKUXOUUBUUPWOUWMUWLUYKCWPZUYMCWPZIZNU
|
|
WLVYKNZUWLVYLNZWBUYNUVHVYMUWLCDUVFUVGWLWMUWLVYKVYLWOVYNUYKVYOUYMUYKCWQUYM
|
|
CWQWKRWEXBXCRVHUWNCXDRXFCUVCUVHXEUWHUUSUXLXGXHWJVHXIVHUBAUVCXJUBUVAUVCUYD
|
|
XKWEXL $.
|
|
$}
|
|
|
|
$( The ` Swap ` function is a set. (Contributed by SF, 6-Jan-2015.) $)
|
|
swapex $p |- Swap e. _V $=
|
|
( cssetk cins2k cins3k cin cpw1 cimak ccompl csik cun csymdif ins2kex pw1ex
|
|
cnnc cvv imakex vvex inex unex sikex ins3kex cswap c1c cdif cxpk cidk ccnvk
|
|
cimagek ccomk c0c dfswap2 ssetkex addcexlem 1cex imagekex nncex xpkex idkex
|
|
csn complex cnvkex cokex snex symdifex eqeltri ) UAABZBZAACZVEDUBEZEZFGCVFV
|
|
GBAHHCIJVIEZEZFUCZVIFZUGZMNUDZDZUEMGZNUDZDZIZUGZUFZHZUHZCZBZVEVEWBAUHZUIURZ
|
|
URZNUDZIZCZJZVIFZGZHZCZDZVIFZHZHZCZIZBZWAHZHZHZHZHZCZDZVKEZEZFZWDHZHZCZWSCZ
|
|
BZIZBZWPHZHZHZHZCZDZXMFZIZJZVKFZGZVHFZNFNUJYMNYLVHYKYJVKVFYIVEAUKKZKXNYHXKX
|
|
MXDXJXCWFXBWEWDAWCUKWBWAVTVPVSVNVOVMVLVIULVHUBUMLZLZOUNMNUOPUPQUEVRUQVQNMUO
|
|
USPUPQRUNZUTZSVAZTKXAWTWSWRVIVEWQYNWPWOWNWMVIVEWLYNWKWGWJWBAYRUKVAWINWHVBPU
|
|
PRTVCYPOUSSZTQYPOZSSTRKXIXHXGXFXEWAYQSSSSSTQXLVKVJVIYPLLZLLZOYGXMYAYFXTXQXS
|
|
XPXOWDYSSSTXRWSUUATKRKYEYDYCYBWPYTSSSSTQUUCORVCUUBOUSYOOPOVD $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Express the ` _S ` relationship via the set construction functors.
|
|
(Contributed by SF, 7-Jan-2015.) $)
|
|
dfsset2 $p |- _S = U.1 U.1 ( ( ( ( _V X._k _V ) X._k _V ) i^i
|
|
`'_k ~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k _S_k ) $=
|
|
( vx vy cv cssetk wcel copab cvv cxpk csik cins3k cins2k cin cimak ccompl
|
|
cpw1 cun csymdif cimagek cnnc ccnvk copk wss c1c cdif ccomk c0c csn cuni1
|
|
cidk csset wb vex opkelssetkg mp2an opabbii setconslem4 df-sset 3eqtr4ri
|
|
) ACZBCZUADEZABFUSUTUBZABFGGHGHDIIJZDDJZDKZLUCOOZMNJVEKVDKVCPQVFOOZMUDVFM
|
|
RSGHLUISNGHLPRTZIUEJVEVEVHDUEUFUGUGGHPJQVFMNIJLVFMKPKQVGMNTLDMUHUHUJVAVBA
|
|
BUSGEUTGEVAVBUKAULBULUSUTGGUMUNUOABDUPABUQUR $.
|
|
$}
|
|
|
|
$( The subset relationship is a set. (Contributed by SF, 6-Jan-2015.) $)
|
|
ssetex $p |- _S e. _V $=
|
|
( cvv cxpk cssetk csik cins3k cins2k cin cpw1 cimak ccompl cun csymdif cnnc
|
|
cimagek ccnvk ccomk csn cuni1 vvex xpkex csset c1c cdif dfsset2 setconslem5
|
|
cidk c0c cnvkex inex ssetkex imakex uni1ex eqeltri ) UAAABZABZCDDEZCCEZCFZG
|
|
UBHHZIJEURFUQFUPKLUSHHZIUCUSINMABGUFMJABGKNOZDPEURURVACPUGQQABKELUSIJDEGUSI
|
|
FKFLUTIJZOZGZCIZRZRAUDVFVEVDCUOVCUNAAASSTSTVBUEUHUIUJUKULULUM $.
|
|
|
|
${
|
|
$d A x y z t w $. $d B x y z t w $.
|
|
$( Express the image functor in terms of the set construction functions.
|
|
(Contributed by SF, 7-Jan-2015.) $)
|
|
dfima2 $p |- ( A " B ) =
|
|
( ( ( ( _V X._k ( _V X._k _V ) ) i^i
|
|
~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k ~P1 ~P1 A ) "_k B ) $=
|
|
( vx vy vw vt cv cvv cxpk cssetk csik cins3k cins2k cin cpw1 cimak ccompl
|
|
cun csymdif wcel cima wbr wrex cab c1c cdif cimagek cnnc cidk ccnvk ccomk
|
|
vz c0c csn df-ima copk vex elimak cop setconslem6 opeq1 eleq1d opkelopkab
|
|
weq opeq2 df-br bitr4i rexbii bitri abbi2i eqtr4i ) ABUACGZDGZAUBZCBUCZDU
|
|
DHHHIIJKKLZJJLZJMZNUEOOZPQLVRMVQMVPRSVSOOZPUFVSPUGUHHINUIUHQHINRUGUJZKUKL
|
|
VRVRWAJUKUMUNUNHIRLSVSPQKLNVSPMRMSVTPQNAOOPZBPZDCABUOVODWCVMWCTVLVMUPWBTZ
|
|
CBUCVOCWBBVMDUQZURWDVNCBWDVLVMUSZATZVNEGZFGZUSZATVLWIUSZATWGULEFWBVLVMEFU
|
|
LAUTECVDWJWKAWHVLWIVAVBFDVDWKWFAWIVMVLVEVBCUQWEVCVLVMAVFVGVHVIVJVK $.
|
|
$}
|
|
|
|
$( The image of a set under a set is a set. (Contributed by SF,
|
|
7-Jan-2015.) $)
|
|
imaexg $p |- ( ( A e. V /\ B e. W ) -> ( A " B ) e. _V ) $=
|
|
( wcel cvv cxpk cssetk csik cins3k cins2k cin cpw1 cimak ccompl cun csymdif
|
|
cimagek cnnc vvex wa cima c1c cdif cidk ccnvk ccomk c0c dfima2 pw1exg xpkex
|
|
csn setconslem5 inex imakexg mpan 3syl sylan syl5eqel ) ACEZBDEZUAABUBFFFGZ
|
|
GZHIIJZHHJZHKZLUCMMZNOJVFKVEKVDPQVGMMZNUDVGNRSFGLUESOFGLPRUFZIUGJVFVFVIHUGU
|
|
HULULFGPJQVGNOIJLVGNKPKQVHNOZLZAMZMZNZBNZFABUIUTVNFEZVAVOFEUTVLFEVMFEZVPACU
|
|
JVLFUJVKFEVQVPVCVJFVBTFFTTUKUKUMUNVKVMFFUOUPUQVNBFDUOURUS $.
|
|
|
|
${
|
|
imaex.1 $e |- A e. _V $.
|
|
imaex.2 $e |- B e. _V $.
|
|
$( The image of a set under a set is a set. (Contributed by SF,
|
|
7-Jan-2015.) $)
|
|
imaex $p |- ( A " B ) e. _V $=
|
|
( cvv wcel cima imaexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c x y z $. $d B a b c x y z $.
|
|
$( Express Quine composition via Kuratowski composition. (Contributed by
|
|
SF, 7-Jan-2015.) $)
|
|
dfco1 $p |- ( A o. B ) =
|
|
U.1 U.1 ( ( ( ( _V X._k _V ) X._k _V ) i^i
|
|
`'_k ~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k
|
|
( ( ( ( _V X._k ( _V X._k _V ) ) i^i
|
|
~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k ~P1 ~P1 A )
|
|
o._k
|
|
( ( ( _V X._k ( _V X._k _V ) ) i^i
|
|
~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k ~P1 ~P1 B ) ) ) $=
|
|
( vx vy vz va vb vc cv cvv cxpk cssetk csik cins3k cins2k cpw1 cimak wcel
|
|
cin cop copk c1c ccompl cun csymdif cdif cimagek cnnc ccnvk ccomk c0c csn
|
|
cidk copab wbr wa wex cuni1 ccom vex opkelcok setconslem6 weq opeq1 opeq2
|
|
eleq1d opkelopkab df-br bitr4i anbi12i exbii opabbii setconslem4 3eqtr4ri
|
|
bitri df-co ) CIZDIZUAJJJKZKLMMNZLLNZLOZSUBPPZQUCNWBOWAOVTUDUEWCPPZQUFWCQ
|
|
UGUHJKSUMUHUCJKSUDUGUIZMUJNWBWBWELUJUKULULJKUDNUEWCQUCMNSWCQOUDOUEWDQUCZS
|
|
ZAPPQZWGBPPQZUJZRZCDUNVQEIZBUOZWLVRAUOZUPZEUQZCDUNVSJKWFUISWJQURURABUSWKW
|
|
PCDWKVQWLUAWIRZWLVRUAWHRZUPZEUQWPEVQVRWHWICUTZDUTZVAWSWOEWQWMWRWNWQVQWLTZ
|
|
BRZWMFIZGIZTZBRVQXETZBRXCHFGWIVQWLFGHBVBFCVCXFXGBXDVQXEVDVFGEVCXGXBBXEWLV
|
|
QVEVFWTEUTZVGVQWLBVHVIWRWLVRTZARZWNXFARWLXETZARXJHFGWHWLVRFGHAVBFEVCXFXKA
|
|
XDWLXEVDVFGDVCXKXIAXEVRWLVEVFXHXAVGWLVRAVHVIVJVKVOVLCDWJVMCDEABVPVN $.
|
|
$}
|
|
|
|
$( The composition of two sets is a set. (Contributed by SF, 7-Jan-2015.) $)
|
|
coexg $p |- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V ) $=
|
|
( wcel cvv cxpk cssetk csik cins3k cins2k cin cpw1 cimak ccompl cun csymdif
|
|
ccomk vvex pw1exg ccom c1c cdif cimagek cnnc cidk ccnvk c0c csn cuni1 dfco1
|
|
wa xpkex setconslem5 cnvkex inex imakexg sylancr cokexg syl2an uni1exg 3syl
|
|
syl syl5eqel ) ACEZBDEZULZABUAFFGZFGZHIIJZHHJZHKZLUBMMZNOJVLKVKKVJPQVMMMZNU
|
|
CVMNUDUEFGLUFUEOFGLPUDUGZIRJVLVLVOHRUHUIUIFGPJQVMNOIJLVMNKPKQVNNOZUGZLZFVHG
|
|
ZVPLZAMZMZNZVTBMZMZNZRZNZUJZUJZFABUKVGWHFEZWIFEWJFEVGVRFEWGFEZWKVIVQVHFFFSS
|
|
UMZSUMVPUNUOUPVEWCFEZWFFEZWLVFVEVTFEZWBFEZWNVSVPFVHSWMUMUNUPZVEWAFEWQACTWAF
|
|
TVCVTWBFFUQURVFWPWEFEZWOWRVFWDFEWSBDTWDFTVCVTWEFFUQURWCWFFFUSUTVRWGFFUQURWH
|
|
FVAWIFVAVBVD $.
|
|
|
|
${
|
|
coex.1 $e |- A e. _V $.
|
|
coex.2 $e |- B e. _V $.
|
|
$( The composition of two sets is a set. (Contributed by SF,
|
|
7-Jan-2015.) $)
|
|
coex $p |- ( A o. B ) e. _V $=
|
|
( cvv wcel ccom coexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z w a b c $.
|
|
$( Express singleton image in terms of the Kuratowski singleton image.
|
|
(Contributed by SF, 7-Jan-2015.) $)
|
|
dfsi2 $p |- SI A =
|
|
U.1 U.1 ( ( ( ( _V X._k _V ) X._k _V ) i^i
|
|
`'_k ~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k
|
|
SI_k ( ( ( _V X._k ( _V X._k _V ) ) i^i
|
|
~ ( ( Ins3_k SI_k SI_k _S_k (+)
|
|
Ins2_k
|
|
( Ins3_k ( _S_k o._k SI_k `'_k Image_k ( ( Image_k ( ( Ins3_k ~ (
|
|
( Ins3_k _S_k i^i Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) ) u.
|
|
Ins2_k ( ( Ins2_k _S_k i^i
|
|
Ins3_k SI_k ~ ( ( Ins2_k _S_k (+)
|
|
Ins3_k ( ( `'_k Image_k ( ( Image_k ( ( Ins3_k ~ ( ( Ins3_k _S_k i^i
|
|
Ins2_k _S_k ) "_k ~P1 ~P1 1c ) \
|
|
( ( Ins2_k Ins2_k _S_k
|
|
(+)
|
|
( Ins2_k Ins3_k _S_k u.
|
|
Ins3_k SI_k SI_k _S_k ) )
|
|
"_k ~P1 ~P1 ~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) i^i ( Nn X._k _V ) ) u.
|
|
( _I_k i^i ( ~ Nn X._k _V ) ) ) o._k _S_k ) u. ( { { 0c } } X._k _V ) ) )
|
|
"_k
|
|
~P1 ~P1 1c ) )
|
|
"_k ~P1 ~P1 1c ) ) ) "_k
|
|
~P1 ~P1 ~P1 ~P1 1c ) ) "_k ~P1 ~P1 A ) ) $=
|
|
( vz vw vx vy va vb cv cvv cxpk cssetk csik cins3k cins2k cin cpw1 ccompl
|
|
cimak cun wcel vc copk c1c csymdif cdif cimagek cnnc cidk ccnvk ccomk c0c
|
|
csn copab wceq wbr w3a wex cuni1 csi wb vex opkelsikg cop setconslem6 weq
|
|
mp2an opeq1 eleq1d opeq2 opkelopkab df-br bitr4i 2exbii bitri setconslem4
|
|
3anbi3i opabbii df-si 3eqtr4ri ) BHZCHZUBIIIJZJKLLMZKKMZKNZOUCPPZRQMWENWD
|
|
NWCSUDWFPPZRUEWFRUFUGIJOUHUGQIJOSUFUIZLUJMWEWEWHKUJUKULULIJSMUDWFRQLMOWFR
|
|
NSNUDWGRQZOAPPRZLZTZBCUMVTDHZULUNZWAEHZULUNZWMWOAUOZUPZEUQDUQZBCUMWBIJWIU
|
|
IOWKRURURAUSWLWSBCWLWNWPWMWOUBWJTZUPZEUQDUQZWSVTITWAITWLXBUTBVACVADEVTWAW
|
|
JIIVBVFXAWRDEWTWQWNWPWTWMWOVCZATZWQFHZGHZVCZATWMXFVCZATXDUAFGWJWMWOFGUAAV
|
|
DFDVEXGXHAXEWMXFVGVHGEVEXHXCAXFWOWMVIVHDVAEVAVJWMWOAVKVLVPVMVNVQBCWKVOBCD
|
|
EAVRVS $.
|
|
$}
|
|
|
|
$( The singleton image of a set is a set. (Contributed by SF,
|
|
7-Jan-2015.) $)
|
|
siexg $p |- ( A e. V -> SI A e. _V ) $=
|
|
( wcel cvv cxpk cssetk csik cins3k cins2k cin cpw1 cimak ccompl cun csymdif
|
|
cimagek cnnc vvex xpkex 3syl csi c1c cdif cidk ccnvk ccomk c0c cuni1 pw1exg
|
|
csn dfsi2 setconslem5 inex imakexg mpan sikexg cnvkex uni1exg syl5eqel ) AB
|
|
CZAUADDEZDEZFGGHZFFHZFIZJUBKKZLMHVEIVDIVCNOVFKKZLUCVFLPQDEJUDQMDEJNPUEZGUFH
|
|
VEVEVHFUFUGUJUJDENHOVFLMGHJVFLINIOVGLMZUEZJZDVAEZVIJZAKZKZLZGZLZUHZUHZDAUKU
|
|
TVRDCZVSDCVTDCUTVPDCZVQDCZWAUTVNDCVODCZWBABUIVNDUIVMDCWDWBVLVIDVARDDRRSZSUL
|
|
UMVMVODDUNUOTVPDUPVKDCWCWAVBVJVADWERSVIULUQUMVKVQDDUNUOTVRDURVSDURTUS $.
|
|
|
|
${
|
|
siex.1 $e |- A e. _V $.
|
|
$( The singleton image of a set is a set. (Contributed by SF,
|
|
7-Jan-2015.) $)
|
|
siex $p |- SI A e. _V $=
|
|
( cvv wcel csi siexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by
|
|
SF, 19-Apr-2004.) $)
|
|
elima $p |- ( A e. ( B " C ) <-> E. x e. C x B A ) $=
|
|
( vy cima wcel cvv wbr wrex elex brex simprd rexlimivw wceq breq2 rexbidv
|
|
cv df-ima elab2g pm5.21nii ) BCDFZGBHGZARZBCIZADJZBUBKUEUCADUEUDHGUCUDBCL
|
|
MNUDERZCIZADJUFEBUBHUGBOUHUEADUGBUDCPQEACDSTUA $.
|
|
|
|
$( Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by
|
|
SF, 11-Aug-2004.) $)
|
|
elima2 $p |- ( A e. ( B " C ) <-> E. x ( x e. C /\ x B A ) ) $=
|
|
( cima wcel cv wbr wrex wa wex elima df-rex bitri ) BCDEFAGZBCHZADIODFPJA
|
|
KABCDLPADMN $.
|
|
|
|
$( Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by
|
|
SF, 14-Aug-1994.) $)
|
|
elima3 $p |- ( A e. ( B " C ) <-> E. x ( x e. C /\ <. x , A >. e. B ) ) $=
|
|
( cima wcel cv cop wrex wa wex wbr elima df-br rexbii bitri df-rex ) BCDE
|
|
FZAGZBHCFZADIZSDFTJAKRSBCLZADIUAABCDMUBTADSBCNOPTADQP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Binary relationship form of the subset relationship. (Contributed by
|
|
SF, 11-Feb-2015.) $)
|
|
brssetg $p |- ( ( A e. V /\ B e. W ) -> ( A _S B <-> A C_ B ) ) $=
|
|
( vx vy cv wss csset sseq1 sseq2 df-sset brabg ) EGZFGZHAOHABHEFABCDINAOJ
|
|
OBAKEFLM $.
|
|
$}
|
|
|
|
${
|
|
brsset.1 $e |- A e. _V $.
|
|
brsset.2 $e |- B e. _V $.
|
|
$( Binary relationship form of the subset relationship. (Contributed by
|
|
SF, 11-Feb-2015.) $)
|
|
brsset $p |- ( A _S B <-> A C_ B ) $=
|
|
( cvv wcel csset wbr wss wb brssetg mp2an ) AEFBEFABGHABIJCDABEEKL $.
|
|
$}
|
|
|
|
${
|
|
brssetsn.1 $e |- A e. _V $.
|
|
brssetsn.2 $e |- B e. _V $.
|
|
$( Set membership in terms of the subset relationship. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
brssetsn $p |- ( { A } _S B <-> A e. B ) $=
|
|
( csn csset wbr wss wcel snex brsset snss bitr4i ) AEZBFGNBHABINBAJDKABCL
|
|
M $.
|
|
|
|
$( Set membership in terms of the subset relationship. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
opelssetsn $p |- ( <. { A } , B >. e. _S <-> A e. B ) $=
|
|
( csn cop csset wcel wbr df-br brssetsn bitr3i ) AEZBFGHMBGIABHMBGJABCDKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A w $. $d A x $. $d A y $. $d A z $. $d B w $. $d B x $. $d B y $.
|
|
$d B z $. $d R w $. $d R x $. $d R y $. $d R z $. $d w x $. $d w y $.
|
|
$d w z $. $d x y $. $d x z $. $d y z $.
|
|
|
|
$( Binary relationship over a singleton image. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
brsi $p |- ( A SI R B <->
|
|
E. x E. y ( A = { x } /\ B = { y } /\ x R y ) ) $=
|
|
( vz vw wbr cvv wcel wa cv csn wceq w3a wex snex eleq1 eqeq1 2exbidv brex
|
|
csi pm3.2i bi2anan9 mpbiri 3adant3 exlimivv 3anbi1d df-si brabg pm5.21nii
|
|
3anbi2d ) CDEUBZHCIJZDIJZKZCALZMZNZDBLZMZNZUQUTEHZOZBPAPZCDUMUAVDUPABUSVB
|
|
UPVCUSVBKUPURIJZVAIJZKVFVGUQQUTQUCUSUNVFVBUOVGCURIRDVAIRUDUEUFUGFLZURNZGL
|
|
ZVANZVCOZBPAPUSVKVCOZBPAPVEFGCDIIUMVHCNZVLVMABVNVIUSVKVCVHCURSUHTVJDNZVMV
|
|
DABVOVKVBUSVCVJDVASULTFGABEUIUJUK $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Epsilon and identity relations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new constant symbols. $)
|
|
$c _E $. $( Letter E (for epsilon relation) $)
|
|
$c _I $. $( Letter I (for identity relation) $)
|
|
|
|
$( Extend class notation to include the epsilon relation. $)
|
|
cep $a class _E $.
|
|
|
|
$( Extend the definition of a class to include identity relation. $)
|
|
cid $a class _I $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the epsilon relation. Similar to Definition 6.22 of
|
|
[TakeutiZaring] p. 30. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-eprel $a |- _E = { <. x , y >. | x e. y } $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
epelc.1 $e |- B e. _V $.
|
|
$( The epsilon relation and the membership relation are the same.
|
|
(Contributed by NM, 13-Aug-1995.) $)
|
|
epelc $p |- ( A _E B <-> A e. B ) $=
|
|
( vx vy cep wbr cvv wcel brex simpld wb wel cv eleq1 eleq2 df-eprel brabg
|
|
elex mpan2 pm5.21nii ) ABFGZAHIZABIZUBUCBHIZABFJKABSUCUEUBUDLCDEMAENZIUDD
|
|
EABHHFDNAUFOUFBAPDEQRTUA $.
|
|
$}
|
|
|
|
$( The epsilon relation and the membership relation are the same.
|
|
(Contributed by NM, 13-Aug-1995.) $)
|
|
epel $p |- ( x _E y <-> x e. y ) $=
|
|
( cv vex epelc ) ACBCBDE $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the identity relation. Definition 9.15 of [Quine] p. 64.
|
|
(Contributed by SF, 5-Jan-2015.) $)
|
|
df-id $a |- _I = { <. x , y >. | x = y } $.
|
|
$}
|
|
|
|
${
|
|
$d w z x $. $d w z y $.
|
|
$( A stronger version of ~ df-id that doesn't require ` x ` and ` y ` to be
|
|
distinct. Ordinarily, we wouldn't use this as a definition, since
|
|
non-distinct dummy variables would make soundness verification more
|
|
difficult (as the proof here shows). The proof can be instructive in
|
|
showing how distinct variable requirements may be eliminated, a task
|
|
that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.)
|
|
(Revised by Mario Carneiro, 18-Nov-2016.) $)
|
|
dfid3 $p |- _I = { <. x , y >. | x = y } $=
|
|
( vz vw cv wceq copab cop wa wex cab wb exbii opeq2 equequ2 anbi12d nfnae
|
|
eqeq2d nfcvd nfeqd cid df-id wal ancom equcom bitri ceqsexv equid biantru
|
|
anbi1i vex 3bitri nfe1 19.9 bitr4i sps drex1 drex2 syl5bb wn nfcvf2 nfopd
|
|
nfand wi a1i cbvexd exbid pm2.61i abbii df-opab 3eqtr4i eqtri ) UAAEZCEZF
|
|
ZACGZVMBEZFZABGZACUBDEZVMVNHZFZVOIZCJZAJZDKVTVMVQHZFZVRIZBJZAJZDKVPVSWEWJ
|
|
DVRAUCZWEWJLWEVTVMVMHZFZVMVMFZIZAJZAJZWKWJWEWPWQWDWOAWDVNVMFZWBIZCJWMWOWC
|
|
WSCWCVOWBIWSWBVOUDVOWRWBACUEUJUFMWBWMCVMAUKWRWAWLVTVNVMVMNRUGWNWMAUHUIULM
|
|
WPAWOAUMUNUOWPWIABAWOWHABVRWOWHLAVRWMWGWNVRVRWLWFVTVMVQVMNRABAOPUPUQURUSW
|
|
KUTZWDWIAABAQWTWCWHCBABBQWTWBVOBWTBVTWAWTBVTSWTBVMVNABVAZWTBVNSZVBTWTBVMV
|
|
NXAXBTVCVNVQFZWCWHLVDWTXCWBWGVOVRXCWAWFVTVNVQVMNRCBAOPVEVFVGVHVIVOACDVJVR
|
|
ABDVJVKVL $.
|
|
$}
|
|
|
|
$( Alternate definition of the identity relation. (Contributed by NM,
|
|
15-Mar-2007.) $)
|
|
dfid2 $p |- _I = { <. x , x >. | x = x } $=
|
|
( dfid3 ) AAB $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Functions and relations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c X. $. $( Times symbol (cross product symbol) (read: 'cross') $)
|
|
$c `' $. $( Small elevated smiley (converse operation) $)
|
|
$c dom $. $( Domain $)
|
|
$c ran $. $( Range $)
|
|
$c |` $. $( Right hook (restriction symbol) $)
|
|
$c Fun $. $( Function predicate $)
|
|
$c Fn $. $( Function connective $)
|
|
$c : $. $( Colon $)
|
|
$c --> $. $( Domain-codomain connective $)
|
|
$c -1-1-> $. $( 'One-to-one' domain-codomain connective $)
|
|
$c -onto-> $. $( 'Onto' domain-codomain connective $)
|
|
$c -1-1-onto-> $. $( 'One-to-one' and 'onto' domain-codomain connective $)
|
|
$c ` $. $( Left apostrophe (function value symbol) $)
|
|
$c Isom $. $( Isomorphism $)
|
|
$c 2nd $. $( Second function. $)
|
|
|
|
$( Extend the definition of a class to include the cross product. $)
|
|
cxp $a class ( A X. B ) $.
|
|
|
|
$( Extend the definition of a class to include the converse of a class. $)
|
|
ccnv $a class `' A $.
|
|
|
|
$( Extend the definition of a class to include the domain of a class. $)
|
|
cdm $a class dom A $.
|
|
|
|
$( Extend the definition of a class to include the range of a class. $)
|
|
crn $a class ran A $.
|
|
|
|
$( Extend the definition of a class to include the restriction of a class.
|
|
(Read: The restriction of ` A ` to ` B ` .) $)
|
|
cres $a class ( A |` B ) $.
|
|
|
|
$( Extend the definition of a wff to include the function predicate. (Read:
|
|
` A ` is a function.) $)
|
|
wfun $a wff Fun A $.
|
|
|
|
$( Extend the definition of a wff to include the function predicate with a
|
|
domain. (Read: ` A ` is a function on ` B ` .) $)
|
|
wfn $a wff A Fn B $.
|
|
|
|
$( Extend the definition of a wff to include the function predicate with
|
|
domain and codomain. (Read: ` F ` maps ` A ` into ` B ` .) $)
|
|
wf $a wff F : A --> B $.
|
|
|
|
$( Extend the definition of a wff to include one-to-one functions. (Read:
|
|
` F ` maps ` A ` one-to-one into ` B ` .) The notation ("1-1" above the
|
|
arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27. $)
|
|
wf1 $a wff F : A -1-1-> B $.
|
|
|
|
$( Extend the definition of a wff to include onto functions. (Read: ` F `
|
|
maps ` A ` onto ` B ` .) The notation ("onto" below the arrow) is from
|
|
Definition 6.15(4) of [TakeutiZaring] p. 27. $)
|
|
wfo $a wff F : A -onto-> B $.
|
|
|
|
$( Extend the definition of a wff to include one-to-one onto functions.
|
|
(Read: ` F ` maps ` A ` one-to-one onto ` B ` .) The notation ("1-1"
|
|
above the arrow and "onto" below the arrow) is from Definition 6.15(6) of
|
|
[TakeutiZaring] p. 27. $)
|
|
wf1o $a wff F : A -1-1-onto-> B $.
|
|
|
|
$( Extend the definition of a class to include the value of a function.
|
|
(Read: The value of ` F ` at ` A ` , or " ` F ` of ` A ` .") $)
|
|
cfv $a class ( F ` A ) $.
|
|
|
|
$( Extend the definition of a wff to include the isomorphism property.
|
|
(Read: ` H ` is an ` R ` , ` S ` isomorphism of ` A ` onto ` B ` .) $)
|
|
wiso $a wff H Isom R , S ( A , B ) $.
|
|
|
|
$( Extend the definition of a class to include the second function. $)
|
|
c2nd $a class 2nd $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( Define the cross product of two classes. Definition 9.11 of [Quine]
|
|
p. 64. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-xp $a |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) } $.
|
|
|
|
$( Define the converse of a class. Definition 9.12 of [Quine] p. 64. We
|
|
use Quine's breve accent (smile) notation. Like Quine, we use it as a
|
|
prefix, which eliminates the need for parentheses. Many authors use the
|
|
postfix superscript "to the minus one." "Converse" is Quine's
|
|
terminology; some authors call it "inverse," especially when the
|
|
argument is a function. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-cnv $a |- `' A = { <. x , y >. | y A x } $.
|
|
|
|
$( Define the range of a class. The notation " ` ran ` " is used by
|
|
Enderton; other authors sometimes use script R or script W. (Contributed
|
|
by SF, 5-Jan-2015.) $)
|
|
df-rn $a |- ran A = ( A " _V ) $.
|
|
|
|
$( Define the domain of a class. The notation " ` dom ` " is used by
|
|
Enderton; other authors sometimes use script D. (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
df-dm $a |- dom A = ran `' A $.
|
|
|
|
$( Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring]
|
|
p. 24. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-res $a |- ( A |` B ) = ( A i^i ( B X. _V ) ) $.
|
|
|
|
$( Define a function. Definition 10.1 of [Quine] p. 65. For alternate
|
|
definitions, see ~ dffun2 , ~ dffun3 , ~ dffun4 , ~ dffun5 , ~ dffun6 ,
|
|
~ dffun7 , ~ dffun8 , and ~ dffun9 . (Contributed by SF,
|
|
5-Jan-2015.) (Revised by Scott Fenton, 14-Apr-2021.) $)
|
|
df-fun $a |- ( Fun A <-> ( A o. `' A ) C_ _I ) $.
|
|
|
|
$( Define a function with domain. Definition 6.15(1) of [TakeutiZaring]
|
|
p. 27. For alternate definitions, see ~ dffn2 , ~ dffn3 , ~ dffn4 , and
|
|
~ dffn5 . (Contributed by SF, 5-Jan-2015.) $)
|
|
df-fn $a |- ( A Fn B <-> ( Fun A /\ dom A = B ) ) $.
|
|
|
|
$( Define a function (mapping) with domain and codomain. Definition
|
|
6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see
|
|
~ dff2 , ~ dff3 , and ~ dff4 . (Contributed by SF, 5-Jan-2015.) $)
|
|
df-f $a |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) ) $.
|
|
|
|
$( Define a one-to-one function. For equivalent definitions see ~ dff12
|
|
and ~ dff13 . Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We
|
|
use their notation ("1-1" above the arrow). (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
df-f1 $a |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) ) $.
|
|
|
|
$( Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27.
|
|
We use their notation ("onto" under the arrow). For alternate
|
|
definitions, see ~ dffo2 , ~ dffo3 , ~ dffo4 , and ~ dffo5 .
|
|
(Contributed by SF, 5-Jan-2015.) $)
|
|
df-fo $a |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) ) $.
|
|
|
|
$( Define a one-to-one onto function. For equivalent definitions see
|
|
~ dff1o2 , ~ dff1o3 , ~ dff1o4 , and ~ dff1o5 . Compare Definition
|
|
6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above
|
|
the arrow and "onto" below the arrow). (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
df-f1o $a |- ( F : A -1-1-onto-> B <->
|
|
( F : A -1-1-> B /\ F : A -onto-> B ) ) $.
|
|
|
|
$( Define the value of a function. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-fv $a |- ( F ` A ) = ( iota x A F x ) $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$( Define the isomorphism predicate. We read this as " ` H ` is an ` R ` ,
|
|
` S ` isomorphism of ` A ` onto ` B ` ." Normally, ` R ` and ` S ` are
|
|
ordering relations on ` A ` and ` B ` respectively. Definition 6.28 of
|
|
[TakeutiZaring] p. 32, whose notation is the same as ours except that
|
|
` R ` and ` S ` are subscripts. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-iso $a |- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\
|
|
A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) ) $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Define the ` 2nd ` function. This function extracts the second member
|
|
of an ordered pair. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-2nd $a |- 2nd = { <. x , y >. | E. z x = <. z , y >. } $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $.
|
|
$( Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) $)
|
|
xpeq1 $p |- ( A = B -> ( A X. C ) = ( B X. C ) ) $=
|
|
( vx vy wceq cv wcel wa copab cxp eleq2 anbi1d opabbidv df-xp 3eqtr4g ) A
|
|
BFZDGZAHZEGCHZIZDEJRBHZTIZDEJACKBCKQUAUCDEQSUBTABRLMNDEACODEBCOP $.
|
|
|
|
$( Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) $)
|
|
xpeq2 $p |- ( A = B -> ( C X. A ) = ( C X. B ) ) $=
|
|
( vx vy wceq cv wcel wa copab cxp eleq2 anbi2d opabbidv df-xp 3eqtr4g ) A
|
|
BFZDGCHZEGZAHZIZDEJRSBHZIZDEJCAKCBKQUAUCDEQTUBRABSLMNDECAODECBOP $.
|
|
|
|
$( Membership in a cross product. Uses fewer axioms than ~ elxp .
|
|
(Contributed by NM, 4-Jul-1994.) $)
|
|
elxpi $p |- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\
|
|
( x e. B /\ y e. C ) ) ) $=
|
|
( vz cv cop wceq wcel wa wex cab cxp eqeq1 anbi1d 2exbidv elabg ibi copab
|
|
df-xp df-opab eqtri eleq2s ) CAGZBGZHZIZUEDJUFEJKZKZBLALZCFGZUGIZUIKZBLAL
|
|
ZFMZDENZCUPJUKUOUKFCUPULCIZUNUJABURUMUHUIULCUGOPQRSUQUIABTUPABDEUAUIABFUB
|
|
UCUD $.
|
|
|
|
$( Membership in a cross product. (Contributed by NM, 4-Jul-1994.) $)
|
|
elxp $p |- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\
|
|
( x e. B /\ y e. C ) ) ) $=
|
|
( cxp wcel cv wa copab cop wceq wex df-xp eleq2i elopab bitri ) CDEFZGCAH
|
|
ZDGBHZEGIZABJZGCSTKLUAIBMAMRUBCABDENOUAABCPQ $.
|
|
|
|
$( Membership in a cross product. (Contributed by NM, 23-Feb-2004.) $)
|
|
elxp2 $p |- ( A e. ( B X. C ) <-> E. x e. B E. y e. C A = <. x , y >. ) $=
|
|
( cv wcel cop wceq wrex wa wex cxp df-rex r19.42v an13 exbii 3bitr3i elxp
|
|
3bitr4ri ) AFZDGZCUABFZHIZBEJZKZALUDUBUCEGZKKZBLZALUEADJCDEMGUFUIAUBUDKZB
|
|
EJUGUJKZBLUFUIUJBENUBUDBEOUKUHBUGUBUDPQRQUEADNABCDEST $.
|
|
$}
|
|
|
|
$( Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) $)
|
|
xpeq12 $p |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) ) $=
|
|
( wceq cxp xpeq1 xpeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.
|
|
|
|
${
|
|
xpeq1i.1 $e |- A = B $.
|
|
$( Equality inference for cross product. (Contributed by NM,
|
|
21-Dec-2008.) $)
|
|
xpeq1i $p |- ( A X. C ) = ( B X. C ) $=
|
|
( wceq cxp xpeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for cross product. (Contributed by NM,
|
|
21-Dec-2008.) $)
|
|
xpeq2i $p |- ( C X. A ) = ( C X. B ) $=
|
|
( wceq cxp xpeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
xpeq12i.1 $e |- A = B $.
|
|
xpeq12i.2 $e |- C = D $.
|
|
$( Equality inference for cross product. (Contributed by FL,
|
|
31-Aug-2009.) $)
|
|
xpeq12i $p |- ( A X. C ) = ( B X. D ) $=
|
|
( wceq cxp xpeq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
xpeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for cross product. (Contributed by Jeff Madsen,
|
|
17-Jun-2010.) $)
|
|
xpeq1d $p |- ( ph -> ( A X. C ) = ( B X. C ) ) $=
|
|
( wceq cxp xpeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for cross product. (Contributed by Jeff Madsen,
|
|
17-Jun-2010.) $)
|
|
xpeq2d $p |- ( ph -> ( C X. A ) = ( C X. B ) ) $=
|
|
( wceq cxp xpeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
xpeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for cross product. (Contributed by NM,
|
|
8-Dec-2013.) $)
|
|
xpeq12d $p |- ( ph -> ( A X. C ) = ( B X. D ) ) $=
|
|
( wceq cxp xpeq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d x y z $.
|
|
nfxp.1 $e |- F/_ x A $.
|
|
nfxp.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for cross product. (Contributed by
|
|
NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfxp $p |- F/_ x ( A X. B ) $=
|
|
( vy vz cxp cv wcel wa copab df-xp nfcri nfan nfopab nfcxfr ) ABCHFIBJZGI
|
|
CJZKZFGLFGBCMTFGARSAAFBDNAGCENOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $.
|
|
$( Ordered pair membership in a cross product. (The proof was shortened by
|
|
Andrew Salmon, 12-Aug-2011.) (Contributed by NM, 15-Nov-1994.)
|
|
(Revised by set.mm contributors, 12-Aug-2011.) $)
|
|
opelxp $p |- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) ) $=
|
|
( vx vy cop cv wceq wcel wa wex cxp eqcom bitri anbi1i an4 2exbii df-clel
|
|
opth elxp anbi12i eeanv bitr4i 3bitr4i ) ABGZEHZFHZGZIZUGCJZUHDJZKZKZFLEL
|
|
UGAIZUKKZUHBIZULKZKZFLELZUFCDMJACJZBDJZKZUNUSEFUNUOUQKZUMKUSUJVDUMUJUIUFI
|
|
VDUFUINUGUHABTOPUOUQUKULQOREFUFCDUAVCUPELZURFLZKUTVAVEVBVFEACSFBDSUBUPURE
|
|
FUCUDUE $.
|
|
|
|
$( Binary relation on a cross product. (Contributed by NM,
|
|
22-Apr-2004.) $)
|
|
brxp $p |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) ) $=
|
|
( cxp wbr cop wcel wa df-br opelxp bitri ) ABCDEZFABGMHACHBDHIABMJABCDKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A w y z $. $d B w y z $. $d C w y z $. $d D w y z $. $d w x y z $.
|
|
$( Distribute proper substitution through the cross product of two
|
|
classes. (Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbxpg $p |- ( A e. D -> [_ A / x ]_ ( B X. C ) =
|
|
( [_ A / x ]_ B X. [_ A / x ]_ C ) ) $=
|
|
( vz vw vy wcel cv wa wex cab csb cxp wsbc sbcexg sbcang sbcel2g bitrd
|
|
wceq csbabg anbi12d exbidv abbidv eqtrd copab df-xp df-opab eqtri csbeq2i
|
|
cop sbcg 3eqtr4g ) BEIZABFJGJZHJZULUAZUPCIZUQDIZKZKZHLZGLZFMZNZURUPABCNZI
|
|
ZUQABDNZIZKZKZHLZGLZFMZABCDOZNVGVIOZUOVFVDABPZFMVOVDAFBEUBUOVRVNFUOVRVCAB
|
|
PZGLVNVCGABEQUOVSVMGUOVSVBABPZHLVMVBHABEQUOVTVLHUOVTURABPZVAABPZKVLURVAAB
|
|
ERUOWAURWBVKURABEUMUOWBUSABPZUTABPZKVKUSUTABERUOWCVHWDVJABUPCESABUQDESUCT
|
|
UCTUDTUDTUEUFABVPVEVPVAGHUGVEGHCDUHVAGHFUIUJUKVQVKGHUGVOGHVGVIUHVKGHFUIUJ
|
|
UN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d y z ph $. $d x ps $.
|
|
rabxp.1 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
|
|
$( Membership in a class builder restricted to a cross product.
|
|
(Contributed by NM, 20-Feb-2014.) $)
|
|
rabxp $p |- { x e. ( A X. B ) | ph }
|
|
= { <. y , z >. | ( y e. A /\ z e. B /\ ps ) } $=
|
|
( cv cxp wcel wa cab cop wceq w3a wex crab copab elxp anbi1i anass anbi2d
|
|
19.41vv df-3an syl6bbr pm5.32i bitri 2exbii 3bitr2i abbii df-opab 3eqtr4i
|
|
df-rab ) CIZFGJZKZALZCMUODIZEIZNOZUSFKZUTGKZBPZLZEQDQZCMACUPRVDDESURVFCUR
|
|
VAVBVCLZLZEQDQZALVHALZEQDQVFUQVIADEUOFGTUAVHADEUDVJVEDEVJVAVGALZLVEVAVGAU
|
|
BVAVKVDVAVKVGBLVDVAABVGHUCVBVCBUEUFUGUHUIUJUKACUPUNVDDECULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Representation of a constant function using ordered pairs. (Contributed
|
|
by NM, 12-Oct-1999.) $)
|
|
fconstopab $p |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y = B ) } $=
|
|
( csn cxp cv wcel wa copab wceq df-xp df-sn abeq2i anbi2i opabbii eqtri )
|
|
CDEZFAGCHZBGZRHZIZABJSTDKZIZABJABCRLUBUDABUAUCSUCBRBDMNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y z C $. $d x y z R $.
|
|
vtoclr.2 $e |- ( ( x R y /\ y R z ) -> x R z ) $.
|
|
$( Variable to class conversion of transitive relation. (Contributed by
|
|
NM, 9-Jun-1998.) $)
|
|
vtoclr $p |- ( ( A R B /\ B R C ) -> A R C ) $=
|
|
( cvv wcel wa wbr brex wi cv wceq breq1 imbi12d imbi2d breq2 simprd mpcom
|
|
anim12i anbi1d anbi12d imbi1d anbi2d vtoclg vtocl2g imp ) DIJEIJZKZFIJZKD
|
|
EGLZEFGLZKZDFGLZUNULUOUMDEGMUOUKUMEFGMUAUCULUMUPUQNZUMAOZBOZGLZUTFGLZKZUS
|
|
FGLZNZNUMDUTGLZVBKZUQNZNUMURNABDEIIUSDPZVEVHUMVIVCVGVDUQVIVAVFVBUSDUTGQUD
|
|
USDFGQRSUTEPZVHURUMVJVGUPUQVJVFUNVBUOUTEDGTUTEFGQUEUFSVAUTCOZGLZKZUSVKGLZ
|
|
NVECFIVKFPZVMVCVNVDVOVLVBVAVKFUTGTUGVKFUSGTRHUHUIUJUB $.
|
|
$}
|
|
|
|
|
|
${
|
|
$d w y z A $. $d w y z B $. $d w x y z C $.
|
|
$( Distributive law for cross product over indexed union. (Contributed by
|
|
set.mm contributors, 26-Apr-2014.) (Revised by Mario Carneiro,
|
|
27-Apr-2014.) $)
|
|
xpiundi $p |- ( C X. U_ x e. A B ) = U_ x e. A ( C X. B ) $=
|
|
( vz vw vy ciun cxp cv wrex wcel wa wex eliun exbii df-rex rexbii 3bitr4i
|
|
elxp2 cop wceq rexcom anbi1i rexcom4 r19.41v 3bitri eqriv ) EDABCHZIZABDC
|
|
IZHZEJZFJGJZUAUBZGUIKZFDKZUMUKLZABKZUMUJLUMULLUOGCKZABKZFDKUTFDKZABKUQUSU
|
|
TFADBUCUPVAFDUNUILZUOMZGNUNCLZABKZUOMZGNZUPVAVDVGGVCVFUOAUNBCOUDPUOGUIQVA
|
|
VEUOMZGNZABKVIABKZGNVHUTVJABUOGCQRVIAGBUEVKVGGVEUOABUFPUGSRURVBABFGUMDCTR
|
|
SFGUMDUITAUMBUKOSUH $.
|
|
|
|
$( Distributive law for cross product over indexed union. (Contributed by
|
|
set.mm contributors, 26-Apr-2014.) (Revised by Mario Carneiro,
|
|
27-Apr-2014.) $)
|
|
xpiundir $p |- ( U_ x e. A B X. C ) = U_ x e. A ( B X. C ) $=
|
|
( vz vy vw ciun cxp cv cop wrex wcel wa df-rex rexbii eliun elxp2 3bitr4i
|
|
wex wceq rexcom4 anbi1i r19.41v bitr4i exbii 3bitr4ri eqriv ) EABCHZDIZAB
|
|
CDIZHZEJZFJZGJKUAGDLZFUILZUMUKMZABLZUMUJMUMULMUNUIMZUONZFTZUOFCLZABLZUPUR
|
|
UNCMZUONZFTZABLVEABLZFTVCVAVEAFBUBVBVFABUOFCOPUTVGFUTVDABLZUONVGUSVHUOAUN
|
|
BCQUCVDUOABUDUEUFUGUOFUIOUQVBABFGUMCDRPSFGUMUIDRAUMBUKQSUH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Membership in a union of cross products when the second factor is
|
|
constant. (Contributed by Mario Carneiro, 29-Dec-2014.) $)
|
|
iunxpconst $p |- U_ x e. A ( { x } X. B ) = ( A X. B ) $=
|
|
( cv csn ciun cxp xpiundir iunid xpeq1i eqtr3i ) ABADEZFZCGABLCGFBCGABLCH
|
|
MBCABIJK $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d y z C $. $d x y z $.
|
|
$( Membership in a union of Cartesian products. (Contributed by Mario
|
|
Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.) $)
|
|
opeliunxp $p |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <->
|
|
( x e. A /\ C e. B ) ) $=
|
|
( vz vy cv csn cxp wcel cvv wa elex syl wex wceq weq eleq2d anbi12d bitri
|
|
cop ciun opexb simprbi adantl wsb csb wb vex opexg mpan wrex df-rex nfs1v
|
|
nfv nfcv nfcsb1v nfxp nfcri nfan sbequ12 sneq csbeq1a xpeq12d cbvex eleq1
|
|
anbi2d exbidv syl5bb df-iun elab2g opelxp anbi2i an12 equcom anbi1i exbii
|
|
elsn 3bitri sbequ12r equcoms eqcomd ceqsexv syl6bb pm5.21nii ) AGZDUAZABW
|
|
FHZCIZUBZJZDKJZWFBJZDCJZLZWKWGKJZWLWGWJMWPWFKJZWLWFDUCUDNWNWLWMDCMUEWLWKW
|
|
MAEUFZWGEGZHZAWSCUGZIZJZLZEOZWOWLWPWKXEUHWQWLWPAUIZWFDKKUJUKFGZWIJZABULZX
|
|
EFWGWJKXIWRXGXBJZLZEOZXGWGPZXEXIWMXHLZAOXLXHABUMXNXKAEXNEUOWRXJAWMAEUNAFX
|
|
BAWTXAAWTUPAWSCUQURUSUTAEQZWMWRXHXJWMAEVAXOWIXBXGXOWHWTCXAWFWSVBAWSCVCZVD
|
|
RSVETXMXKXDEXMXJXCWRXGWGXBVFVGVHVIAFBWIVJVKNXEEAQZWRDXAJZLZLZEOWOXDXTEXDW
|
|
RWFWTJZXRLZLYAXSLXTXCYBWRWFDWTXAVLVMWRYAXRVNYAXQXSYAXOXQAWSVRAEVOTVPVSVQX
|
|
SWOEWFXFXQWRWMXRWNWMEAVTXQXACDXQCXACXAPAEXPWAWBRSWCTWDWE $.
|
|
$}
|
|
${
|
|
$d y A $. $d y B $. $d x y C $. $d x y D $. $d x E $.
|
|
$( Membership in a union of Cartesian products. Analogue of ~ elxp for
|
|
nonconstant ` B ( x ) ` . (Contributed by Mario Carneiro,
|
|
29-Dec-2014.) $)
|
|
eliunxp $p |- ( C e. U_ x e. A ( { x } X. B ) <->
|
|
E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) ) $=
|
|
( cv csn cxp ciun wcel cop wceq wex wa elex pm4.71ri opeqexb anbi1i exbii
|
|
cvv bitri nfiu1 nfel2 19.41 19.41v eleq1 opeliunxp syl6bb pm5.32i 3bitr2i
|
|
bitr3i ) EACAFZGDHZIZJZEULBFZKZLZBMZAMZUONZUSUONZAMURULCJUPDJNZNZBMZAMUOE
|
|
TJZUONVAUOVFEUNOPVFUTUOABEQRUAUSUOAAEUNACUMUBUCUDVBVEAVBURUONZBMVEURUOBUE
|
|
VGVDBURUOVCURUOUQUNJVCEUQUNUFACDUPUGUHUISUKSUJ $.
|
|
|
|
$d x A $.
|
|
opeliunxp2.1 $e |- ( x = C -> B = E ) $.
|
|
$( Membership in a union of Cartesian products. (Contributed by Mario
|
|
Carneiro, 14-Feb-2015.) $)
|
|
opeliunxp2 $p |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <->
|
|
( C e. A /\ D e. E ) ) $=
|
|
( cop cv csn cxp ciun wcel cvv wa elex opexb sylib simpld wb adantr nfiu1
|
|
nfcv nfel2 nfbi wceq opeq1 eleq1d eleq1 anbi12d bibi12d opeliunxp vtoclgf
|
|
nfv eleq2d pm5.21nii ) DEHZABAIZJCKZLZMZDNMZDBMZEFMZOZVAVBENMZVAUQNMVBVFO
|
|
UQUTPDEQRSVCVBVDDBPUAUREHZUTMZURBMZECMZOZTVAVETADNADUCVAVEAAUQUTABUSUBUDV
|
|
EAUNUEURDUFZVHVAVKVEVLVGUQUTURDEUGUHVLVIVCVJVDURDBUIVLCFEGUOUJUKABCEULUMU
|
|
P $.
|
|
$}
|
|
|
|
|
|
${
|
|
$d x y z A $. $d x z B $. $d y z ph $. $d x ps $.
|
|
raliunxp.1 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
|
|
$( Write a double restricted quantification as one universal quantifier.
|
|
In this version of ~ ralxp , ` B ( y ) ` is not assumed to be constant.
|
|
(Contributed by Mario Carneiro, 29-Dec-2014.) $)
|
|
raliunxp $p |- ( A. x e. U_ y e. A ( { y } X. B ) ph <->
|
|
A. y e. A A. z e. B ps ) $=
|
|
( cv csn cxp wcel wi wal wa wral wex albii vex bitri ciun cop wceq imbi1i
|
|
eliunxp 19.23vv bitr4i alrot3 impexp imbi2d ceqsalv 2albii df-ral 3bitr4i
|
|
opex r2al ) CIZDFDIZJGKUAZLZAMZCNZURFLEIZGLOZBMZENDNZACUSPBEGPDFPVBUQURVC
|
|
UBZUCZVDOZAMZENDNZCNZVFVAVKCVAVIEQDQZAMVKUTVMADEFGUQUEUDVIADEUFUGRVLVJCNZ
|
|
ENDNVFVJCDEUHVNVEDEVNVHVDAMZMZCNVEVJVPCVHVDAUIRVOVECVGURVCDSESUOVHABVDHUJ
|
|
UKTULTTACUSUMBDEFGUPUN $.
|
|
|
|
$( Write a double restricted quantification as one universal quantifier.
|
|
In this version of ~ rexxp , ` B ( y ) ` is not assumed to be constant.
|
|
(Contributed by Mario Carneiro, 14-Feb-2015.) $)
|
|
rexiunxp $p |- ( E. x e. U_ y e. A ( { y } X. B ) ph <->
|
|
E. y e. A E. z e. B ps ) $=
|
|
( wn cv csn cxp ciun wral wrex cop wceq notbid raliunxp dfrex2 3bitr4i
|
|
ralnex ralbii bitri notbii ) AIZCDFDJZKGLMZNZIBEGOZIZDFNZIACUHOUJDFOUIULU
|
|
IBIZEGNZDFNULUFUMCDEFGCJUGEJPQABHRSUNUKDFBEGUBUCUDUEACUHTUJDFTUA $.
|
|
|
|
$d y B $.
|
|
$( Universal quantification restricted to a Cartesian product is equivalent
|
|
to a double restricted quantification. The hypothesis specifies an
|
|
implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by
|
|
Mario Carneiro, 29-Dec-2014.) $)
|
|
ralxp $p |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps ) $=
|
|
( cxp wral cv csn ciun iunxpconst raleqi raliunxp bitr3i ) ACFGIZJACDFDKL
|
|
GIMZJBEGJDFJACSRDFGNOABCDEFGHPQ $.
|
|
|
|
$( Existential quantification restricted to a Cartesian product is
|
|
equivalent to a double restricted quantification. (Contributed by NM,
|
|
11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.) $)
|
|
rexxp $p |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps ) $=
|
|
( cxp wrex cv csn ciun iunxpconst rexeqi rexiunxp bitr3i ) ACFGIZJACDFDKL
|
|
GIMZJBEGJDFJACSRDFGNOABCDEFGHPQ $.
|
|
$}
|
|
|
|
${
|
|
$d u v w x y A $. $d u v w x y z B $. $d u v w ph $. $d u v w ps $.
|
|
ralxpf.1 $e |- F/ y ph $.
|
|
ralxpf.2 $e |- F/ z ph $.
|
|
ralxpf.3 $e |- F/ x ps $.
|
|
ralxpf.4 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
|
|
$( Version of ~ ralxp with bound-variable hypotheses. (Contributed by NM,
|
|
18-Aug-2006.) (Revised by set.mm contributors, 20-Dec-2008.) $)
|
|
ralxpf $p |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps ) $=
|
|
( vw vu vv wral wsb nfv cv wceq nfsb cxp cbvralsv nfcv nfs1 nfral sbequ12
|
|
ralbii weq ralbidv cbvral cop wa wex wb eqvinop nfbi sbhypf opth sylan9bb
|
|
vex sylbi exlimi ralxp 3bitr4ri bitri ) ACFGUAZOACLPZLVFOZBEGOZDFOZACLVFU
|
|
BBDMPZEGOZMFOVKENPZNGOZMFOVJVHVLVNMFVKENGUBUGVIVLDMFVIMQVKDEGDGUCBDMBMQUD
|
|
ZUEDMUHZBVKEGBDMUFZUIUJVGVMLMNFGLRZMRZNRZUKZSVRDRZERZUKZSZWDWASZULZEUMZDU
|
|
MVGVMUNZDEVRVSVTMUTNUTUOWHWIDVGVMDACLDHTVKENDVOTUPWGWIEVGVMEACLEITVKENVKN
|
|
QUDUPWEVGBWFVMABCLWDJKUQWFVPENUHZULBVMUNWBWCVSVTURVPBVKWJVMVQVKENUFUSVAUS
|
|
VBVBVAVCVDVE $.
|
|
|
|
$( Version of ~ rexxp with bound-variable hypotheses. (Contributed by NM,
|
|
19-Dec-2008.) $)
|
|
rexxpf $p |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps ) $=
|
|
( wn cxp wral wrex nfn cv cop wb dfrex2 notbi ralxpf notbii rexbii rexnal
|
|
wceq sylib bitri 3bitr4i ) ALZCFGMZNZLBLZEGNZDFNZLZACUKOBEGOZDFOZULUOUJUM
|
|
CDEFGADHPAEIPBCJPCQDQEQRUFABSUJUMSKABUAUGUBUCACUKTURUNLZDFOUPUQUSDFBEGTUD
|
|
UNDFUEUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d w x y A $. $d w x y z B $. $d w C $. $d w D $.
|
|
iunxpf.1 $e |- F/_ y C $.
|
|
iunxpf.2 $e |- F/_ z C $.
|
|
iunxpf.3 $e |- F/_ x D $.
|
|
iunxpf.4 $e |- ( x = <. y , z >. -> C = D ) $.
|
|
$( Indexed union on a cross product is equals a double indexed union. The
|
|
hypothesis specifies an implicit substitution. (Contributed by NM,
|
|
19-Dec-2008.) $)
|
|
iunxpf $p |- U_ x e. ( A X. B ) C = U_ y e. A U_ z e. B D $=
|
|
( vw cxp ciun cv wcel wrex nfel2 cop eliun wceq eleq2d rexxpf bitri eqriv
|
|
rexbii 3bitr4i ) LADEMZFNZBDCEGNZNZLOZFPZAUHQULGPZCEQZBDQZULUIPULUKPZUMUN
|
|
ABCDEBULFHRCULFIRAULGJRAOBOCOSUAFGULKUBUCAULUHFTUQULUJPZBDQUPBULDUJTURUOB
|
|
DCULEGTUFUDUGUE $.
|
|
$}
|
|
|
|
${
|
|
brelg.1 $e |- R C_ ( C X. D ) $.
|
|
$( Two things in a binary relation belong to the relation's domain.
|
|
(Contributed by NM, 17-May-1996.) $)
|
|
brel $p |- ( A R B -> ( A e. C /\ B e. D ) ) $=
|
|
( wbr cxp wcel wa ssbri brxp sylib ) ABEGABCDHZGACIBDIJENABFKABCDLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Membership in a cross product. (Contributed by NM, 5-Mar-1995.) $)
|
|
elxp3 $p |- ( A e. ( B X. C ) <->
|
|
E. x E. y ( <. x , y >. = A /\ <. x , y >. e. ( B X. C ) ) ) $=
|
|
( cxp wcel cv cop wceq wa wex elxp eqcom opelxp anbi12i 2exbii bitr4i ) C
|
|
DEFZGCAHZBHZIZJZTDGUAEGKZKZBLALUBCJZUBSGZKZBLALABCDEMUHUEABUFUCUGUDUBCNTU
|
|
ADEOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Distributive law for cross product over union. Theorem 103 of [Suppes]
|
|
p. 52. (Contributed by NM, 12-Aug-2004.) $)
|
|
xpundi $p |- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) ) $=
|
|
( vx vy cv wcel cun wa copab cxp wo elun anbi2i andi bitri opabbii unopab
|
|
eqtr4i df-xp uneq12i 3eqtr4i ) DFAGZEFZBCHZGZIZDEJZUCUDBGZIZDEJZUCUDCGZIZ
|
|
DEJZHZAUEKABKZACKZHUHUJUMLZDEJUOUGURDEUGUCUIULLZIURUFUSUCUDBCMNUCUIULOPQU
|
|
JUMDERSDEAUETUPUKUQUNDEABTDEACTUAUB $.
|
|
|
|
$( Distributive law for cross product over union. Similar to Theorem 103
|
|
of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.) $)
|
|
xpundir $p |- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) ) $=
|
|
( vx vy cv cun wcel wa copab cxp wo elun anbi1i andir bitri unopab eqtr4i
|
|
opabbii df-xp uneq12i 3eqtr4i ) DFZABGZHZEFCHZIZDEJZUCAHZUFIZDEJZUCBHZUFI
|
|
ZDEJZGZUDCKACKZBCKZGUHUJUMLZDEJUOUGURDEUGUIULLZUFIURUEUSUFUCABMNUIULUFOPS
|
|
UJUMDEQRDEUDCTUPUKUQUNDEACTDEBCTUAUB $.
|
|
$}
|
|
|
|
$( The cross product of two unions. (Contributed by NM, 12-Aug-2004.) $)
|
|
xpun $p |- ( ( A u. B ) X. ( C u. D ) ) =
|
|
( ( ( A X. C ) u. ( A X. D ) ) u. ( ( B X. C ) u. ( B X. D ) ) ) $=
|
|
( cun cxp xpundi xpundir uneq12i un4 3eqtri ) ABEZCDEFLCFZLDFZEACFZBCFZEZAD
|
|
FZBDFZEZEOREPSEELCDGMQNTABCHABDHIOPRSJK $.
|
|
|
|
$( Intersection of binary relation with cross product. (Contributed by NM,
|
|
3-Mar-2007.) $)
|
|
brinxp2 $p |- ( A ( R i^i ( C X. D ) ) B <->
|
|
( A e. C /\ B e. D /\ A R B ) ) $=
|
|
( wbr cxp wa wcel cin w3a ancom brxp anbi1i bitri brin df-3an 3bitr4i ) ABE
|
|
FZABCDGZFZHZACIZBDIZHZSHZABETJFUCUDSKUBUASHUFSUALUAUESABCDMNOABETPUCUDSQR
|
|
$.
|
|
|
|
$( Intersection of binary relation with cross product. (Contributed by NM,
|
|
9-Mar-1997.) $)
|
|
brinxp $p |- ( ( A e. C /\ B e. D ) ->
|
|
( A R B <-> A ( R i^i ( C X. D ) ) B ) ) $=
|
|
( cxp cin wbr wcel wa w3a brinxp2 df-3an bitri baibr ) ABECDFGHZACIZBDIZJZA
|
|
BEHZPQRTKSTJABCDELQRTMNO $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( An abstraction relation is a subset of a related cross product.
|
|
(Contributed by NM, 16-Jul-1995.) $)
|
|
opabssxp $p |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ph ) }
|
|
C_ ( A X. B ) $=
|
|
( cv wcel wa copab cxp simpl ssopab2i df-xp sseqtr4i ) BFDGCFEGHZAHZBCIOB
|
|
CIDEJPOBCOAKLBCDEMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y ps $.
|
|
optocl.1 $e |- D = ( B X. C ) $.
|
|
optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
|
|
optocl.3 $e |- ( ( x e. B /\ y e. C ) -> ph ) $.
|
|
$( Implicit substitution of class for ordered pair. (Contributed by NM,
|
|
5-Mar-1995.) $)
|
|
optocl $p |- ( A e. D -> ps ) $=
|
|
( cxp wcel cv cop wceq wa wex elxp3 sylbi opelxp syl5ib exlimivv eleq2s
|
|
imp ) BEFGLZHEUFMCNZDNZOZEPZUIUFMZQZDRCRBCDEFGSULBCDUJUKBUKAUJBUKUGFMUHGM
|
|
QAUGUHFGUAKTJUBUEUCTIUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d z w B $. $d x y z w C $. $d x y z w D $.
|
|
$d x y ps $. $d z w ch $. $d z w R $.
|
|
2optocl.1 $e |- R = ( C X. D ) $.
|
|
2optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
|
|
2optocl.3 $e |- ( <. z , w >. = B -> ( ps <-> ch ) ) $.
|
|
2optocl.4 $e |- ( ( ( x e. C /\ y e. D ) /\ ( z e. C /\ w e. D ) ) ->
|
|
ph ) $.
|
|
$( Implicit substitution of classes for ordered pairs. (Contributed by NM,
|
|
12-Mar-1995.) $)
|
|
2optocl $p |- ( ( A e. R /\ B e. R ) -> ch ) $=
|
|
( wcel wi cv cop wceq imbi2d wa ex optocl com12 impcom ) ILQHLQZCUHBRUHCR
|
|
FGIJKLMFSZGSZTIUABCUHOUBUHUIJQUJKQUCZBUKARUKBRDEHJKLMDSZESZTHUAABUKNUBULJ
|
|
QUMKQUCUKAPUDUEUFUEUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d z w v u B $. $d v u C $. $d x y z w v u D $.
|
|
$d x y z w v u F $. $d z w v u R $. $d x y ps $. $d z w ch $.
|
|
$d v u th $.
|
|
3optocl.1 $e |- R = ( D X. F ) $.
|
|
3optocl.2 $e |- ( <. x , y >. = A -> ( ph <-> ps ) ) $.
|
|
3optocl.3 $e |- ( <. z , w >. = B -> ( ps <-> ch ) ) $.
|
|
3optocl.4 $e |- ( <. v , u >. = C -> ( ch <-> th ) ) $.
|
|
3optocl.5 $e |- ( ( ( x e. D /\ y e. F ) /\ ( z e. D /\ w e. F )
|
|
/\ ( v e. D /\ u e. F ) ) -> ph ) $.
|
|
$( Implicit substitution of classes for ordered pairs. (Contributed by NM,
|
|
12-Mar-1995.) $)
|
|
3optocl $p |- ( ( A e. R /\ B e. R /\ C e. R ) -> th ) $=
|
|
( wcel wa wi cv cop wceq imbi2d 3expia 2optocl com12 optocl impcom 3impa
|
|
) KOUBZLOUBZMOUBZDUQUOUPUCZDURCUDURDUDIJMNPOQIUEZJUEZUFMUGCDURTUHURUSNUBU
|
|
TPUBUCZCVAAUDVABUDVACUDEFGHKLNPOQEUEZFUEZUFKUGABVARUHGUEZHUEZUFLUGBCVASUH
|
|
VBNUBVCPUBUCVDNUBVEPUBUCVAAUAUIUJUKULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u D $. $d x y z w v u S $. $d x y ph $. $d z w v u ps $.
|
|
opbrop.1 $e |- ( ( ( z = A /\ w = B ) /\ ( v = C /\ u = D ) ) ->
|
|
( ph <-> ps ) ) $.
|
|
opbrop.2 $e |- R = { <. x , y >. | ( ( x e. ( S X. S ) /\
|
|
y e. ( S X. S ) ) /\
|
|
E. z E. w E. v E. u ( ( x = <. z , w >. /\
|
|
y = <. v , u >. ) /\ ph ) ) } $.
|
|
$( Ordered pair membership in a relation. Special case. (Contributed by
|
|
NM, 5-Aug-1995.) $)
|
|
opbrop $p |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) ->
|
|
( <. A , B >. R <. C , D >. <-> ps ) ) $=
|
|
( wcel wa cv wex cop cxp wbr copsex4g anbi2d cvv opexg eleq1 anbi1d eqeq1
|
|
wb 4exbidv anbi12d brabg syl2an opelxp anbi12i biimpri biantrurd 3bitr4d
|
|
wceq ) INQJNQRZKNQLNQRZRZIJUAZNNUBZQZKLUAZVFQZRZVEESFSUAZVAZVHGSHSUAZVAZR
|
|
ZARZHTGTFTETZRZVJBRVEVHMUCZBVDVQBVJABEFGHIJKLNNOUDUEVBVEUFQVHUFQVSVRUKVCI
|
|
JNNUGKLNNUGCSZVFQZDSZVFQZRZVTVKVAZWBVMVAZRZARZHTGTFTETZRVGWCRZVLWFRZARZHT
|
|
GTFTETZRVRCDVEVHUFUFMVTVEVAZWDWJWIWMWNWAVGWCVTVEVFUHUIWNWHWLEFGHWNWGWKAWN
|
|
WEVLWFVTVEVKUJUIUIULUMWBVHVAZWJVJWMVQWOWCVIVGWBVHVFUHUEWOWLVPEFGHWOWKVOAW
|
|
OWFVNVLWBVHVMUJUEUIULUMPUNUOVDVJBVJVDVGVBVIVCIJNNUPKLNNUPUQURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( The cross product with the empty set is empty. Part of Theorem 3.13(ii)
|
|
of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.) $)
|
|
xp0r $p |- ( (/) X. A ) = (/) $=
|
|
( vz vx vy c0 cxp cv wcel cop wceq wa wex elxp noel simprl mto nex 2false
|
|
bitri eqriv ) BEAFZEBGZUAHUBCGZDGZIJZUCEHZUDAHZKKZDLZCLZUBEHZCDUBEAMUJUKU
|
|
ICUHDUHUFUCNUEUFUGOPQQUBNRST $.
|
|
$}
|
|
|
|
$( The cross product of the universe with itself is the universe.
|
|
(Contributed by Scott Fenton, 14-Apr-2021.) $)
|
|
xpvv $p |- ( _V X. _V ) = _V $=
|
|
( vx cvv cxp wceq cv wcel eqv cproj1 cproj2 cop vex proj1ex proj2ex opelxp
|
|
opeq mpbir2an eqeltri mpgbir ) BBCZBDAEZSFAASGTTHZTIZJZSTOUCSFUABFUBBFTAKZ
|
|
LTUDMUAUBBBNPQR $.
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $.
|
|
$( A subclass relationship depends only on a relation's ordered pairs.
|
|
Theorem 3.2(i) of [Monk1] p. 33. (The proof was shortened by Andrew
|
|
Salmon, 27-Aug-2011.) (Contributed by NM, 2-Aug-1994.) (Revised by
|
|
set.mm contributors, 27-Aug-2011.) $)
|
|
ssrel $p |- ( A C_ B <->
|
|
A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) ) $=
|
|
( vz wss cv cop wcel wi wal ssel alrimivv cproj1 cproj2 wceq imbi12d spcv
|
|
eleq1d eleq1i vex proj1ex opeq1 albidv proj2ex opeq2 3imtr4g ssrdv impbii
|
|
syl opeq ) CDFZAGZBGZHZCIZUODIZJZBKZAKZULURABCDUOLMUTECDUTEGZNZVAOZHZCIZV
|
|
DDIZVACIVADIUTVBUNHZCIZVGDIZJZBKZVEVFJZUSVKAVBVAEUAZUBUMVBPZURVJBVNUPVHUQ
|
|
VIVNUOVGCUMVBUNUCZSVNUOVGDVOSQUDRVJVLBVCVAVMUEUNVCPZVHVEVIVFVPVGVDCUNVCVB
|
|
UFZSVPVGVDDVQSQRUJVAVDCVAUKZTVAVDDVRTUGUHUI $.
|
|
|
|
$( Extensionality principle for relations. Theorem 3.2(ii) of [Monk1]
|
|
p. 33. (Contributed by NM, 2-Aug-1994.) (Revised by Scott Fenton,
|
|
14-Apr-2021.) $)
|
|
eqrel $p |- ( A = B <->
|
|
A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) $=
|
|
( wss wa cv cop wcel wi wal wceq wb ssrel anbi12i eqss 2albiim 3bitr4i )
|
|
CDEZDCEZFAGBGHZCIZUADIZJBKAKZUCUBJBKAKZFCDLUBUCMBKAKSUDTUEABCDNABDCNOCDPU
|
|
BUCABQR $.
|
|
|
|
$( Subclass principle for operators. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
ssopr $p |- ( A C_ B <->
|
|
A. x A. y A. z ( <. <. x , y >. , z >. e. A ->
|
|
<. <. x , y >. , z >. e. B ) ) $=
|
|
( vw wss cv cop wcel wi wal ssrel wex cvv vex albii alrot3 eleq1d 3bitri
|
|
alcom bitri wceq opeqex ax-mp 19.23vv bitr4i opeq1 imbi12d ceqsalv 2albii
|
|
a1bi opex ) DEGZFHZCHZIZDJZUQEJZKZFLZCLZAHZBHZIZUPIZDJZVFEJZKZBLALZCLVICL
|
|
BLALUNUTCLFLVBFCDEMUTFCUAUBVAVJCVAUOVEUCZUTKZBLALZFLVLFLZBLALVJUTVMFUTVKB
|
|
NANZUTKVMVOUTUOOJVOFPABUOOUDUEULVKUTABUFUGQVLFABRVNVIABUTVIFVEVCVDAPBPUMV
|
|
KURVGUSVHVKUQVFDUOVEUPUHZSVKUQVFEVPSUIUJUKTQVICABRT $.
|
|
|
|
$( Extensionality principle for operators. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
eqopr $p |- ( A = B <->
|
|
A. x A. y A. z ( <. <. x , y >. , z >. e. A <->
|
|
<. <. x , y >. , z >. e. B ) ) $=
|
|
( wss wa cv cop wcel wi wal wceq ssopr anbi12i eqss 2albiim albii 19.26
|
|
wb bitri 3bitr4i ) DEFZEDFZGAHBHICHIZDJZUEEJZKCLBLZALZUGUFKCLBLZALZGZDEMU
|
|
FUGTCLBLZALZUCUIUDUKABCDENABCEDNODEPUNUHUJGZALULUMUOAUFUGBCQRUHUJASUAUB
|
|
$.
|
|
$}
|
|
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
relssi.1 $e |- ( <. x , y >. e. A -> <. x , y >. e. B ) $.
|
|
$( Inference from subclass principle for relations. (Contributed by NM,
|
|
31-Mar-1998.) (Revised by Scott Fenton, 15-Apr-2021.) $)
|
|
relssi $p |- A C_ B $=
|
|
( wss cv cop wcel wi wal ssrel ax-gen mpgbir ) CDFAGBGHZCIODIJZBKAABCDLPB
|
|
EMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y ph $.
|
|
relssdv.1 $e |- ( ph -> ( <. x , y >. e. A -> <. x , y >. e. B ) ) $.
|
|
|
|
$( Deduction from subclass principle for relations. (Contributed by set.mm
|
|
contributors, 11-Sep-2004.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
relssdv $p |- ( ph -> A C_ B ) $=
|
|
( cv cop wcel wi wal wss alrimivv ssrel sylibr ) ABGCGHZDIPEIJZCKBKDELAQB
|
|
CFMBCDENO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
eqrelriv.1 $e |- ( <. x , y >. e. A <-> <. x , y >. e. B ) $.
|
|
$( Inference from extensionality principle for relations. (Contributed by
|
|
FL, 15-Oct-2012.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
eqrelriv $p |- A = B $=
|
|
( wceq cv cop wcel wb wal eqrel ax-gen mpgbir ) CDFAGBGHZCIODIJZBKAABCDLP
|
|
BEMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
eqbrriv.1 $e |- ( x A y <-> x B y ) $.
|
|
$( Inference from extensionality principle for relations. (Contributed by
|
|
NM, 12-Dec-2006.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
eqbrriv $p |- A = B $=
|
|
( cv wbr cop wcel df-br 3bitr3i eqrelriv ) ABCDAFZBFZCGMNDGMNHZCIODIEMNCJ
|
|
MNDJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d ph x $. $d ph y $.
|
|
eqrelrdv.1 $e |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) ) $.
|
|
$( Deduce equality of relations from equivalence of membership.
|
|
(Contributed by Rodolfo Medina, 10-Oct-2010.) (Revised by Scott Fenton,
|
|
16-Apr-2021.) $)
|
|
eqrelrdv $p |- ( ph -> A = B ) $=
|
|
( cv cop wcel wb wal wceq alrimivv eqrel sylibr ) ABGCGHZDIPEIJZCKBKDELAQ
|
|
BCFMBCDENO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
eqoprriv.1 $e |- ( <. <. x , y >. , z >. e. A <->
|
|
<. <. x , y >. , z >. e. B ) $.
|
|
|
|
$( Equality inference for operators. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
eqoprriv $p |- A = B $=
|
|
( wceq cv cop wcel wb wal eqopr gen2 mpgbir ) DEGAHBHICHIZDJPEJKZCLBLAABC
|
|
DEMQBCFNO $.
|
|
$}
|
|
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z ph $.
|
|
eqoprrdv.1 $e |- ( ph -> ( <. <. x , y >. , z >. e. A <->
|
|
<. <. x , y >. , z >. e. B ) ) $.
|
|
|
|
$( Equality deduction for operators. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
eqoprrdv $p |- ( ph -> A = B ) $=
|
|
( cv cop wcel wb wal wceq alrimiv alrimivv eqopr sylibr ) ABHCHIDHIZEJRFJ
|
|
KZDLZCLBLEFMATBCASDGNOBCDEFPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $.
|
|
$( Subset theorem for cross product. Generalization of Theorem 101 of
|
|
[Suppes] p. 52. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by NM, 26-Aug-1995.) (Revised by set.mm
|
|
contributors, 27-Aug-2011.) $)
|
|
xpss12 $p |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) ) $=
|
|
( vx vy wss wa cv wcel copab cxp ssel im2anan9 ssopab2dv df-xp 3sstr4g )
|
|
ABGZCDGZHZEIZAJZFIZCJZHZEFKUABJZUCDJZHZEFKACLBDLTUEUHEFRUBUFSUDUGABUAMCDU
|
|
CMNOEFACPEFBDPQ $.
|
|
$}
|
|
|
|
$( Subset relation for cross product. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
xpss1 $p |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) ) $=
|
|
( wss cxp ssid xpss12 mpan2 ) ABDCCDACEBCEDCFABCCGH $.
|
|
|
|
$( Subset relation for cross product. (Contributed by Jeff Hankins,
|
|
30-Aug-2009.) $)
|
|
xpss2 $p |- ( A C_ B -> ( C X. A ) C_ ( C X. B ) ) $=
|
|
( wss cxp ssid xpss12 mpan ) CCDABDCAECBEDCFCCABGH $.
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $.
|
|
br1st.1 $e |- B e. _V $.
|
|
$( Binary relationship equivalence for the ` 1st ` function. (Contributed
|
|
by set.mm contributors, 8-Jan-2015.) $)
|
|
br1st $p |- ( A 1st B <-> E. x A = <. B , x >. ) $=
|
|
( vy vz c1st wbr cvv wcel cop wceq wex brex simpld vex opex eleq1 exbidv
|
|
cv mpbiri exlimiv wb eqeq1 opeq1 eqeq2d df-1st brabg mpan2 pm5.21nii ) BC
|
|
GHZBIJZBCATZKZLZAMZUKULCIJZBCGNOUOULAUOULUNIJCUMDAPQBUNIRUAUBULUQUKUPUCDE
|
|
TZFTZUMKZLZAMBUTLZAMUPEFBCIIGURBLVAVBAURBUTUDSUSCLZVBUOAVCUTUNBUSCUMUEUFS
|
|
EFAUGUHUIUJ $.
|
|
|
|
$( Binary relationship equivalence for the ` 2nd ` function. (Contributed
|
|
by set.mm contributors, 8-Jan-2015.) $)
|
|
br2nd $p |- ( A 2nd B <-> E. x A = <. x , B >. ) $=
|
|
( vy vz c2nd wbr cvv wcel cop wceq wex brex simpld vex opex eleq1 exbidv
|
|
cv mpbiri exlimiv wb eqeq1 opeq2 eqeq2d df-2nd brabg mpan2 pm5.21nii ) BC
|
|
GHZBIJZBATZCKZLZAMZUKULCIJZBCGNOUOULAUOULUNIJUMCAPDQBUNIRUAUBULUQUKUPUCDE
|
|
TZUMFTZKZLZAMBUTLZAMUPEFBCIIGURBLVAVBAURBUTUDSUSCLZVBUOAVCUTUNBUSCUMUEUFS
|
|
EFAUGUHUIUJ $.
|
|
|
|
${
|
|
$d A w $. $d B w $. $d C x y z w $.
|
|
brswap.2 $e |- C e. _V $.
|
|
$( Binary relationship equivalence for the ` Swap ` function.
|
|
(Contributed by set.mm contributors, 8-Jan-2015.) $)
|
|
brswap2 $p |- ( A Swap <. B , C >. <-> A = <. C , B >. ) $=
|
|
( vx vz vw vy cop cswap cvv wcel wceq opex cv wa wex eqeq1 2exbidv brex
|
|
wbr simpld eleq1 mpbiri anbi1d w3a anbi2d eqcom opth bitri anbi1i ancom
|
|
df-3an 3bitr4ri syl6bbr excom syl6bb opeq2 opeq1 ceqsex2v df-swap brabg
|
|
wb eqeq2d mpan2 pm5.21nii ) ABCJZKUBZALMZACBJZNZVIVJVHLMZAVHKUAUCVLVJVK
|
|
LMCBEDOAVKLUDUEVJVMVIVLVDBCDEOFPZGPZHPZJZNZIPZVPVOJZNZQZHRGRAVQNZWAQZHR
|
|
GRZVLFIAVHLLKVNANZWBWDGHWFVRWCWAVNAVQSUFTVSVHNZWEVPBNZVOCNZWCUGZGRHRZVL
|
|
WGWEWJHRGRWKWGWDWJGHWGWDWCVHVTNZQZWJWGWAWLWCVSVHVTSUHWLWCQWHWIQZWCQWMWJ
|
|
WLWNWCWLVTVHNWNVHVTUIVPVOBCUJUKULWCWLUMWHWIWCUNUOUPTWJGHUQURWCAVOBJZNVL
|
|
HGBCDEWHVQWOAVPBVOUSVEWIWOVKAVOCBUTVEVAURFIGHVBVCVFVG $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d x y B $. $d x y C $. $d x y D $. $d ph z w $.
|
|
$d ps z w $.
|
|
|
|
$( A relation expressed as an ordered pair abstraction. (Contributed by
|
|
set.mm contributors, 11-Dec-2006.) $)
|
|
opabid2 $p |- { <. x , y >. | <. x , y >. e. A } = A $=
|
|
( vz vw cv cop wcel copab vex weq opeq1 eleq1d opeq2 opelopab eqrelriv )
|
|
DEAFZBFZGZCHZABICTDFZRGZCHUAEFZGZCHABUAUCDJEJADKSUBCQUARLMBEKUBUDCRUCUANM
|
|
OP $.
|
|
|
|
$( Intersection of two ordered pair class abstractions. (Contributed by
|
|
NM, 30-Sep-2002.) $)
|
|
inopab $p |- ( { <. x , y >. | ph } i^i { <. x , y >. | ps } ) =
|
|
{ <. x , y >. | ( ph /\ ps ) } $=
|
|
( vz vw copab cin wa cv cop wcel sbcan sbcbii opelopabsb anbi12i 3bitr4ri
|
|
wsbc elin 3bitr4i eqrelriv ) EFACDGZBCDGZHZABIZCDGZEJZFJZKZUBLZUIUCLZIZUE
|
|
DUHRZCUGRZUIUDLUIUFLADUHRZBDUHRZIZCUGRUOCUGRZUPCUGRZIUNULUOUPCUGMUMUQCUGA
|
|
BDUHMNUJURUKUSACDUGUHOBCDUGUHOPQUIUBUCSUECDUGUHOTUA $.
|
|
|
|
$( The intersection of two cross products. Exercise 9 of [TakeutiZaring]
|
|
p. 25. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|
|
(Contributed by NM, 3-Aug-1994.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
inxp $p |- ( ( A X. B ) i^i ( C X. D ) ) =
|
|
( ( A i^i C ) X. ( B i^i D ) ) $=
|
|
( vx vy cv wcel wa copab cin cxp inopab elin anbi12i bitr4i opabbii eqtri
|
|
an4 df-xp ineq12i 3eqtr4i ) EGZAHZFGZBHZIZEFJZUCCHZUEDHZIZEFJZKZUCACKZHZU
|
|
EBDKZHZIZEFJZABLZCDLZKUNUPLUMUGUKIZEFJUSUGUKEFMVBUREFVBUDUIIZUFUJIZIURUDU
|
|
FUIUJSUOVCUQVDUCACNUEBDNOPQRUTUHVAULEFABTEFCDTUAEFUNUPTUB $.
|
|
|
|
$( Distributive law for cross product over intersection. Theorem 102 of
|
|
[Suppes] p. 52. (Contributed by NM, 26-Sep-2004.) $)
|
|
xpindi $p |- ( A X. ( B i^i C ) ) = ( ( A X. B ) i^i ( A X. C ) ) $=
|
|
( cxp cin inxp inidm xpeq1i eqtr2i ) ABDACDEAAEZBCEZDAKDABACFJAKAGHI $.
|
|
|
|
$( Distributive law for cross product over intersection. Similar to
|
|
Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.) $)
|
|
xpindir $p |- ( ( A i^i B ) X. C ) = ( ( A X. C ) i^i ( B X. C ) ) $=
|
|
( cxp cin inxp inidm xpeq2i eqtr2i ) ACDBCDEABEZCCEZDJCDACBCFKCJCGHI $.
|
|
$}
|
|
|
|
|
|
${
|
|
$d x y A $.
|
|
opabbi2i.1 $e |- ( <. x , y >. e. A <-> ph ) $.
|
|
$( Equality of a class variable and an ordered pair abstractions
|
|
(inference rule). Compare ~ abbi2i . (Contributed by Scott Fenton,
|
|
18-Apr-2021.) $)
|
|
opabbi2i $p |- A = { <. x , y >. | ph } $=
|
|
( cv cop wcel copab opabid2 opabbii eqtr3i ) BFCFGDHZBCIDABCIBCDJMABCEKL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y ph $.
|
|
opabbi2dv.1 $e |- ( ph -> ( <. x , y >. e. A <-> ps ) ) $.
|
|
$( Deduce equality of a relation and an ordered-pair class builder.
|
|
Compare ~ abbi2dv . (Contributed by NM, 24-Feb-2014.) $)
|
|
opabbi2dv $p |- ( ph -> A = { <. x , y >. | ps } ) $=
|
|
( cv cop wcel copab opabid2 opabbidv syl5eqr ) AECGDGHEIZCDJBCDJCDEKANBCD
|
|
FLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( For sets, the identity relation is the same as equality. (Contributed
|
|
by NM, 30-Apr-2004.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
ideqg $p |- ( B e. V -> ( A _I B <-> A = B ) ) $=
|
|
( vx vy wcel cid wbr wceq cvv wa brex adantl simpr adantr eqeltrd jca weq
|
|
elex cv eqeq1 eqeq2 df-id brabg pm5.21nd ) BCFZABGHZABIZAJFZBJFZKZUGUKUFA
|
|
BGLMUFUHKZUIUJULABJUFUHNUFUJUHBCSOZPUMQDERAETZIUHDEABJJGDTAUNUAUNBAUBDEUC
|
|
UDUE $.
|
|
|
|
$( For sets, the identity relation is the same as equality. (Contributed
|
|
by SF, 8-Jan-2015.) $)
|
|
ideqg2 $p |- ( A e. V -> ( A _I B <-> A = B ) ) $=
|
|
( vx vy wcel cid wbr wceq cvv wa brex adantl elex simpl eleq1 biimpac jca
|
|
sylan cv weq eqeq1 eqeq2 df-id brabg pm5.21nd ) ACFZABGHZABIZAJFZBJFZKZUH
|
|
ULUGABGLMUGUJUIULACNUJUIKUJUKUJUIOUIUJUKABJPQRSDEUAAETZIUIDEABJJGDTAUMUBU
|
|
MBAUCDEUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
ideq.1 $e |- B e. _V $.
|
|
$( For sets, the identity relation is the same as equality. (Contributed
|
|
by NM, 13-Aug-1995.) (Revised by set.mm contributors, 1-Jun-2008.) $)
|
|
ideq $p |- ( A _I B <-> A = B ) $=
|
|
( cvv wcel cid wbr wceq wb ideqg ax-mp ) BDEABFGABHICABDJK $.
|
|
$}
|
|
|
|
$( A set is identical to itself. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by NM, 28-May-2008.) (Revised by set.mm
|
|
contributors, 27-Aug-2011.) $)
|
|
ididg $p |- ( A e. V -> A _I A ) $=
|
|
( wcel cid wbr wceq eqid ideqg mpbiri ) ABCAADEAAFAGAABHI $.
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $.
|
|
$( Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.) $)
|
|
coss1 $p |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) ) $=
|
|
( vx vy vz wss cv wbr wa wex copab id ssbrd anim2d eximdv ssopab2dv df-co
|
|
ccom 3sstr4g ) ABGZDHEHZCIZUBFHZAIZJZEKZDFLUCUBUDBIZJZEKZDFLACSBCSUAUGUJD
|
|
FUAUFUIEUAUEUHUCUAABUBUDUAMNOPQDFEACRDFEBCRT $.
|
|
|
|
$( Subclass theorem for composition. (Contributed by set.mm contributors,
|
|
5-Apr-2013.) $)
|
|
coss2 $p |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) ) $=
|
|
( vx vy vz wss cv wbr wa wex copab id ssbrd anim1d eximdv ssopab2dv df-co
|
|
ccom 3sstr4g ) ABGZDHZEHZAIZUCFHCIZJZEKZDFLUBUCBIZUEJZEKZDFLCASCBSUAUGUJD
|
|
FUAUFUIEUAUDUHUEUAABUBUCUAMNOPQDFECARDFECBRT $.
|
|
$}
|
|
|
|
$( Equality theorem for composition of two classes. (Contributed by set.mm
|
|
contributors, 3-Jan-1997.) $)
|
|
coeq1 $p |- ( A = B -> ( A o. C ) = ( B o. C ) ) $=
|
|
( wss wa ccom wceq coss1 anim12i eqss 3imtr4i ) ABDZBADZEACFZBCFZDZONDZEABG
|
|
NOGLPMQABCHBACHIABJNOJK $.
|
|
|
|
$( Equality theorem for composition of two classes. (Contributed by set.mm
|
|
contributors, 3-Jan-1997.) $)
|
|
coeq2 $p |- ( A = B -> ( C o. A ) = ( C o. B ) ) $=
|
|
( wss wa ccom wceq coss2 anim12i eqss 3imtr4i ) ABDZBADZECAFZCBFZDZONDZEABG
|
|
NOGLPMQABCHBACHIABJNOJK $.
|
|
|
|
${
|
|
coeq1i.1 $e |- A = B $.
|
|
$( Equality inference for composition of two classes. (Contributed by
|
|
set.mm contributors, 16-Nov-2000.) $)
|
|
coeq1i $p |- ( A o. C ) = ( B o. C ) $=
|
|
( wceq ccom coeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for composition of two classes. (Contributed by
|
|
set.mm contributors, 16-Nov-2000.) $)
|
|
coeq2i $p |- ( C o. A ) = ( C o. B ) $=
|
|
( wceq ccom coeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
coeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for composition of two classes. (Contributed by
|
|
set.mm contributors, 16-Nov-2000.) $)
|
|
coeq1d $p |- ( ph -> ( A o. C ) = ( B o. C ) ) $=
|
|
( wceq ccom coeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for composition of two classes. (Contributed by
|
|
set.mm contributors, 16-Nov-2000.) $)
|
|
coeq2d $p |- ( ph -> ( C o. A ) = ( C o. B ) ) $=
|
|
( wceq ccom coeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
coeq12i.1 $e |- A = B $.
|
|
coeq12i.2 $e |- C = D $.
|
|
$( Equality inference for composition of two classes. (Contributed by FL,
|
|
7-Jun-2012.) $)
|
|
coeq12i $p |- ( A o. C ) = ( B o. D ) $=
|
|
( ccom coeq1i coeq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
|
|
${
|
|
coeq12d.1 $e |- ( ph -> A = B ) $.
|
|
coeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for composition of two classes. (Contributed by FL,
|
|
7-Jun-2012.) $)
|
|
coeq12d $p |- ( ph -> ( A o. C ) = ( B o. D ) ) $=
|
|
( ccom coeq1d coeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z $. $d y z w A $. $d y z w B $.
|
|
nfco.1 $e |- F/_ x A $.
|
|
nfco.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for function value. (Contributed by
|
|
NM, 1-Sep-1999.) $)
|
|
nfco $p |- F/_ x ( A o. B ) $=
|
|
( vy vw vz ccom cv wbr wa wex copab df-co nfcv nfbr nfan nfex nfopab
|
|
nfcxfr ) ABCIFJZGJZCKZUCHJZBKZLZGMZFHNFHGBCOUHFHAUGAGUDUFAAUBUCCAUBPEAUCP
|
|
ZQAUCUEBUIDAUEPQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $.
|
|
$( Binary relation on a composition. (Contributed by set.mm contributors,
|
|
21-Sep-2004.) $)
|
|
brco $p |- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) $=
|
|
( vy vz ccom wbr cvv wcel wa wex brex simpld simprd anim12i wceq exbidv
|
|
cv exlimiv breq1 anbi1d breq2 anbi2d df-co brabg pm5.21nii ) BCDEHZIBJKZC
|
|
JKZLZBATZEIZUMCDIZLZAMZBCUINUPULAUNUJUOUKUNUJUMJKZBUMENOUOURUKUMCDNPQUAFT
|
|
ZUMEIZUMGTZDIZLZAMUNVBLZAMUQFGBCJJUIUSBRZVCVDAVEUTUNVBUSBUMEUBUCSVACRZVDU
|
|
PAVFVBUOUNVACUMDUDUESFGADEUFUGUH $.
|
|
|
|
$( Ordered pair membership in a composition. (The proof was shortened by
|
|
Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors,
|
|
27-Dec-1996.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
opelco $p |- ( <. A , B >. e. ( C o. D ) <-> E. x ( A D x /\ x C B ) ) $=
|
|
( cop ccom wcel wbr cv wa wex df-br brco bitr3i ) BCFDEGZHBCPIBAJZEIQCDIK
|
|
ALBCPMABCDENO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Subset theorem for converse. (Contributed by set.mm contributors,
|
|
22-Mar-1998.) $)
|
|
cnvss $p |- ( A C_ B -> `' A C_ `' B ) $=
|
|
( vy vx wss wbr copab ccnv cop wcel ssel 3imtr4g ssopab2dv df-cnv 3sstr4g
|
|
cv df-br ) ABEZCPZDPZAFZDCGSTBFZDCGAHBHRUAUBDCRSTIZAJUCBJUAUBABUCKSTAQSTB
|
|
QLMDCANDCBNO $.
|
|
$}
|
|
|
|
$( Equality theorem for converse. (Contributed by set.mm contributors,
|
|
13-Aug-1995.) $)
|
|
cnveq $p |- ( A = B -> `' A = `' B ) $=
|
|
( wss wa ccnv wceq cnvss anim12i eqss 3imtr4i ) ABCZBACZDAEZBEZCZNMCZDABFMN
|
|
FKOLPABGBAGHABIMNIJ $.
|
|
|
|
${
|
|
cnveqi.1 $e |- A = B $.
|
|
$( Equality inference for converse. (Contributed by set.mm contributors,
|
|
23-Dec-2008.) $)
|
|
cnveqi $p |- `' A = `' B $=
|
|
( wceq ccnv cnveq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
cnveqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for converse. (Contributed by set.mm contributors,
|
|
6-Dec-2013.) $)
|
|
cnveqd $p |- ( ph -> `' A = `' B ) $=
|
|
( wceq ccnv cnveq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
|
|
by set.mm contributors, 24-Mar-1998.) $)
|
|
elcnv $p |- ( A e. `' R <-> E. x E. y ( A = <. x , y >. /\ y R x ) ) $=
|
|
( ccnv wcel cv wbr copab cop wceq wa wex df-cnv eleq2i elopab bitri ) CDE
|
|
ZFCBGZAGZDHZABIZFCTSJKUALBMAMRUBCABDNOUAABCPQ $.
|
|
|
|
$( Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
|
|
by set.mm contributors, 11-Aug-2004.) $)
|
|
elcnv2 $p |- ( A e. `' R <->
|
|
E. x E. y ( A = <. x , y >. /\ <. y , x >. e. R ) ) $=
|
|
( ccnv wcel cv cop wceq wbr wa wex elcnv df-br anbi2i 2exbii bitri ) CDEF
|
|
CAGZBGZHIZSRDJZKZBLALTSRHDFZKZBLALABCDMUBUDABUAUCTSRDNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d x y z $.
|
|
nfcnv.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for converse. (Contributed by NM,
|
|
31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfcnv $p |- F/_ x `' A $=
|
|
( vz vy ccnv cv wbr copab df-cnv nfcv nfbr nfopab nfcxfr ) ABFDGZEGZBHZED
|
|
IEDBJQEDAAOPBAOKCAPKLMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d R x y $.
|
|
$( The converse of a binary relation swaps arguments. Theorem 11 of
|
|
[Suppes] p. 61. (Contributed by set.mm contributors, 13-Aug-1995.) $)
|
|
brcnv $p |- ( A `' R B <-> B R A ) $=
|
|
( vy vx ccnv wbr cvv wcel wa ancomd cv breq2 breq1 df-cnv brabg pm5.21nii
|
|
brex ) ABCFZGAHIZBHIZJBACGZABSRUBUATBACRKDLZELZCGUCACGUBEDABHHSUDAUCCMUCB
|
|
ACNEDCOPQ $.
|
|
|
|
$( Ordered-pair membership in converse. (Contributed by set.mm
|
|
contributors, 13-Aug-1995.) $)
|
|
opelcnv $p |- ( <. A , B >. e. `' R <-> <. B , A >. e. R ) $=
|
|
( ccnv wbr cop wcel brcnv df-br 3bitr3i ) ABCDZEBACEABFKGBAFCGABCHABKIBAC
|
|
IJ $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( Distributive law of converse over class composition. Theorem 26 of
|
|
[Suppes] p. 64. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by set.mm contributors, 19-Mar-1998.)
|
|
(Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
cnvco $p |- `' ( A o. B ) = ( `' B o. `' A ) $=
|
|
( vx vy vz cv ccom wbr copab ccnv wa wex brcnv anbi12i ancom bitr3i exbii
|
|
brco bitri opabbii df-cnv df-co 3eqtr4i ) CFZDFZABGZHZDCIUEEFZAJZHZUHUDBJ
|
|
ZHZKZELZDCIUFJUKUIGUGUNDCUGUDUHBHZUHUEAHZKZELUNEUDUEABRUQUMEUQULUJKUMULUO
|
|
UJUPUHUDBMUEUHAMNULUJOPQSTDCUFUADCEUKUIUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $.
|
|
$( The converse of a class union is the (indexed) union of the converses of
|
|
its members. (Contributed by set.mm contributors, 11-Aug-2004.) $)
|
|
cnvuni $p |- `' U. A = U_ x e. A `' x $=
|
|
( vy vz vw cuni ccnv cv ciun wcel wrex cop wa elcnv2 eluni2 anbi2i bitr4i
|
|
wceq wex rexcom4 r19.42v 2exbii rexbii exbii 3bitrri 3bitri eliun eqriv )
|
|
CBFZGZABAHZGZIZCHZUJJZUNULJZABKZUNUMJUOUNDHZEHZLRZUSURLZUIJZMZESDSUTVAUKJ
|
|
ZMZABKZESZDSZUQDEUNUINVCVFDEVCUTVDABKZMVFVBVIUTAVABOPUTVDABUAQUBUQVEESZDS
|
|
ZABKVJABKZDSVHUPVKABDEUNUKNUCVJADBTVLVGDVEAEBTUDUEUFAUNBULUGQUH $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( Membership in a range. (Contributed by set.mm contributors,
|
|
2-Apr-2004.) $)
|
|
elrn $p |- ( A e. ran B <-> E. x x B A ) $=
|
|
( crn wcel cvv cima cv wbr wrex wex df-rn eleq2i elima rexv 3bitri ) BCDZ
|
|
EBCFGZEAHBCIZAFJSAKQRBCLMABCFNSAOP $.
|
|
|
|
$( Membership in a range. (Contributed by set.mm contributors,
|
|
10-Jul-1994.) $)
|
|
elrn2 $p |- ( A e. ran B <-> E. x <. x , A >. e. B ) $=
|
|
( crn wcel cv wbr wex cop elrn df-br exbii bitri ) BCDEAFZBCGZAHNBICEZAHA
|
|
BCJOPANBCKLM $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $.
|
|
$( Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by
|
|
set.mm contributors, 2-Apr-2004.) $)
|
|
eldm $p |- ( A e. dom B <-> E. y A B y ) $=
|
|
( cdm wcel cv ccnv wbr wex crn df-dm eleq2i elrn bitri brcnv exbii ) BCDZ
|
|
EZAFZBCGZHZAIZBSCHZAIRBTJZEUBQUDBCKLABTMNUAUCASBCOPN $.
|
|
|
|
$( Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by
|
|
set.mm contributors, 1-Aug-1994.) $)
|
|
eldm2 $p |- ( A e. dom B <-> E. y <. A , y >. e. B ) $=
|
|
( cdm wcel cv wbr wex cop eldm df-br exbii bitri ) BCDEBAFZCGZAHBNICEZAHA
|
|
BCJOPABNCKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of domain. (Contributed by set.mm contributors,
|
|
5-Feb-2015.) $)
|
|
dfdm2 $p |- dom A = { x | E. y x A y } $=
|
|
( cv wbr wex cdm eldm abbi2i ) ADZBDCEBFACGBJCHI $.
|
|
|
|
$( Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring]
|
|
p. 24. (Contributed by set.mm contributors, 28-Dec-1996.) $)
|
|
dfdm3 $p |- dom A = { x | E. y <. x , y >. e. A } $=
|
|
( cv cop wcel wex cdm eldm2 abbi2i ) ADZBDECFBGACHBKCIJ $.
|
|
|
|
$( Alternate definition of range. Definition 4 of [Suppes] p. 60.
|
|
(Contributed by set.mm contributors, 27-Dec-1996.) $)
|
|
dfrn2 $p |- ran A = { y | E. x x A y } $=
|
|
( cv wbr wex crn elrn abbi2i ) ADBDZCEAFBCGAJCHI $.
|
|
|
|
$( Alternate definition of range. Definition 6.5(2) of [TakeutiZaring]
|
|
p. 24. (Contributed by set.mm contributors, 28-Dec-1996.) $)
|
|
dfrn3 $p |- ran A = { y | E. x <. x , y >. e. A } $=
|
|
( crn cv wbr wex cab cop wcel dfrn2 df-br exbii abbii eqtri ) CDAEZBEZCFZ
|
|
AGZBHPQICJZAGZBHABCKSUABRTAPQCLMNO $.
|
|
|
|
$( Alternate definition of range. (Contributed by set.mm contributors,
|
|
5-Feb-2015.) $)
|
|
dfrn4 $p |- ran A = dom `' A $=
|
|
( vx vy crn ccnv cdm cv wbr wex wcel brcnv exbii eldm elrn 3bitr4ri eqriv
|
|
) BADZAEZFZBGZCGZRHZCIUATAHZCITSJTQJUBUCCTUAAKLCTRMCTANOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y w v $. $d w v A $.
|
|
dfdmf.1 $e |- F/_ x A $.
|
|
dfdmf.2 $e |- F/_ y A $.
|
|
$( Definition of domain, using bound-variable hypotheses instead of
|
|
distinct variable conditions. (Contributed by NM, 8-Mar-1995.)
|
|
(Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
dfdmf $p |- dom A = { x | E. y x A y } $=
|
|
( vw vv cdm cv wbr wex cab dfdm2 nfcv nfbr nfv breq2 cbvex abbii nfex weq
|
|
breq1 exbidv cbvab 3eqtri ) CHFIZGIZCJZGKZFLUFBIZCJZBKZFLAIZUJCJZBKZALFGC
|
|
MUIULFUHUKGBBUFUGCBUFNEBUGNOUKGPUGUJUFCQRSULUOFAUKABAUFUJCAUFNDAUJNOTUOFP
|
|
FAUAUKUNBUFUMUJCUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Subset theorem for domain. (Contributed by set.mm contributors,
|
|
11-Aug-1994.) $)
|
|
dmss $p |- ( A C_ B -> dom A C_ dom B ) $=
|
|
( vx vy wss cdm cv cop wcel wex ssel eximdv eldm2 3imtr4g ssrdv ) ABEZCAF
|
|
ZBFZPCGZDGHZAIZDJTBIZDJSQISRIPUAUBDABTKLDSAMDSBMNO $.
|
|
$}
|
|
|
|
$( Equality theorem for domain. (Contributed by set.mm contributors,
|
|
11-Aug-1994.) $)
|
|
dmeq $p |- ( A = B -> dom A = dom B ) $=
|
|
( wss wa cdm wceq dmss anim12i eqss 3imtr4i ) ABCZBACZDAEZBEZCZNMCZDABFMNFK
|
|
OLPABGBAGHABIMNIJ $.
|
|
|
|
${
|
|
dmeqi.1 $e |- A = B $.
|
|
$( Equality inference for domain. (Contributed by set.mm contributors,
|
|
4-Mar-2004.) $)
|
|
dmeqi $p |- dom A = dom B $=
|
|
( wceq cdm dmeq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
dmeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for domain. (Contributed by set.mm contributors,
|
|
4-Mar-2004.) $)
|
|
dmeqd $p |- ( ph -> dom A = dom B ) $=
|
|
( wceq cdm dmeq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $.
|
|
$( Membership of first of an ordered pair in a domain. (Contributed by
|
|
set.mm contributors, 30-Jul-1995.) $)
|
|
opeldm $p |- ( <. A , B >. e. C -> A e. dom C ) $=
|
|
( vy cop wcel wex cdm cvv elex opexb simprbi syl wceq opeq2 eleq1d spcegv
|
|
cv mpcom eldm2 sylibr ) ABEZCFZADRZEZCFZDGZACHFBIFZUCUGUCUBIFZUHUBCJUIAIF
|
|
UHABKLMUFUCDBIUDBNUEUBCUDBAOPQSDACTUA $.
|
|
$}
|
|
|
|
$( Membership of first of a binary relation in a domain. (Contributed by
|
|
set.mm contributors, 8-Jan-2015.) $)
|
|
breldm $p |- ( A R B -> A e. dom R ) $=
|
|
( wbr cop wcel cdm df-br opeldm sylbi ) ABCDABECFACGFABCHABCIJ $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The domain of a union is the union of domains. Exercise 56(a) of
|
|
[Enderton] p. 65. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by set.mm contributors, 12-Aug-1994.)
|
|
(Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
dmun $p |- dom ( A u. B ) = ( dom A u. dom B ) $=
|
|
( vx vy cun cdm cv cop wcel wex cab dfdm3 wo eldm orbi12i elun df-br brun
|
|
wbr bitr3i exbii 19.43 bitri 3bitr4i abbi2i eqtr4i ) ABEZFCGZDGZHUGIZDJZC
|
|
KAFZBFZEZCDUGLUKCUNUHULIZUHUMIZMUHUIASZDJZUHUIBSZDJZMZUHUNIUKUOURUPUTDUHA
|
|
NDUHBNOUHULUMPUKUQUSMZDJVAUJVBDUJUHUIUGSVBUHUIUGQUHUIABRTUAUQUSDUBUCUDUEU
|
|
F $.
|
|
|
|
$( The domain of an intersection belong to the intersection of domains.
|
|
Theorem 6 of [Suppes] p. 60. (Contributed by set.mm contributors,
|
|
15-Sep-2004.) $)
|
|
dmin $p |- dom ( A i^i B ) C_ ( dom A i^i dom B ) $=
|
|
( vx vy cin cdm cv cop wcel wa wex 19.40 eldm2 elin exbii anbi12i 3imtr4i
|
|
bitri ssriv ) CABEZFZAFZBFZEZCGZDGHZAIZUFBIZJZDKZUGDKZUHDKZJZUEUAIZUEUDIZ
|
|
UGUHDLUNUFTIZDKUJDUETMUPUIDUFABNORUOUEUBIZUEUCIZJUMUEUBUCNUQUKURULDUEAMDU
|
|
EBMPRQS $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
|
|
(Contributed by set.mm contributors, 3-Feb-2004.) $)
|
|
dmuni $p |- dom U. A = U_ x e. A dom x $=
|
|
( vy vz cuni cdm cv ciun cop wcel wex wrex eluni exbii excom 19.41v ancom
|
|
wa eldm2 bitri anbi1i bicomi 3bitri df-rex bitr4i eliun 3bitr4i eqriv ) C
|
|
BEZFZABAGZFZHZCGZDGIZUIJZDKZUNULJZABLZUNUJJUNUMJUQUKBJZURRZAKZUSUQUOUKJZU
|
|
TRZAKZDKVDDKZAKVBUPVEDAUOBMNVDDAOVFVAAVFVCDKZUTRZVAVCUTDPVAVHVAURUTRVHUTU
|
|
RQURVGUTDUNUKSUATUBTNUCURABUDUEDUNUISAUNBULUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The domain of a class of ordered pairs. (Contributed by NM,
|
|
16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.) $)
|
|
dmopab $p |- dom { <. x , y >. | ph } = { x | E. y ph } $=
|
|
( copab cdm wbr wex cab nfopab1 nfopab2 dfdmf cop wcel df-br opabid bitri
|
|
cv exbii abbii eqtri ) ABCDZEBQZCQZUAFZCGZBHACGZBHBCUAABCIABCJKUEUFBUDACU
|
|
DUBUCLUAMAUBUCUANABCOPRST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Upper bound for the domain of a restricted class of ordered pairs.
|
|
(Contributed by set.mm contributors, 31-Jan-2004.) $)
|
|
dmopabss $p |- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A $=
|
|
( cv wcel wa copab cdm wex cab dmopab 19.42v abbii ssab2 eqsstri ) BEDFZA
|
|
GZBCHIRCJZBKZDRBCLTQACJZGZBKDSUBBQACMNUABDOPP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The domain of a restricted class of ordered pairs. (Contributed by
|
|
set.mm contributors, 31-Jan-2004.) $)
|
|
dmopab3 $p |- ( A. x e. A E. y ph <->
|
|
dom { <. x , y >. | ( x e. A /\ ph ) } = A ) $=
|
|
( wex wral cv wcel wi wal wa wb copab cdm wceq df-ral pm4.71 albii dmopab
|
|
cab 19.42v abbii eqtri eqeq1i eqcom abeq2 3bitr2ri 3bitri ) ACEZBDFBGDHZU
|
|
IIZBJUJUJUIKZLZBJZUJAKZBCMNZDOZUIBDPUKUMBUJUIQRUQULBTZDODUROUNUPURDUPUOCE
|
|
ZBTURUOBCSUSULBUJACUAUBUCUDDURUEULBDUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1]
|
|
p. 36. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|
|
(Contributed by set.mm contributors, 4-Jul-1994.) (Revised by set.mm
|
|
contributors, 27-Aug-2011.) $)
|
|
dm0 $p |- dom (/) = (/) $=
|
|
( vx vy c0 cdm wceq cv wcel wn eq0 cop wex noel nex eldm2 mtbir mpgbir )
|
|
CDZCEAFZQGZHAAQISRBFJZCGZBKUABTLMBRCNOP $.
|
|
|
|
$( The domain of the identity relation is the universe. (The proof was
|
|
shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm
|
|
contributors, 30-Apr-1998.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
dmi $p |- dom _I = _V $=
|
|
( vx vy cid cdm cvv wceq cv wcel eqv wbr wex weq a9e vex ideq bitri exbii
|
|
equcom mpbir eldm mpgbir ) CDZEFAGZUBHZAAUBIUDUCBGZCJZBKZUGBALZBKBAMUFUHB
|
|
UFABLUHUCUEBNOABRPQSBUCCTSUA $.
|
|
|
|
$( The domain of the universe is the universe. (Contributed by set.mm
|
|
contributors, 8-Aug-2003.) $)
|
|
dmv $p |- dom _V = _V $=
|
|
( cvv cdm ssv cid dmi wss dmss ax-mp eqsstr3i eqssi ) ABZAKCADBZKEDAFLKFD
|
|
CDAGHIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( An empty domain implies an empty range. (Contributed by set.mm
|
|
contributors, 21-May-1998.) $)
|
|
dm0rn0 $p |- ( dom A = (/) <-> ran A = (/) ) $=
|
|
( vx vy cv wbr wex cab c0 wceq wcel wb wal alnex noel albii abeq1 3bitr4i
|
|
wn nbn eqeq1i cdm crn excom xchbinx bitr4i 3bitr3i dfdm2 dfrn2 ) BDZCDZAE
|
|
ZCFZBGZHIZUKBFZCGZHIZAUAZHIAUBZHIULUIHJZKZBLZUOUJHJZKZCLZUNUQULRZBLZUORZC
|
|
LZVBVEVGUOCFZRVIVGULBFVJULBMUKBCUCUDUOCMUEVFVABUTULUINSOVHVDCVCUOUJNSOUFU
|
|
LBHPUOCHPQURUMHBCAUGTUSUPHBCAUHTQ $.
|
|
|
|
$( A class is empty iff its domain is empty. (Contributed by set.mm
|
|
contributors, 15-Sep-2004.) (Revised by Scott Fenton, 17-Apr-2021.) $)
|
|
dmeq0 $p |- ( A = (/) <-> dom A = (/) ) $=
|
|
( vx vy cv cdm wcel wn wal cop c0 wb wceq wex eldm2 notbii alnex noel nbn
|
|
albii 3bitr2i eq0 eqrel 3bitr4ri ) BDZAEZFZGZBHUDCDIZAFZUHJFZKZCHZBHUEJLA
|
|
JLUGULBUGUICMZGUIGZCHULUFUMCUDANOUICPUNUKCUJUIUHQRSTSBUEUABCAJUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The domain of a cross product. Part of Theorem 3.13(x) of [Monk1]
|
|
p. 37. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|
|
(Contributed by set.mm contributors, 28-Jul-1995.) (Revised by set.mm
|
|
contributors, 27-Aug-2011.) $)
|
|
dmxp $p |- ( B =/= (/) -> dom ( A X. B ) = A ) $=
|
|
( vy vx c0 wne cxp cdm cv wcel wa copab df-xp dmeqi wex wral n0 ralrimivw
|
|
wceq biimpi dmopab3 sylib syl5eq ) BEFZABGZHCIAJDIBJZKCDLZHZAUEUGCDABMNUD
|
|
UFDOZCAPUHASUDUICAUDUIDBQTRUFCDAUAUBUC $.
|
|
$}
|
|
|
|
$( The domain of a square cross product. (Contributed by set.mm
|
|
contributors, 28-Jul-1995.) $)
|
|
dmxpid $p |- dom ( A X. A ) = A $=
|
|
( cxp cdm wceq c0 dm0 xpeq1 xp0r syl6eq dmeqd id 3eqtr4a dmxp pm2.61ine ) A
|
|
ABZCZADAEAEDZECEPAFQOEQOEABEAEAGAHIJQKLAAMN $.
|
|
|
|
$( The domain of the intersection of two square cross products. Unlike
|
|
~ dmin , equality holds. (Contributed by set.mm contributors,
|
|
29-Jan-2008.) $)
|
|
dmxpin $p |- dom ( ( A X. A ) i^i ( B X. B ) ) = ( A i^i B ) $=
|
|
( cxp cin cdm inxp dmeqi dmxpid eqtri ) AACBBCDZEABDZKCZEKJLAABBFGKHI $.
|
|
|
|
$( The cross product of a class with itself is one-to-one. (The proof was
|
|
shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm
|
|
contributors, 5-Nov-2006.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
xpid11 $p |- ( ( A X. A ) = ( B X. B ) <-> A = B ) $=
|
|
( cxp wceq cdm dmeq dmxpid 3eqtr3g xpeq12 anidms impbii ) AACZBBCZDZABDZNLE
|
|
MEABLMFAGBGHONABABIJK $.
|
|
|
|
$( The first member of an ordered pair in a class belongs to the domain
|
|
of the class. (Contributed by set.mm contributors, 28-Jul-2004.)
|
|
(Revised by Scott Fenton, 18-Apr-2021.) $)
|
|
proj1eldm $p |- ( B e. A -> Proj1 B e. dom A ) $=
|
|
( wcel cproj1 cproj2 cop cdm opeq eleq1i opeldm sylbi ) BACBDZBEZFZACLAGC
|
|
BNABHILMAJK $.
|
|
|
|
$( Equality theorem for restrictions. (Contributed by set.mm contributors,
|
|
7-Aug-1994.) $)
|
|
reseq1 $p |- ( A = B -> ( A |` C ) = ( B |` C ) ) $=
|
|
( wceq cvv cxp cin cres ineq1 df-res 3eqtr4g ) ABDACEFZGBLGACHBCHABLIACJBCJ
|
|
K $.
|
|
|
|
$( Equality theorem for restrictions. (Contributed by set.mm contributors,
|
|
8-Aug-1994.) $)
|
|
reseq2 $p |- ( A = B -> ( C |` A ) = ( C |` B ) ) $=
|
|
( wceq cvv cxp cin cres xpeq1 ineq2d df-res 3eqtr4g ) ABDZCAEFZGCBEFZGCAHCB
|
|
HMNOCABEIJCAKCBKL $.
|
|
|
|
${
|
|
reseqi.1 $e |- A = B $.
|
|
$( Equality inference for restrictions. (Contributed by set.mm
|
|
contributors, 21-Oct-2014.) $)
|
|
reseq1i $p |- ( A |` C ) = ( B |` C ) $=
|
|
( wceq cres reseq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality inference for restrictions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
reseq2i $p |- ( C |` A ) = ( C |` B ) $=
|
|
( wceq cres reseq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
|
|
reseqi.2 $e |- C = D $.
|
|
$( Equality inference for restrictions. (Contributed by set.mm
|
|
contributors, 21-Oct-2014.) $)
|
|
reseq12i $p |- ( A |` C ) = ( B |` D ) $=
|
|
( cres reseq1i reseq2i eqtri ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$}
|
|
|
|
${
|
|
reseqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for restrictions. (Contributed by set.mm
|
|
contributors, 21-Oct-2014.) $)
|
|
reseq1d $p |- ( ph -> ( A |` C ) = ( B |` C ) ) $=
|
|
( wceq cres reseq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality deduction for restrictions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
reseq2d $p |- ( ph -> ( C |` A ) = ( C |` B ) ) $=
|
|
( wceq cres reseq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
reseqd.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for restrictions. (Contributed by set.mm
|
|
contributors, 21-Oct-2014.) $)
|
|
reseq12d $p |- ( ph -> ( A |` C ) = ( B |` D ) ) $=
|
|
( cres reseq1d reseq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
nfres.1 $e |- F/_ x A $.
|
|
nfres.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for restriction. (Contributed by NM,
|
|
15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.) $)
|
|
nfres $p |- F/_ x ( A |` B ) $=
|
|
( cres cvv cxp cin df-res nfcv nfxp nfin nfcxfr ) ABCFBCGHZIBCJABODACGEAG
|
|
KLMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $.
|
|
$( Equality theorem for image. (Contributed by set.mm contributors,
|
|
14-Aug-1994.) $)
|
|
imaeq1 $p |- ( A = B -> ( A " C ) = ( B " C ) ) $=
|
|
( vy vx wceq cv wbr wrex cab cima breq rexbidv abbidv df-ima 3eqtr4g ) AB
|
|
FZDGZEGZAHZDCIZEJRSBHZDCIZEJACKBCKQUAUCEQTUBDCRSABLMNEDACOEDBCOP $.
|
|
|
|
$( Equality theorem for image. (Contributed by set.mm contributors,
|
|
14-Aug-1994.) $)
|
|
imaeq2 $p |- ( A = B -> ( C " A ) = ( C " B ) ) $=
|
|
( vy vx wceq cv wbr wrex cab cima rexeq abbidv df-ima 3eqtr4g ) ABFZDGEGC
|
|
HZDAIZEJQDBIZEJCAKCBKPRSEQDABLMEDCANEDCBNO $.
|
|
$}
|
|
|
|
${
|
|
imaeq1i.1 $e |- A = B $.
|
|
$( Equality theorem for image. (Contributed by set.mm contributors,
|
|
21-Dec-2008.) $)
|
|
imaeq1i $p |- ( A " C ) = ( B " C ) $=
|
|
( wceq cima imaeq1 ax-mp ) ABEACFBCFEDABCGH $.
|
|
|
|
$( Equality theorem for image. (Contributed by set.mm contributors,
|
|
21-Dec-2008.) $)
|
|
imaeq2i $p |- ( C " A ) = ( C " B ) $=
|
|
( wceq cima imaeq2 ax-mp ) ABECAFCBFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
imaeq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality theorem for image. (Contributed by FL, 15-Dec-2006.) $)
|
|
imaeq1d $p |- ( ph -> ( A " C ) = ( B " C ) ) $=
|
|
( wceq cima imaeq1 syl ) ABCFBDGCDGFEBCDHI $.
|
|
|
|
$( Equality theorem for image. (Contributed by FL, 15-Dec-2006.) $)
|
|
imaeq2d $p |- ( ph -> ( C " A ) = ( C " B ) ) $=
|
|
( wceq cima imaeq2 syl ) ABCFDBGDCGFEBCDHI $.
|
|
|
|
imaeq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality theorem for image. (Contributed by SF, 8-Jan-2018.) $)
|
|
imaeq12d $p |- ( ph -> ( A " C ) = ( B " D ) ) $=
|
|
( cima imaeq1d imaeq2d eqtrd ) ABDHCDHCEHABCDFIADECGJK $.
|
|
$}
|
|
|
|
${
|
|
$d A t x $. $d B t x $. $d C t x $.
|
|
$( Membership in an image under a unit power class. (Contributed by set.mm
|
|
contributors, 19-Feb-2015.) $)
|
|
elimapw1 $p |- ( A e. ( B " ~P1 C ) <-> E. x e. C <. { x } , A >. e. B ) $=
|
|
( vt cpw1 cima wcel cv wbr wrex csn cop elima wceq wa bitr4i bitri rexbii
|
|
wex df-rex elpw1 anbi1i r19.41v exbii rexcom4 snex breq1 ceqsexv df-br )
|
|
BCDFZGHEIZBCJZEUKKZAIZLZBMCHZADKZEBCUKNUNUPBCJZADKZURUNULUPOZUMPZETZADKZU
|
|
TUNULUKHZUMPZETZVDUMEUKUAVGVBADKZETVDVFVHEVFVAADKZUMPVHVEVIUMAULDUBUCVAUM
|
|
ADUDQUEVBAEDUFQRVCUSADUMUSEUPUOUGULUPBCUHUISRUSUQADUPBCUJSRR $.
|
|
|
|
$( Membership in an image under two unit power classes. (Contributed by
|
|
set.mm contributors, 18-Mar-2015.) $)
|
|
elimapw12 $p |- ( A e. ( B " ~P1 ~P1 C ) <->
|
|
E. x e. C <. { { x } } , A >. e. B ) $=
|
|
( vt cpw1 cima wcel cv csn cop wrex elimapw1 wex df-rex wceq elpw1 anbi1i
|
|
wa bitri r19.41v bitr4i exbii rexcom4 opeq1d eleq1d ceqsexv rexbii bitr3i
|
|
snex sneq ) BCDFZFGHEIZJZBKZCHZEULLZAIZJZJZBKZCHZADLZEBCULMUQUMULHZUPSZEN
|
|
ZVCUPEULOVFUMUSPZUPSZADLZENZVCVEVIEVEVGADLZUPSVIVDVKUPAUMDQRVGUPADUAUBUCV
|
|
JVHENZADLVCVHAEDUDVLVBADUPVBEUSURUJVGUOVACVGUNUTBUMUSUKUEUFUGUHUITTT $.
|
|
|
|
$( Membership in an image under three unit power classes. (Contributed by
|
|
set.mm contributors, 18-Mar-2015.) $)
|
|
elimapw13 $p |- ( A e. ( B " ~P1 ~P1 ~P1 C ) <->
|
|
E. x e. C <. { { { x } } } , A >. e. B ) $=
|
|
( vt cpw1 cima wcel cv csn cop wrex elimapw12 wa df-rex wceq elpw1 bitr4i
|
|
wex bitri anbi1i r19.41v exbii rexcom4 sneqd opeq1d eleq1d ceqsexv rexbii
|
|
snex sneq ) BCDFZFFGHEIZJZJZBKZCHZEULLZAIZJZJZJZBKZCHZADLZEBCULMURUMULHZU
|
|
QNZESZVEUQEULOVHUMUTPZUQNZESZADLZVEVHVJADLZESVLVGVMEVGVIADLZUQNVMVFVNUQAU
|
|
MDQUAVIUQADUBRUCVJAEDUDRVKVDADUQVDEUTUSUJVIUPVCCVIUOVBBVIUNVAUMUTUKUEUFUG
|
|
UHUITTT $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d A x t $. $d B x t $.
|
|
$( Membership in an image under cardinal one. (Contributed by set.mm
|
|
contributors, 6-Feb-2015.) $)
|
|
elima1c $p |- ( A e. ( B " 1c ) <-> E. x <. { x } , A >. e. B ) $=
|
|
( c1c cima wcel cvv cpw1 cv csn cop wrex wex imaeq2i eleq2i elimapw1 rexv
|
|
df1c2 3bitri ) BCDEZFBCGHZEZFAIJBKCFZAGLUCAMTUBBDUACRNOABCGPUCAQS $.
|
|
|
|
$( Membership in an image under the unit power class of cardinal one.
|
|
(Contributed by set.mm contributors, 25-Feb-2015.) $)
|
|
elimapw11c $p |- ( A e. ( B " ~P1 1c ) <-> E. x <. { { x } } , A >. e. B )
|
|
$=
|
|
( vt c1c cpw1 cima wcel cv csn cop wrex wex elimapw1 wa wceq df-rex exbii
|
|
el1c bitri anbi1i 19.41v bitr4i excom snex opeq1d eleq1d ceqsexv 3bitri
|
|
sneq ) BCEFGHDIZJZBKZCHZDELZAIZJZJZBKZCHZAMZDBCENUOUKEHZUNOZDMZUKUQPZUNOZ
|
|
DMZAMZVAUNDEQVDVFAMZDMVHVCVIDVCVEAMZUNOVIVBVJUNAUKSUAVEUNAUBUCRVFDAUDTVGU
|
|
TAUNUTDUQUPUEVEUMUSCVEULURBUKUQUJUFUGUHRUIT $.
|
|
$}
|
|
|
|
$( Binary relation on a restriction. (Contributed by set.mm contributors,
|
|
12-Dec-2006.) $)
|
|
brres $p |- ( A ( C |` D ) B <-> ( A C B /\ A e. D ) ) $=
|
|
( cres wbr cvv cxp cin wa wcel df-res breqi brin brex simprd adantr pm4.71i
|
|
anass brxp anbi2i 3bitr4ri 3bitri ) ABCDEZFABCDGHZIZFABCFZABUEFZJZUGADKZJZA
|
|
BUDUFCDLMABCUENUKBGKZJUGUJULJZJUKUIUGUJULSUKULUGULUJUGAGKULABCOPQRUHUMUGABD
|
|
GTUAUBUC $.
|
|
|
|
$( Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring]
|
|
p. 25. (Contributed by set.mm contributors, 13-Nov-1995.) $)
|
|
opelres $p |- ( <. A , B >. e. ( C |` D ) <->
|
|
( <. A , B >. e. C /\ A e. D ) ) $=
|
|
( cres wbr wcel wa cop brres df-br anbi1i 3bitr3i ) ABCDEZFABCFZADGZHABIZNG
|
|
QCGZPHABCDJABNKORPABCKLM $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Alternate definition of image. (Contributed by set.mm contributors,
|
|
19-Apr-2004.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
dfima3 $p |- ( A " B ) = ran ( A |` B ) $=
|
|
( vx vy cima cres crn cv cop wcel wex wa opelres ancom bitri exbii elima3
|
|
elrn2 3bitr4ri eqriv ) CABEZABFZGZDHZCHZIZUBJZDKUDBJZUFAJZLZDKUEUCJUEUAJU
|
|
GUJDUGUIUHLUJUDUEABMUIUHNOPDUEUBRDUEABQST $.
|
|
|
|
$( Alternate definition of image. Compare definition (d) of [Enderton]
|
|
p. 44. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|
|
(Contributed by set.mm contributors, 14-Aug-1994.) (Revised by set.mm
|
|
contributors, 27-Aug-2011.) $)
|
|
dfima4 $p |- ( A " B ) = { y | E. x ( x e. B /\ <. x , y >. e. A ) } $=
|
|
( cima cv wbr wrex cab wcel cop wa df-ima df-br rexbii df-rex bitri abbii
|
|
wex eqtri ) CDEAFZBFZCGZADHZBIUADJUAUBKCJZLASZBIBACDMUDUFBUDUEADHUFUCUEAD
|
|
UAUBCNOUEADPQRT $.
|
|
$}
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d x y z $.
|
|
nfima.1 $e |- F/_ x A $.
|
|
nfima.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for image. (Contributed by NM,
|
|
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
|
|
nfima $p |- F/_ x ( A " B ) $=
|
|
( vz vy cima cv wbr wrex cab df-ima nfcv nfbr nfrex nfab nfcxfr ) ABCHFIZ
|
|
GIZBJZFCKZGLGFBCMUBAGUAAFCEASTBASNDATNOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d B z $. $d A z $.
|
|
nfimad.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfimad.3 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nfima .
|
|
(Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
nfimad $p |- ( ph -> F/_ x ( A " B ) ) $=
|
|
( vz cv wcel wal cab cima wnfc nfaba1 nfima wb wa nfnfc1 nfan abidnf
|
|
imaeq1d imaeq2d sylan9eq nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUGDIZBJGKZL
|
|
ZMZBCDLZMZBUIUKUHBGNUJBGNOABCMZBDMZUMUOPEFUPUQQBULUNUPUQBBCRBDRSUPUQULCUK
|
|
LUNUPUICUKBGCTUAUQUKDCBGDTUBUCUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d C y $. $d x y $. $d F y $.
|
|
$( Move class substitution in and out of the image of a function.
|
|
(Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro,
|
|
4-Dec-2016.) $)
|
|
csbima12g $p |- ( A e. C -> [_ A / x ]_ ( F " B ) =
|
|
( [_ A / x ]_ F " [_ A / x ]_ B ) ) $=
|
|
( vy cv cima csb csbeq1 imaeq12d eqeq12d vex nfcsb1v nfima csbeq1a csbief
|
|
wceq vtoclg ) AFGZECHZIZATEIZATCIZHZRABUAIZABEIZABCIZHZRFBDTBRZUBUFUEUIAT
|
|
BUAJUJUCUGUDUHATBEJATBCJKLATUAUEFMAUCUDATENATCNOAGTREUCCUDATEPATCPKQS $.
|
|
$}
|
|
|
|
$( Equality theorem for range. (Contributed by set.mm contributors,
|
|
29-Dec-1996.) $)
|
|
rneq $p |- ( A = B -> ran A = ran B ) $=
|
|
( wceq cvv cima crn imaeq1 df-rn 3eqtr4g ) ABCADEBDEAFBFABDGAHBHI $.
|
|
|
|
${
|
|
rneqi.1 $e |- A = B $.
|
|
$( Equality inference for range. (Contributed by set.mm contributors,
|
|
4-Mar-2004.) $)
|
|
rneqi $p |- ran A = ran B $=
|
|
( wceq crn rneq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
rneqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for range. (Contributed by set.mm contributors,
|
|
4-Mar-2004.) $)
|
|
rneqd $p |- ( ph -> ran A = ran B ) $=
|
|
( wceq crn rneq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( Subset theorem for range. (Contributed by set.mm contributors,
|
|
22-Mar-1998.) $)
|
|
rnss $p |- ( A C_ B -> ran A C_ ran B ) $=
|
|
( wss ccnv cdm crn cnvss dmss syl dfrn4 3sstr4g ) ABCZADZEZBDZEZAFBFLMOCNPC
|
|
ABGMOHIAJBJK $.
|
|
|
|
$( The second argument of a binary relation belongs to its range.
|
|
(Contributed by set.mm contributors, 29-Jun-2008.) $)
|
|
brelrn $p |- ( A C B -> B e. ran C ) $=
|
|
( ccnv wbr cdm wcel crn breldm brcnv bicomi dfrn4 eleq2i 3imtr4i ) BACDZEZB
|
|
OFZGABCEZBCHZGBAOIPRBACJKSQBCLMN $.
|
|
|
|
$( Membership of second member of an ordered pair in a range. (Contributed
|
|
by set.mm contributors, 8-Jan-2015.) $)
|
|
opelrn $p |- ( <. A , B >. e. C -> B e. ran C ) $=
|
|
( cop wcel wbr crn df-br brelrn sylbir ) ABDCEABCFBCGEABCHABCIJ $.
|
|
|
|
${
|
|
$d x y w v $. $d w v A $.
|
|
dfrnf.1 $e |- F/_ x A $.
|
|
dfrnf.2 $e |- F/_ y A $.
|
|
$( Definition of range, using bound-variable hypotheses instead of distinct
|
|
variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by
|
|
Mario Carneiro, 15-Oct-2016.) $)
|
|
dfrnf $p |- ran A = { y | E. x x A y } $=
|
|
( vv vw crn cv wbr wex cab dfrn2 nfcv nfbr nfv breq1 cbvex abbii nfex
|
|
wceq breq2 exbidv cbvab 3eqtri ) CHFIZGIZCJZFKZGLAIZUGCJZAKZGLUJBIZCJZAKZ
|
|
BLFGCMUIULGUHUKFAAUFUGCAUFNDAUGNOUKFPUFUJUGCQRSULUOGBUKBABUJUGCBUJNEBUGNO
|
|
TUOGPUGUMUAUKUNAUGUMUJCUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
nfrn.1 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for range. (Contributed by NM,
|
|
1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfrn $p |- F/_ x ran A $=
|
|
( crn cvv cima df-rn nfcv nfima nfcxfr ) ABDBEFBGABECAEHIJ $.
|
|
|
|
$( Bound-variable hypothesis builder for domain. (Contributed by NM,
|
|
30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nfdm $p |- F/_ x dom A $=
|
|
( cdm ccnv crn df-dm nfcnv nfrn nfcxfr ) ABDBEZFBGAKABCHIJ $.
|
|
$}
|
|
|
|
$( Domain of an intersection. (Contributed by FL, 15-Oct-2012.) $)
|
|
dmiin $p |- dom |^|_ x e. A B C_ |^|_ x e. A dom B $=
|
|
( ciin cdm wss nfii1 nfdm ssiinf cv wcel iinss2 dmss syl mprgbir ) ABCDZEZA
|
|
BCEZDFQRFZABABRQAPABCGHIAJBKPCFSABCLPCMNO $.
|
|
|
|
${
|
|
$d A w y $. $d B w y $. $d V w y $. $d x w y $.
|
|
$( Distribute proper substitution through the range of a class.
|
|
(Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbrng $p |- ( A e. V -> [_ A / x ]_ ran B = ran [_ A / x ]_ B ) $=
|
|
( vw vy wcel cv cop wex cab csb crn wsbc csbabg sbcexg exbidv bitrd dfrn3
|
|
sbcel2g abbidv eqtrd csbeq2i 3eqtr4g ) BDGZABEHFHIZCGZEJZFKZLZUFABCLZGZEJ
|
|
ZFKZABCMZLUKMUEUJUHABNZFKUNUHAFBDOUEUPUMFUEUPUGABNZEJUMUGEABDPUEUQULEABUF
|
|
CDTQRUAUBABUOUIEFCSUCEFUKSUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The range of a class of ordered pairs. (Contributed by NM,
|
|
14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.) $)
|
|
rnopab $p |- ran { <. x , y >. | ph } = { y | E. x ph } $=
|
|
( copab crn wbr wex cab nfopab1 nfopab2 dfrnf cop wcel df-br opabid bitri
|
|
cv exbii abbii eqtri ) ABCDZEBQZCQZUAFZBGZCHABGZCHBCUAABCIABCJKUEUFCUDABU
|
|
DUBUCLUAMAUBUCUANABCOPRST $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The range of a function expressed as a class abstraction. (Contributed
|
|
by set.mm contributors, 23-Mar-2006.) $)
|
|
rnopab2 $p |- ran { <. x , y >. | ( x e. A /\ y = B ) } =
|
|
{ y | E. x e. A y = B } $=
|
|
( cv wcel wceq wa copab crn wex cab wrex rnopab df-rex abbii eqtr4i ) AEC
|
|
FBEDGZHZABIJSAKZBLRACMZBLSABNUATBRACOPQ $.
|
|
$}
|
|
|
|
$( The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1]
|
|
p. 36. (Contributed by set.mm contributors, 4-Jul-1994.) $)
|
|
rn0 $p |- ran (/) = (/) $=
|
|
( c0 cdm wceq crn dm0 dm0rn0 mpbi ) ABACADACEAFG $.
|
|
|
|
$( A relation is empty iff its range is empty. (Contributed by set.mm
|
|
contributors, 15-Sep-2004.) (Revised by Scott Fenton, 17-Apr-2021.) $)
|
|
rneq0 $p |- ( A = (/) <-> ran A = (/) ) $=
|
|
( c0 wceq cdm crn dmeq0 dm0rn0 bitri ) ABCADBCAEBCAFAGH $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( Domain of a composition. Theorem 21 of [Suppes] p. 63. (The proof was
|
|
shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm
|
|
contributors, 19-Mar-1998.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
dmcoss $p |- dom ( A o. B ) C_ dom B $=
|
|
( vx vy vz ccom cdm cv wbr wex wcel wa brco exbii excom bitri simpl eximi
|
|
exlimiv eldm sylbi 3imtr4i ssriv ) CABFZGZBGZCHZDHZUDIZDJZUGEHZBIZEJZUGUE
|
|
KUGUFKUJULUKUHAIZLZDJZEJZUMUJUOEJZDJUQUIURDEUGUHABMNUODEOPUPULEUOULDULUNQ
|
|
SRUADUGUDTEUGBTUBUC $.
|
|
$}
|
|
|
|
$( Range of a composition. (Contributed by set.mm contributors,
|
|
19-Mar-1998.) $)
|
|
rncoss $p |- ran ( A o. B ) C_ ran A $=
|
|
( ccnv ccom cdm crn dmcoss dfrn4 cnvco dmeqi eqtri 3sstr4i ) BCZACZDZEZNEAB
|
|
DZFZAFMNGRQCZEPQHSOABIJKAHL $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( Domain of a composition. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by set.mm contributors, 28-May-1998.)
|
|
(Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
dmcosseq $p |- ( ran B C_ dom A -> dom ( A o. B ) = dom B ) $=
|
|
( vx vy vz crn cdm wss ccom dmcoss a1i cv wbr wcel wa brelrn ssel syl6ibr
|
|
wex eldm syl6ib ancld 19.42v eximdv exbii excom bitri 3imtr4g ssrdv eqssd
|
|
syl5 brco ) BFZAGZHZABIZGZBGZUQURHUOABJKUOCURUQUOCLZDLZBMZDSZUSELZUPMZESZ
|
|
USURNUSUQNUOVBVAUTVCAMZOZESZDSZVEUOVAVHDUOVAVAVFESZOVHUOVAVJUOVAUTUNNZVJV
|
|
AUTUMNUOVKUSUTBPUMUNUTQUKEUTATUAUBVAVFEUCRUDVEVGDSZESVIVDVLEDUSVCABULUEVG
|
|
EDUFUGRDUSBTEUSUPTUHUIUJ $.
|
|
|
|
$( Domain of a composition. (Contributed by set.mm contributors,
|
|
19-Mar-1998.) $)
|
|
dmcoeq $p |- ( dom A = ran B -> dom ( A o. B ) = dom B ) $=
|
|
( cdm crn wceq wss ccom eqimss2 dmcosseq syl ) ACZBDZELKFABGCBCELKHABIJ
|
|
$.
|
|
$}
|
|
|
|
$( Range of a composition. (Contributed by set.mm contributors,
|
|
19-Mar-1998.) $)
|
|
rncoeq $p |- ( dom A = ran B -> ran ( A o. B ) = ran A ) $=
|
|
( ccnv cdm crn wceq ccom dmcoeq df-dm dfrn4 eqeq12i eqcom bitri cnvco dmeqi
|
|
eqtri 3imtr4i ) BCZDZACZEZFZRTGZDZTDZFADZBEZFZABGZEZAEZFRTHUHUASFUBUFUAUGSA
|
|
IBJKUASLMUJUDUKUEUJUICZDUDUIJULUCABNOPAJKQ $.
|
|
|
|
$( Distribute proper substitution through the restriction of a class.
|
|
~ csbresg is derived from the virtual deduction proof csbresgVD in
|
|
set.mm. (Contributed by Alan Sare, 10-Nov-2012.) $)
|
|
csbresg $p |- ( A e. V -> [_ A / x ]_ ( B |` C ) =
|
|
( [_ A / x ]_ B |` [_ A / x ]_ C ) ) $=
|
|
( wcel cvv cxp cin cres csbing csbxpg csbconstg xpeq2d eqtrd ineq2d csbeq2i
|
|
csb df-res 3eqtr4g ) BEFZABCDGHZIZRZABCRZABDRZGHZIZABCDJZRUEUFJUAUDUEABUBRZ
|
|
IUHABECUBKUAUJUGUEUAUJUFABGRZHUGABDGELUAUKGUFABGEMNOPOABUIUCCDSQUEUFST $.
|
|
|
|
$( A restriction to the empty set is empty. (Contributed by set.mm
|
|
contributors, 12-Nov-1994.) $)
|
|
res0 $p |- ( A |` (/) ) = (/) $=
|
|
( c0 cres cvv cxp cin df-res xp0r ineq2i in0 3eqtri ) ABCABDEZFABFBABGLBADH
|
|
IAJK $.
|
|
|
|
$( Ordered pair membership in a restriction when the first member belongs to
|
|
the restricting class. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by set.mm contributors, 30-Apr-2004.)
|
|
(Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
opres $p |- ( A e. D ->
|
|
( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) ) $=
|
|
( wcel cop wa cres iba opelres syl6rbbr ) ADEZABFZCEZNLGMCDHELNIABCDJK $.
|
|
|
|
$( A restricted identity relation is equivalent to equality in its domain.
|
|
(Contributed by set.mm contributors, 30-Apr-2004.) $)
|
|
resieq $p |- ( B e. A -> ( B ( _I |` A ) C <-> B = C ) ) $=
|
|
( cid cres wbr wcel wa wceq brres iba ideqg2 bitr3d syl5bb ) BCDAEFBCDFZBAG
|
|
ZHZPBCIZBCDAJPOQRPOKBCALMN $.
|
|
|
|
$( The restriction of a restriction. (Contributed by set.mm contributors,
|
|
27-Mar-2008.) $)
|
|
resres $p |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) ) $=
|
|
( cres cvv cxp cin df-res ineq1i xpindir ineq2i inass 3eqtr4ri 3eqtri ) ABD
|
|
ZCDOCEFZGABEFZGZPGZABCGZDZOCHORPABHIATEFZGAQPGZGUASUBUCABCEJKATHAQPLMN $.
|
|
|
|
$( Distributive law for restriction over union. Theorem 31 of [Suppes]
|
|
p. 65. (Contributed by set.mm contributors, 30-Sep-2002.) $)
|
|
resundi $p |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) ) $=
|
|
( cun cvv cxp cin cres xpundir ineq2i indi eqtri df-res uneq12i 3eqtr4i ) A
|
|
BCDZEFZGZABEFZGZACEFZGZDZAPHABHZACHZDRASUADZGUCQUFABCEIJASUAKLAPMUDTUEUBABM
|
|
ACMNO $.
|
|
|
|
$( Distributive law for restriction over union. (Contributed by set.mm
|
|
contributors, 23-Sep-2004.) $)
|
|
resundir $p |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) ) $=
|
|
( cun cvv cxp cin cres indir df-res uneq12i 3eqtr4i ) ABDZCEFZGANGZBNGZDMCH
|
|
ACHZBCHZDABNIMCJQORPACJBCJKL $.
|
|
|
|
$( Class restriction distributes over intersection. (Contributed by FL,
|
|
6-Oct-2008.) $)
|
|
resindi $p |- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) ) $=
|
|
( cin cvv cxp cres xpindir ineq2i inindi eqtri df-res ineq12i 3eqtr4i ) ABC
|
|
DZEFZDZABEFZDZACEFZDZDZAOGABGZACGZDQARTDZDUBPUEABCEHIARTJKAOLUCSUDUAABLACLM
|
|
N $.
|
|
|
|
$( Class restriction distributes over intersection. (Contributed by set.mm
|
|
contributors, 18-Dec-2008.) $)
|
|
resindir $p |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) ) $=
|
|
( cin cvv cxp cres inindir df-res ineq12i 3eqtr4i ) ABDZCEFZDAMDZBMDZDLCGAC
|
|
GZBCGZDABMHLCIPNQOACIBCIJK $.
|
|
|
|
$( Move intersection into class restriction. (Contributed by set.mm
|
|
contributors, 18-Dec-2008.) $)
|
|
inres $p |- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C ) $=
|
|
( cin cvv cxp cres inass df-res ineq2i 3eqtr4ri ) ABDZCEFZDABMDZDLCGABCGZDA
|
|
BMHLCIONABCIJK $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25.
|
|
(Contributed by set.mm contributors, 1-Aug-1994.) $)
|
|
dmres $p |- dom ( A |` B ) = ( B i^i dom A ) $=
|
|
( vx vy cdm cin cres cv wbr wcel wex 19.41v eldm brres exbii bitri anbi1i
|
|
wa 3bitr4ri ineqri incom eqtr3i ) AEZBFABGZEZBUCFCUCBUECHZDHZAIZUFBJZRZDK
|
|
ZUHDKZUIRUFUEJZUFUCJZUIRUHUIDLUMUFUGUDIZDKUKDUFUDMUOUJDUFUGABNOPUNULUIDUF
|
|
AMQSTUCBUAUB $.
|
|
$}
|
|
|
|
$( A domain restricted to a subclass equals the subclass. (Contributed by
|
|
set.mm contributors, 2-Mar-1997.) (Revised by set.mm contributors,
|
|
28-Aug-2004.) $)
|
|
ssdmres $p |- ( A C_ dom B <-> dom ( B |` A ) = A ) $=
|
|
( cdm wss cin wceq cres df-ss dmres eqeq1i bitr4i ) ABCZDALEZAFBAGCZAFALHNM
|
|
ABAIJK $.
|
|
|
|
$( A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25.
|
|
(Contributed by set.mm contributors, 2-Aug-1994.) $)
|
|
resss $p |- ( A |` B ) C_ A $=
|
|
( cres cvv cxp cin df-res inss1 eqsstri ) ABCABDEZFAABGAJHI $.
|
|
|
|
$( Commutative law for restriction. (Contributed by set.mm contributors,
|
|
27-Mar-1998.) $)
|
|
rescom $p |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B ) $=
|
|
( cin cres incom reseq2i resres 3eqtr4i ) ABCDZEACBDZEABECEACEBEJKABCFGABCH
|
|
ACBHI $.
|
|
|
|
$( Subclass theorem for restriction. (Contributed by set.mm contributors,
|
|
16-Aug-1994.) $)
|
|
ssres $p |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) ) $=
|
|
( wss cvv cxp cin cres ssrin df-res 3sstr4g ) ABDACEFZGBLGACHBCHABLIACJBCJK
|
|
$.
|
|
|
|
$( Subclass theorem for restriction. (The proof was shortened by Andrew
|
|
Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 22-Mar-1998.)
|
|
(Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
ssres2 $p |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) ) $=
|
|
( wss cvv cxp cin cres xpss1 sslin syl df-res 3sstr4g ) ABDZCAEFZGZCBEFZGZC
|
|
AHCBHNOQDPRDABEIOQCJKCALCBLM $.
|
|
|
|
$( Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.
|
|
(Contributed by set.mm contributors, 9-Aug-1994.) $)
|
|
resabs1 $p |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) ) $=
|
|
( wss cres cin resres wceq df-ss incom eqeq1i bitri biimpi reseq2d syl5eq )
|
|
BCDZACEBEACBFZEABEACBGPQBAPQBHZPBCFZBHRBCISQBBCJKLMNO $.
|
|
|
|
$( Absorption law for restriction. (Contributed by set.mm contributors,
|
|
27-Mar-1998.) $)
|
|
resabs2 $p |- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) ) $=
|
|
( wss cres rescom resabs1 syl5eq ) BCDABEZCEACEBEIABCFABCGH $.
|
|
|
|
$( Idempotent law for restriction. (Contributed by set.mm contributors,
|
|
27-Mar-1998.) $)
|
|
residm $p |- ( ( A |` B ) |` B ) = ( A |` B ) $=
|
|
( wss cres wceq ssid resabs2 ax-mp ) BBCABDZBDIEBFABBGH $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Membership in a restriction. (Contributed by Scott Fenton,
|
|
17-Mar-2011.) $)
|
|
elres $p |- ( A e. ( B |` C )
|
|
<-> E. x e. C E. y ( A = <. x , y >. /\ <. x , y >. e. B ) ) $=
|
|
( cv cop wceq cres wcel wex wrex eleq1 opelres ancom bitri syl6bb pm5.32i
|
|
wa bitr4i an12 2exbii opeqex pm4.71ri 19.41vv df-rex exdistr 3bitr4i ) CA
|
|
FZBFZGZHZCDEIZJZSZBKAKZUIEJZULUKDJZSZSZBKAKZUNUSBKZAELZUOUTABUOULUQURSZSU
|
|
TULUNVDULUNUKUMJZVDCUKUMMVEURUQSVDUIUJDENURUQOPQRULUQURUAPUBUNULBKAKZUNSU
|
|
PUNVFABCUMUCUDULUNABUETVCUQVBSAKVAVBAEUFUQUSABUGTUH $.
|
|
|
|
${
|
|
elsnres.1 $e |- C e. _V $.
|
|
$( Memebership in restriction to a singleton. (Contributed by Scott
|
|
Fenton, 17-Mar-2011.) $)
|
|
elsnres $p |- ( A e. ( B |` { C } )
|
|
<-> E. y ( A = <. C , y >. /\ <. C , y >. e. B ) ) $=
|
|
( vx csn cres wcel cv cop wceq wa wex elres rexcom4 opeq1 eqeq2d eleq1d
|
|
wrex anbi12d rexsn exbii 3bitri ) BCDGZHIBFJZAJZKZLZUHCIZMZANFUETUKFUET
|
|
ZANBDUGKZLZUMCIZMZANFABCUEOUKFAUEPULUPAUKUPFDEUFDLZUIUNUJUOUQUHUMBUFDUG
|
|
QZRUQUHUMCURSUAUBUCUD $.
|
|
$}
|
|
|
|
$( Simplification law for restriction. (Contributed by set.mm
|
|
contributors, 16-Aug-1994.) (Revised by set.mm contributors,
|
|
15-Mar-2004.) (Revised by Scott Fenton, 18-Apr-2021.) $)
|
|
ssreseq $p |- ( dom A C_ B -> ( A |` B ) = A ) $=
|
|
( vx vy cdm wss cres resss a1i cv cop wcel opeldm ssel syl5 ancld opelres
|
|
wa syl6ibr relssdv eqssd ) AEZBFZABGZAUDAFUCABHIUCCDAUDUCCJZDJZKZALZUHUEB
|
|
LZRUGUDLUCUHUIUHUEUBLUCUIUEUFAMUBBUENOPUEUFABQSTUA $.
|
|
$}
|
|
|
|
$( A class restricted to its domain equals itself. (Contributed by set.mm
|
|
contributors, 12-Dec-2006.) (Revised by Scott Fenton, 18-Apr-2021.) $)
|
|
resdm $p |- ( A |` dom A ) = A $=
|
|
( cdm wss cres wceq ssid ssreseq ax-mp ) ABZICAIDAEIFAIGH $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Restriction of a class abstraction of ordered pairs. (Contributed by
|
|
set.mm contributors, 5-Nov-2002.) $)
|
|
resopab $p |- ( { <. x , y >. | ph } |` A ) =
|
|
{ <. x , y >. | ( x e. A /\ ph ) } $=
|
|
( copab cres cvv cxp cin cv wa df-res df-xp biantru opabbii eqtr4i ineq2i
|
|
wcel vex incom eqtri inopab 3eqtri ) ABCEZDFUDDGHZIZBJDRZBCEZUDIZUGAKBCEU
|
|
DDLUFUDUHIUIUEUHUDUEUGCJGRZKZBCEUHBCDGMUGUKBCUJUGCSNOPQUDUHTUAUGABCUBUC
|
|
$.
|
|
|
|
$( A subclass of the identity function is the identity function restricted
|
|
to its domain. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by set.mm contributors, 13-Dec-2003.)
|
|
(Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
iss $p |- ( A C_ _I <-> A = ( _I |` dom A ) ) $=
|
|
( vx vy cid wss cdm cres wceq cv cop wcel wa ssel opeldm a1i jcad weq wbr
|
|
wi syl5bi df-br vex ideq bitr3i anbi1i wex eldm2 syl6ib opeq2 eleq1d syli
|
|
biimprd exlimdv biimpd imp3a impbid opelres syl6bbr eqrelrdv resss mpbiri
|
|
syl9 sseq1 impbii ) ADEZADAFZGZHZVEBCAVGVEBIZCIZJZAKZVKDKZVIVFKZLZVKVGKVE
|
|
VLVOVEVLVMVNADVKMZVLVNSVEVIVJANOPVOBCQZVNLVEVLVMVQVNVMVIVJDRVQVIVJDUAVIVJ
|
|
CUBUCUDZUEVEVQVNVLVEVNVIVIJZAKZVQVLVNVLCUFVEVTCVIAUGVEVLVTCVLVEVQVTVEVLVM
|
|
VQVPVRUHVQVTVLVQVSVKAVIVJVIUIUJZULUKUMTVQVTVLWAUNVBUOTUPVIVJDVFUQURUSVHVE
|
|
VGDEDVFUTAVGDVCVAVD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Restriction of a class abstraction of ordered pairs. (Contributed by
|
|
set.mm contributors, 24-Aug-2007.) $)
|
|
resopab2 $p |- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) =
|
|
{ <. x , y >. | ( x e. A /\ ph ) } ) $=
|
|
( wss cv wcel wa copab cres resopab wi wb ssel pm4.71 sylib anass syl6rbb
|
|
anbi1d opabbidv syl5eq ) DEFZBGZEHZAIZBCJDKUDDHZUFIZBCJUGAIZBCJUFBCDLUCUH
|
|
UIBCUCUIUGUEIZAIUHUCUGUJAUCUGUEMUGUJNDEUDOUGUEPQTUGUEARSUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z R $.
|
|
$( Alternate definition of the restriction operation. (Contributed by
|
|
Mario Carneiro, 5-Nov-2013.) $)
|
|
dfres2 $p |- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) } $=
|
|
( vz vw cres cv wbr wa copab cop vex weq eleq1 breq1 anbi12d breq2 anbi2d
|
|
wcel opelopab brres ancom bitri df-br 3bitr2ri eqrelriv ) EFDCGZAHZCTZUIB
|
|
HZDIZJZABKZEHZFHZLZUNTUOCTZUOUPDIZJZUOUPUHIZUQUHTUMURUOUKDIZJUTABUOUPEMFM
|
|
AENUJURULVBUIUOCOUIUOUKDPQBFNVBUSURUKUPUODRSUAVAUSURJUTUOUPDCUBUSURUCUDUO
|
|
UPUHUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The restricted identity expressed with the class builder. (Contributed
|
|
by FL, 25-Apr-2012.) $)
|
|
opabresid $p |- { <. x , y >. | ( x e. A /\ y = x ) } = ( _I |` A ) $=
|
|
( weq copab cres cv wcel cid resopab equcom opabbii eqtr4i reseq1i eqtr3i
|
|
wa dfid3 ) BADZABEZCFAGCHRPABEICFRABCJSICSABDZABEIRTABBAKLABQMNO $.
|
|
$}
|
|
|
|
$( The domain of a restricted identity function. (Contributed by set.mm
|
|
contributors, 27-Aug-2004.) $)
|
|
dmresi $p |- dom ( _I |` A ) = A $=
|
|
( cid cdm wss cres wceq cvv ssv dmi sseqtr4i ssdmres mpbi ) ABCZDBAECAFAGMA
|
|
HIJABKL $.
|
|
|
|
$( Any class restricted to the universe is itself. (Contributed by set.mm
|
|
contributors, 16-Mar-2004.) (Revised by Scott Fenton, 18-Apr-2021.) $)
|
|
resid $p |- ( A |` _V ) = A $=
|
|
( cdm cvv wss cres wceq ssv ssreseq ax-mp ) ABZCDACEAFJGACHI $.
|
|
|
|
$( A restriction to an image. (Contributed by set.mm contributors,
|
|
29-Sep-2004.) $)
|
|
resima $p |- ( ( A |` B ) " B ) = ( A " B ) $=
|
|
( cres crn cima residm rneqi dfima3 3eqtr4i ) ABCZBCZDJDJBEABEKJABFGJBHABHI
|
|
$.
|
|
|
|
$( Image under a restricted class. (Contributed by FL, 31-Aug-2009.) $)
|
|
resima2 $p |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) ) $=
|
|
( wss cin cres crn cima wceq sseqin2 biimpi rneqd dfima3 resres rneqi eqtri
|
|
reseq2d 3eqtr4g ) BCDZACBEZFZGZABFZGACFZBHZABHSUAUCSTBASTBIBCJKQLUEUDBFZGUB
|
|
UDBMUFUAACBNOPABMR $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The image of the domain of a class is the range of the class.
|
|
(Contributed by set.mm contributors, 14-Aug-1994.) $)
|
|
imadmrn $p |- ( A " dom A ) = ran A $=
|
|
( vx vy cv wbr cdm wrex cab wex cima wcel wa df-rex breldm pm4.71ri exbii
|
|
crn bitr4i abbii df-ima dfrn2 3eqtr4i ) BDZCDZAEZBAFZGZCHUEBIZCHAUFJAQUGU
|
|
HCUGUCUFKZUELZBIUHUEBUFMUEUJBUEUIUCUDANOPRSCBAUFTBCAUAUB $.
|
|
|
|
$( The image of a class is a subset of its range. Theorem 3.16(xi) of
|
|
[Monk1] p. 39. (Contributed by set.mm contributors, 31-Mar-1995.) $)
|
|
imassrn $p |- ( A " B ) C_ ran A $=
|
|
( vx vy cv wcel cop wa wex cab cima crn simpr eximi ss2abi dfima4 3sstr4i
|
|
dfrn3 ) CEZBFZSDEGAFZHZCIZDJUACIZDJABKALUCUDDUBUACTUAMNOCDABPCDARQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Image under the identity relation. Theorem 3.16(viii) of [Monk1]
|
|
p. 38. (Contributed by set.mm contributors, 30-Apr-1998.) $)
|
|
imai $p |- ( _I " A ) = A $=
|
|
( vx vy cid cima cv wcel cop wa wex cab dfima4 wceq wbr df-br ideq bitr3i
|
|
vex anbi2i bitri ancom exbii eleq1 ceqsexv abbii abid2 3eqtri ) DAEBFZAGZ
|
|
UHCFZHDGZIZBJZCKUJAGZCKABCDALUMUNCUMUHUJMZUIIZBJUNULUPBULUIUOIUPUKUOUIUKU
|
|
HUJDNUOUHUJDOUHUJCRZPQSUIUOUATUBUIUNBUJUQUHUJAUCUDTUECAUFUG $.
|
|
$}
|
|
|
|
$( The range of the restricted identity function. (Contributed by set.mm
|
|
contributors, 27-Aug-2004.) $)
|
|
rnresi $p |- ran ( _I |` A ) = A $=
|
|
( cid cima cres crn dfima3 imai eqtr3i ) BACBADEABAFAGH $.
|
|
|
|
$( The image of a restriction of the identity function. (Contributed by FL,
|
|
31-Dec-2006.) $)
|
|
resiima $p |- ( B C_ A -> ( ( _I |` A ) " B ) = B ) $=
|
|
( wss cid cres cima crn dfima3 resabs1 rneqd rnresi syl6eq syl5eq ) BACZDAE
|
|
ZBFOBEZGZBOBHNQDBEZGBNPRDBAIJBKLM $.
|
|
|
|
$( Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed
|
|
by set.mm contributors, 20-May-1998.) $)
|
|
ima0 $p |- ( A " (/) ) = (/) $=
|
|
( c0 cima cres crn dfima3 res0 rneqi rn0 3eqtri ) ABCABDZEBEBABFKBAGHIJ $.
|
|
|
|
$( Image under the empty relation. (Contributed by FL, 11-Jan-2007.) $)
|
|
0ima $p |- ( (/) " A ) = (/) $=
|
|
( c0 cima crn imassrn rn0 sseqtri 0ss eqssi ) BACZBJBDBBAEFGJHI $.
|
|
|
|
$( A class whose image under another is empty is disjoint with the other's
|
|
domain. (Contributed by FL, 24-Jan-2007.) $)
|
|
imadisj $p |- ( ( A " B ) = (/) <-> ( dom A i^i B ) = (/) ) $=
|
|
( cima wceq cres crn cdm cin dfima3 eqeq1i dm0rn0 dmres incom eqtri 3bitr2i
|
|
c0 ) ABCZPDABEZFZPDRGZPDAGZBHZPDQSPABIJRKTUBPTBUAHUBABLBUAMNJO $.
|
|
|
|
$( A preimage under any class is included in the domain of the class.
|
|
(Contributed by FL, 29-Jan-2007.) $)
|
|
cnvimass $p |- ( `' A " B ) C_ dom A $=
|
|
( ccnv cima crn cdm imassrn df-dm sseqtr4i ) ACZBDJEAFJBGAHI $.
|
|
|
|
$( The preimage of the range of a class is the domain of the class.
|
|
(Contributed by Jeff Hankins, 15-Jul-2009.) $)
|
|
cnvimarndm $p |- ( `' A " ran A ) = dom A $=
|
|
( ccnv cdm cima crn imadmrn dfrn4 imaeq2i df-dm 3eqtr4i ) ABZKCZDKEKAEZDACK
|
|
FMLKAGHAIJ $.
|
|
|
|
${
|
|
$d A x y $. $d R x y $.
|
|
$( The image of a singleton. (Contributed by set.mm contributors,
|
|
9-Jan-2015.) $)
|
|
imasn $p |- ( R " { A } ) = { y | A R y } $=
|
|
( vx cvv wcel csn cima cv wbr wceq wrex df-ima breq1 rexsng abbidv syl5eq
|
|
cab wn c0 ima0 snprc biimpi imaeq2d wex brex exlimiv con3i wne abn0 df-ne
|
|
simpld bitr3i con2bii sylibr 3eqtr4a pm2.61i ) BEFZCBGZHZBAIZCJZARZKURUTD
|
|
IZVACJZDUSLZARVCADCUSMURVFVBAVEVBDBEVDBVACNOPQURSZCTHTUTVCCUAVGUSTCVGUSTK
|
|
BUBUCUDVGVBAUEZSVCTKZVHURVBURAVBURVAEFBVACUFULUGUHVHVIVHVCTUIVISVBAUJVCTU
|
|
KUMUNUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $.
|
|
$( Membership in an image of a singleton. (The proof was shortened by
|
|
Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors,
|
|
15-Mar-2004.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
elimasn $p |- ( C e. ( A " { B } ) <-> <. B , C >. e. A ) $=
|
|
( vx csn cima wcel cvv cop elex wbr df-br brex simprd sylbir cv cab breq2
|
|
elabg imasn eleq2i bicomi 3bitr4g pm5.21nii ) CABEFZGZCHGZBCIAGZCUEJUHBCA
|
|
KZUGBCALZUIBHGUGBCAMNOUGCBDPZAKZDQZGUIUFUHULUIDCHUKCBARSUEUMCDBATUAUIUHUJ
|
|
UBUCUD $.
|
|
$}
|
|
|
|
$( Membership in an initial segment. The idiom ` ( ``' A " { B } ) ` ,
|
|
meaning ` { x | x A B } ` , is used to specify an initial segment in (for
|
|
example) Definition 6.21 of [TakeutiZaring] p. 30. (The proof was
|
|
shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm
|
|
contributors, 28-Apr-2004.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
eliniseg $p |- ( C e. ( `' A " { B } ) <-> C A B ) $=
|
|
( ccnv csn cima wcel cop wbr elimasn df-br brcnv 3bitr2i ) CADZBEFGBCHNGBCN
|
|
ICBAINBCJBCNKBCALM $.
|
|
|
|
${
|
|
$d A x $.
|
|
epini.1 $e |- A e. _V $.
|
|
$( Any set is equal to its preimage under the converse epsilon relation.
|
|
(Contributed by Mario Carneiro, 9-Mar-2013.) $)
|
|
epini $p |- ( `' _E " { A } ) = A $=
|
|
( vx cep ccnv csn cima wbr cab imasn brcnv epelc bitri abbii abid2 3eqtri
|
|
cv wcel ) DEZAFGACQZSHZCITARZCIACASJUAUBCUATADHUBATDKTABLMNCAOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An idiom that signifies an initial segment of an ordering, used, for
|
|
example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
|
|
set.mm contributors, 28-Apr-2004.) $)
|
|
iniseg $p |- ( `' A " { B } ) = { x | x A B } $=
|
|
( ccnv csn cima cv wbr cab imasn brcnv abbii eqtri ) BDZCEFCAGZNHZAIOCBHZ
|
|
AIACNJPQACOBKLM $.
|
|
$}
|
|
|
|
$( Subset theorem for image. (Contributed by set.mm contributors,
|
|
16-Mar-2004.) $)
|
|
imass1 $p |- ( A C_ B -> ( A " C ) C_ ( B " C ) ) $=
|
|
( wss cres crn cima ssres rnss syl dfima3 3sstr4g ) ABDZACEZFZBCEZFZACGBCGM
|
|
NPDOQDABCHNPIJACKBCKL $.
|
|
|
|
$( Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
|
|
(Contributed by set.mm contributors, 22-Mar-1998.) $)
|
|
imass2 $p |- ( A C_ B -> ( C " A ) C_ ( C " B ) ) $=
|
|
( wss cres crn cima ssres2 rnss syl dfima3 3sstr4g ) ABDZCAEZFZCBEZFZCAGCBG
|
|
MNPDOQDABCHNPIJCAKCBKL $.
|
|
|
|
$( The image of a singleton outside the domain is empty. (Contributed by
|
|
set.mm contributors, 22-May-1998.) $)
|
|
ndmima $p |- ( -. A e. dom B -> ( B " { A } ) = (/) ) $=
|
|
( cdm wcel wn csn cima cres crn c0 dfima3 wceq cin dmres incom eqtri disjsn
|
|
biimpri syl5eq dm0rn0 sylib ) ABCZDEZBAFZGBUDHZIZJBUDKUCUECZJLUFJLUCUGUBUDM
|
|
ZJUGUDUBMUHBUDNUDUBOPUHJLUCUBAQRSUETUAS $.
|
|
|
|
${
|
|
$d x y z R $.
|
|
$( Two ways of saying a relation is transitive. Definition of transitivity
|
|
in [Schechter] p. 51. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by set.mm contributors, 27-Dec-1996.)
|
|
(Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
cotr $p |- ( ( R o. R ) C_ R <->
|
|
A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) ) $=
|
|
( ccom wss cv cop wcel wi wal wbr wa ssrel alcom 19.23v df-br bitri albii
|
|
wex brco bitr3i imbi12i bitr4i ) DDEZDFAGZCGZHZUEIZUHDIZJZCKZAKUFBGZDLUMU
|
|
GDLMZUFUGDLZJZCKBKZAKACUEDNUQULAUQUPBKZCKULUPBCOURUKCURUNBTZUOJUKUNUOBPUS
|
|
UIUOUJUSUFUGUELUIBUFUGDDUAUFUGUEQUBUFUGDQUCRSRSUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y R $.
|
|
$( Two ways of saying a relation is symmetric. Similar to definition of
|
|
symmetry in [Schechter] p. 51. (The proof was shortened by Andrew
|
|
Salmon, 27-Aug-2011.) (Contributed by set.mm contributors,
|
|
28-Dec-1996.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
cnvsym $p |- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) ) $=
|
|
( cv cop ccnv wcel wi wal wss wbr alcom ssrel brcnv bitr3i imbi12i 2albii
|
|
df-br 3bitr4i ) BDZADZEZCFZGZUBCGZHZAIBIUFBIAIUCCJUATCKZTUACKZHZBIAIUFBAL
|
|
BAUCCMUIUFABUGUDUHUEUGTUAUCKUDTUACNTUAUCROTUACRPQS $.
|
|
|
|
$( Two ways of saying a relation is antisymmetric. Definition of
|
|
antisymmetry in [Schechter] p. 51. (The proof was shortened by Andrew
|
|
Salmon, 27-Aug-2011.) (Contributed by set.mm contributors,
|
|
9-Sep-2004.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
intasym $p |- ( ( R i^i `' R ) C_ _I <->
|
|
A. x A. y ( ( x R y /\ y R x ) -> x = y ) ) $=
|
|
( ccnv cin cid wss cv cop wcel wi wal wbr wa weq ssrel df-br bitri bitr3i
|
|
brin brcnv anbi2i vex ideq imbi12i 2albii ) CCDZEZFGAHZBHZIZUHJZUKFJZKZBL
|
|
ALUIUJCMZUJUICMZNZABOZKZBLALABUHFPUNUSABULUQUMURULUIUJUHMZUQUIUJUHQUTUOUI
|
|
UJUGMZNUQUIUJCUGTVAUPUOUIUJCUAUBRSUMUIUJFMURUIUJFQUIUJBUCUDSUEUFR $.
|
|
|
|
$( Two ways of saying a relation is irreflexive. Definition of
|
|
irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.)
|
|
(Revised by Andrew Salmon, 27-Aug-2011.) $)
|
|
intirr $p |- ( ( R i^i _I ) = (/) <-> A. x -. x R x ) $=
|
|
( vy cid cin c0 wceq cv cop wcel ccompl wi wal weq wbr wn wss df-br albii
|
|
vex incom eqeq1i disj5 ssrel 3bitri ideq bitr3i notbii opex bitr4i 2albii
|
|
elcompl imbi12i equcom imbi1i breq2 notbid ceqsalv bitri 3bitr2i ) BDEZFG
|
|
ZAHZCHZIZDJZVEBKZJZLZCMAMZACNZVCVDBOZPZLZCMZAMVCVCBOZPZAMVBDBEZFGDVGQVJVA
|
|
VRFBDUAUBDBUCACDVGUDUEVNVIACVKVFVMVHVKVCVDDOVFVCVDCTZUFVCVDDRUGVMVEBJZPVH
|
|
VLVTVCVDBRUHVEBVCVDATZVSUIULUJUMUKVOVQAVOCANZVMLZCMVQVNWCCVKWBVMACUNUOSVM
|
|
VQCVCWAWBVLVPVDVCVCBUPUQURUSSUT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d z w ph $.
|
|
$( The converse of a class abstraction of ordered pairs. (The proof was
|
|
shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm
|
|
contributors, 11-Dec-2003.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
cnvopab $p |- `' { <. x , y >. | ph } = { <. y , x >. | ph } $=
|
|
( vz vw copab ccnv cop wcel wsbc opelopabsb sbccom bitri opelcnv eqrelriv
|
|
cv 3bitr4i ) DEABCFZGZACBFZEPZDPZHRIZABUAJCUBJZUBUAHZSIUETIUCACUBJBUAJUDA
|
|
BCUAUBKABCUAUBLMUBUARNACBUBUAKQO $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The converse of the empty set. (Contributed by set.mm contributors,
|
|
6-Apr-1998.) $)
|
|
cnv0 $p |- `' (/) = (/) $=
|
|
( vx vy c0 ccnv cv cop wcel noel opelcnv mtbir 2false eqrelriv ) ABCDZCAE
|
|
ZBEZFZMGZPCGQONFZCGRHNOCIJPHKL $.
|
|
|
|
$( The converse of the identity relation. Theorem 3.7(ii) of [Monk1]
|
|
p. 36. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|
|
(Contributed by set.mm contributors, 26-Apr-1998.) (Revised by set.mm
|
|
contributors, 27-Aug-2011.) $)
|
|
cnvi $p |- `' _I = _I $=
|
|
( vy vx cv cid wbr copab wceq ccnv ideq equcom bitri opabbii df-cnv df-id
|
|
vex 3eqtr4i ) ACZBCZDEZBAFRQGZBAFDHDSTBASQRGTQRBOIABJKLBADMBANP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The converse of a union is the union of converses. Theorem 16 of
|
|
[Suppes] p. 62. (The proof was shortened by Andrew Salmon,
|
|
27-Aug-2011.) (Contributed by set.mm contributors, 25-Mar-1998.)
|
|
(Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
cnvun $p |- `' ( A u. B ) = ( `' A u. `' B ) $=
|
|
( vy vx cv wbr copab cun ccnv unopab brun opabbii eqtr4i uneq12i 3eqtr4ri
|
|
wo df-cnv ) CEZDEZAFZDCGZRSBFZDCGZHZRSABHZFZDCGZAIZBIZHUEIUDTUBPZDCGUGTUB
|
|
DCJUFUJDCRSABKLMUHUAUIUCDCAQDCBQNDCUEQO $.
|
|
|
|
$( Distributive law for converse over set difference. (Contributed by
|
|
set.mm contributors, 26-Jun-2014.) $)
|
|
cnvdif $p |- `' ( A \ B ) = ( `' A \ `' B ) $=
|
|
( vx vy cdif ccnv cv cop wcel wn wa opelcnv notbii anbi12i bitr4i 3bitr4i
|
|
eldif eqrelriv ) CDABEZFZAFZBFZEZDGZCGZHZSIZUEUDHZUAIZUHUBIZJZKZUHTIUHUCI
|
|
UGUFAIZUFBIZJZKULUFABQUIUMUKUOUEUDALUJUNUEUDBLMNOUEUDSLUHUAUBQPR $.
|
|
|
|
$( Distributive law for converse over intersection. Theorem 15 of [Suppes]
|
|
p. 62. (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by
|
|
set.mm contributors, 26-Jun-2014.) $)
|
|
cnvin $p |- `' ( A i^i B ) = ( `' A i^i `' B ) $=
|
|
( cdif ccnv cin cnvdif difeq2i eqtri dfin4 cnveqi 3eqtr4i ) AABCZCZDZADZO
|
|
BDZCZCZABEZDOPENOLDZCRALFTQOABFGHSMABIJOPIK $.
|
|
$}
|
|
|
|
$( Distributive law for range over union. Theorem 8 of [Suppes] p. 60.
|
|
(Contributed by set.mm contributors, 24-Mar-1998.) $)
|
|
rnun $p |- ran ( A u. B ) = ( ran A u. ran B ) $=
|
|
( cun ccnv cdm crn cnvun dmeqi dmun eqtri dfrn4 uneq12i 3eqtr4i ) ABCZDZEZA
|
|
DZEZBDZEZCZNFAFZBFZCPQSCZEUAOUDABGHQSIJNKUBRUCTAKBKLM $.
|
|
|
|
$( The range of an intersection belongs the intersection of ranges. Theorem
|
|
9 of [Suppes] p. 60. (Contributed by set.mm contributors,
|
|
15-Sep-2004.) $)
|
|
rnin $p |- ran ( A i^i B ) C_ ( ran A i^i ran B ) $=
|
|
( cin ccnv cdm crn cnvin dmeqi dmin eqsstri dfrn4 ineq12i 3sstr4i ) ABCZDZE
|
|
ZADZEZBDZEZCZNFAFZBFZCPQSCZEUAOUDABGHQSIJNKUBRUCTAKBKLM $.
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( The range of a union. Part of Exercise 8 of [Enderton] p. 41.
|
|
(Contributed by set.mm contributors, 17-Mar-2004.) $)
|
|
rnuni $p |- ran U. A = U_ x e. A ran x $=
|
|
( vz vy cuni crn cv ciun cop wcel wex wrex eluni exbii excom elrn2 anbi1i
|
|
wa ancom 19.41v 3bitr4ri 3bitri df-rex bitr4i eliun 3bitr4i eqriv ) CBEZF
|
|
ZABAGZFZHZDGCGZIZUHJZDKZUMUKJZABLZUMUIJUMULJUPUJBJZUQRZAKZURUPUNUJJZUSRZA
|
|
KZDKVCDKZAKVAUOVDDAUNBMNVCDAOVEUTAUQUSRVBDKZUSRUTVEUQVFUSDUMUJPQUSUQSVBUS
|
|
DTUANUBUQABUCUDDUMUHPAUMBUKUEUFUG $.
|
|
$}
|
|
|
|
$( Distributive law for image over union. Theorem 35 of [Suppes] p. 65.
|
|
(Contributed by set.mm contributors, 30-Sep-2002.) $)
|
|
imaundi $p |- ( A " ( B u. C ) ) = ( ( A " B ) u. ( A " C ) ) $=
|
|
( cun cres crn cima resundi rneqi rnun eqtri dfima3 uneq12i 3eqtr4i ) ABCDZ
|
|
EZFZABEZFZACEZFZDZAOGABGZACGZDQRTDZFUBPUEABCHIRTJKAOLUCSUDUAABLACLMN $.
|
|
|
|
$( The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.) $)
|
|
imaundir $p |- ( ( A u. B ) " C ) = ( ( A " C ) u. ( B " C ) ) $=
|
|
( cun cima cres crn dfima3 resundir rneqi rnun 3eqtri uneq12i eqtr4i ) ABDZ
|
|
CEZACFZGZBCFZGZDZACEZBCEZDPOCFZGQSDZGUAOCHUDUEABCIJQSKLUBRUCTACHBCHMN $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $.
|
|
$( An upper bound for intersection with a domain. Theorem 40 of [Suppes]
|
|
p. 66, who calls it "somewhat surprising." (Contributed by set.mm
|
|
contributors, 11-Aug-2004.) $)
|
|
dminss $p |- ( dom R i^i A ) C_ ( `' R " ( R " A ) ) $=
|
|
( vx vy cdm cin ccnv cima cv wbr wcel wa wex wrex rspe elima sylibr brcnv
|
|
ancoms biimpri adantr jca eximi eldm anbi1i 19.41v 3bitr4i elima2 3imtr4i
|
|
elin ssriv ) CBEZAFZBGZBAHZHZCIZDIZBJZUQAKZLZDMZURUOKZURUQUNJZLZDMUQUMKZU
|
|
QUPKVAVEDVAVCVDUTUSVCUTUSLUSCANVCUSCAOCURBAPQSUSVDUTVDUSURUQBRTUAUBUCUQUL
|
|
KZUTLUSDMZUTLVFVBVGVHUTDUQBUDUEUQULAUJUSUTDUFUGDUQUNUOUHUIUK $.
|
|
|
|
$( An upper bound for intersection with an image. Theorem 41 of [Suppes]
|
|
p. 66. (Contributed by set.mm contributors, 11-Aug-2004.) $)
|
|
imainss $p |- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) ) $=
|
|
( vy vx cima cin ccnv cv wcel wbr wa wex simpll brcnv 19.8a sylan2br elin
|
|
elima2 anbi1i ancoms adantll jca simplr anbi2i bitri eximi 19.41v 3bitr4i
|
|
sylanbrc 3imtr4i ssriv ) DCAFZBGZCACHZBFZGZFZEIZAJZUSDIZCKZLZVABJZLZEMZUS
|
|
UQJZVBLZEMVAUNJZVAURJVEVHEVEUTVDVAUSUOKZLZDMZLZVBVHVEUTVLUTVBVDNVBVDVLUTV
|
|
DVBVLVBVDVJVLVAUSCOVKDPQUAUBUCUTVBVDUDVGVMVBVGUTUSUPJZLVMUSAUPRVNVLUTDUSU
|
|
OBSUEUFTUJUGVAUMJZVDLVCEMZVDLVIVFVOVPVDEVACASTVAUMBRVCVDEUHUIEVACUQSUKUL
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The converse of a cross product. Exercise 11 of [Suppes] p. 67. (The
|
|
proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by
|
|
set.mm contributors, 14-Aug-1999.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
cnvxp $p |- `' ( A X. B ) = ( B X. A ) $=
|
|
( vy vx cv wcel copab ccnv cxp cnvopab ancom opabbii eqtri cnveqi 3eqtr4i
|
|
wa df-xp ) CEAFZDEBFZPZCDGZHZSRPZDCGZABIZHBAIUBTDCGUDTCDJTUCDCRSKLMUEUACD
|
|
ABQNDCBAQO $.
|
|
$}
|
|
|
|
$( The cross product with the empty set is empty. Part of Theorem 3.13(ii)
|
|
of [Monk1] p. 37. (Contributed by set.mm contributors, 12-Apr-2004.) $)
|
|
xp0 $p |- ( A X. (/) ) = (/) $=
|
|
( c0 cxp ccnv xp0r cnveqi cnvxp cnv0 3eqtr3i ) BACZDBDABCBJBAEFBAGHI $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The cross product of nonempty classes is nonempty. (Variation of a
|
|
theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by
|
|
set.mm contributors, 30-Jun-2006.) (Revised by set.mm contributors,
|
|
19-Apr-2007.) $)
|
|
xpnz $p |- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) ) $=
|
|
( vx vy c0 wne wa cxp cv wcel wex n0 anbi12i eeanv bitr4i cop opelxp wceq
|
|
syl6eq necon3i ne0i sylbir exlimivv sylbi xpeq1 xp0r xpeq2 xp0 jca impbii
|
|
) AEFZBEFZGZABHZEFZUMCIZAJZDIZBJZGZDKCKZUOUMUQCKZUSDKZGVAUKVBULVCCALDBLMU
|
|
QUSCDNOUTUOCDUTUPURPZUNJUOUPURABQUNVDUAUBUCUDUOUKULAEUNEAERUNEBHEAEBUEBUF
|
|
STBEUNEBERUNAEHEBEAUGAUHSTUIUJ $.
|
|
$}
|
|
|
|
$( At least one member of an empty cross product is empty. (Contributed by
|
|
set.mm contributors, 27-Aug-2006.) $)
|
|
xpeq0 $p |- ( ( A X. B ) = (/) <-> ( A = (/) \/ B = (/) ) ) $=
|
|
( cxp c0 wceq wne wa wn wo xpnz necon2bbii ianor nne orbi12i 3bitri ) ABCZD
|
|
EADFZBDFZGZHQHZRHZIADEZBDEZISPDABJKQRLTUBUAUCADMBDMNO $.
|
|
|
|
$( Cross products with disjoint sets are disjoint. (Contributed by set.mm
|
|
contributors, 13-Sep-2004.) $)
|
|
xpdisj1 $p |- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) ) $=
|
|
( cin c0 wceq cxp inxp xpeq1 xp0r syl6eq syl5eq ) ABEZFGZACHBDHENCDEZHZFACB
|
|
DIOQFPHFNFPJPKLM $.
|
|
|
|
$( Cross products with disjoint sets are disjoint. (Contributed by set.mm
|
|
contributors, 13-Sep-2004.) $)
|
|
xpdisj2 $p |- ( ( A i^i B ) = (/) -> ( ( C X. A ) i^i ( D X. B ) ) = (/) ) $=
|
|
( cin c0 wceq cxp inxp xpeq2 xp0 syl6eq syl5eq ) ABEZFGZCAHDBHECDEZNHZFCADB
|
|
IOQPFHFNFPJPKLM $.
|
|
|
|
$( Cross products with two different singletons are disjoint. (Contributed
|
|
by set.mm contributors, 28-Jul-2004.) (Revised by set.mm contributors,
|
|
3-Jun-2007.) $)
|
|
xpsndisj $p |- ( B =/= D -> ( ( A X. { B } ) i^i ( C X. { D } ) ) = (/) ) $=
|
|
( wne csn cin c0 wceq cxp disjsn2 xpdisj2 syl ) BDEBFZDFZGHIANJCOJGHIBDKNOA
|
|
CLM $.
|
|
|
|
$( A double restriction to disjoint classes is the empty set. (The proof was
|
|
shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm
|
|
contributors, 7-Oct-2004.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
resdisj $p |- ( ( A i^i B ) = (/) -> ( ( C |` A ) |` B ) = (/) ) $=
|
|
( cin c0 wceq cres resres reseq2 res0 syl6eq syl5eq ) ABDZEFZCAGBGCMGZECABH
|
|
NOCEGEMECICJKL $.
|
|
|
|
$( The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37.
|
|
(Contributed by set.mm contributors, 12-Apr-2004.) (Revised by set.mm
|
|
contributors, 9-Apr-2007.) $)
|
|
rnxp $p |- ( A =/= (/) -> ran ( A X. B ) = B ) $=
|
|
( c0 wne cxp crn cdm ccnv dfrn4 cnvxp dmeqi eqtri dmxp syl5eq ) ACDABEZFZBA
|
|
EZGZBPOHZGROISQABJKLBAMN $.
|
|
|
|
$( The domain of a cross product is a subclass of the first factor.
|
|
(Contributed by set.mm contributors, 19-Mar-2007.) $)
|
|
dmxpss $p |- dom ( A X. B ) C_ A $=
|
|
( cxp cdm wss c0 wceq 0ss xpeq2 xp0 syl6eq dmeqd dm0 sseq1d mpbiri wne dmxp
|
|
eqimss syl pm2.61ine ) ABCZDZAEZBFBFGZUCFAEAHUDUBFAUDUBFDFUDUAFUDUAAFCFBFAI
|
|
AJKLMKNOBFPUBAGUCABQUBARST $.
|
|
|
|
$( The range of a cross product is a subclass of the second factor. (The
|
|
proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by
|
|
set.mm contributors, 16-Jan-2006.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
rnxpss $p |- ran ( A X. B ) C_ B $=
|
|
( cxp crn cdm ccnv dfrn4 cnvxp dmeqi eqtri dmxpss eqsstri ) ABCZDZBACZEZBNM
|
|
FZEPMGQOABHIJBAKL $.
|
|
|
|
$( The range of a square cross product. (Contributed by FL, 17-May-2010.) $)
|
|
rnxpid $p |- ran ( A X. A ) = A $=
|
|
( cxp crn wceq c0 rn0 xpeq2 xp0 syl6eq rneqd id 3eqtr4a rnxp pm2.61ine ) AA
|
|
BZCZADAEAEDZECEPAFQOEQOAEBEAEAGAHIJQKLAAMN $.
|
|
|
|
$( A cross-product subclass relationship is equivalent to the relationship
|
|
for it components. (Contributed by set.mm contributors, 17-Dec-2008.) $)
|
|
ssxpb $p |- ( ( A X. B ) =/= (/) -> ( ( A X. B ) C_ ( C X. D ) <->
|
|
( A C_ C /\ B C_ D ) ) ) $=
|
|
( cxp c0 wne wss wa cdm wceq xpnz dmxp adantl sylbir adantr eqsstr3d syl6ss
|
|
dmss crn dmxpss rnxp rnss rnxpss jca ex xpss12 impbid1 ) ABEZFGZUICDEZHZACH
|
|
ZBDHZIZUJULUOUJULIZUMUNUPAUKJZCUPAUIJZUQUJURAKZULUJAFGZBFGZIZUSABLZVAUSUTAB
|
|
MNOPULURUQHUJUIUKSNQCDUARUPBUKTZDUPBUITZVDUJVEBKZULUJVBVFVCUTVFVAABUBPOPULV
|
|
EVDHUJUIUKUCNQCDUDRUEUFACBDUGUH $.
|
|
|
|
$( The cross product of non-empty classes is one-to-one. (Contributed by
|
|
set.mm contributors, 31-May-2008.) $)
|
|
xp11 $p |- ( ( A =/= (/) /\ B =/= (/) )
|
|
-> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) ) $=
|
|
( c0 wne wa cxp wceq wi xpnz anidm neeq1 anbi2d syl5bbr wss ssxpb syl5ibcom
|
|
eqimss eqss eqimss2 anim12d anbi12i bitr4i syl6ib sylbid com12 sylbi xpeq12
|
|
an4 impbid1 ) AEFBEFGZABHZCDHZIZACIZBDIZGZULUMEFZUOURJABKUOUSURUOUSUSUNEFZG
|
|
ZURUSUSUSGUOVAUSLUOUSUTUSUMUNEMNOUOVAACPZBDPZGZCAPZDBPZGZGZURUOUSVDUTVGUOUM
|
|
UNPUSVDUMUNSABCDQRUOUNUMPUTVGUNUMUACDABQRUBVHVBVEGZVCVFGZGURVBVCVEVFUJUPVIU
|
|
QVJACTBDTUCUDUEUFUGUHACBDUIUK $.
|
|
|
|
$( Cancellation law for cross-product. (Contributed by set.mm contributors,
|
|
30-Aug-2011.) $)
|
|
xpcan $p |- ( C =/= (/) -> ( ( C X. A ) = ( C X. B ) <-> A = B ) ) $=
|
|
( c0 wne cxp wceq wb wa xp11 biantrur syl6bbr wn wi nne xpeq2 syl6eq eqeq1d
|
|
eqid xp0 eqcom syl6bb adantl df-ne wo xpeq0 orel1 syl5bi sylbi adantr simpr
|
|
sylbid jctild eqtr3 syl6 sylan2b impbid1 pm2.61dan ) CDEZADEZCAFZCBFZGZABGZ
|
|
HUSUTIVCCCGZVDIVDCACBJVEVDCSKLUSUTMZIVCVDVFUSADGZVCVDNADOUSVGIZVCVGBDGZIVDV
|
|
HVCVIVGVHVCVBDGZVIVGVCVJHUSVGVCDVBGVJVGVADVBVGVACDFDADCPCTQRDVBUAUBUCUSVJVI
|
|
NZVGUSCDGZMZVKCDUDVJVLVIUEVMVICBUFVLVIUGUHUIUJULUSVGUKUMABDUNUOUPABCPUQUR
|
|
$.
|
|
|
|
$( Cancellation law for cross-product. (Contributed by set.mm contributors,
|
|
30-Aug-2011.) $)
|
|
xpcan2 $p |- ( C =/= (/) -> ( ( A X. C ) = ( B X. C ) <-> A = B ) ) $=
|
|
( c0 wne cxp wceq wb wa xp11 eqid biantru syl6bbr wn nne xp0r syl6eq eqeq1d
|
|
xpeq1 eqcom syl6bb adantr wi df-ne wo xpeq0 orel2 syl5bi sylbi adantl simpl
|
|
sylbid jctild eqtr3 syl6 impbid1 sylanb pm2.61ian ) ADEZCDEZACFZBCFZGZABGZH
|
|
ZUSUTIVCVDCCGZIVDACBCJVFVDCKLMUSNADGZUTVEADOVGUTIZVCVDVHVCVGBDGZIVDVHVCVIVG
|
|
VHVCVBDGZVIVGVCVJHUTVGVCDVBGVJVGVADVBVGVADCFDADCSCPQRDVBTUAUBUTVJVIUCZVGUTC
|
|
DGZNZVKCDUDVJVIVLUEVMVIBCUFVLVIUGUHUIUJULVGUTUKUMABDUNUOABCSUPUQUR $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Subset of the range of a restriction. (Contributed by set.mm
|
|
contributors, 16-Jan-2006.) $)
|
|
ssrnres $p |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B ) $=
|
|
( vy vx cxp cin crn wceq wss wa cres rnss ax-mp cvv wcel wex elrn2 3bitri
|
|
cv eqss inss2 rnxpss sstri biantrur ssv xpss2 sslin df-res sseqtr4i mpan2
|
|
sstr cop ssel syl6ib ancrd elin opelxp anbi2i opelres anbi1i anass bitr2i
|
|
exbii 19.41v syl6ibr ssrdv impbii 3bitr2ri ) CABFZGZHZBIVLBJZBVLJZKVNBCAL
|
|
ZHZJZVLBUAVMVNVLVJHZBVKVJJVLVRJCVJUBVKVJMNABUCUDUEVNVQVNVLVPJZVQVKVOJVSVK
|
|
CAOFZGZVOVJVTJZVKWAJBOJWBBUFBOAUGNVJVTCUHNCAUIUJVKVOMNBVLVPULUKVQDBVLVQDT
|
|
ZBPZETZWCUMZVOPZEQZWDKZWCVLPZVQWDWHVQWDWCVPPWHBVPWCUNEWCVORUOUPWJWFVKPZEQ
|
|
WGWDKZEQWIEWCVKRWKWLEWKWFCPZWFVJPZKWMWEAPZWDKZKZWLWFCVJUQWNWPWMWEWCABURUS
|
|
WLWMWOKZWDKWQWGWRWDWEWCCAUTVAWMWOWDVBVCSVDWGWDEVESVFVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $.
|
|
$( Range of the intersection with a cross product. (The proof was
|
|
shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm
|
|
contributors, 17-Jan-2006.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
rninxp $p |- ( ran ( C i^i ( A X. B ) ) = B <->
|
|
A. y e. B E. x e. A x C y ) $=
|
|
( cres crn wss wcel wral cxp cin wceq wbr wrex dfss3 ssrnres cima dfima3
|
|
cv eleq2i elima bitr3i ralbii 3bitr3i ) DECFGZHBTZUFIZBDJECDKLGDMATUGENAC
|
|
OZBDJBDUFPCDEQUHUIBDUHUGECRZIUIUJUFUGECSUAAUGECUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x y C $.
|
|
$( Domain of the intersection with a cross product. (Contributed by set.mm
|
|
contributors, 17-Jan-2006.) $)
|
|
dminxp $p |- ( dom ( C i^i ( A X. B ) ) = A <->
|
|
A. x e. A E. y e. B x C y ) $=
|
|
( cxp cin cdm wceq ccnv crn wbr wrex wral df-dm cnvin cnvxp ineq2i eqtri
|
|
cv rneqi eqeq1i rninxp brcnv rexbii ralbii 3bitri ) ECDFZGZHZCIEJZDCFZGZK
|
|
ZCIBTZATZUKLZBDMZACNUPUOELZBDMZACNUJUNCUJUIJZKUNUIOVAUMVAUKUHJZGUMEUHPVBU
|
|
LUKCDQRSUASUBBADCUKUCURUTACUQUSBDUOUPEUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y R $.
|
|
$( The double converse of a class is the original class. (Contributed by
|
|
Scott Fenton, 17-Apr-2021.) $)
|
|
cnvcnv $p |- `' `' R = R $=
|
|
( vx vy ccnv cv wbr brcnv bitri eqbrriv ) BCADZDZABEZCEZKFMLJFLMAFLMJGMLA
|
|
GHI $.
|
|
$}
|
|
|
|
$( Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
|
|
(Revised by Scott Fenton, 17-Apr-2021.) $)
|
|
cnveqb $p |- ( A = B <-> `' A = `' B ) $=
|
|
( wceq ccnv cnveq cnvcnv 3eqtr3g impbii ) ABCADZBDZCZABEKIDJDABIJEAFBFGH $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( The domain of a singleton is nonzero iff the singleton argument is a
|
|
set. (Contributed by NM, 14-Dec-2008.) (Proof shortened by
|
|
Andrew Salmon, 27-Aug-2011.) (Revised by Scott Fenton, 19-Apr-2021.) $)
|
|
dmsnn0 $p |- ( A e. _V <-> dom { A } =/= (/) ) $=
|
|
( vx vy cv csn cdm wcel wex cop wceq wne cvv eldm2 opex elsnc eqcom bitri
|
|
c0 vex exbii n0 opeqexb 3bitr4ri ) BDZAEZFZGZBHAUDCDZIZJZCHZBHUFRKALGUGUK
|
|
BUGUIUEGZCHUKCUDUEMULUJCULUIAJUJUIAUDUHBSCSNOUIAPQTQTBUFUABCAUBUC $.
|
|
$}
|
|
|
|
$( The range of a singleton is nonzero iff the singleton argument is a
|
|
set. (Contributed by set.mm contributors, 14-Dec-2008.)
|
|
(Revised by Scott Fenton, 19-Apr-2021.) $)
|
|
rnsnn0 $p |- ( A e. _V <-> ran { A } =/= (/) ) $=
|
|
( cvv wcel csn cdm c0 wne crn dmsnn0 dm0rn0 necon3bii bitri ) ABCADZEZFGMHZ
|
|
FGAINFOFMJKL $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x z V $.
|
|
$( The domain of a singleton of an ordered pair is the singleton of the
|
|
first member. (Contributed by Mario Carneiro, 26-Apr-2015.) $)
|
|
dmsnopg $p |- ( B e. V -> dom { <. A , B >. } = { A } ) $=
|
|
( vy vx vz cv cop csn cdm wceq opeq2 sneqd dmeqd eqeq1d wex wcel wa bitri
|
|
vex wbr weq df-br opex elsnc opth ancom 3bitri exbii ceqsexv eldm 3bitr4i
|
|
biidd elsn eqriv vtoclg ) ADGZHZIZJZAIZKABHZIZJZVAKDBCUQBKZUTVDVAVEUSVCVE
|
|
URVBUQBALMNOEUTVAEGZFGZUSUAZFPZVFAKZVFUTQVFVAQVIFDUBZVJRZFPVJVHVLFVHVFVGH
|
|
ZUSQVMURKZVLVFVGUSUCVMURVFVGETFTUDUEVNVJVKRVLVFVGAUQUFVJVKUGSUHUIVJVJFUQD
|
|
TVKVJUMUJSFVFUSUKEAUNULUOUP $.
|
|
|
|
$( The domain of a singleton of an ordered pair is a subset of the
|
|
singleton of the first member (with no sethood assumptions on ` B ` ).
|
|
(Contributed by Mario Carneiro, 30-Apr-2015.) $)
|
|
dmsnopss $p |- dom { <. A , B >. } C_ { A } $=
|
|
( cvv wcel cop csn cdm wss wceq dmsnopg eqimss syl wn opexb simprbi con3i
|
|
c0 snprc sylib dmeqd dm0 syl6eq 0ss syl6eqss pm2.61i ) BCDZABEZFZGZAFZHZU
|
|
FUIUJIUKABCJUIUJKLUFMZUIQUJULUIQGQULUHQULUGCDZMUHQIUMUFUMACDUFABNOPUGRSTU
|
|
AUBUJUCUDUE $.
|
|
|
|
$( The domain of an unordered pair of ordered pairs. (Contributed by Mario
|
|
Carneiro, 26-Apr-2015.) $)
|
|
dmpropg $p |- ( ( B e. V /\ D e. W ) ->
|
|
dom { <. A , B >. , <. C , D >. } = { A , C } ) $=
|
|
( wcel wa cop csn cdm cun cpr wceq dmsnopg uneq12 syl2an df-pr dmeqi dmun
|
|
eqtri 3eqtr4g ) BEGZDFGZHABIZJZKZCDIZJZKZLZAJZCJZLZUEUHMZKZACMUCUGULNUJUM
|
|
NUKUNNUDABEOCDFOUGULUJUMPQUPUFUILZKUKUOUQUEUHRSUFUITUAACRUB $.
|
|
$}
|
|
|
|
${
|
|
dmsnop.1 $e |- B e. _V $.
|
|
$( The domain of a singleton of an ordered pair is the singleton of the
|
|
first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by
|
|
Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro,
|
|
26-Apr-2015.) $)
|
|
dmsnop $p |- dom { <. A , B >. } = { A } $=
|
|
( cvv wcel cop csn cdm wceq dmsnopg ax-mp ) BDEABFGHAGICABDJK $.
|
|
|
|
dmprop.1 $e |- D e. _V $.
|
|
$( The domain of an unordered pair of ordered pairs. (Contributed by NM,
|
|
13-Sep-2011.) $)
|
|
dmprop $p |- dom { <. A , B >. , <. C , D >. } = { A , C } $=
|
|
( cvv wcel cop cpr cdm wceq dmpropg mp2an ) BGHDGHABICDIJKACJLEFABCDGGMN
|
|
$.
|
|
|
|
dmtpop.1 $e |- F e. _V $.
|
|
$( The domain of an unordered triple of ordered pairs. (Contributed by NM,
|
|
14-Sep-2011.) $)
|
|
dmtpop $p |- dom { <. A , B >. , <. C , D >. , <. E , F >. }
|
|
= { A , C , E } $=
|
|
( cop ctp cdm cpr csn cun df-tp dmeqi dmun dmprop dmsnop uneq12i 3eqtri
|
|
eqtr4i ) ABJZCDJZEFJZKZLZACMZENZOZACEKUHUDUEMZUFNZOZLULLZUMLZOUKUGUNUDUEU
|
|
FPQULUMRUOUIUPUJABCDGHSEFITUAUBACEPUC $.
|
|
$}
|
|
|
|
${
|
|
op1sta.1 $e |- A e. _V $.
|
|
op1sta.2 $e |- B e. _V $.
|
|
$( Extract the first member of an ordered pair. (Contributed by Raph
|
|
Levien, 4-Dec-2003.) $)
|
|
op1sta $p |- U. dom { <. A , B >. } = A $=
|
|
( cop csn cdm cuni dmsnop unieqi unisn eqtri ) ABEFGZHAFZHAMNABDIJACKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
cnvsn.1 $e |- A e. _V $.
|
|
cnvsn.2 $e |- B e. _V $.
|
|
$( Converse of a singleton of an ordered pair. (Contributed by NM,
|
|
11-May-1998.) $)
|
|
cnvsn $p |- `' { <. A , B >. } = { <. B , A >. } $=
|
|
( vx vy cop csn ccnv cv wcel wceq vex opex elsnc ancom opth 3bitr4i bitri
|
|
wa opelcnv eqrelriv ) EFABGZHZIZBAGZHZFJZEJZGZUDKZUIUHGZUFLZULUEKULUGKUKU
|
|
JUCLZUMUJUCUHUIFMZEMZNOUHALZUIBLZTURUQTUNUMUQURPUHUIABQUIUHBAQRSUIUHUDUAU
|
|
LUFUIUHUPUONORUB $.
|
|
|
|
$( Swap the members of an ordered pair. (Contributed by set.mm
|
|
contributors, 14-Dec-2008.) $)
|
|
opswap $p |- U. `' { <. A , B >. } = <. B , A >. $=
|
|
( cop csn ccnv cuni cnvsn unieqi opex unisn eqtri ) ABEFGZHBAEZFZHONPABCD
|
|
IJOBADCKLM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
rnsnop.1 $e |- A e. _V $.
|
|
$( The range of a singleton of an ordered pair is the singleton of the
|
|
second member. (Contributed by set.mm contributors, 24-Jul-2004.) $)
|
|
rnsnop $p |- ran { <. A , B >. } = { B } $=
|
|
( vy vx cop csn crn cv wbr wex wceq wcel df-br vex opex elsnc opth bitri
|
|
wa exbii biidd ceqsexv elrn 3bitr4i eqriv ) DABFZGZHZBGZEIZDIZUHJZEKZULBL
|
|
ZULUIMULUJMUNUKALZUOTZEKUOUMUQEUMUKULFZUHMZUQUKULUHNUSURUGLUQURUGUKULEODO
|
|
ZPQUKULABRSSUAUOUOEACUPUOUBUCSEULUHUDULBUTQUEUF $.
|
|
$}
|
|
|
|
${
|
|
op2nda.1 $e |- A e. _V $.
|
|
op2nda.2 $e |- B e. _V $.
|
|
$( Extract the second member of an ordered pair. (Contributed by set.mm
|
|
contributors, 9-Jan-2015.) $)
|
|
op2nda $p |- U. ran { <. A , B >. } = B $=
|
|
( cop csn crn cuni rnsnop unieqi unisn eqtri ) ABEFGZHBFZHBMNABCIJBDKL $.
|
|
$}
|
|
|
|
${
|
|
$d s t A $. $d s t B $. $d s t F $.
|
|
$( An image under the converse of a restriction. (Contributed by Jeff
|
|
Hankins, 12-Jul-2009.) $)
|
|
cnvresima $p |- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A ) $=
|
|
( vt vs cres ccnv cima cin cv wcel cop wa wex elima3 anass opelres anbi1i
|
|
opelcnv bitr2i 3bitr4ri anbi2i exbii 19.41v bitri elin 3bitri eqriv ) DCA
|
|
FZGZBHZCGZBHZAIZDJZUKKEJZBKZUPUOLZUJKZMZENZUQURULKZMZENZUOAKZMZUOUNKZEUOU
|
|
JBOVAVCVEMZENVFUTVHEVHUQVBVEMZMUTUQVBVEPVIUSUQUOUPLZUIKVJCKZVEMUSVIUOUPCA
|
|
QUPUOUISVBVKVEUPUOCSRUAUBTUCVCVEEUDUEVGUOUMKZVEMVFUOUMAUFVLVDVEEUOULBORTU
|
|
GUH $.
|
|
$}
|
|
|
|
$( Restriction to the domain of a restriction. (Contributed by set.mm
|
|
contributors, 8-Apr-2007.) $)
|
|
resdmres $p |- ( A |` dom ( A |` B ) ) = ( A |` B ) $=
|
|
( cvv cxp cdm cin cres df-res resdm eqtr3i ineq2i incom 3eqtri dmres xpeq1i
|
|
in12 xpindir eqtri 3eqtr4i ) ABCDZAEZCDZFZFZATFZAABGZEZGZUFUDTAUBFZFTAFUEAT
|
|
UBPUIATAUAGUIAAUAHAIJKTALMUHAUGCDZFUDAUGHUJUCAUJBUAFZCDUCUGUKCABNOBUACQRKRA
|
|
BHS $.
|
|
|
|
$( The image of the domain of a restriction. (Contributed by set.mm
|
|
contributors, 8-Apr-2007.) $)
|
|
imadmres $p |- ( A " dom ( A |` B ) ) = ( A " B ) $=
|
|
( cres cdm crn cima resdmres rneqi dfima3 3eqtr4i ) AABCZDZCZEKEALFABFMKABG
|
|
HALIABIJ $.
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z C $.
|
|
$( Alternate definition of a class composition, using only one bound
|
|
variable. (Contributed by set.mm contributors, 19-Dec-2008.) $)
|
|
dfco2 $p |- ( A o. B )
|
|
= U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) $=
|
|
( vy vz ccom cvv ccnv cv csn cima cxp ciun cop wcel wbr wex opelco bitri
|
|
wa wrex eliun opelxp eliniseg elimasn df-br bitr4i anbi12i exbii eqrelriv
|
|
rexv 3bitrri ) DEBCFZAGCHAIZJZKZBUOKZLZMZDIZEIZNZUMOUTUNCPZUNVABPZTZAQZVB
|
|
USOZAUTVABCRVGVBUROZAGUAVHAQVFAVBGURUBVHAUKVHVEAVHUTUPOZVAUQOZTVEUTVAUPUQ
|
|
UCVIVCVJVDCUNUTUDVJUNVANBOVDBUNVAUEUNVABUFUGUHSUIULSUJ $.
|
|
|
|
$( Generalization of ~ dfco2 , where ` C ` can have any value between
|
|
` dom A i^i ran B ` and ` _V ` . (The proof was shortened by Andrew
|
|
Salmon, 27-Aug-2011.) (Contributed by set.mm contributors,
|
|
21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
dfco2a $p |- ( ( dom A i^i ran B ) C_ C -> ( A o. B )
|
|
= U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) ) $=
|
|
( vy vz vw cdm cvv cv cima ciun wcel wex wa cop sylbi wrex eliun bitri
|
|
crn cin wss ccom ccnv csn cxp wceq elimasn opeldm eliniseg brelrn anim12i
|
|
dfco2 wbr ancoms adantl exlimivv elxp elin ssel syl5 pm4.71rd exbidv rexv
|
|
3imtr4i df-rex 3bitr4g eqrdv syl5eq ) BHZCUAZUBZDUCZBCUDAICUEAJZUFZKZBVPK
|
|
ZUGZLZADVSLZABCUNVNEVTWAVNEJZVSMZANZVODMZWCOZANZWBVTMZWBWAMZVNWCWFAVNWCWE
|
|
WCVOVMMZVNWEWBFJZGJZPUHZWKVQMZWLVRMZOZOZGNFNVOVKMZVOVLMZOZWCWJWQWTFGWPWTW
|
|
MWOWNWTWOWRWNWSWOVOWLPBMWRBVOWLUIVOWLBUJQWNWKVOCUOWSCVOWKUKWKVOCULQUMUPUQ
|
|
URFGWBVQVRUSVOVKVLUTVFVMDVOVAVBVCVDWHWCAIRWDAWBIVSSWCAVETWIWCADRWGAWBDVSS
|
|
WCADVGTVHVIVJ $.
|
|
|
|
$( Class composition distributes over union. (The proof was shortened by
|
|
Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors,
|
|
21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
coundi $p |- ( A o. ( B u. C ) ) = ( ( A o. B ) u. ( A o. C ) ) $=
|
|
( vx vz vy cv wbr wa wex copab cun ccom wo unopab brun anbi1i andir bitri
|
|
df-co exbii 19.43 bitr2i opabbii eqtri uneq12i 3eqtr4ri ) DGZEGZBHZUIFGAH
|
|
ZIZEJZDFKZUHUICHZUKIZEJZDFKZLZUHUIBCLZHZUKIZEJZDFKZABMZACMZLAUTMUSUMUQNZD
|
|
FKVDUMUQDFOVGVCDFVCULUPNZEJVGVBVHEVBUJUONZUKIVHVAVIUKUHUIBCPQUJUOUKRSUAUL
|
|
UPEUBUCUDUEVEUNVFURDFEABTDFEACTUFDFEAUTTUG $.
|
|
|
|
$( Class composition distributes over union. (The proof was shortened by
|
|
Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors,
|
|
21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.) $)
|
|
coundir $p |- ( ( A u. B ) o. C ) = ( ( A o. C ) u. ( B o. C ) ) $=
|
|
( vx vy vz cv wbr wa wex copab cun ccom wo unopab brun anbi2i bitri df-co
|
|
andi exbii 19.43 bitr2i opabbii eqtri uneq12i 3eqtr4ri ) DGEGZCHZUHFGZAHZ
|
|
IZEJZDFKZUIUHUJBHZIZEJZDFKZLZUIUHUJABLZHZIZEJZDFKZACMZBCMZLUTCMUSUMUQNZDF
|
|
KVDUMUQDFOVGVCDFVCULUPNZEJVGVBVHEVBUIUKUONZIVHVAVIUIUHUJABPQUIUKUOTRUAULU
|
|
PEUBUCUDUEVEUNVFURDFEACSDFEBCSUFDFEUTCSUG $.
|
|
|
|
$( Restricted first member of a class composition. (The proof was
|
|
shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm
|
|
contributors, 12-Oct-2004.) (Revised by set.mm contributors,
|
|
27-Aug-2011.) $)
|
|
cores $p |- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) ) $=
|
|
( vz vy vx crn wss cv wbr cres wa wex copab ccom wcel brelrn ssel df-co
|
|
wb iba brres syl6rbbr syl56 pm5.32d exbidv opabbidv 3eqtr4g ) BGZCHZDIZEI
|
|
ZBJZULFIZACKZJZLZEMZDFNUMULUNAJZLZEMZDFNUOBOABOUJURVADFUJUQUTEUJUMUPUSUMU
|
|
LUIPUJULCPZUPUSTUKULBQUICULRVBUSUSVBLUPVBUSUAULUNACUBUCUDUEUFUGDFEUOBSDFE
|
|
ABSUH $.
|
|
|
|
$( Associative law for the restriction of a composition. (Contributed by
|
|
set.mm contributors, 12-Dec-2006.) $)
|
|
resco $p |- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) ) $=
|
|
( vx vy vz ccom cres cv wbr wcel wex brco anbi1i 19.41v an32 brres bitr4i
|
|
wa exbii 3bitr2i 3bitr4i eqbrriv ) DEABGZCHZABCHZGZDIZEIZUDJZUHCKZSZUHFIZ
|
|
UFJZUMUIAJZSZFLZUHUIUEJUHUIUGJULUHUMBJZUOSZFLZUKSUSUKSZFLUQUJUTUKFUHUIABM
|
|
NUSUKFOVAUPFVAURUKSZUOSUPURUOUKPUNVBUOUHUMBCQNRTUAUHUIUDCQFUHUIAUFMUBUC
|
|
$.
|
|
|
|
$( Image of the composition of two classes. (Contributed by Jason
|
|
Orendorff, 12-Dec-2006.) $)
|
|
imaco $p |- ( ( A o. B ) " C ) = ( A " ( B " C ) ) $=
|
|
( vx vy vz ccom cima cv wbr wrex wcel wex df-rex elima r19.41v exbii brco
|
|
wa rexbii rexcom4 3bitri anbi1i 3bitr4i 3bitr4ri eqriv ) DABGZCHZABCHZHZE
|
|
IZDIZAJZEUIKUKUILZUMSZEMZULUJLULUHLZUMEUINEULAUIOFIZUKBJZUMSZFCKZEMZUSFCK
|
|
ZUMSZEMUQUPVAVDEUSUMFCPQUQURULUGJZFCKUTEMZFCKVBFULUGCOVEVFFCEURULABRTUTFE
|
|
CUAUBUOVDEUNVCUMFUKBCOUCQUDUEUF $.
|
|
|
|
$( The range of the composition of two classes. (Contributed by set.mm
|
|
contributors, 12-Dec-2006.) $)
|
|
rnco $p |- ran ( A o. B ) = ran ( A |` ran B ) $=
|
|
( vy vx vz ccom crn cres cv wbr wex wcel wa brco exbii excom ancom 19.41v
|
|
elrn 3bitr4i anbi2i brres bitr4i 3bitri eqriv ) CABFZGZABGZHZGZDIZCIZUFJZ
|
|
DKZEIZULUIJZEKZULUGLULUJLUNUKUOBJZUOULAJZMZEKZDKUTDKZEKUQUMVADEUKULABNOUT
|
|
DEPVBUPEVBUSUOUHLZMZUPURDKZUSMUSVEMVBVDVEUSQURUSDRVCVEUSDUOBSUATUOULAUHUB
|
|
UCOUDDULUFSEULUISTUE $.
|
|
$}
|
|
|
|
$( The range of the composition of two classes. (Contributed by set.mm
|
|
contributors, 27-Mar-2008.) $)
|
|
rnco2 $p |- ran ( A o. B ) = ( A " ran B ) $=
|
|
( ccom crn cres cima rnco dfima3 eqtr4i ) ABCDABDZEDAJFABGAJHI $.
|
|
|
|
$( The domain of a composition. Exercise 27 of [Enderton] p. 53.
|
|
(Contributed by set.mm contributors, 4-Feb-2004.) $)
|
|
dmco $p |- dom ( A o. B ) = ( `' B " dom A ) $=
|
|
( ccom cdm ccnv crn cima df-dm cnvco rneqi rnco2 imaeq2i eqtr4i 3eqtri ) AB
|
|
CZDOEZFBEZAEZCZFZQADZGZOHPSABIJTQRFZGUBQRKUAUCQAHLMN $.
|
|
|
|
${
|
|
$d w x y z A $. $d w y z B $. $d w y z C $.
|
|
$( Composition with an indexed union. (Contributed by set.mm contributors,
|
|
21-Dec-2008.) $)
|
|
coiun $p |- ( A o. U_ x e. C B ) = U_ x e. C ( A o. B ) $=
|
|
( vy vz vw ciun ccom cv wbr wex wrex cop wcel eliun df-br rexbii 3bitr4i
|
|
wa anbi1i r19.41v bitr4i exbii rexcom4 opelco bitri eqrelriv ) EFBADCHZIZ
|
|
ADBCIZHZEJZGJZUIKZUNFJZBKZTZGLZUMUNCKZUQTZGLZADMZUMUPNZUJOVDULOZUSVAADMZG
|
|
LVCURVFGURUTADMZUQTVFUOVGUQUMUNNZUIOVHCOZADMUOVGAVHDCPUMUNUIQUTVIADUMUNCQ
|
|
RSUAUTUQADUBUCUDVAAGDUEUCGUMUPBUIUFVEVDUKOZADMVCAVDDUKPVJVBADGUMUPBCUFRUG
|
|
SUH $.
|
|
$}
|
|
|
|
$( Absorption of a reverse (preimage) restriction of the second member of a
|
|
class composition. (Contributed by set.mm contributors, 11-Dec-2006.) $)
|
|
cores2 $p |- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) ) $=
|
|
( cdm wss ccnv cres ccom cnvcnv coeq1i wceq df-dm sseq1i cores sylbi syl5eq
|
|
crn cnvco 3eqtr4g cnveqd 3eqtr3g ) ADZCEZABFZCGZFZHZFZFABHZFZFUGUIUCUHUJUCU
|
|
FFZAFZHZUDULHZUHUJUCUMUEULHZUNUKUEULUEIJUCULQZCEUOUNKUBUPCALMUDULCNOPAUFRAB
|
|
RSTUGIUIIUA $.
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( Composition with the empty set. Theorem 20 of [Suppes] p. 63.
|
|
(Contributed by set.mm contributors, 24-Apr-2004.) $)
|
|
co02 $p |- ( A o. (/) ) = (/) $=
|
|
( vx vy vz c0 ccom cv cop wcel wbr wa wex noel df-br mtbir intnanr opelco
|
|
nex 2false eqrelriv ) BCAEFZEBGZCGZHZUAIZUDEIUEUBDGZEJZUFUCAJZKZDLUIDUGUH
|
|
UGUBUFHZEIUJMUBUFENOPRDUBUCAEQOUDMST $.
|
|
|
|
$( Composition with the empty set. (Contributed by set.mm contributors,
|
|
24-Apr-2004.) $)
|
|
co01 $p |- ( (/) o. A ) = (/) $=
|
|
( vx vy c0 ccom wceq cv wcel wn eq0 cproj1 wbr cproj2 wex cop df-br mtbir
|
|
wa noel intnan nex opeq eleq1i opelco bitri mpgbir ) DAEZDFBGZUGHZIBBUGJU
|
|
IUHKZCGZALZUKUHMZDLZRZCNZUOCUNULUNUKUMOZDHUQSUKUMDPQTUAUIUJUMOZUGHUPUHURU
|
|
GUHUBUCCUJUMDAUDUEQUF $.
|
|
|
|
$( Composition with the identity relation. Part of Theorem 3.7(i) of
|
|
[Monk1] p. 36. (Contributed by set.mm contributors, 22-Apr-2004.)
|
|
(Revised by Scott Fenton, 14-Apr-2021.) $)
|
|
coi1 $p |- ( A o. _I ) = A $=
|
|
( vx vy vz cid ccom cv wbr wa wex weq brco ideq equcom bitri anbi1i exbii
|
|
vex breq1 ceqsexv 3bitri eqbrriv ) BCAEFZABGZCGZUCHUDDGZEHZUFUEAHZIZDJDBK
|
|
ZUHIZDJUDUEAHZDUDUEAELUIUKDUGUJUHUGBDKUJUDUFDRMBDNOPQUHULDUDBRUFUDUEASTUA
|
|
UB $.
|
|
|
|
$( Composition with the identity relation. Part of Theorem 3.7(i) of
|
|
[Monk1] p. 36. (Contributed by set.mm contributors, 22-Apr-2004.)
|
|
(Revised by Scott Fenton, 17-Apr-2021.) $)
|
|
coi2 $p |- ( _I o. A ) = A $=
|
|
( cid ccom wceq ccnv cnvco cnvi coeq2i coi1 3eqtri cnveqb mpbir ) BACZADM
|
|
EZAEZDNOBEZCOBCOBAFPBOGHOIJMAKL $.
|
|
$}
|
|
|
|
$( Composition with a restricted identity relation. (Contributed by FL,
|
|
19-Jun-2011.) (Revised by Scott Fenton, 17-Apr-2021.) $)
|
|
coires1 $p |- ( A o. ( _I |` B ) ) = ( A |` B ) $=
|
|
( cid ccom cres resco coi1 reseq1i eqtr3i ) ACDZBEACBEDABEACBFJABAGHI $.
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $.
|
|
$( Associative law for class composition. Theorem 27 of [Suppes] p. 64.
|
|
Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for
|
|
any classes whatsoever, not just functions or even relations.
|
|
(Contributed by set.mm contributors, 27-Jan-1997.) $)
|
|
coass $p |- ( ( A o. B ) o. C ) = ( A o. ( B o. C ) ) $=
|
|
( vx vy vz vw ccom cv wbr wex excom anass 2exbii bitr2i brco bitr4i exbii
|
|
wa 3bitr4i anbi2i 19.42v anbi1i 19.41v eqbrriv ) DEABHZCHZABCHZHZDIZFIZCJ
|
|
ZUKEIZUFJZSZFKZUJGIZUHJZUQUMAJZSZGKZUJUMUGJUJUMUIJULUKUQBJZUSSZSZGKZFKZUL
|
|
VBSZUSSZFKZGKZUPVAVJVHGKFKVFVHGFLVHVDFGULVBUSMNOUOVEFUOULVCGKZSVEUNVKULGU
|
|
KUMABPUAULVCGUBQRUTVIGUTVGFKZUSSVIURVLUSFUJUQBCPUCVGUSFUDQRTFUJUMUFCPGUJU
|
|
MAUHPTUE $.
|
|
$}
|
|
|
|
$( A class is transitive iff its converse is transitive. (Contributed by
|
|
FL, 19-Sep-2011.) (Revised by Scott Fenton, 18-Apr-2021.) $)
|
|
cnvtr $p |- ( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) $=
|
|
( ccom wss ccnv cnvco cnvss syl5eqssr cnveqi cnvcnv eqtr3i 3sstr3g impbii )
|
|
AABZACZADZOBZOCZNPMDZOAAEZMAFGQPDZODMAPOFRDTMRPSHMIJAIKL $.
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A class is included in the cross product of its domain and range.
|
|
Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by set.mm
|
|
contributors, 3-Aug-1994.) (Revised by Scott Fenton, 15-Apr-2021.) $)
|
|
ssdmrn $p |- A C_ ( dom A X. ran A ) $=
|
|
( vx vy cdm crn cxp wss cv cop wi wal ssrel opeldm opelrn opelxp sylanbrc
|
|
wcel ax-gen mpgbir ) AADZAEZFZGBHZCHZIZAQZUEUBQZJZCKBBCAUBLUHCUFUCTQUDUAQ
|
|
UGUCUDAMUCUDANUCUDTUAOPRS $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Definition of converse in terms of image and ` Swap ` . (Contributed by
|
|
set.mm contributors, 8-Jan-2015.) $)
|
|
dfcnv2 $p |- `' A = ( Swap " A ) $=
|
|
( vx vy vz ccnv cswap cima cv cop wbr wcel wa wex wceq vex brswap2 anbi1i
|
|
exbii opex bitri eleq1 ceqsexv wrex elima df-rex exancom opelcnv 3bitr4ri
|
|
eqrelriv ) BCAEZFAGZDHZBHZCHZIZFJZULAKZLZDMZUNUMIZAKZUOUKKZUOUJKUSULUTNZU
|
|
QLZDMVAURVDDUPVCUQULUMUNBOZCOZPQRUQVADUTUNUMVFVESULUTAUAUBTVBUPDAUCZUSDUO
|
|
FAUDVGUQUPLDMUSUPDAUEUQUPDUFTTUMUNAUGUHUI $.
|
|
$}
|
|
|
|
$( The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring]
|
|
p. 26. (Contributed by set.mm contributors, 17-Mar-1998.) $)
|
|
cnvexg $p |- ( A e. V -> `' A e. _V ) $=
|
|
( wcel ccnv cswap cima cvv dfcnv2 swapex imaexg mpan syl5eqel ) ABCZADEAFZG
|
|
AHEGCMNGCIEAGBJKL $.
|
|
|
|
${
|
|
cnvex.1 $e |- A e. _V $.
|
|
$( The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring]
|
|
p. 26. (Contributed by set.mm contributors, 19-Dec-2003.) $)
|
|
cnvex $p |- `' A e. _V $=
|
|
( cvv wcel ccnv cnvexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
$( A class is a set iff its converse is a set. (Contributed by FL,
|
|
3-Mar-2007.) (Revised by Scott Fenton, 18-Apr-2021.) $)
|
|
cnvexb $p |- ( R e. _V <-> `' R e. _V ) $=
|
|
( cvv wcel ccnv cnvexg cnvcnv syl5eqelr impbii ) ABCADZBCZABEJAIDBAFIBEGH
|
|
$.
|
|
|
|
$( The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26.
|
|
Similar to Lemma 3D of [Enderton] p. 41. (Contributed by set.mm
|
|
contributors, 8-Jan-2015.) $)
|
|
rnexg $p |- ( A e. V -> ran A e. _V ) $=
|
|
( wcel crn cvv cima df-rn vvex imaexg mpan2 syl5eqel ) ABCZADAEFZEAGLEECMEC
|
|
HAEBEIJK $.
|
|
|
|
$( The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26.
|
|
(Contributed by set.mm contributors, 8-Jan-2015.) $)
|
|
dmexg $p |- ( A e. V -> dom A e. _V ) $=
|
|
( wcel cdm ccnv crn cvv df-dm cnvexg rnexg syl syl5eqel ) ABCZADAEZFZGAHMNG
|
|
COGCABINGJKL $.
|
|
|
|
${
|
|
dmex.1 $e |- A e. _V $.
|
|
$( The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring]
|
|
p. 26. (Contributed by set.mm contributors, 7-Jul-2008.) $)
|
|
dmex $p |- dom A e. _V $=
|
|
( cvv wcel cdm dmexg ax-mp ) ACDAECDBACFG $.
|
|
|
|
$( The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring]
|
|
p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by set.mm
|
|
contributors, 7-Jul-2008.) $)
|
|
rnex $p |- ran A e. _V $=
|
|
( cvv wcel crn rnexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Membership in a cross product. This version requires no quantifiers or
|
|
dummy variables. (Contributed by set.mm contributors, 17-Feb-2004.) $)
|
|
elxp4 $p |- ( A e. ( B X. C ) <-> ( A = <. U. dom { A } , U. ran { A } >.
|
|
/\ ( U. dom { A } e. B /\ U. ran { A } e. C ) ) ) $=
|
|
( vx vy wcel cv cop wceq wa wex csn cdm cuni crn sneq unieqd vex pm4.71ri
|
|
syl6req elxp rneqd op2nda adantr exbii snex rnex uniex opeq2 eqeq2d eleq1
|
|
anbi2d anbi12d ceqsexv bitri dmeqd op1sta anbi1i anass 3bitri dmex anbi1d
|
|
cxp opeq1 ) ABCVCFADGZEGZHZIZVEBFZVFCFZJZJZEKZDKVEALZMZNZIZAVEVNOZNZHZIZV
|
|
IVSCFZJZJZJZDKAVPVSHZIZVPBFZWBJZJZDEABCUAVMWEDVMWDVQWAJZWCJWEVMVFVSIZVLJZ
|
|
EKWDVLWMEVLWLVHWLVKVHVSVGLZOZNVFVHVRWOVHVNWNAVGPUBQVEVFDRZERUCTUDSUEVLWDE
|
|
VSVRVNAUFZUGUHZWLVHWAVKWCWLVGVTAVFVSVEUIUJWLVJWBVIVFVSCUKULUMUNUOWAWKWCWA
|
|
VQWAVPVTLZMZNVEWAVOWTWAVNWSAVTPUPQVEVSWPWRUQTSURVQWAWCUSUTUEWDWJDVPVOVNWQ
|
|
VAUHVQWAWGWCWIVQVTWFAVEVPVSVDUJVQVIWHWBVEVPBUKVBUMUNUT $.
|
|
$}
|
|
|
|
$( If a cross product is a set, one of its components must be a set.
|
|
(Contributed by set.mm contributors, 27-Aug-2006.) $)
|
|
xpexr $p |- ( ( A X. B ) e. C -> ( A e. _V \/ B e. _V ) ) $=
|
|
( cxp wcel cvv wn wi wceq 0ex eleq1 mpbiri pm2.24d a1d wne crn rnexg eleq1d
|
|
c0 rnxp syl5ib a1dd pm2.61ine orrd ) ABDZCEZAFEZBFEZUFUGGZUHHZHASASIZUJUFUK
|
|
UGUHUKUGSFEJASFKLMNASOZUFUHUIUFUEPZFEULUHUECQULUMBFABTRUAUBUCUD $.
|
|
|
|
$( If a nonempty cross product is a set, so are both of its components.
|
|
(Contributed by set.mm contributors, 27-Aug-2006.) (Revised by set.mm
|
|
contributors, 5-May-2007.) $)
|
|
xpexr2 $p |- ( ( ( A X. B ) e. C /\ ( A X. B ) =/= (/) ) ->
|
|
( A e. _V /\ B e. _V ) ) $=
|
|
( cxp c0 wne wcel wa cvv xpnz cdm wceq dmxp adantl adantr eqeltrrd crn rnxp
|
|
dmexg rnexg anim12dan ancom2s sylan2br ) ABDZEFUDCGZAEFZBEFZHAIGZBIGZHZABJU
|
|
EUGUFUJUEUGUHUFUIUEUGHUDKZAIUGUKALUEABMNUEUKIGUGUDCSOPUEUFHUDQZBIUFULBLUEAB
|
|
RNUEULIGUFUDCTOPUAUBUC $.
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( Alternate definition of the ` 2nd ` function. (Contributed by SF,
|
|
8-Jan-2015.) $)
|
|
df2nd2 $p |- 2nd = ( 1st o. Swap ) $=
|
|
( vx vz vy vw cv cop wceq wex copab cswap wbr c1st wa c2nd ccom vex br1st
|
|
anbi1i ancom exbii 19.41v 3bitr4i excom opex breq2 ceqsexv bitri 3bitr2ri
|
|
brswap2 opabbii df-2nd df-co 3eqtr4i ) AEZBEZCEZFGZBHZACIUNDEZJKZUSUPLKZM
|
|
ZDHZACINLJOURVCACVCUSUPUOFZGZUTMZBHZDHVFDHZBHURVBVGDVAUTMVEBHZUTMVBVGVAVI
|
|
UTBUSUPCPZQRUTVASVEUTBUAUBTVFBDUCVHUQBVHUNVDJKZUQUTVKDVDUPUOVJBPZUDUSVDUN
|
|
JUEUFUNUPUOVJVLUIUGTUHUJACBUKACDLJULUM $.
|
|
$}
|
|
|
|
$( The ` 2nd ` function is a set. (Contributed by SF, 8-Jan-2015.) $)
|
|
2ndex $p |- 2nd e. _V $=
|
|
( c2nd c1st cswap ccom cvv df2nd2 1stex swapex coex eqeltri ) ABCDEFBCGHIJ
|
|
$.
|
|
|
|
${
|
|
$d A x y z w v $. $d B x y z w v $.
|
|
$( Define cross product via the set construction functions. (Contributed
|
|
by SF, 8-Jan-2015.) $)
|
|
dfxp2 $p |- ( A X. B ) = ( ( `' 1st " A ) i^i ( `' 2nd " B ) ) $=
|
|
( vx vy vz vw vv c1st ccnv c2nd cv cop wceq wrex wbr wcel wex weq 3bitr4i
|
|
wa cxp cima cin eeanv vex opeq2 eqeq2d opeq1 bi2anan9 spc2ev anidms simpl
|
|
eqtr2 opth adantl sylbi syl eqtrd exlimivv impbii brcnv br1st bitri br2nd
|
|
anbi12i 2rexbii elxp2 elima elin reeanv eqriv ) CABUAZHIZAUBZJIZBUBZUCZCK
|
|
ZDKZEKZLZMZEBNDANVSVRVMOZVTVRVOOZTZEBNDANZVRVLPVRVQPZWBWEDEABVRVSFKZLZMZV
|
|
RGKZVTLZMZTZGQFQZWJFQZWMGQZTWBWEWJWMFGUDWBWOWBWOWNWBWBTFGVTVSEUEZDUEZFERZ
|
|
WJWBGDRZWMWBWTWIWAVRWHVTVSUFZUGXAWLWAVRWKVSVTUHUGUIUJUKWNWBFGWNVRWIWAWJWM
|
|
ULWNWIWLMZWIWAMZVRWIWLUMXCDGRZWTTXDVSWHWKVTUNWTXDXEXBUOUPUQURUSUTWCWPWDWQ
|
|
WCVRVSHOWPVSVRHVAFVRVSWSVBVCWDVRVTJOWQVTVRJVAGVRVTWRVDVCVESVFDEVRABVGVRVN
|
|
PZVRVPPZTWCDANZWDEBNZTWGWFXFXHXGXIDVRVMAVHEVRVOBVHVEVRVNVPVIWCWDDEABVJSSV
|
|
K $.
|
|
$}
|
|
|
|
$( The cross product of two sets is a set. Proposition 6.2 of
|
|
[TakeutiZaring] p. 23. (Contributed by set.mm contributors,
|
|
14-Aug-1994.) $)
|
|
xpexg $p |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V ) $=
|
|
( wcel cxp c1st ccnv cima c2nd cin cvv dfxp2 1stex cnvex imaexg 2ndex inexg
|
|
wa mpan syl2an syl5eqel ) ACEZBDEZSABFGHZAIZJHZBIZKZLABMUCUFLEZUHLEZUILEUDU
|
|
ELEUCUJGNOUEALCPTUGLEUDUKJQOUGBLDPTUFUHLLRUAUB $.
|
|
|
|
${
|
|
xpex.1 $e |- A e. _V $.
|
|
xpex.2 $e |- B e. _V $.
|
|
$( The cross product of two sets is a set. Proposition 6.2 of
|
|
[TakeutiZaring] p. 23. (Contributed by set.mm contributors,
|
|
14-Aug-1994.) $)
|
|
xpex $p |- ( A X. B ) e. _V $=
|
|
( cvv wcel cxp xpexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
$( The restriction of a set to a set is a set. (Contributed by set.mm
|
|
contributors, 8-Jan-2015.) $)
|
|
resexg $p |- ( ( A e. V /\ B e. W ) -> ( A |` B ) e. _V ) $=
|
|
( wcel wa cres cvv cxp cin df-res vvex xpexg mpan2 inexg sylan2 syl5eqel )
|
|
ACEZBDEZFABGABHIZJZHABKSRTHEZUAHESHHEUBLBHDHMNATCHOPQ $.
|
|
|
|
${
|
|
resex.1 $e |- A e. _V $.
|
|
resex.2 $e |- B e. _V $.
|
|
$( The restriction of a set to a set is a set. (Contributed by set.mm
|
|
contributors, 8-Jan-2015.) $)
|
|
resex $p |- ( A |` B ) e. _V $=
|
|
( cvv wcel cres resexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d A a b x $. $d B a b $.
|
|
$( The converse of an intersection is the intersection of the converse.
|
|
(Contributed by FL, 15-Oct-2012.) (Revised by Scott Fenton,
|
|
18-Apr-2021.) $)
|
|
cnviin $p |- `' |^|_ x e. A B = |^|_ x e. A `' B $=
|
|
( va vb ciin ccnv cv cop wcel wral cvv wb opex eliin ax-mp opelcnv ralbii
|
|
vex bitri 3bitr4i eqrelriv ) DEABCFZGZABCGZFZEHZDHZIZUCJZUICJZABKZUHUGIZU
|
|
DJUMUFJZUILJUJULMUGUHESZDSZNAUIBCLOPUHUGUCQUNUMUEJZABKZULUMLJUNURMUHUGUPU
|
|
ONAUMBUELOPUQUKABUHUGCQRTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
$( Alternate definition of a function. (Contributed by set.mm
|
|
contributors, 29-Dec-1996.) (Revised by set.mm contributors,
|
|
23-Apr-2004.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
dffun2 $p |- ( Fun A <->
|
|
A. x A. y A. z ( ( x A y /\ x A z ) -> y = z ) ) $=
|
|
( wfun ccnv ccom cid wss cv cop wcel wi wal wbr wa weq df-fun wex bitr4i
|
|
ssrel opelco brcnv anbi1i exbii bitri df-br vex ideq bitr3i 19.23v 2albii
|
|
imbi12i alrot3 3bitri ) DEDDFZGZHIBJZCJZKZUQLZUTHLZMZCNBNZAJZURDOZVEUSDOZ
|
|
PZBCQZMZCNBNANZDRBCUQHUAVDVJANZCNBNVKVCVLBCVCVHASZVIMVLVAVMVBVIVAURVEUPOZ
|
|
VGPZASVMAURUSDUPUBVOVHAVNVFVGURVEDUCUDUEUFVBURUSHOVIURUSHUGURUSCUHUIUJUMV
|
|
HVIAUKTULVJABCUNTUO $.
|
|
|
|
$( Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
|
|
(Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
dffun3 $p |- ( Fun A <-> A. x E. z A. y ( x A y -> y = z ) ) $=
|
|
( wfun cv wbr wa weq wi wal wex dffun2 wmo breq2 mo4 nfv mo2 bitr3i albii
|
|
bitri ) DEAFZBFZDGZUBCFZDGZHBCIZJCKBKZAKUDUGJBKCLZAKABCDMUHUIAUHUDBNUIUDU
|
|
FBCUCUEUBDOPUDBCUDCQRSTUA $.
|
|
|
|
$( Alternate definition of a function. Definition 6.4(4) of
|
|
[TakeutiZaring] p. 24. (Contributed by set.mm contributors,
|
|
29-Dec-1996.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
dffun4 $p |- ( Fun A <->
|
|
A. x A. y A. z ( ( <. x , y >. e. A /\ <. x , z >. e. A )
|
|
-> y = z ) ) $=
|
|
( wfun cv wbr wa weq wi wal wcel dffun2 df-br anbi12i imbi1i albii 2albii
|
|
cop bitri ) DEAFZBFZDGZUACFZDGZHZBCIZJZCKZBKAKUAUBSDLZUAUDSDLZHZUGJZCKZBK
|
|
AKABCDMUIUNABUHUMCUFULUGUCUJUEUKUAUBDNUAUDDNOPQRT $.
|
|
|
|
$( Alternate definition of function. (Contributed by set.mm contributors,
|
|
29-Dec-1996.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
dffun5 $p |- ( Fun A <-> A. x E. z A. y ( <. x , y >. e. A -> y = z ) ) $=
|
|
( wfun cv wbr weq wal wex cop wcel dffun3 df-br imbi1i albii exbii bitri
|
|
wi ) DEAFZBFZDGZBCHZSZBIZCJZAITUAKDLZUCSZBIZCJZAIABCDMUFUJAUEUICUDUHBUBUG
|
|
UCTUADNOPQPR $.
|
|
$}
|
|
|
|
${
|
|
$d x y w v u $. $d A w v u $.
|
|
dffun6f.1 $e |- F/_ x A $.
|
|
dffun6f.2 $e |- F/_ y A $.
|
|
$( Definition of function, using bound-variable hypotheses instead of
|
|
distinct variable conditions. (Contributed by NM, 9-Mar-1995.)
|
|
(Revised by Mario Carneiro, 15-Oct-2016.) (Revised by Scott Fenton,
|
|
16-Apr-2021.) $)
|
|
dffun6f $p |- ( Fun A <-> A. x E* y x A y ) $=
|
|
( vw vv vu wfun cv wbr weq wi wal wex wmo nfcv nfbr nfv albii breq2 cbvmo
|
|
dffun3 mo2 nfmo breq1 mobidv cbval 3bitr3ri bitr4i ) CIFJZGJZCKZGHLMGNHOZ
|
|
FNZAJZBJZCKZBPZANZFGHCUCUMGPZFNUKUQCKZBPZFNUOUTVAVCFUMVBGBBUKULCBUKQEBULQ
|
|
RVBGSULUQUKCUAUBTVAUNFUMGHUMHSUDTVCUSFAVBABAUKUQCAUKQDAUQQRUEUSFSFALVBURB
|
|
UKUPUQCUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( Alternate definition of a function using "at most one" notation.
|
|
(Contributed by NM, 9-Mar-1995.) (Revised by Scott Fenton,
|
|
16-Apr-2021.) $)
|
|
dffun6 $p |- ( Fun F <-> A. x E* y x F y ) $=
|
|
( nfcv dffun6f ) ABCACDBCDE $.
|
|
|
|
$( A function has at most one value for each argument. (Contributed by NM,
|
|
24-May-1998.) $)
|
|
funmo $p |- ( Fun F -> E* y A F y ) $=
|
|
( vx cv wbr cvv wcel wa wi wal wfun wmo brreldmex ancri ax-gen wceq breq1
|
|
mobidv spcgv com12 dffun6 moanimv 3imtr4i moim mpsyl ) BAEZCFZBGHZUHIZJZA
|
|
KCLZUJAMZUHAMZUKAUHUIBUGCNOPDEZUGCFZAMZDKZUIUNJULUMUIURUNUQUNDBGUOBQUPUHA
|
|
UOBUGCRSTUADACUBUIUHAUCUDUHUJAUEUF $.
|
|
$}
|
|
|
|
$( Subclass theorem for function predicate. (The proof was shortened by
|
|
Mario Carneiro, 24-Jun-2014.) (Contributed by set.mm contributors,
|
|
16-Aug-1994.) (Revised by set.mm contributors, 24-Jun-2014.) $)
|
|
funss $p |- ( A C_ B -> ( Fun B -> Fun A ) ) $=
|
|
( wss ccnv ccom cid wfun coss1 cnvss coss2 syl sstrd sstr2 df-fun 3imtr4g
|
|
wi ) ABCZBBDZEZFCZAADZEZFCZBGAGQUBSCTUCPQUBBUAEZSABUAHQUARCUDSCABIUARBJKLUB
|
|
SFMKBNANO $.
|
|
|
|
$( Equality theorem for function predicate. (Contributed by set.mm
|
|
contributors, 16-Aug-1994.) $)
|
|
funeq $p |- ( A = B -> ( Fun A <-> Fun B ) ) $=
|
|
( wss wa wfun wi wceq wb funss anim12i ancoms eqss dfbi2 3imtr4i ) ABCZBACZ
|
|
DAEZBEZFZRQFZDZABGQRHPOUAPSOTBAIABIJKABLQRMN $.
|
|
|
|
${
|
|
funeqi.1 $e |- A = B $.
|
|
$( Equality inference for the function predicate. (Contributed by Jonathan
|
|
Ben-Naim, 3-Jun-2011.) $)
|
|
funeqi $p |- ( Fun A <-> Fun B ) $=
|
|
( wceq wfun wb funeq ax-mp ) ABDAEBEFCABGH $.
|
|
$}
|
|
|
|
${
|
|
funeqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for the function predicate. (Contributed by set.mm
|
|
contributors, 23-Feb-2013.) $)
|
|
funeqd $p |- ( ph -> ( Fun A <-> Fun B ) ) $=
|
|
( wceq wfun wb funeq syl ) ABCEBFCFGDBCHI $.
|
|
$}
|
|
|
|
${
|
|
nffun.1 $e |- F/_ x F $.
|
|
$( Bound-variable hypothesis builder for a function. (Contributed by NM,
|
|
30-Jan-2004.) $)
|
|
nffun $p |- F/ x Fun F $=
|
|
( wfun ccnv ccom cid wss df-fun nfcnv nfco nfcv nfss nfxfr ) BDBBEZFZGHAB
|
|
IAPGABOCABCJKAGLMN $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y F $.
|
|
$( There is exactly one value of a function. (Contributed by NM,
|
|
22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
funeu $p |- ( ( Fun F /\ A F B ) -> E! y A F y ) $=
|
|
( wfun wbr wa cv wex wmo weu cdm wcel breldm eldm sylib adantl adantr jca
|
|
funmo eu5 sylibr ) DEZBCDFZGZBAHDFZAIZUFAJZGUFAKUEUGUHUDUGUCUDBDLMUGBCDNA
|
|
BDOPQUCUHUDABDTRSUFAUAUB $.
|
|
|
|
$( There is exactly one value of a function. (Contributed by NM,
|
|
3-Aug-1994.) $)
|
|
funeu2 $p |- ( ( Fun F /\ <. A , B >. e. F ) -> E! y <. A , y >. e. F ) $=
|
|
( cop wcel wfun wbr cv weu df-br wa funeu eubii sylib sylan2br ) BCEDFDGZ
|
|
BCDHZBAIZEDFZAJZBCDKQRLBSDHZAJUAABCDMUBTABSDKNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( Alternate definition of a function. One possibility for the definition
|
|
of a function in [Enderton] p. 42. (Enderton's definition is ambiguous
|
|
because "there is only one" could mean either "there is at most one" or
|
|
"there is exactly one." However, ~ dffun8 shows that it doesn't matter
|
|
which meaning we pick.) (Contributed by set.mm contributors,
|
|
4-Nov-2002.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
dffun7 $p |- ( Fun A <-> A. x e. dom A E* y x A y ) $=
|
|
( cv wbr wmo wal cdm wcel wi wfun wral wex moabs eldm imbi1i bitr4i albii
|
|
dffun6 df-ral 3bitr4i ) ADZBDCEZBFZAGUBCHZIZUDJZAGCKUDAUELUDUGAUDUCBMZUDJ
|
|
UGUCBNUFUHUDBUBCOPQRABCSUDAUETUA $.
|
|
|
|
$( Alternate definition of a function. One possibility for the definition
|
|
of a function in [Enderton] p. 42. Compare ~ dffun7 . (The proof was
|
|
shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm
|
|
contributors, 4-Nov-2002.) (Revised by set.mm contributors,
|
|
18-Sep-2011.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
dffun8 $p |- ( Fun A <-> A. x e. dom A E! y x A y ) $=
|
|
( wfun cv wbr wmo cdm wral weu dffun7 wcel wex eldm exmoeu2 sylbi ralbiia
|
|
wb bitri ) CDAEZBECFZBGZACHZIUABJZAUCIABCKUBUDAUCTUCLUABMUBUDRBTCNUABOPQS
|
|
$.
|
|
|
|
$( Alternate definition of a function. (Contributed by set.mm
|
|
contributors, 28-Mar-2007.) (Revised by Scott Fenton, 16-Apr-2021.) $)
|
|
dffun9 $p |- ( Fun A <-> A. x e. dom A E* y ( y e. ran A /\ x A y ) ) $=
|
|
( wfun cv wbr wmo cdm wral crn wcel wa dffun7 brelrn pm4.71ri mobii bitri
|
|
ralbii ) CDAEZBEZCFZBGZACHZITCJKZUALZBGZAUCIABCMUBUFAUCUAUEBUAUDSTCNOPRQ
|
|
$.
|
|
$}
|
|
|
|
$( An equivalence for the function predicate. (Contributed by set.mm
|
|
contributors, 13-Aug-2004.) $)
|
|
funfn $p |- ( Fun A <-> A Fn dom A ) $=
|
|
( wfun cdm wceq wa wfn eqid biantru df-fn bitr4i ) ABZKACZLDZEALFMKLGHALIJ
|
|
$.
|
|
|
|
$( The identity relation is a function. Part of Theorem 10.4 of [Quine]
|
|
p. 65. (Contributed by set.mm contributors, 30-Apr-1998.) $)
|
|
funi $p |- Fun _I $=
|
|
( cid wfun ccnv ccom wss cnvi coeq2i coi1 eqtri eqimssi df-fun mpbir ) ABAA
|
|
CZDZAENANAADAMAAFGAHIJAKL $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The universe is not a function. (Contributed by Raph Levien,
|
|
27-Jan-2004.) $)
|
|
nfunv $p |- -. Fun _V $=
|
|
( vx vy vz cvv wfun cv cop wcel wa weq wal wex wne vex pm3.2i exbii exnal
|
|
wn opex bitri ccompl complex necompl necomi wceq opeq2 eleq1d df-ne neeq2
|
|
anbi2d syl5bbr anbi12d spcev 19.8a 19.23bi mp2b exanali mpbi dffun4 mtbir
|
|
wi ) DEAFZBFZGDHZVBCFZGZDHZIZBCJZVACKZBKZAKZVHVIRZIZCLZBLZALZVLRZVDVBVCUA
|
|
ZGZDHZIZVCVSMZIZVOVQWBWCVDWAVBVCANZBNZSVBVSWEVCWFUBZSOVSVCVCUCUDOVNWDCVSW
|
|
GVEVSUEZVHWBVMWCWHVGWAVDWHVFVTDVEVSVBUFUGUJVMVCVEMWHWCVCVEUHVEVSVCUIUKULU
|
|
MVOVQBVPAUNUOUPVQVKRZALVRVPWIAVPVJRZBLWIVOWJBVHVICUQPVJBQTPVKAQTURABCDUSU
|
|
T $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( A class of ordered pairs is a function when there is at most one second
|
|
member for each pair. (Contributed by NM, 16-May-1995.) $)
|
|
funopab $p |- ( Fun { <. x , y >. | ph } <-> A. x E* y ph ) $=
|
|
( copab wfun cv wbr wmo wal nfopab1 nfopab2 dffun6f cop wcel df-br opabid
|
|
bitri mobii albii ) ABCDZEBFZCFZTGZCHZBIACHZBIBCTABCJABCKLUDUEBUCACUCUAUB
|
|
MTNAUAUBTOABCPQRSQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y A $.
|
|
$( A class of ordered pairs of values is a function. (Contributed by
|
|
set.mm contributors, 14-Nov-1995.) $)
|
|
funopabeq $p |- Fun { <. x , y >. | y = A } $=
|
|
( cv wceq copab wfun wmo funopab moeq mpgbir ) BDCEZABFGLBHALABIBCJK $.
|
|
|
|
$( A class of ordered pairs of values in the form used by ~ fvopab4 is a
|
|
function. (Contributed by set.mm contributors, 17-Feb-2013.) $)
|
|
funopab4 $p |- Fun { <. x , y >. | ( ph /\ y = A ) } $=
|
|
( cv wceq wa copab wss wfun simpr ssopab2i funopabeq funss mp2 ) ACEDFZGZ
|
|
BCHZPBCHZISJRJQPBCAPKLBCDMRSNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z F $. $d x y z G $.
|
|
$( The composition of two functions is a function. Exercise 29 of
|
|
[TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof
|
|
shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
funco $p |- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) ) $=
|
|
( vx vz vy wfun wa cv wbr wex copab wmo wal funmo alrimiv moexexv syl2anr
|
|
ccom funopab sylibr df-co funeqi ) AFZBFZGZCHZDHZBIZUGEHAIZGDJZCEKZFZABRZ
|
|
FUEUJELZCMULUEUNCUDUHDLUIELZDMUNUCDUFBNUCUODEUGANOUHUIDEPQOUJCESTUMUKCEDA
|
|
BUAUBT $.
|
|
$}
|
|
|
|
$( A restriction of a function is a function. Compare Exercise 18 of
|
|
[TakeutiZaring] p. 25. (Contributed by set.mm contributors,
|
|
16-Aug-1994.) $)
|
|
funres $p |- ( Fun F -> Fun ( F |` A ) ) $=
|
|
( cres wss wfun wi resss funss ax-mp ) BACZBDBEJEFBAGJBHI $.
|
|
|
|
${
|
|
$d x y F $. $d x y G $.
|
|
$( The restriction of a function to the domain of a subclass equals the
|
|
subclass. (Contributed by NM, 15-Aug-1994.) $)
|
|
funssres $p |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G ) $=
|
|
( vx vy wfun wss wa cdm cres cop wcel ssel adantl opeldm a1i jcad wex imp
|
|
cv wi weu funeu2 eldm2 ancrd eximdv syl5bi eupick exp43 com23 com34 imp3a
|
|
syl2an pm2.43d impbid opelres syl6rbbr eqrelrdv ) AEZBAFZGZCDABHZIZBUTCSZ
|
|
DSZJZBKZVEAKZVCVAKZGZVEVBKUTVFVIUTVFVGVHUSVFVGTURBAVELZMVFVHTUTVCVDBNOPUT
|
|
VGVHVFUTVGVHVFTUTVGVHVGVFURUSVGVHVGVFTZTZTURVGUSVLURVGUSVHVKURVGGVGDUAVGV
|
|
FGZDQZVKUSVHGDVCVDAUBUSVHVNVHVFDQUSVNDVCBUCUSVFVMDUSVFVGVJUDUEUFRVGVFDUGU
|
|
LUHUIRUJUMUKUNVCVDAVAUOUPUQ $.
|
|
$}
|
|
|
|
$( Equality of restrictions of a function and a subclass. (Contributed by
|
|
set.mm contributors, 16-Aug-1994.) (Revised by set.mm contributors,
|
|
2-Jun-2007.) $)
|
|
fun2ssres $p |- ( ( Fun F /\ G C_ F /\ A C_ dom G ) ->
|
|
( F |` A ) = ( G |` A ) ) $=
|
|
( wfun wss cdm cres wceq wa resabs1 eqcomd funssres reseq1d sylan9eqr 3impa
|
|
) BDZCBEZACFZEZBAGZCAGZHSPQIZTBRGZAGZUASUDTBARJKUBUCCABCLMNO $.
|
|
|
|
${
|
|
$d x y z F $. $d x y z G $.
|
|
$( The union of functions with disjoint domains is a function. Theorem 4.6
|
|
of [Monk1] p. 43. (Contributed by set.mm contributors, 12-Aug-1994.) $)
|
|
funun $p |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) ->
|
|
Fun ( F u. G ) ) $=
|
|
( vx vy vz wfun wa cdm cin c0 wceq cv cop wcel wi wal wo wn opeldm dffun4
|
|
cun weq elun anbi12i anddi bitri disj1 imnan bicomi 3imtr4i anim12i orel2
|
|
sp nsyl incom eqeq1i orel1 orim12d syl5bi biimpi 19.21bi 19.21bbi sylan9r
|
|
syl jaao alrimiv alrimivv sylibr ) AFZBFZGZAHZBHZIZJKZGZCLZDLZMZABUAZNZVQ
|
|
ELZMZVTNZGZDEUBZOZEPZDPCPVTFVPWHCDVPWGEVOWEVSANZWCANZGZVSBNZWCBNZGZQZVKWF
|
|
WEWKWIWMGZQZWLWJGZWNQZQZVOWOWEWIWLQZWJWMQZGWTWAXAWDXBVSABUCWCABUCUDWIWLWJ
|
|
WMUEUFVOWQWKWSWNVOWPRWQWKOVOVQVLNZVQVMNZGZWPXCXDROZCPXFVOXERZXFCUMCVLVMUG
|
|
XFXGXCXDUHUIUJWIXCWMXDVQVRASVQWBBSUKUNWPWKULVDVOWRRWSWNOVOXDXCGZWRXDXCROZ
|
|
CPZXIVOXHRZXICUMVOVMVLIZJKXJVNXLJVLVMUOUPCVMVLUGUFXIXKXDXCUHUIUJWLXDWJXCV
|
|
QVRBSVQWBASUKUNWRWNUQVDURUSVIWKWFVJWNVIWKWFOZDEVIXMEPDPZCVIXNCPCDEATUTVAV
|
|
BVJWNWFOZDEVJXOEPDPZCVJXPCPCDEBTUTVAVBVEVCVFVGCDEVTTVH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( A singleton of an ordered pair is a function. Theorem 10.5 of [Quine]
|
|
p. 65. (Contributed by NM, 12-Aug-1994.) (Revised by Scott Fenton,
|
|
16-Apr-2021.) $)
|
|
funsn $p |- Fun { <. A , B >. } $=
|
|
( vx vy cop csn wfun cv wbr wmo dffun6 wceq wi moeq a1i wa wcel df-br vex
|
|
bitri opex elsnc opth mobii moanimv mpbir mpgbir ) ABEZFZGCHZDHZUIIZDJZCC
|
|
DUIKUMUJALZUKBLZDJZMZUPUNDBNOUMUNUOPZDJUQULURDULUJUKEZUIQZURUJUKUIRUTUSUH
|
|
LURUSUHUJUKCSDSUAUBUJUKABUCTTUDUNUODUETUFUG $.
|
|
$}
|
|
|
|
$( A singleton of an ordered pair is a function. Theorem 10.5 of [Quine]
|
|
p. 65. (Contributed by set.mm contributors, 28-Jun-2011.) (Revised by
|
|
set.mm contributors, 1-Oct-2013.) $)
|
|
funsngOLD $p |- ( ( A e. V /\ B e. W ) -> Fun { <. A , B >. } ) $=
|
|
( cop csn wfun wcel wa funsn a1i ) ABEFGACHBDHIABJK $.
|
|
|
|
|
|
$( A set of two pairs is a function if their first members are different.
|
|
(Contributed by FL, 26-Jun-2011.) (Revised by Scott Fenton,
|
|
16-Apr-2021.) $)
|
|
funprg $p |- ( ( A =/= B /\ C e. V /\ D e. W )
|
|
-> Fun { <. A , C >. , <. B , D >. } ) $=
|
|
( wne wcel w3a cop csn cun wfun cpr cdm cin c0 wceq dmsnopg funsn eqtrd syl
|
|
3ad2ant2 3ad2ant3 ineq12d disjsn2 3ad2ant1 funun mpanl12 funeqi sylibr
|
|
df-pr ) ABGZCEHZDFHZIZACJZKZBDJZKZLZMZUQUSNZMUPUROZUTOZPZQRZVBUPVFAKZBKZPZQ
|
|
UPVDVHVEVIUNUMVDVHRUOACESUCUOUMVEVIRUNBDFSUDUEUMUNVJQRUOABUFUGUAURMUTMVGVBA
|
|
CTBDTURUTUHUIUBVCVAUQUSULUJUK $.
|
|
|
|
|
|
$( A set of two pairs is a function if their first members are different.
|
|
(Contributed by FL, 26-Jun-2011.) $)
|
|
funprgOLD $p |- ( ( A =/= B /\ ( A e. V /\ B e. W ) /\ ( C e. T /\ D e. U ) )
|
|
-> Fun { <. A , C >. , <. B , D >. } ) $=
|
|
( wcel wa cop csn wfun cdm cin c0 wceq funsngOLD syl2anc dmsnopg simp2l simp3l
|
|
wne w3a cun cpr simp2r simp3r ineq12d disjsn2 3ad2ant1 eqtrd funun syl21anc
|
|
syl df-pr funeqi sylibr ) ABUCZAGIZBHIZJZCEIZDFIZJZUDZACKZLZBDKZLZUEZMZVGVI
|
|
UFZMVFVHMZVJMZVHNZVJNZOZPQVLVFUTVCVNUSUTVAVEUAUSVBVCVDUBZACGERSVFVAVDVOUSUT
|
|
VAVEUGUSVBVCVDUHZBDHFRSVFVRALZBLZOZPVFVPWAVQWBVFVCVPWAQVSACETUOVFVDVQWBQVTB
|
|
DFTUOUIUSVBWCPQVEABUJUKULVHVJUMUNVMVKVGVIUPUQUR $.
|
|
|
|
${
|
|
funpr.1 $e |- C e. _V $.
|
|
funpr.2 $e |- D e. _V $.
|
|
$( A function with a domain of two elements. (Contributed by Jeff Madsen,
|
|
20-Jun-2010.) $)
|
|
funpr $p |- ( A =/= B -> Fun { <. A , C >. , <. B , D >. } ) $=
|
|
( wne cvv wcel cop cpr wfun funprg mp3an23 ) ABGCHIDHIACJBDJKLEFABCDHHMN
|
|
$.
|
|
$}
|
|
|
|
${
|
|
fnsn.1 $e |- A e. _V $.
|
|
fnsn.2 $e |- B e. _V $.
|
|
$( Functionality and domain of the singleton of an ordered pair.
|
|
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $)
|
|
fnsn $p |- { <. A , B >. } Fn { A } $=
|
|
( cop csn wfn wfun cdm wceq funsn dmsnop df-fn mpbir2an ) ABEFZAFZGOHOIPJ
|
|
ABKABDLOPMN $.
|
|
$}
|
|
|
|
$( Domain of a function with a domain of two different values. (Contributed
|
|
by FL, 26-Jun-2011.) $)
|
|
fnprg $p |- ( ( A =/= B /\ ( A e. V /\ B e. W ) /\ ( C e. T /\ D e. U ) )
|
|
-> { <. A , C >. , <. B , D >. } Fn { A , B } ) $=
|
|
( wne wcel wa w3a cop cpr wfun cdm wceq wfn funprgOLD dmpropg 3ad2ant3
|
|
df-fn sylanbrc ) ABIZAGJBHJKZCEJDFJKZLACMBDMNZOUGPABNZQZUGUHRABCDEFGHSUFUDU
|
|
IUEACBDEFTUAUGUHUBUC $.
|
|
|
|
$( The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed
|
|
by set.mm contributors, 7-Apr-1998.) $)
|
|
fun0 $p |- Fun (/) $=
|
|
( c0 wfun ccnv ccom cid wss co01 0ss eqsstri df-fun mpbir ) ABAACZDZEFMAELG
|
|
EHIAJK $.
|
|
|
|
${
|
|
$d f g x y z w v A $. $d x y B $. $d x y R $.
|
|
$( A simpler equivalence for single-rooted (see ~ funcnv ). (Contributed
|
|
by set.mm contributors, 9-Aug-2004.) $)
|
|
funcnv2 $p |- ( Fun `' A <-> A. y E* x x A y ) $=
|
|
( ccnv wfun cv wbr wmo wal dffun6 brcnv mobii albii bitri ) CDZEBFZAFZOGZ
|
|
AHZBIQPCGZAHZBIBAOJSUABRTAPQCKLMN $.
|
|
|
|
$( The converse of a class is a function iff the class is single-rooted,
|
|
which means that for any ` y ` in the range of ` A ` there is at most
|
|
one ` x ` such that ` x A y ` . Definition of single-rooted in
|
|
[Enderton] p. 43. See ~ funcnv2 for a simpler version. (Contributed by
|
|
set.mm contributors, 13-Aug-2004.) $)
|
|
funcnv $p |- ( Fun `' A <-> A. y e. ran A E* x x A y ) $=
|
|
( cv wbr wmo wal crn wcel wi ccnv wfun wral brelrn pm4.71ri mobii moanimv
|
|
wa bitri albii funcnv2 df-ral 3bitr4i ) ADZBDZCEZAFZBGUECHZIZUGJZBGCKLUGB
|
|
UHMUGUJBUGUIUFRZAFUJUFUKAUFUIUDUECNOPUIUFAQSTABCUAUGBUHUBUC $.
|
|
|
|
$( A condition showing a class is single-rooted. (See ~ funcnv ).
|
|
(Contributed by set.mm contributors, 26-May-2006.) $)
|
|
funcnv3 $p |- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y ) $=
|
|
( cv wbr wmo crn wral wex wa ccnv wfun wreu wcel biimpi biantrurd ralbiia
|
|
cdm elrn weu funcnv df-reu breldm pm4.71ri eubii 3bitr2i ralbii 3bitr4i
|
|
eu5 ) ADZBDZCEZAFZBCGZHULAIZUMJZBUNHCKLULACRZMZBUNHUMUPBUNUKUNNZUOUMUSUOA
|
|
UKCSOPQABCUAURUPBUNURUJUQNZULJZATULATUPULAUQUBULVAAULUTUJUKCUCUDUEULAUIUF
|
|
UGUH $.
|
|
|
|
$( Single-rootedness (see ~ funcnv ) of a class cut down by a cross
|
|
product. (Contributed by NM, 5-Mar-2007.) $)
|
|
fncnv $p |- ( `' ( R i^i ( A X. B ) ) Fn B <->
|
|
A. y e. B E! x e. A x R y ) $=
|
|
( cxp cin ccnv wceq wa cv wbr wral anbi2i wmo wcel wi ancom bitri 3bitr4i
|
|
wfn wfun cdm wreu df-fn dfrn4 eqeq1i wrex wrmo rninxp anbi1i funcnv raleq
|
|
biimt moanimv brin brxp anass mobii df-rmo imbi2i syl6rbbr ralbiia syl6bb
|
|
crn syl5bb pm5.32i r19.26 reu5 ralbii 3bitr2i ) ECDFZGZHZDUAVNUBZVNUCZDIZ
|
|
JVOVMVEZDIZJZAKZBKZELZACUDZBDMZVNDUEVSVQVOVRVPDVMUFUGNVSVOJZWCACUHZWCACUI
|
|
ZJZBDMZVTWEVSWHBDMZJWGBDMZWKJWFWJVSWLWKABCDEUJUKVSVOWKVOWAWBVMLZAOZBVRMZV
|
|
SWKABVMULVSWOWNBDMWKWNBVRDUMWNWHBDWBDPZWHWPWHQZWNWPWHUNWPWACPZWCJZJZAOWPW
|
|
SAOZQWNWQWPWSAUOWMWTAWMWCWAWBVLLZJZWTWAWBEVLUPXCWPWRJZWCJZWTXCWCXDJXEXBXD
|
|
WCXBWRWPJXDWAWBCDUQWRWPRSNWCXDRSWPWRWCURSSUSWHXAWPWCACUTVATVBVCVDVFVGWGWH
|
|
BDVHTVOVSRWDWIBDWCACVIVJTVK $.
|
|
|
|
$( Two ways of stating that ` A ` is one-to-one.
|
|
Each side is equivalent to Definition 6.4(3) of
|
|
[TakeutiZaring] p. 24, who use the notation "Un_2 (A)" for one-to-one.
|
|
(Contributed by NM, 17-Jan-2006.) (Revised by Scott Fenton,
|
|
18-Apr-2021.) $)
|
|
fun11 $p |- ( ( Fun A /\ Fun `' A ) <->
|
|
A. x A. y A. z A. w ( ( x A y /\ z A w ) -> ( x = z <-> y = w ) ) ) $=
|
|
( cv wbr wa weq wi wal wfun bi2.04 anbi12i 2albii 19.26-2 alcom nfv albii
|
|
anbi1d wb ccnv dfbi2 imbi2i pm4.76 breq1 imbi1d equsal bitri breq2 3bitri
|
|
3bitr2i bitr2i dffun2 brcnv imbi1i alrot3 3bitr4i alrot4 ) CFZBFZEGZUTDFZ
|
|
EGZHZBDIZJZBKZDKZCKZAFZVCEGZVDHZACIZJZAKZDKCKZHZVKVAEGZVDHZVNVFUAZJZBKAKZ
|
|
DKCKZELZEUBZLZHWBDKCKBKAKWDVHVPHZDKCKVRWCWHCDWCVNVTVFJZJZVFVTVNJZJZHZBKAK
|
|
WJBKAKZWLBKZAKZHWHWBWMABWBVTVNVFJZVFVNJZHZJVTWQJZVTWRJZHWMWAWSVTVNVFUCUDV
|
|
TWQWRUEWTWJXAWLVTVNVFMVTVFVNMNULOWJWLABPWNVHWPVPWNWJAKZBKVHWJABQXBVGBWIVG
|
|
ACVGARVNVTVEVFVNVSVBVDVKUTVAEUFTUGUHSUIWOVOAWKVOBDVOBRVFVTVMVNVFVSVLVDVAV
|
|
CVKEUJTUGUHSNUKOVHVPCDPUMWEVJWGVQWEVGDKBKZCKVJCBDEUNXCVICVGBDQSUIVCVKWFGZ
|
|
VCUTWFGZHZVNJZCKZAKDKVOCKZAKDKWGVQXHXIDAXGVOCXFVMVNXDVLXEVDVCVKEUOVCUTEUO
|
|
NUPSODACWFUNVOCDAUQURNWBABCDUSUR $.
|
|
|
|
$( The union of a chain (with respect to inclusion) of functions is a
|
|
function. (Contributed by set.mm contributors, 10-Aug-2004.) $)
|
|
fununi $p |- ( A. f e. A ( Fun f /\ A. g e. A ( f C_ g \/ g C_ f ) ) ->
|
|
Fun U. A ) $=
|
|
( vx vy vz vw vv cv wfun wss wo wral wa wcel weq wi wal sps wex cuni ssel
|
|
cop r19.28av ralimi anim1d dffun4 sp sylbi syl9r adantl anim2d adantr imp
|
|
jaod funeq sseq1 sseq2 orbi12d anbi12d anbi2d cbvral2v ralcom orcom bitri
|
|
syl5bb anbi12i anidm anandir 2ralbii bitr2i 3bitr3i eluni eeanv an4 ancom
|
|
r19.26-2 2exbii 3bitr2i imbi1i 19.23v albii impexp 2albii bitr4i 3bitr3ri
|
|
r2al 3imtr4i alrimiv alrimivv syl sylibr ) BIZJZWMCIZKZWOWMKZLZCAMNZBAMZD
|
|
IZEIUCZAUAZOZXAFIUCZXCOZNZEFPZQZFRZERDRZXCJWTWNWRNZCAMZBAMZXKWSXMBAWNWRCA
|
|
UDUEXNXJDEXNXIFGIZJZHIZJZNZXOXQKZXQXOKZLZNZHAMZGAMZXBXOOZXEXQOZNZXHQZHAMZ
|
|
GAMZXNXIYDYJGAYCYIHAXSYBYIXSXTYIYAXRXTYIQXPXTYHXBXQOZYGNZXRXHXTYFYLYGXOXQ
|
|
XBUBUFXRYMXHQZFRZERZDRYNDEFXQUGYPYNDYOYNEYNFUHSSUIUJUKXPYAYIQXRYAYHYFXEXO
|
|
OZNZXPXHYAYGYQYFXQXOXEUBULXPYRXHQZFRZERZDRYSDEFXOUGUUAYSDYTYSEYSFUHSSUIUJ
|
|
UMUOUNUEUEXNXNNXPYBNZHAMGAMZXRYBNZHAMGAMZNZXNYEXNUUCXNUUEXLUUBXPXOWOKZWOX
|
|
OKZLZNBCGHAABGPZWNXPWRUUIWMXOUPUUJWPUUGWQUUHWMXOWOUQWMXOWOURUSUTCHPZUUIYB
|
|
XPUUKUUGXTUUHYAWOXQXOURWOXQXOUQUSVAVBXNXLBAMCAMUUEXLBCAAVCXLUUDWNXOWMKZWM
|
|
XOKZLZNCBGHAACGPZWRUUNWNWRWQWPLUUOUUNWPWQVDUUOWQUULWPUUMWOXOWMUQWOXOWMURU
|
|
SVFVABHPZWNXRUUNYBWMXQUPUUPUULXTUUMYAWMXQXOURWMXQXOUQUSUTVBVEVGXNVHYEUUBU
|
|
UDNZHAMGAMUUFYCUUQGHAAXPXRYBVIVJUUBUUDGHAAVQVKVLXIXOAOZXQAOZNZYHNZHTZGTZX
|
|
HQZYKXGUVCXHXGYFUURNZGTZYGUUSNZHTZNUVEUVGNZHTGTUVCXDUVFXFUVHGXBAVMHXEAVMV
|
|
GUVEUVGGHVNUVIUVAGHUVIYHUUTNUVAYFUURYGUUSVOYHUUTVPVEVRVSVTUVAXHQZHRZGRZUV
|
|
BXHQZGRYKUVDUVKUVMGUVAXHHWAWBUVLUUTYIQZHRGRYKUVJUVNGHUUTYHXHWCWDYIGHAAWGW
|
|
EUVBXHGWAWFVEWHWIWJWKDEFXCUGWL $.
|
|
|
|
$( The union of a chain (with respect to inclusion) of single-rooted sets
|
|
is single-rooted. (See ~ funcnv for "single-rooted" definition.)
|
|
(Contributed by set.mm contributors, 11-Aug-2004.) $)
|
|
funcnvuni $p |- ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) )
|
|
-> Fun `' U. A ) $=
|
|
( vy vx vz vw vv cv ccnv wfun wss wo wral wa wceq wrex wi wal wcel eqeq2d
|
|
cuni cnveq cbvrexv funeqd sseq1 sseq2 orbi12d ralbidv anbi12d rspcv funeq
|
|
biimprcd cnvss orim12i wb sseq12 ancoms syl5ibrcom exp3a syl6com rexlimdv
|
|
cab com23 alrimdv anim12ii syl5bi alrimiv df-ral eqeq1 rexbidv elab ralab
|
|
vex anbi2i imbi12i albii bitr2i sylib fununi syl ciun cnvuni cnvex dfiun2
|
|
eqtri funeqi sylibr ) BIZJZKZWICIZLZWLWILZMZCANZOZBANZDIZEIZJZPZEAQZDVCZU
|
|
BZKZAUBJZKWRFIZKZXHGIZLZXJXHLZMZGXDNZOZFXDNZXFWRXHXAPZEAQZXIXJXAPZEAQZXMR
|
|
ZGSZOZRZFSZXPWRYDFXRXHHIZJZPZHAQWRYCXQYHEHAWTYFPXAYGXHWTYFUCUAUDWRYHYCHAY
|
|
FATWRYGKZYFWLLZWLYFLZMZCANZOZYHYCRWQYNBYFAWIYFPZWKYIWPYMYOWJYGWIYFUCUEYOW
|
|
OYLCAYOWMYJWNYKWIYFWLUFWIYFWLUGUHUIUJUKYIYHXIYMYBYHXIYIXHYGULUMYMYHYAGYMX
|
|
TYHXMYMXSYHXMRZEAWTATYMYFWTLZWTYFLZMZXSYPRYLYSCWTAWLWTPYJYQYKYRWLWTYFUGWL
|
|
WTYFUFUHUKYSXSYHXMYSXMXSYHOZYGXALZXAYGLZMYQUUAYRUUBYFWTUNWTYFUNUOYTXKUUAX
|
|
LUUBYHXSXKUUAUPXHYGXJXAUQURXJXAXHYGUQUHUSUTVAVBVDVEVFVAVBVGVHXPXHXDTZXORZ
|
|
FSYEXOFXDVIUUDYDFUUCXRXOYCXCXRDXHFVNWSXHPXBXQEAWSXHXAVJVKVLXNYBXIXCXTXMGD
|
|
WSXJPXBXSEAWSXJXAVJVKVMVOVPVQVRVSXDFGVTWAXGXEXGEAXAWBXEEAWCEDAXAWTEVNWDWE
|
|
WFWGWH $.
|
|
|
|
$( The union of a chain (with respect to inclusion) of one-to-one functions
|
|
is a one-to-one function. (Contributed by set.mm contributors,
|
|
11-Aug-2004.) $)
|
|
fun11uni $p |- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\
|
|
A. g e. A ( f C_ g \/ g C_ f ) ) ->
|
|
( Fun U. A /\ Fun `' U. A ) ) $=
|
|
( cv wfun ccnv wa wo wral cuni simpl anim1i ralimi fununi simpr funcnvuni
|
|
wss syl jca ) BDZEZTFEZGZTCDZQUDTQHCAIZGZBAIZAJZEZUHFEZUGUAUEGZBAIUIUFUKB
|
|
AUCUAUEUAUBKLMABCNRUGUBUEGZBAIUJUFULBAUCUBUEUAUBOLMABCPRS $.
|
|
$}
|
|
|
|
$( The intersection with a function is a function. Exercise 14(a) of
|
|
[Enderton] p. 53. (The proof was shortened by Andrew Salmon,
|
|
17-Sep-2011.) (Contributed by set.mm contributors, 19-Mar-2004.)
|
|
(Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
funin $p |- ( Fun F -> Fun ( F i^i G ) ) $=
|
|
( cin wss wfun wi inss1 funss ax-mp ) ABCZADAEJEFABGJAHI $.
|
|
|
|
$( The restriction of a one-to-one function is one-to-one. (Contributed by
|
|
set.mm contributors, 25-Mar-1998.) $)
|
|
funres11 $p |- ( Fun `' F -> Fun `' ( F |` A ) ) $=
|
|
( cres wss ccnv wfun wi resss cnvss funss mp2b ) BACZBDLEZBEZDNFMFGBAHLBIMN
|
|
JK $.
|
|
|
|
$( The converse of a restricted function. (Contributed by set.mm
|
|
contributors, 27-Mar-1998.) $)
|
|
funcnvres $p |- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) ) $=
|
|
( ccnv wfun cima cres cdm dfima3 dfrn4 eqtri reseq2i wceq resss cnvss ax-mp
|
|
crn wss funssres mpan2 syl5req ) BCZDZUABAEZFUABAFZCZGZFZUEUCUFUAUCUDPUFBAH
|
|
UDIJKUBUEUAQZUGUELUDBQUHBAMUDBNOUAUERST $.
|
|
|
|
$( Converse of a restricted identity function. (Contributed by FL,
|
|
4-Mar-2007.) $)
|
|
cnvresid $p |- `' ( _I |` A ) = ( _I |` A ) $=
|
|
( cid ccnv wfun cres wceq cnvi eqcomi funi funeq mpbii ax-mp cima funcnvres
|
|
imai reseq12i syl6eq ) BCZDZBAEZCZTFBRFZSRBGHUBBDSIBRJKLSUARBAMZETABNRBUCAG
|
|
AOPQL $.
|
|
|
|
$( The converse of a restriction of the converse of a function equals the
|
|
function restricted to the image of its converse. (Contributed by set.mm
|
|
contributors, 4-May-2005.) $)
|
|
funcnvres2 $p |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) ) $=
|
|
( wfun ccnv cres cima wceq cnvcnv funeqi funcnvres sylbir reseq1i syl6eq )
|
|
BCZBDZAEDZODZOAFZEZBRENQCPSGQBBHZIAOJKQBRTLM $.
|
|
|
|
$( The image of the preimage of a function. (Contributed by set.mm
|
|
contributors, 25-May-2004.) $)
|
|
funimacnv $p |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) ) $=
|
|
( wfun ccnv cima cres crn cin funcnvres2 rneqd dfima3 syl6reqr dfrn4 ineq2i
|
|
cdm dmres df-dm 3eqtr2ri syl6eq ) BCZBBDZAEZEZUAAFZDZGZABGZHZTUFBUBFZGUCTUE
|
|
UIABIJBUBKLUHAUAOZHUDOUFUGUJABMNUAAPUDQRS $.
|
|
|
|
$( A kind of contraposition law that infers a subclass of an image from a
|
|
preimage subclass. (Contributed by set.mm contributors, 25-May-2004.) $)
|
|
funimass1 $p |- ( ( Fun F /\ A C_ ran F ) ->
|
|
( ( `' F " A ) C_ B -> A C_ ( F " B ) ) ) $=
|
|
( ccnv cima wss wfun crn wa imass2 funimacnv wceq dfss biimpi eqcomd sseq1d
|
|
cin sylan9eq syl5ib ) CDAEZBFCTEZCBEZFCGZACHZFZIZAUBFTBCJUFUAAUBUCUEUAAUDQZ
|
|
AACKUEAUGUEAUGLAUDMNORPS $.
|
|
|
|
$( A kind of contraposition law that infers an image subclass from a subclass
|
|
of a preimage. (Contributed by set.mm contributors, 25-May-2004.)
|
|
(Revised by set.mm contributors, 4-May-2007.) $)
|
|
funimass2 $p |- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B ) $=
|
|
( ccnv cima wss wfun imass2 crn cin funimacnv sseq2d inss1 sstr2 mpi syl6bi
|
|
imp sylan2 ) ACDBEZFCGZCAEZCSEZFZUABFZASCHTUCUDTUCUABCIZJZFZUDTUBUFUABCKLUG
|
|
UFBFUDBUEMUAUFBNOPQR $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( The image of a difference is the difference of images. (Contributed by
|
|
NM, 24-May-1998.) $)
|
|
imadif $p |- ( Fun `' F ->
|
|
( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) ) $=
|
|
( vy vx cdif cima cv wcel wn wa wbr wex exbii wi wmo sylib bitri elima2
|
|
wo ccnv wfun anandir 19.40 sylbi nfe1 nfan funmo brcnv mobii mopick sylan
|
|
wal nfv con2d imnan alrimi ex exancom 3imtr3g anim2d syl5 19.29r sylan2br
|
|
alnex ianor anbi2i pm3.24 intnan anass mtbir biorfi 3bitr4i impbid1 eldif
|
|
andi an32 anbi1i notbii anbi12i 3bitr4g eqrdv ) CUAZUBZDCABFZGZCAGZCBGZFZ
|
|
WDEHZAIZWJBIZJZKZWJDHZCLZKZEMZWKWPKZEMZWLWPKZEMZJZKZWOWFIZWOWIIZWDWRXDWRW
|
|
TWMWPKZEMZKZWDXDWRWSXGKZEMXIWQXJEWKWMWPUCNWSXGEUDUEWDXHXCWTWDWPWMKZEMZXAJ
|
|
ZEUMZXHXCWDXLXNWDXLKZXMEWDXLEWDEUNXKEUFUGXOWLWPJZOXMXOWPWLWDWPEPZXLWPWMOW
|
|
DWOWJWCLZEPXQEWOWCUHXRWPEWOWJCUIUJQWPWMEUKULUOWLWPUPQUQURWPWMEUSXAEVEZUTV
|
|
AVBXDWSXMKZEMZWRXCWTXNYAXSWSXMEVCVDXTWQEWSWMXPTZKWSWMKZWSXPKZTZXTWQWSWMXP
|
|
VPXMYBWSWLWPVFVGWQYCYEWKWMWPVQYDYCYDWKWPXPKZKYFWKWPVHVIWKWPXPVJVKVLRVMNQV
|
|
NXEWJWEIZWPKZEMWREWOCWESYHWQEYGWNWPWJABVOVRNRXFWOWGIZWOWHIZJZKXDWOWGWHVOY
|
|
IWTYKXCEWOCASYJXBEWOCBSVSVTRWAWB $.
|
|
$}
|
|
|
|
$( The image of an intersection is the intersection of images. (Contributed
|
|
by Paul Chapman, 11-Apr-2009.) $)
|
|
imain $p |- ( Fun `' F ->
|
|
( F " ( A i^i B ) ) = ( ( F " A ) i^i ( F " B ) ) ) $=
|
|
( ccnv wfun cdif cima cin imadif difeq2d eqtrd dfin4 imaeq2i 3eqtr4g ) CDEZ
|
|
CAABFZFZGZCAGZSCBGZFZFZCABHZGSTHORSCPGZFUBAPCIOUDUASABCIJKUCQCABLMSTLN $.
|
|
|
|
$( Equality theorem for function predicate with domain. (Contributed by
|
|
set.mm contributors, 1-Aug-1994.) $)
|
|
fneq1 $p |- ( F = G -> ( F Fn A <-> G Fn A ) ) $=
|
|
( wceq wfun cdm wa wfn funeq dmeq eqeq1d anbi12d df-fn 3bitr4g ) BCDZBEZBFZ
|
|
ADZGCEZCFZADZGBAHCAHOPSRUABCIOQTABCJKLBAMCAMN $.
|
|
|
|
$( Equality theorem for function predicate with domain. (Contributed by
|
|
set.mm contributors, 1-Aug-1994.) $)
|
|
fneq2 $p |- ( A = B -> ( F Fn A <-> F Fn B ) ) $=
|
|
( wceq wfun cdm wa wfn eqeq2 anbi2d df-fn 3bitr4g ) ABDZCEZCFZADZGNOBDZGCAH
|
|
CBHMPQNABOIJCAKCBKL $.
|
|
|
|
${
|
|
fneq1d.1 $e |- ( ph -> F = G ) $.
|
|
$( Equality deduction for function predicate with domain. (Contributed by
|
|
Paul Chapman, 22-Jun-2011.) $)
|
|
fneq1d $p |- ( ph -> ( F Fn A <-> G Fn A ) ) $=
|
|
( wceq wfn wb fneq1 syl ) ACDFCBGDBGHEBCDIJ $.
|
|
$}
|
|
|
|
${
|
|
fneq2d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for function predicate with domain. (Contributed by
|
|
Paul Chapman, 22-Jun-2011.) $)
|
|
fneq2d $p |- ( ph -> ( F Fn A <-> F Fn B ) ) $=
|
|
( wceq wfn wb fneq2 syl ) ABCFDBGDCGHEBCDIJ $.
|
|
$}
|
|
|
|
${
|
|
fneq12d.1 $e |- ( ph -> F = G ) $.
|
|
fneq12d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for function predicate with domain. (Contributed by
|
|
set.mm contributors, 26-Jun-2011.) $)
|
|
fneq12d $p |- ( ph -> ( F Fn A <-> G Fn B ) ) $=
|
|
( wfn fneq1d fneq2d bitrd ) ADBHEBHECHABDEFIABCEGJK $.
|
|
$}
|
|
|
|
${
|
|
fneq1i.1 $e |- F = G $.
|
|
$( Equality inference for function predicate with domain. (Contributed by
|
|
Paul Chapman, 22-Jun-2011.) $)
|
|
fneq1i $p |- ( F Fn A <-> G Fn A ) $=
|
|
( wceq wfn wb fneq1 ax-mp ) BCEBAFCAFGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
fneq2i.1 $e |- A = B $.
|
|
$( Equality inference for function predicate with domain. (Contributed by
|
|
set.mm contributors, 4-Sep-2011.) $)
|
|
fneq2i $p |- ( F Fn A <-> F Fn B ) $=
|
|
( wceq wfn wb fneq2 ax-mp ) ABECAFCBFGDABCHI $.
|
|
$}
|
|
|
|
${
|
|
nffn.1 $e |- F/_ x F $.
|
|
nffn.2 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for a function with domain.
|
|
(Contributed by NM, 30-Jan-2004.) $)
|
|
nffn $p |- F/ x F Fn A $=
|
|
( wfn wfun cdm wceq wa df-fn nffun nfdm nfeq nfan nfxfr ) CBFCGZCHZBIZJAC
|
|
BKQSAACDLARBACDMENOP $.
|
|
$}
|
|
|
|
$( A function with domain is a function. (Contributed by set.mm
|
|
contributors, 1-Aug-1994.) $)
|
|
fnfun $p |- ( F Fn A -> Fun F ) $=
|
|
( wfn wfun cdm wceq df-fn simplbi ) BACBDBEAFBAGH $.
|
|
|
|
$( The domain of a function. (Contributed by set.mm contributors,
|
|
2-Aug-1994.) $)
|
|
fndm $p |- ( F Fn A -> dom F = A ) $=
|
|
( wfn wfun cdm wceq df-fn simprbi ) BACBDBEAFBAGH $.
|
|
|
|
${
|
|
funfni.1 $e |- ( ( Fun F /\ B e. dom F ) -> ph ) $.
|
|
$( Inference to convert a function and domain antecedent. (Contributed by
|
|
set.mm contributors, 22-Apr-2004.) $)
|
|
funfni $p |- ( ( F Fn A /\ B e. A ) -> ph ) $=
|
|
( wfn wcel wa wfun cdm fnfun adantr fndm eleq2d biimpar syl2anc ) DBFZCBG
|
|
ZHDIZCDJZGZAQSRBDKLQUARQTBCBDMNOEP $.
|
|
$}
|
|
|
|
$( A function has a unique domain. (Contributed by set.mm contributors,
|
|
11-Aug-1994.) $)
|
|
fndmu $p |- ( ( F Fn A /\ F Fn B ) -> A = B ) $=
|
|
( wfn cdm fndm sylan9req ) CADCBDACEBACFBCFG $.
|
|
|
|
$( The first argument of binary relation on a function belongs to the
|
|
function's domain. (Contributed by set.mm contributors, 7-May-2004.) $)
|
|
fnbr $p |- ( ( F Fn A /\ B F C ) -> B e. A ) $=
|
|
( wfn cdm wceq wbr wcel fndm wa breldm adantl simpl eleqtrd sylan ) DAEDFZA
|
|
GZBCDHZBAIADJRSKBQASBQIRBCDLMRSNOP $.
|
|
|
|
$( The first argument of an ordered pair in a function belongs to the
|
|
function's domain. (Contributed by set.mm contributors, 8-Aug-1994.)
|
|
(Revised by set.mm contributors, 25-Mar-2007.) $)
|
|
fnop $p |- ( ( F Fn A /\ <. B , C >. e. F ) -> B e. A ) $=
|
|
( cop wcel wfn wbr df-br fnbr sylan2br ) BCEDFDAGBCDHBAFBCDIABCDJK $.
|
|
|
|
${
|
|
$d x y F $. $d x y B $. $d x A $.
|
|
$( There is exactly one value of a function. (The proof was shortened by
|
|
Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors,
|
|
22-Apr-2004.) (Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
fneu $p |- ( ( F Fn A /\ B e. A ) -> E! y B F y ) $=
|
|
( vx cv wbr weu cdm wcel wfun wceq breq1 eubidv imbi2d wex eldm wmo funmo
|
|
wi exmoeu2 syl5ib sylbi vtoclga impcom funfni ) CAFZDGZAHZBCDCDIZJDKZUIUK
|
|
EFZUGDGZAHZTZUKUITECUJULCLZUNUIUKUPUMUHAULCUGDMNOULUJJUMAPZUOAULDQUKUMARU
|
|
QUNAULDSUMAUAUBUCUDUEUF $.
|
|
|
|
$( There is exactly one value of a function. (Contributed by set.mm
|
|
contributors, 7-Nov-1995.) $)
|
|
fneu2 $p |- ( ( F Fn A /\ B e. A ) -> E! y <. B , y >. e. F ) $=
|
|
( wfn wcel wa cv wbr weu cop fneu df-br eubii sylib ) DBECBFGCAHZDIZAJCPK
|
|
DFZAJABCDLQRACPDMNO $.
|
|
$}
|
|
|
|
$( The union of two functions with disjoint domains. (Contributed by set.mm
|
|
contributors, 22-Sep-2004.) $)
|
|
fnun $p |- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) ->
|
|
( F u. G ) Fn ( A u. B ) ) $=
|
|
( wfn wa cin c0 wceq cun wfun cdm wi df-fn ineq12 eqeq1d funun syl6bir dmun
|
|
anbi2d uneq12 syl5eq jctird syl6ibr exp3a impcom an4s syl2anb imp ) CAEZDBE
|
|
ZFABGZHIZCDJZABJZEZUJCKZCLZAIZFDKZDLZBIZFUMUPMZUKCANDBNUQUTUSVBVCUSVBFZUQUT
|
|
FZVCVDVEUMUPVDVEUMFZUNKZUNLZUOIZFUPVDVFVGVIVDVFVEURVAGZHIZFVGVDVKUMVEVDVJUL
|
|
HURAVABOPTCDQRVDVHURVAJUOCDSURAVABUAUBUCUNUONUDUEUFUGUHUI $.
|
|
|
|
${
|
|
$d X x y $. $d Y y $.
|
|
fnunop.x $e |- ( ph -> X e. _V ) $.
|
|
fnunop.y $e |- ( ph -> Y e. _V ) $.
|
|
fnunop.f $e |- ( ph -> F Fn D ) $.
|
|
fnunop.g $e |- G = ( F u. { <. X , Y >. } ) $.
|
|
fnunop.e $e |- E = ( D u. { X } ) $.
|
|
fnunop.d $e |- ( ph -> -. X e. D ) $.
|
|
$( Extension of a function with a new ordered pair. (Contributed by NM,
|
|
28-Sep-2013.) $)
|
|
fnunsn $p |- ( ph -> G Fn E ) $=
|
|
( vx vy cop csn wfn wceq cvv cun cin c0 wcel cv opeq1 sneqd fneq12d opeq2
|
|
sneq fneq1d vex fnsn vtocl2g syl2anc disjsn sylibr syl21anc fneq1i fneq2i
|
|
wn fnun bitri ) ADFGPZQZUAZBFQZUAZRZECRZADBRVEVGRZBVGUBUCSZVIJAFTUDGTUDVK
|
|
HINUEZOUEZPZQZVMQZRFVNPZQZVGRVKNOFGTTVMFSZVQVGVPVSVTVOVRVMFVNUFUGVMFUJUHV
|
|
NGSZVGVSVEWAVRVDVNGFUIUGUKVMVNNULOULUMUNUOAFBUDVAVLMBFUPUQBVGDVEVBURVJVFC
|
|
RVICEVFKUSCVHVFLUTVCUQ $.
|
|
$}
|
|
|
|
$( Composition of two functions. (Contributed by set.mm contributors,
|
|
22-May-2006.) $)
|
|
fnco $p |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B ) $=
|
|
( wfn crn wss w3a ccom wfun cdm wceq fnfun funco syl2an 3adant3 fndm sseq2d
|
|
wa biimpar dmcosseq syl 3adant2 3ad2ant2 eqtrd df-fn sylanbrc ) CAEZDBEZDFZ
|
|
AGZHZCDIZJZUMKZBLUMBEUHUIUNUKUHCJDJUNUIACMBDMCDNOPULUODKZBUHUKUOUPLZUIUHUKS
|
|
UJCKZGZUQUHUSUKUHURAUJACQRTCDUAUBUCUIUHUPBLUKBDQUDUEUMBUFUG $.
|
|
|
|
$( A function does not change when restricted to its domain. (Contributed by
|
|
set.mm contributors, 5-Sep-2004.) $)
|
|
fnresdm $p |- ( F Fn A -> ( F |` A ) = F ) $=
|
|
( wfn cdm wceq wss cres fndm eqimss ssreseq 3syl ) BACBDZAELAFBAGBEABHLAIBA
|
|
JK $.
|
|
|
|
$( A function restricted to a class disjoint with its domain is empty.
|
|
(Contributed by set.mm contributors, 23-Sep-2004.) $)
|
|
fnresdisj $p |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) ) $=
|
|
( cres c0 wceq cdm wfn dmeq0 dmres incom eqtri ineq1d syl5eq eqeq1d syl5rbb
|
|
cin fndm ) CBDZEFSGZEFCAHZABQZEFSIUATUBEUATCGZBQZUBTBUCQUDCBJBUCKLUAUCABACR
|
|
MNOP $.
|
|
|
|
$( Membership in two functions restricted by each other's domain.
|
|
(Contributed by set.mm contributors, 8-Aug-1994.) $)
|
|
2elresin $p |- ( ( F Fn A /\ G Fn B ) ->
|
|
( ( <. x , y >. e. F /\ <. x , z >. e. G ) <->
|
|
( <. x , y >. e. ( F |` ( A i^i B ) ) /\
|
|
<. x , z >. e. ( G |` ( A i^i B ) ) ) ) ) $=
|
|
( wfn wa cv cop wcel cin cres fnop anim12i opelres simplbi2com resss sseli
|
|
wi an4 elin 3imtr4i anim12d syl ex pm2.43d impbid1 ) FDHZGEHZIZAJZBJZKZFLZU
|
|
MCJZKZGLZIZUOFDEMZNZLZURGVANZLZIZULUTVFULUTUTVFUAZULUTIZUMVALZVGUJUPIZUKUSI
|
|
ZIUMDLZUMELZIVHVIVJVLVKVMDUMUNFOEUMUQGOPUJUKUPUSUBUMDEUCUDVIUPVCUSVEVCUPVIU
|
|
MUNFVAQRVEUSVIUMUQGVAQRUEUFUGUHVCUPVEUSVBFUOFVASTVDGURGVASTPUI $.
|
|
|
|
$( Restriction of a function with a subclass of its domain. (Contributed by
|
|
set.mm contributors, 10-Oct-2007.) $)
|
|
fnssresb $p |- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) ) $=
|
|
( cres wfn wfun cdm wceq wa wss df-fn fnfun funres biantrurd ssdmres sseq2d
|
|
syl fndm syl5bbr bitr3d syl5bb ) CBDZBEUBFZUBGBHZIZCAEZBAJZUBBKUFUDUEUGUFUC
|
|
UDUFCFUCACLBCMQNUDBCGZJUFUGBCOUFUHABACRPSTUA $.
|
|
|
|
$( Restriction of a function with a subclass of its domain. (Contributed by
|
|
set.mm contributors, 2-Aug-1994.) (Revised by set.mm contributors,
|
|
25-Sep-2004.) $)
|
|
fnssres $p |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B ) $=
|
|
( wfn cres wss fnssresb biimpar ) CADCBEBDBAFABCGH $.
|
|
|
|
$( Restriction of a function's domain with an intersection. (Contributed by
|
|
set.mm contributors, 9-Aug-1994.) $)
|
|
fnresin1 $p |- ( F Fn A -> ( F |` ( A i^i B ) ) Fn ( A i^i B ) ) $=
|
|
( wfn cin wss cres inss1 fnssres mpan2 ) CADABEZAFCKGKDABHAKCIJ $.
|
|
|
|
$( Restriction of a function's domain with an intersection. (Contributed by
|
|
set.mm contributors, 9-Aug-1994.) $)
|
|
fnresin2 $p |- ( F Fn A -> ( F |` ( B i^i A ) ) Fn ( B i^i A ) ) $=
|
|
( wfn cin wss cres inss2 fnssres mpan2 ) CADBAEZAFCKGKDBAHAKCIJ $.
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( An equivalence for functionality of a restriction. Compare ~ dffun8 .
|
|
(Contributed by Mario Carneiro, 20-May-2015.) $)
|
|
fnres $p |- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y ) $=
|
|
( cres wfun cdm wceq wa wbr wex wmo wral ancom wal wcel bitri 3bitr4i wss
|
|
cv wfn weu wi brres mobii moanimv albii dffun6 df-ral dmres inss1 eqsstri
|
|
cin eqss mpbiran dfss3 elin2 baib eldm syl6bb ralbiia 3bitri r19.26 df-fn
|
|
anbi12i eu5 ralbii ) DCEZFZVHGZCHZIZATZBTZDJZBKZVOBLZIZACMZVHCUAVOBUBZACM
|
|
VQACMZVPACMZIWBWAIVLVSWAWBNVIWAVKWBVMVNVHJZBLZAOVMCPZVQUCZAOVIWAWDWFAWDWE
|
|
VOIZBLWFWCWGBWCVOWEIWGVMVNDCUDVOWENQUEWEVOBUFQUGABVHUHVQACUIRVKCVJSZVMVJP
|
|
ZACMWBVKVJCSWHVJCDGZUMCDCUJZCWJUKULVJCUNUOACVJUPWIVPACWEWIVMWJPZVPWIWEWLV
|
|
MCWJVJWKUQURBVMDUSUTVAVBVEVPVQACVCRVHCVDVTVRACVOBVFVGR $.
|
|
$}
|
|
|
|
$( Functionality and domain of restricted identity. (Contributed by set.mm
|
|
contributors, 27-Aug-2004.) $)
|
|
fnresi $p |- ( _I |` A ) Fn A $=
|
|
( cid cres wfn wfun cdm wceq funi funres ax-mp dmresi df-fn mpbir2an ) BACZ
|
|
ADNEZNFAGBEOHABIJAKNALM $.
|
|
|
|
$( The image of a function's domain is its range. (The proof was shortened
|
|
by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors,
|
|
4-Nov-2004.) (Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
fnima $p |- ( F Fn A -> ( F " A ) = ran F ) $=
|
|
( wfn cima cres crn dfima3 fnresdm rneqd syl5eq ) BACZBADBAEZFBFBAGKLBABHIJ
|
|
$.
|
|
|
|
$( A function with empty domain is empty. (The proof was shortened by Andrew
|
|
Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 15-Apr-1998.)
|
|
(Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
fn0 $p |- ( F Fn (/) <-> F = (/) ) $=
|
|
( wfn wceq cdm fndm dmeq0 sylibr wfun fun0 dm0 df-fn mpbir2an mpbiri impbii
|
|
c0 fneq1 ) AOBZAOCZQADOCROAEAFGRQOOBZSOHODOCIJOOKLOAOPMN $.
|
|
|
|
$( A class that is disjoint with the domain of a function has an empty image
|
|
under the function. (Contributed by FL, 24-Jan-2007.) $)
|
|
fnimadisj $p |- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( F " C ) = (/) ) $=
|
|
( wfn cin c0 wceq wa cdm cima fndm ineq1d eqeq1d biimpar imadisj sylibr ) C
|
|
ADZABEZFGZHCIZBEZFGZCBJFGQUBSQUARFQTABACKLMNCBOP $.
|
|
|
|
${
|
|
$d y z A $. $d y z B $. $d x y z $.
|
|
iunfopab.1 $e |- B e. _V $.
|
|
$( Two ways to express a function as a class of ordered pairs. (The proof
|
|
was shortened by Andrew Salmon, 17-Sep-2011.) (Unnecessary distinct
|
|
variable restrictions were removed by David Abernethy, 19-Sep-2011.)
|
|
(Contributed by set.mm contributors, 19-Dec-2008.) $)
|
|
iunfopab $p |- U_ x e. A { <. x , B >. }
|
|
= { <. x , y >. | ( x e. A /\ y = B ) } $=
|
|
( vz cv cop csn wcel wrex cab wceq wa wex ciun copab df-rex vex exbii
|
|
elsnc anbi2i opeq2 eqeq2d anbi2d ceqsexv an13 3bitr2i bitri abbii df-opab
|
|
df-iun 3eqtr4i ) FGZAGZDHZIZJZACKZFLUNUOBGZHZMZUOCJZUTDMZNZNZBOZAOZFLACUQ
|
|
PVEABQUSVHFUSVCURNZAOVHURACRVIVGAVIVCUNUPMZNZVDVCVBNZNZBOVGURVJVCUNUPFSUA
|
|
UBVLVKBDEVDVBVJVCVDVAUPUNUTDUOUCUDUEUFVMVFBVDVCVBUGTUHTUIUJAFCUQULVEABFUK
|
|
UM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
fnopabg.1 $e |- F = { <. x , y >. | ( x e. A /\ ph ) } $.
|
|
$( Functionality and domain of an ordered-pair class abstraction.
|
|
(Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro,
|
|
4-Dec-2016.) $)
|
|
fnopabg $p |- ( A. x e. A E! y ph <-> F Fn A ) $=
|
|
( wmo wex wa wral cv wcel copab wfn weu wfun cdm wceq wal 3bitr4i moanimv
|
|
wi albii funopab df-ral 3bitr4ri dmopab3 anbi12i r19.26 df-fn ancom bitri
|
|
eu5 ralbii fneq1i ) ACGZACHZIZBDJZBKDLZAIZBCMZDNZACOZBDJEDNUPBDJZUQBDJZIV
|
|
BPZVBQDRZIUSVCVEVGVFVHVACGZBSUTUPUBZBSVGVEVIVJBUTACUAUCVABCUDUPBDUEUFABCD
|
|
UGUHUPUQBDUIVBDUJTVDURBDVDUQUPIURACUMUQUPUKULUNDEVBFUOT $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
fnopab2g.1 $e |- F = { <. x , y >. | ( x e. A /\ y = B ) } $.
|
|
$( Functionality and domain of an ordered-pair class abstraction.
|
|
(Contributed by set.mm contributors, 23-Mar-2006.) $)
|
|
fnopab2g $p |- ( A. x e. A B e. _V <-> F Fn A ) $=
|
|
( cvv wcel wral cv wceq weu wfn eueq ralbii fnopabg bitri ) DGHZACIBJDKZB
|
|
LZACIECMRTACBDNOSABCEFPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
fnopab.1 $e |- ( x e. A -> E! y ph ) $.
|
|
fnopab.2 $e |- F = { <. x , y >. | ( x e. A /\ ph ) } $.
|
|
$( Functionality and domain of an ordered-pair class abstraction.
|
|
(Contributed by set.mm contributors, 5-Mar-1996.) $)
|
|
fnopab $p |- F Fn A $=
|
|
( weu wral wfn rgen fnopabg mpbi ) ACHZBDIEDJNBDFKABCDEGLM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
fnopab2.1 $e |- B e. _V $.
|
|
fnopab2.2 $e |- F = { <. x , y >. | ( x e. A /\ y = B ) } $.
|
|
$( Functionality and domain of an ordered-pair class abstraction.
|
|
(Contributed by set.mm contributors, 29-Jan-2004.) $)
|
|
fnopab2 $p |- F Fn A $=
|
|
( cv wceq weu wcel eueq1 a1i fnopab ) BHDIZABCEOBJAHCKBDFLMGN $.
|
|
|
|
$( Domain of an ordered-pair class abstraction that specifies a function.
|
|
(Contributed by set.mm contributors, 6-Sep-2005.) $)
|
|
dmopab2 $p |- dom F = A $=
|
|
( wfn cdm wceq fnopab2 fndm ax-mp ) ECHEICJABCDEFGKCELM $.
|
|
$}
|
|
|
|
$( Equality theorem for functions. (Contributed by set.mm contributors,
|
|
1-Aug-1994.) $)
|
|
feq1 $p |- ( F = G -> ( F : A --> B <-> G : A --> B ) ) $=
|
|
( wceq wfn crn wss wa wf fneq1 rneq sseq1d anbi12d df-f 3bitr4g ) CDEZCAFZC
|
|
GZBHZIDAFZDGZBHZIABCJABDJQRUATUCACDKQSUBBCDLMNABCOABDOP $.
|
|
|
|
$( Equality theorem for functions. (Contributed by set.mm contributors,
|
|
1-Aug-1994.) $)
|
|
feq2 $p |- ( A = B -> ( F : A --> C <-> F : B --> C ) ) $=
|
|
( wceq wfn crn wss wa wf fneq2 anbi1d df-f 3bitr4g ) ABEZDAFZDGCHZIDBFZQIAC
|
|
DJBCDJOPRQABDKLACDMBCDMN $.
|
|
|
|
$( Equality theorem for functions. (Contributed by set.mm contributors,
|
|
1-Aug-1994.) $)
|
|
feq3 $p |- ( A = B -> ( F : C --> A <-> F : C --> B ) ) $=
|
|
( wceq wfn crn wss wa wf sseq2 anbi2d df-f 3bitr4g ) ABEZDCFZDGZAHZIPQBHZIC
|
|
ADJCBDJORSPABQKLCADMCBDMN $.
|
|
|
|
$( Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (The
|
|
proof was shortened by Andrew Salmon, 17-Sep-2011.) $)
|
|
feq23 $p |- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) ) $=
|
|
( wceq wf feq2 feq3 sylan9bb ) ACFABEGCBEGBDFCDEGACBEHBDCEIJ $.
|
|
|
|
${
|
|
feq1d.1 $e |- ( ph -> F = G ) $.
|
|
$( Equality deduction for functions. (Contributed by set.mm contributors,
|
|
19-Feb-2008.) $)
|
|
feq1d $p |- ( ph -> ( F : A --> B <-> G : A --> B ) ) $=
|
|
( wceq wf wb feq1 syl ) ADEGBCDHBCEHIFBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
feq2d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq2d $p |- ( ph -> ( F : A --> C <-> F : B --> C ) ) $=
|
|
( wceq wf wb feq2 syl ) ABCGBDEHCDEHIFBCDEJK $.
|
|
$}
|
|
|
|
${
|
|
feq12d.1 $e |- ( ph -> F = G ) $.
|
|
feq12d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq12d $p |- ( ph -> ( F : A --> C <-> G : B --> C ) ) $=
|
|
( wf feq1d feq2d bitrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $.
|
|
$}
|
|
|
|
${
|
|
feq1i.1 $e |- F = G $.
|
|
$( Equality inference for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq1i $p |- ( F : A --> B <-> G : A --> B ) $=
|
|
( wceq wf wb feq1 ax-mp ) CDFABCGABDGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
feq2i.1 $e |- A = B $.
|
|
$( Equality inference for functions. (Contributed by set.mm contributors,
|
|
5-Sep-2011.) $)
|
|
feq2i $p |- ( F : A --> C <-> F : B --> C ) $=
|
|
( wceq wf wb feq2 ax-mp ) ABFACDGBCDGHEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
feq23i.1 $e |- A = C $.
|
|
feq23i.2 $e |- B = D $.
|
|
$( Equality inference for functions. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
feq23i $p |- ( F : A --> B <-> F : C --> D ) $=
|
|
( wceq wf wb feq23 mp2an ) ACHBDHABEICDEIJFGABCDEKL $.
|
|
$}
|
|
|
|
${
|
|
feq23d.1 $e |- ( ph -> A = C ) $.
|
|
feq23d.2 $e |- ( ph -> B = D ) $.
|
|
$( Equality deduction for functions. (Contributed by set.mm contributors,
|
|
8-Jun-2013.) $)
|
|
feq23d $p |- ( ph -> ( F : A --> B <-> F : C --> D ) ) $=
|
|
( wceq wf wb feq23 syl2anc ) ABDICEIBCFJDEFJKGHBCDEFLM $.
|
|
$}
|
|
|
|
${
|
|
nff.1 $e |- F/_ x F $.
|
|
nff.2 $e |- F/_ x A $.
|
|
nff.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for a mapping. (Contributed by NM,
|
|
29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nff $p |- F/ x F : A --> B $=
|
|
( wf wfn crn wss wa df-f nffn nfrn nfss nfan nfxfr ) BCDHDBIZDJZCKZLABCDM
|
|
SUAAABDEFNATCADEOGPQR $.
|
|
$}
|
|
|
|
${
|
|
elimf.1 $e |- G : A --> B $.
|
|
$( Eliminate a mapping hypothesis for the weak deduction theorem ~ dedth ,
|
|
when a special case ` G : A --> B ` is provable, in order to convert
|
|
` F : A --> B ` from a hypothesis to an antecedent. (Contributed by
|
|
set.mm contributors, 24-Aug-2006.) $)
|
|
elimf $p |- if ( F : A --> B , F , G ) : A --> B $=
|
|
( wf cif feq1 elimhyp ) ABCFZABJCDGZFABDFCDABCKHABDKHEI $.
|
|
$}
|
|
|
|
$( A mapping is a function. (Contributed by set.mm contributors,
|
|
2-Aug-1994.) $)
|
|
ffn $p |- ( F : A --> B -> F Fn A ) $=
|
|
( wf wfn crn wss df-f simplbi ) ABCDCAECFBGABCHI $.
|
|
|
|
$( Any function is a mapping into ` _V ` . (The proof was shortened by
|
|
Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors,
|
|
31-Oct-1995.) (Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
dffn2 $p |- ( F Fn A <-> F : A --> _V ) $=
|
|
( wfn crn cvv wss wa wf ssv biantru df-f bitr4i ) BACZMBDZEFZGAEBHOMNIJAEBK
|
|
L $.
|
|
|
|
$( A mapping is a function. (Contributed by set.mm contributors,
|
|
3-Aug-1994.) $)
|
|
ffun $p |- ( F : A --> B -> Fun F ) $=
|
|
( wf wfn wfun ffn fnfun syl ) ABCDCAECFABCGACHI $.
|
|
|
|
$( The domain of a mapping. (Contributed by set.mm contributors,
|
|
2-Aug-1994.) $)
|
|
fdm $p |- ( F : A --> B -> dom F = A ) $=
|
|
( wf wfn cdm wceq ffn fndm syl ) ABCDCAECFAGABCHACIJ $.
|
|
|
|
${
|
|
fdmi.1 $e |- F : A --> B $.
|
|
$( The domain of a mapping. (Contributed by set.mm contributors,
|
|
28-Jul-2008.) $)
|
|
fdmi $p |- dom F = A $=
|
|
( wf cdm wceq fdm ax-mp ) ABCECFAGDABCHI $.
|
|
$}
|
|
|
|
$( The range of a mapping. (Contributed by set.mm contributors,
|
|
3-Aug-1994.) $)
|
|
frn $p |- ( F : A --> B -> ran F C_ B ) $=
|
|
( wf wfn crn wss df-f simprbi ) ABCDCAECFBGABCHI $.
|
|
|
|
$( A function maps to its range. (Contributed by set.mm contributors,
|
|
1-Sep-1999.) $)
|
|
dffn3 $p |- ( F Fn A <-> F : A --> ran F ) $=
|
|
( wfn crn wss wa wf ssid biantru df-f bitr4i ) BACZLBDZMEZFAMBGNLMHIAMBJK
|
|
$.
|
|
|
|
$( Expanding the codomain of a mapping. (The proof was shortened by Andrew
|
|
Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 10-May-1998.)
|
|
(Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
fss $p |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C ) $=
|
|
( wss wf wfn crn wa sstr2 com12 anim2d df-f 3imtr4g impcom ) BCEZABDFZACDFZ
|
|
PDAGZDHZBEZISTCEZIQRPUAUBSUAPUBTBCJKLABDMACDMNO $.
|
|
|
|
$( Composition of two mappings. (The proof was shortened by Andrew Salmon,
|
|
17-Sep-2011.) (Contributed by set.mm contributors, 29-Aug-1999.)
|
|
(Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
fco $p |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C ) $=
|
|
( wfn crn wss wa ccom wf wi fnco 3expib adantr rncoss sstr mpan adantl df-f
|
|
jctird imp anbi12i 3imtr4i ) DBFZDGZCHZIZEAFZEGBHZIZIDEJZAFZULGZCHZIZBCDKZA
|
|
BEKZIACULKUHUKUPUHUKUMUOUEUKUMLUGUEUIUJUMBADEMNOUGUOUEUNUFHUGUODEPUNUFCQRSU
|
|
AUBUQUHURUKBCDTABETUCACULTUD $.
|
|
|
|
$( A mapping is a class of ordered pairs. (The proof was shortened by Andrew
|
|
Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 3-Aug-1994.)
|
|
(Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
fssxp $p |- ( F : A --> B -> F C_ ( A X. B ) ) $=
|
|
( wf cdm crn cxp ssdmrn wss wceq fdm eqimss syl frn xpss12 syl2anc syl5ss )
|
|
ABCDZCCEZCFZGZABGZCHRSAIZTBIUAUBIRSAJUCABCKSALMABCNSATBOPQ $.
|
|
|
|
$( Two ways of specifying a partial function from ` A ` to ` B ` .
|
|
(Contributed by set.mm contributors, 13-Nov-2007.) $)
|
|
funssxp $p |- ( ( Fun F /\ F C_ ( A X. B ) ) <->
|
|
( F : dom F --> B /\ dom F C_ A ) ) $=
|
|
( wfun cxp wss wa cdm wf wfn funfn biimpi rnss rnxpss syl6ss anim12i sylibr
|
|
crn df-f jca dmss dmxpss adantl ffun adantr fssxp xpss1 sylan9ss impbii ) C
|
|
DZCABEZFZGZCHZBCIZUNAFZGZUMUOUPUMCUNJZCRZBFZGUOUJURULUTUJURCKLULUSUKRBCUKMA
|
|
BNOPUNBCSQULUPUJULUNUKHACUKUAABUBOUCTUQUJULUOUJUPUNBCUDUEUOUPCUNBEUKUNBCUFU
|
|
NABUGUHTUI $.
|
|
|
|
$( A mapping is a partial function. (Contributed by set.mm contributors,
|
|
25-Nov-2007.) $)
|
|
ffdm $p |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) ) $=
|
|
( wf cdm wss fdm feq2d ibir wceq eqimss syl jca ) ABCDZCEZBCDZOAFZNPNOABCAB
|
|
CGZHINOAJQROAKLM $.
|
|
|
|
$( The members of an ordered pair element of a mapping belong to the
|
|
mapping's domain and codomain. (Contributed by set.mm contributors,
|
|
9-Jan-2015.) $)
|
|
opelf $p |- ( ( F : A --> B /\ <. C , D >. e. F ) ->
|
|
( C e. A /\ D e. B ) ) $=
|
|
( wf cop wcel wa cxp fssxp sseld opelxp syl6ib imp ) ABEFZCDGZEHZCAHDBHIZPR
|
|
QABJZHSPETQABEKLCDABMNO $.
|
|
|
|
$( The union of two functions with disjoint domains. (Contributed by set.mm
|
|
contributors, 22-Sep-2004.) $)
|
|
fun $p |- ( ( ( F : A --> C /\ G : B --> D ) /\ ( A i^i B ) = (/) ) ->
|
|
( F u. G ) : ( A u. B ) --> ( C u. D ) ) $=
|
|
( cin c0 wceq wf wa cun wfn crn wss fnun expcom wi rnun df-f unss12 anim12d
|
|
syl5eqss a1i anbi12i an4 bitri 3imtr4g impcom ) ABGHIZACEJZBDFJZKZABLZCDLZE
|
|
FLZJZUJEAMZFBMZKZENZCOZFNZDOZKZKZUPUNMZUPNZUOOZKUMUQUJUTVGVEVIUTUJVGABEFPQV
|
|
EVIRUJVEVHVAVCLUOEFSVACVCDUAUCUDUBUMURVBKZUSVDKZKVFUKVJULVKACETBDFTUEURVBUS
|
|
VDUFUGUNUOUPTUHUI $.
|
|
|
|
$( Composition of two functions. (Contributed by set.mm contributors,
|
|
22-May-2006.) $)
|
|
fnfco $p |- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B ) $=
|
|
( wf wfn crn wss wa ccom df-f fnco 3expb sylan2b ) BADECAFZDBFZDGAHZICDJBFZ
|
|
BADKOPQRABCDLMN $.
|
|
|
|
$( Restriction of a function with a subclass of its domain. (Contributed by
|
|
set.mm contributors, 23-Sep-2004.) $)
|
|
fssres $p |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B ) $=
|
|
( wf wss cres wfn crn df-f fnssres resss rnss ax-mp sstr mpan anim12i an32s
|
|
wa sylanb sylibr ) ABDEZCAFZSDCGZCHZUDIZBFZSZCBUDEUBDAHZDIZBFZSUCUHABDJUIUC
|
|
UKUHUIUCSUEUKUGACDKUFUJFZUKUGUDDFULDCLUDDMNUFUJBOPQRTCBUDJUA $.
|
|
|
|
$( Restriction of a restricted function with a subclass of its domain.
|
|
(Contributed by set.mm contributors, 21-Jul-2005.) $)
|
|
fssres2 $p |- ( ( ( F |` A ) : A --> B /\ C C_ A ) ->
|
|
( F |` C ) : C --> B ) $=
|
|
( cres wf wss wa fssres wb resabs1 feq1d adantl mpbid ) ABDAEZFZCAGZHCBOCEZ
|
|
FZCBDCEZFZABCOIQSUAJPQCBRTDCAKLMN $.
|
|
|
|
$( Composition of a mapping and restricted identity. (The proof was
|
|
shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm
|
|
contributors, 13-Dec-2003.) (Revised by set.mm contributors,
|
|
18-Sep-2011.) $)
|
|
fcoi1 $p |- ( F : A --> B -> ( F o. ( _I |` A ) ) = F ) $=
|
|
( cid cres ccom coi1 reseq1i resco eqtr3i wfn wceq ffn fnresdm syl syl5eqr
|
|
wf ) ABCQZCDAEFZCAEZCCDFZAETSUACACGHCDAIJRCAKTCLABCMACNOP $.
|
|
|
|
$( Composition of restricted identity and a mapping. (The proof was
|
|
shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm
|
|
contributors, 13-Dec-2003.) (Revised by set.mm contributors,
|
|
18-Sep-2011.) $)
|
|
fcoi2 $p |- ( F : A --> B -> ( ( _I |` B ) o. F ) = F ) $=
|
|
( wf cid cres ccom crn wss wceq frn cores syl coi2 syl6eq ) ABCDZEBFCGZECGZ
|
|
CPCHBIQRJABCKECBLMCNO $.
|
|
|
|
${
|
|
$d y F $. $d y A $. $d y B $. $d y C $.
|
|
$( There is exactly one value of a function in its codomain. (Contributed
|
|
by set.mm contributors, 10-Dec-2003.) $)
|
|
feu $p |- ( ( F : A --> B /\ C e. A ) -> E! y e. B <. C , y >. e. F ) $=
|
|
( wf wcel wa cv cop weu wreu wfn ffn fneu2 sylan wb opelf simprd ex mpbid
|
|
pm4.71rd eubidv adantr df-reu sylibr ) BCEFZDBGZHZAIZCGZDUJJEGZHZAKZULACL
|
|
UIULAKZUNUGEBMUHUOBCENABDEOPUGUOUNQUHUGULUMAUGULUKUGULUKUGULHUHUKBCDUJERS
|
|
TUBUCUDUAULACUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y F $. $d x y A $. $d x y B $.
|
|
$( The converse of a restriction of a function. (Contributed by set.mm
|
|
contributors, 26-Mar-1998.) $)
|
|
fcnvres $p |- ( F : A --> B -> `' ( F |` A ) = ( `' F |` B ) ) $=
|
|
( vy vx wf cres ccnv cv cop wa wbr wb df-br wfn ffn opelcnv opelres bitri
|
|
wcel fnbr sylan crn wss brelrn ssel2 syl2an 2thd sylan2br pm5.32da anbi1i
|
|
frn 3bitr4g eqrelrdv ) ABCFZDECAGZHZCHZBGZUOEIZDIZJZCTZUTATZKZVCVABTZKZVA
|
|
UTJZUQTZVHUSTZUOVCVDVFVCUOUTVACLZVDVFMUTVACNUOVKKVDVFUOCAOVKVDABCPAUTVACU
|
|
AUBUOCUCZBUDVAVLTVFVKABCULUTVACUEVLBVAUFUGUHUIUJVIVBUPTVEVAUTUPQUTVACARSV
|
|
JVHURTZVFKVGVAUTURBRVMVCVFVAUTCQUKSUMUN $.
|
|
$}
|
|
|
|
$( The preimage of a class disjoint with a mapping's codomain is empty.
|
|
(Contributed by FL, 24-Jan-2007.) $)
|
|
fimacnvdisj $p |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) ->
|
|
( `' F " C ) = (/) ) $=
|
|
( wf cin c0 wceq wa ccnv cdm cima wss crn dfrn4 frn adantr syl5eqssr ssdisj
|
|
sylancom imadisj sylibr ) ABDEZBCFGHZIZDJZKZCFGHZUFCLGHUCUDUGBMUHUEUGDNZBDO
|
|
UCUIBMUDABDPQRUGBCSTUFCUAUB $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
fint.1 $e |- B =/= (/) $.
|
|
$( Function into an intersection. (The proof was shortened by Andrew
|
|
Salmon, 17-Sep-2011.) (Contributed by set.mm contributors,
|
|
14-Oct-1999.) (Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
fint $p |- ( F : A --> |^| B <-> A. x e. B F : A --> x ) $=
|
|
( wfn crn cint wss wa cv wral wf ssint anbi2i c0 wne wb r19.28zv df-f
|
|
ax-mp bitr4i ralbii 3bitr4i ) DBFZDGZCHZIZJZUEUFAKZIZJZACLZBUGDMBUJDMZACL
|
|
UIUEUKACLZJZUMUHUOUEAUFCNOCPQUMUPREUEUKACSUAUBBUGDTUNULACBUJDTUCUD $.
|
|
$}
|
|
|
|
$( Mapping into an intersection. (The proof was shortened by Andrew Salmon,
|
|
17-Sep-2011.) (Contributed by set.mm contributors, 14-Sep-1999.)
|
|
(Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
fin $p |- ( F : A --> ( B i^i C ) <-> ( F : A --> B /\ F : A --> C ) ) $=
|
|
( wfn crn cin wss wa wf ssin anbi2i anandi bitr3i df-f anbi12i 3bitr4i ) DA
|
|
EZDFZBCGZHZIZRSBHZIZRSCHZIZIZATDJABDJZACDJZIUBRUCUEIZIUGUJUARSBCKLRUCUEMNAT
|
|
DOUHUDUIUFABDOACDOPQ $.
|
|
|
|
$( If a mapping is a set, its domain is a set. (The proof was shortened by
|
|
Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors,
|
|
27-Aug-2006.) (Revised by set.mm contributors, 18-Sep-2011.) $)
|
|
dmfex $p |- ( ( F e. C /\ F : A --> B ) -> A e. _V ) $=
|
|
( wf wcel cvv cdm wceq wi fdm dmexg eleq1 syl5ib syl impcom ) ABDEZDCFZAGFZ
|
|
QDHZAIZRSJABDKRTGFUASDCLTAGMNOP $.
|
|
|
|
$( The empty function. (Contributed by set.mm contributors, 14-Aug-1999.) $)
|
|
f0 $p |- (/) : (/) --> A $=
|
|
( c0 wfn crn wss wfun cdm wceq fun0 dm0 df-fn mpbir2an rn0 0ss eqsstri df-f
|
|
wf ) BABQBBCZBDZAERBFBGBHIJBBKLSBAMANOBABPL $.
|
|
|
|
$( A class is a function with empty codomain iff it and its domain are
|
|
empty. (Contributed by set.mm contributors, 10-Dec-2003.) $)
|
|
f00 $p |- ( F : A --> (/) <-> ( F = (/) /\ A = (/) ) ) $=
|
|
( c0 wf wceq wa wfn wfun cdm ffun crn wss frn ss0 syl dm0rn0 df-fn sylanbrc
|
|
sylibr fn0 sylib fdm eqtr3d jca f0 feq1 feq2 sylan9bb mpbiri impbii ) ACBDZ
|
|
BCEZACEZFZUKULUMUKBCGZULUKBHBIZCEZUOACBJUKBKZCEZUQUKURCLUSACBMURNOBPSZBCQRB
|
|
TUAUKUPACACBUBUTUCUDUNUKCCCDZCUEULUKACCDUMVAACBCUFACCCUGUHUIUJ $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
fconst.1 $e |- B e. _V $.
|
|
$( A cross product with a singleton is a constant function. (The proof was
|
|
shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm
|
|
contributors, 14-Aug-1999.) (Revised by set.mm contributors,
|
|
18-Sep-2011.) $)
|
|
fconst $p |- ( A X. { B } ) : A --> { B } $=
|
|
( vx vy csn cxp wf wfn crn wss fconstopab fnopab2 rnxpss df-f mpbir2an )
|
|
ABFZAQGZHRAIRJQKDEABRCDEABLMAQNAQROP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( A cross product with a singleton is a constant function. (Contributed
|
|
by set.mm contributors, 19-Oct-2004.) $)
|
|
fconstg $p |- ( B e. V -> ( A X. { B } ) : A --> { B } ) $=
|
|
( vx cv csn cxp wf wceq sneq xpeq2d feq1 feq3 sylan9bb syl2anc vex fconst
|
|
wb vtoclg ) ADEZFZAUAGZHZABFZAUDGZHZDBCTBIZUBUEIZUAUDIZUCUFRUGUAUDATBJZKU
|
|
JUHUCAUAUEHUIUFAUAUBUELUAUDAUEMNOATDPQS $.
|
|
$( A cross product with a singleton is a constant function. (Contributed
|
|
by set.mm contributors, 24-Jul-2014.) $)
|
|
fnconstg $p |- ( B e. V -> ( A X. { B } ) Fn A ) $=
|
|
( wcel csn cxp wf wfn fconstg ffn syl ) BCDABEZALFZGMAHABCIALMJK $.
|
|
$}
|
|
|
|
$( Equality theorem for one-to-one functions. (Contributed by set.mm
|
|
contributors, 10-Feb-1997.) $)
|
|
f1eq1 $p |- ( F = G -> ( F : A -1-1-> B <-> G : A -1-1-> B ) ) $=
|
|
( wceq wf ccnv wfun wa wf1 feq1 cnveq funeqd anbi12d df-f1 3bitr4g ) CDEZAB
|
|
CFZCGZHZIABDFZDGZHZIABCJABDJQRUATUCABCDKQSUBCDLMNABCOABDOP $.
|
|
|
|
$( Equality theorem for one-to-one functions. (Contributed by set.mm
|
|
contributors, 10-Feb-1997.) $)
|
|
f1eq2 $p |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) ) $=
|
|
( wceq wf ccnv wfun wa wf1 feq2 anbi1d df-f1 3bitr4g ) ABEZACDFZDGHZIBCDFZQ
|
|
IACDJBCDJOPRQABCDKLACDMBCDMN $.
|
|
|
|
$( Equality theorem for one-to-one functions. (Contributed by set.mm
|
|
contributors, 10-Feb-1997.) $)
|
|
f1eq3 $p |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) ) $=
|
|
( wceq wf ccnv wfun wa wf1 feq3 anbi1d df-f1 3bitr4g ) ABEZCADFZDGHZICBDFZQ
|
|
ICADJCBDJOPRQABCDKLCADMCBDMN $.
|
|
|
|
${
|
|
nff1.1 $e |- F/_ x F $.
|
|
nff1.2 $e |- F/_ x A $.
|
|
nff1.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for a one-to-one function.
|
|
(Contributed by NM, 16-May-2004.) $)
|
|
nff1 $p |- F/ x F : A -1-1-> B $=
|
|
( wf1 wf ccnv wfun wa df-f1 nff nfcnv nffun nfan nfxfr ) BCDHBCDIZDJZKZLA
|
|
BCDMSUAAABCDEFGNATADEOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y F $.
|
|
$( Alternate definition of a one-to-one function. (Contributed by set.mm
|
|
contributors, 31-Dec-1996.) (Revised by set.mm contributors,
|
|
22-Sep-2004.) $)
|
|
dff12 $p |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. y E* x x F y ) ) $=
|
|
( wf1 wf ccnv wfun wa cv wbr wmo wal df-f1 funcnv2 anbi2i bitri ) CDEFCDE
|
|
GZEHIZJSAKBKELAMBNZJCDEOTUASABEPQR $.
|
|
$}
|
|
|
|
$( A one-to-one mapping is a mapping. (Contributed by set.mm contributors,
|
|
31-Dec-1996.) $)
|
|
f1f $p |- ( F : A -1-1-> B -> F : A --> B ) $=
|
|
( wf1 wf ccnv wfun df-f1 simplbi ) ABCDABCECFGABCHI $.
|
|
|
|
$( A one-to-one mapping is a function on its domain. (Contributed by set.mm
|
|
contributors, 8-Mar-2014.) $)
|
|
f1fn $p |- ( F : A -1-1-> B -> F Fn A ) $=
|
|
( wf1 wf wfn f1f ffn syl ) ABCDABCECAFABCGABCHI $.
|
|
|
|
$( A one-to-one mapping is a function. (Contributed by set.mm contributors,
|
|
8-Mar-2014.) $)
|
|
f1fun $p |- ( F : A -1-1-> B -> Fun F ) $=
|
|
( wf1 wfn wfun f1fn fnfun syl ) ABCDCAECFABCGACHI $.
|
|
|
|
$( The domain of a one-to-one mapping. (Contributed by set.mm contributors,
|
|
8-Mar-2014.) $)
|
|
f1dm $p |- ( F : A -1-1-> B -> dom F = A ) $=
|
|
( wf1 wfn cdm wceq f1fn fndm syl ) ABCDCAECFAGABCHACIJ $.
|
|
|
|
$( A function that is one-to-one is also one-to-one on some superset of its
|
|
range. (Contributed by Mario Carneiro, 12-Jan-2013.) $)
|
|
f1ss $p |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C ) $=
|
|
( wf1 wss wa wf ccnv wfun f1f fss sylan df-f1 simprbi adantr sylanbrc ) ABD
|
|
EZBCFZGACDHZDIJZACDERABDHZSTABDKABCDLMRUASRUBUAABDNOPACDNQ $.
|
|
|
|
$( Two ways to express that a set ` A ` is one-to-one. Each side is equivalent
|
|
to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation
|
|
"Un_2 (A)" for one-to-one. We do not introduce a separate notation since we
|
|
rarely use it. (Contributed by set.mm contributors, 13-Aug-2004.) (Revised
|
|
by Scott Fenton, 18-Apr-2021.) $)
|
|
f1funfun $p |- ( A : dom A -1-1-> _V <-> ( Fun `' A /\ Fun A ) ) $=
|
|
( cdm cvv wf1 wf ccnv wfun df-f1 ancom wfn crn wss ssv df-f mpbiran2 bitr4i
|
|
wa funfn anbi2i 3bitri ) ABZCADUACAEZAFGZQUCUBQUCAGZQUACAHUBUCIUBUDUCUBAUAJ
|
|
ZUDUBUEAKZCLUFMUACANOARPST $.
|
|
|
|
$( Composition of one-to-one functions. Exercise 30 of [TakeutiZaring]
|
|
p. 25. (Contributed by set.mm contributors, 28-May-1998.) $)
|
|
f1co $p |- ( ( F : B -1-1-> C /\ G : A -1-1-> B ) ->
|
|
( F o. G ) : A -1-1-> C ) $=
|
|
( wf ccnv wfun wa ccom wf1 fco funco cnvco funeqi sylibr anim12i an4s df-f1
|
|
ancoms anbi12i 3imtr4i ) BCDFZDGZHZIZABEFZEGZHZIZIACDEJZFZUKGZHZIZBCDKZABEK
|
|
ZIACUKKUCUGUEUIUOUCUGIULUEUIIUNABCDELUIUEUNUIUEIUHUDJZHUNUHUDMUMURDENOPTQRU
|
|
PUFUQUJBCDSABESUAACUKSUB $.
|
|
|
|
$( Equality theorem for onto functions. (Contributed by set.mm contributors,
|
|
1-Aug-1994.) $)
|
|
foeq1 $p |- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) ) $=
|
|
( wceq wfn crn wa wfo fneq1 rneq eqeq1d anbi12d df-fo 3bitr4g ) CDEZCAFZCGZ
|
|
BEZHDAFZDGZBEZHABCIABDIPQTSUBACDJPRUABCDKLMABCNABDNO $.
|
|
|
|
$( Equality theorem for onto functions. (Contributed by set.mm contributors,
|
|
1-Aug-1994.) $)
|
|
foeq2 $p |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) ) $=
|
|
( wceq wfn crn wa wfo fneq2 anbi1d df-fo 3bitr4g ) ABEZDAFZDGCEZHDBFZPHACDI
|
|
BCDINOQPABDJKACDLBCDLM $.
|
|
|
|
$( Equality theorem for onto functions. (Contributed by set.mm contributors,
|
|
1-Aug-1994.) $)
|
|
foeq3 $p |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) ) $=
|
|
( wceq wfn crn wa wfo eqeq2 anbi2d df-fo 3bitr4g ) ABEZDCFZDGZAEZHOPBEZHCAD
|
|
ICBDINQROABPJKCADLCBDLM $.
|
|
|
|
${
|
|
nffo.1 $e |- F/_ x F $.
|
|
nffo.2 $e |- F/_ x A $.
|
|
nffo.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for an onto function. (Contributed by
|
|
NM, 16-May-2004.) $)
|
|
nffo $p |- F/ x F : A -onto-> B $=
|
|
( wfo wfn crn wceq wa df-fo nffn nfrn nfeq nfan nfxfr ) BCDHDBIZDJZCKZLAB
|
|
CDMSUAAABDEFNATCADEOGPQR $.
|
|
$}
|
|
|
|
$( An onto mapping is a mapping. (Contributed by set.mm contributors,
|
|
3-Aug-1994.) $)
|
|
fof $p |- ( F : A -onto-> B -> F : A --> B ) $=
|
|
( wfn crn wceq wa wss wfo wf eqimss anim2i df-fo df-f 3imtr4i ) CADZCEZBFZG
|
|
PQBHZGABCIABCJRSPQBKLABCMABCNO $.
|
|
|
|
$( An onto mapping is a function. (Contributed by set.mm contributors,
|
|
29-Mar-2008.) $)
|
|
fofun $p |- ( F : A -onto-> B -> Fun F ) $=
|
|
( wfo wf wfun fof ffun syl ) ABCDABCECFABCGABCHI $.
|
|
|
|
$( An onto mapping is a function on its domain. (Contributed by set.mm
|
|
contributors, 16-Dec-2008.) $)
|
|
fofn $p |- ( F : A -onto-> B -> F Fn A ) $=
|
|
( wfo wf wfn fof ffn syl ) ABCDABCECAFABCGABCHI $.
|
|
|
|
$( The codomain of an onto function is its range. (Contributed by set.mm
|
|
contributors, 3-Aug-1994.) $)
|
|
forn $p |- ( F : A -onto-> B -> ran F = B ) $=
|
|
( wfo wfn crn wceq df-fo simprbi ) ABCDCAECFBGABCHI $.
|
|
|
|
$( Alternate definition of an onto function. (Contributed by set.mm
|
|
contributors, 22-Mar-2006.) $)
|
|
dffo2 $p |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) ) $=
|
|
( wfo wf crn wceq wa fof forn jca wfn ffn df-fo biimpri sylan impbii ) ABCD
|
|
ZABCEZCFBGZHRSTABCIABCJKSCALZTRABCMRUATHABCNOPQ $.
|
|
|
|
$( The image of the domain of an onto function. (Contributed by set.mm
|
|
contributors, 29-Nov-2002.) $)
|
|
foima $p |- ( F : A -onto-> B -> ( F " A ) = B ) $=
|
|
( wfo cima crn cdm imadmrn wf wceq fof fdm imaeq2 3syl syl5reqr forn eqtrd
|
|
) ABCDZCAEZCFZBRTCCGZEZSCHRABCIUAAJUBSJABCKABCLUAACMNOABCPQ $.
|
|
|
|
$( A function maps onto its range. (Contributed by set.mm contributors,
|
|
10-May-1998.) $)
|
|
dffn4 $p |- ( F Fn A <-> F : A -onto-> ran F ) $=
|
|
( wfn crn wceq wa wfo eqid biantru df-fo bitr4i ) BACZLBDZMEZFAMBGNLMHIAMBJ
|
|
K $.
|
|
|
|
$( A function maps its domain onto its range. (Contributed by set.mm
|
|
contributors, 23-Jul-2004.) $)
|
|
funforn $p |- ( Fun A <-> A : dom A -onto-> ran A ) $=
|
|
( wfun cdm wfn crn wfo funfn dffn4 bitri ) ABAACZDJAEAFAGJAHI $.
|
|
|
|
$( An onto function has unique domain and range. (Contributed by set.mm
|
|
contributors, 5-Nov-2006.) $)
|
|
fodmrnu $p |- ( ( F : A -onto-> B /\ F : C -onto-> D ) ->
|
|
( A = C /\ B = D ) ) $=
|
|
( wfo wa wceq wfn fofn fndmu syl2an crn forn sylan9req jca ) ABEFZCDEFZGACH
|
|
ZBDHQEAIECISRABEJCDEJACEKLQRBEMDABENCDENOP $.
|
|
|
|
$( Restriction of a function. (Contributed by set.mm contributors,
|
|
4-Mar-1997.) $)
|
|
fores $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( F |` A ) : A -onto-> ( F " A ) ) $=
|
|
( wfun cdm wss cres cima wfo funres anim1i wfn wceq df-fn crn dfima3 eqcomi
|
|
wa df-fo mpbiran2 ssdmres anbi2i 3bitr4i sylibr ) BCZABDEZQBAFZCZUEQZABAGZU
|
|
FHZUDUGUEABIJUFAKZUGUFDALZQUJUHUFAMUJUKUFNZUILUIUMBAOPAUIUFRSUEULUGABTUAUBU
|
|
C $.
|
|
|
|
$( Composition of onto functions. (Contributed by set.mm contributors,
|
|
22-Mar-2006.) $)
|
|
foco $p |- ( ( F : B -onto-> C /\ G : A -onto-> B ) ->
|
|
( F o. G ) : A -onto-> C ) $=
|
|
( wf crn wceq ccom wfo fco ad2ant2r cdm fdm eqtr3 sylan rncoeq eqeq1d dffo2
|
|
wa biimpar an32s adantrl jca anbi12i 3imtr4i ) BCDFZDGZCHZTZABEFZEGZBHZTZTZ
|
|
ACDEIZFZUPGZCHZTBCDJZABEJZTACUPJUOUQUSUGUKUQUIUMABCDEKLUJUMUSUKUGUMUIUSUGUM
|
|
TDMZULHZUIUSUGVBBHUMVCBCDNVBULBOPVCUSUIVCURUHCDEQRUAPUBUCUDUTUJVAUNBCDSABES
|
|
UEACUPSUF $.
|
|
|
|
$( A nonzero constant function is onto. (Contributed by set.mm contributors,
|
|
12-Jan-2007.) $)
|
|
foconst $p |- ( ( F : A --> { B } /\ F =/= (/) ) -> F : A -onto-> { B } ) $=
|
|
( csn wf c0 wne wa crn wceq wfo wn rneq0 necon3abii wss frn sssn sylib ord
|
|
wo syl5bi imdistani dffo2 sylibr ) ABDZCEZCFGZHUFCIZUEJZHAUECKUFUGUIUGUHFJZ
|
|
LUFUIUJCFCMNUFUJUIUFUHUEOUJUITAUECPUHBQRSUAUBAUECUCUD $.
|
|
|
|
$( Equality theorem for one-to-one onto functions. (Contributed by set.mm
|
|
contributors, 10-Feb-1997.) $)
|
|
f1oeq1 $p |- ( F = G -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) ) $=
|
|
( wceq wf1 wfo wa wf1o f1eq1 foeq1 anbi12d df-f1o 3bitr4g ) CDEZABCFZABCGZH
|
|
ABDFZABDGZHABCIABDIOPRQSABCDJABCDKLABCMABDMN $.
|
|
|
|
$( Equality theorem for one-to-one onto functions. (Contributed by set.mm
|
|
contributors, 10-Feb-1997.) $)
|
|
f1oeq2 $p |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) ) $=
|
|
( wceq wf1 wfo wa wf1o f1eq2 foeq2 anbi12d df-f1o 3bitr4g ) ABEZACDFZACDGZH
|
|
BCDFZBCDGZHACDIBCDIOPRQSABCDJABCDKLACDMBCDMN $.
|
|
|
|
$( Equality theorem for one-to-one onto functions. (Contributed by set.mm
|
|
contributors, 10-Feb-1997.) $)
|
|
f1oeq3 $p |- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) ) $=
|
|
( wceq wf1 wfo wa wf1o f1eq3 foeq3 anbi12d df-f1o 3bitr4g ) ABEZCADFZCADGZH
|
|
CBDFZCBDGZHCADICBDIOPRQSABCDJABCDKLCADMCBDMN $.
|
|
|
|
$( Equality theorem for one-to-one onto functions. (Contributed by FL,
|
|
14-Jul-2012.) $)
|
|
f1oeq23 $p |- ( ( A = B /\ C = D ) ->
|
|
( F : A -1-1-onto-> C <-> F : B -1-1-onto-> D ) ) $=
|
|
( wceq wf1o f1oeq2 f1oeq3 sylan9bb ) ABFACEGBCEGCDFBDEGABCEHCDBEIJ $.
|
|
|
|
${
|
|
nff1o.1 $e |- F/_ x F $.
|
|
nff1o.2 $e |- F/_ x A $.
|
|
nff1o.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for a one-to-one onto function.
|
|
(Contributed by NM, 16-May-2004.) $)
|
|
nff1o $p |- F/ x F : A -1-1-onto-> B $=
|
|
( wf1o wf1 wfo wa df-f1o nff1 nffo nfan nfxfr ) BCDHBCDIZBCDJZKABCDLQRAAB
|
|
CDEFGMABCDEFGNOP $.
|
|
$}
|
|
|
|
$( A one-to-one onto mapping is a one-to-one mapping. (Contributed by set.mm
|
|
contributors, 12-Dec-2003.) $)
|
|
f1of1 $p |- ( F : A -1-1-onto-> B -> F : A -1-1-> B ) $=
|
|
( wf1o wf1 wfo df-f1o simplbi ) ABCDABCEABCFABCGH $.
|
|
|
|
$( A one-to-one onto mapping is a mapping. (Contributed by set.mm
|
|
contributors, 12-Dec-2003.) $)
|
|
f1of $p |- ( F : A -1-1-onto-> B -> F : A --> B ) $=
|
|
( wf1o wf1 wf f1of1 f1f syl ) ABCDABCEABCFABCGABCHI $.
|
|
|
|
$( A one-to-one onto mapping is function on its domain. (Contributed by
|
|
set.mm contributors, 12-Dec-2003.) $)
|
|
f1ofn $p |- ( F : A -1-1-onto-> B -> F Fn A ) $=
|
|
( wf1o wf wfn f1of ffn syl ) ABCDABCECAFABCGABCHI $.
|
|
|
|
$( A one-to-one onto mapping is a function. (Contributed by set.mm
|
|
contributors, 12-Dec-2003.) $)
|
|
f1ofun $p |- ( F : A -1-1-onto-> B -> Fun F ) $=
|
|
( wf1o wfn wfun f1ofn fnfun syl ) ABCDCAECFABCGACHI $.
|
|
|
|
$( The domain of a one-to-one onto mapping. (Contributed by set.mm
|
|
contributors, 8-Mar-2014.) $)
|
|
f1odm $p |- ( F : A -1-1-onto-> B -> dom F = A ) $=
|
|
( wf1o wfn cdm wceq f1ofn fndm syl ) ABCDCAECFAGABCHACIJ $.
|
|
|
|
$( Alternate definition of one-to-one onto function. (The proof was
|
|
shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm
|
|
contributors, 10-Feb-1997.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
dff1o2 $p |- ( F : A -1-1-onto-> B
|
|
<-> ( F Fn A /\ Fun `' F /\ ran F = B ) ) $=
|
|
( wf1o wf1 wfo wa wf ccnv wfun wfn crn w3a df-f1o df-f1 df-fo anbi12i ancom
|
|
wceq 3anass an12 bitri anbi1i bitr4i wss eqimss df-f biimpri sylan2 3adant2
|
|
anass pm4.71i 3bitr4i 3bitri ) ABCDABCEZABCFZGABCHZCIJZGZCAKZCLZBSZGZGZUTUR
|
|
VBMZABCNUOUSUPVCABCOABCPQUQURVCGZGZVEUQGZVDVEVGVFUQGVHUQVFRVEVFUQVEUTURVBGG
|
|
VFUTURVBTUTURVBUAUBUCUDUQURVCUKVEUQUTVBUQURVBUTVABUEZUQVABUFUQUTVIGABCUGUHU
|
|
IUJULUMUN $.
|
|
|
|
$( Alternate definition of one-to-one onto function. (The proof was
|
|
shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm
|
|
contributors, 25-Mar-1998.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
dff1o3 $p |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) ) $=
|
|
( wfn ccnv wfun crn wceq w3a wf1o wfo df-3an an32 bitri dff1o2 df-fo anbi1i
|
|
wa 3bitr4i ) CADZCEFZCGBHZIZTUBRZUARZABCJABCKZUARUCTUARUBRUETUAUBLTUAUBMNAB
|
|
COUFUDUAABCPQS $.
|
|
|
|
$( A one-to-one onto function is an onto function. (Contributed by set.mm
|
|
contributors, 28-Apr-2004.) $)
|
|
f1ofo $p |- ( F : A -1-1-onto-> B -> F : A -onto-> B ) $=
|
|
( wf1o wfo ccnv wfun dff1o3 simplbi ) ABCDABCECFGABCHI $.
|
|
|
|
$( Alternate definition of one-to-one onto function. (The proof was
|
|
shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm
|
|
contributors, 25-Mar-1998.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
dff1o4 $p |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) ) $=
|
|
( wf1o wfn ccnv wfun crn w3a wa dff1o2 3anass cdm dfrn4 eqeq1i anbi2i df-fn
|
|
wceq bitr4i 3bitri ) ABCDCAEZCFZGZCHZBRZIUAUCUEJZJUAUBBEZJABCKUAUCUELUFUGUA
|
|
UFUCUBMZBRZJUGUEUIUCUDUHBCNOPUBBQSPT $.
|
|
|
|
$( Alternate definition of one-to-one onto function. (The proof was
|
|
shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm
|
|
contributors, 10-Dec-2003.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
dff1o5 $p |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) ) $=
|
|
( wf1o wf1 wfo wa crn wceq df-f1o wf biantrurd dffo2 syl6rbbr pm5.32i bitri
|
|
f1f ) ABCDABCEZABCFZGRCHBIZGABCJRSTRTABCKZTGSRUATABCQLABCMNOP $.
|
|
|
|
$( A one-to-one function maps onto its range. (Contributed by set.mm
|
|
contributors, 13-Aug-2004.) $)
|
|
f1orn $p |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) ) $=
|
|
( wfn ccnv wfun crn wceq w3a wa wf1o df-3an dff1o2 eqid biantru 3bitr4i ) B
|
|
ACZBDEZBFZRGZHPQIZSIARBJTPQSKARBLSTRMNO $.
|
|
|
|
$( A one-to-one function maps one-to-one onto its range. (Contributed by
|
|
set.mm contributors, 4-Sep-2004.) $)
|
|
f1f1orn $p |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F ) $=
|
|
( wf1 wfn ccnv wfun crn wf1o f1fn wf df-f1 simprbi f1orn sylanbrc ) ABCDZCA
|
|
ECFGZACHCIABCJPABCKQABCLMACNO $.
|
|
|
|
$( A class is a one-to-one onto function iff its converse is a one-to-one
|
|
onto function with domain and range interchanged. (Contributed by set.mm
|
|
contributors, 8-Dec-2003.) (Modified by Scott Fenton, 17-Apr-2021.) $)
|
|
f1ocnvb $p |- ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) $=
|
|
( ccnv wfn wa wf1o cnvcnv fneq1i anbi2i ancom bitri dff1o4 3bitr4ri ) CDZBE
|
|
ZODZAEZFZCAEZPFZBAOGABCGSPTFUARTPAQCCHIJPTKLBAOMABCMN $.
|
|
|
|
$( The converse of a one-to-one onto function is also one-to-one onto. (The
|
|
proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by
|
|
set.mm contributors, 11-Feb-1997.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
f1ocnv $p |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A ) $=
|
|
( wf1o ccnv f1ocnvb biimpi ) ABCDBACEDABCFG $.
|
|
|
|
$( The restriction of a one-to-one function maps one-to-one onto the image.
|
|
(Contributed by set.mm contributors, 25-Mar-1998.) $)
|
|
f1ores $p |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto->
|
|
( F " C ) ) $=
|
|
( wf ccnv wfun wa wss cima cres wfo wf1 wf1o cdm ffun adantr sseq2d biimpar
|
|
fdm fores syl2anc funres11 anim12i an32s df-f1 anbi1i dff1o3 3imtr4i ) ABDE
|
|
ZDFGZHZCAIZHCDCJZDCKZLZUOFGZHZABDMZUMHCUNUONUJUMUKURUJUMHZUPUKUQUTDGZCDOZIZ
|
|
UPUJVAUMABDPQUJVCUMUJVBACABDTRSCDUAUBCDUCUDUEUSULUMABDUFUGCUNUOUHUI $.
|
|
|
|
$( The converse of a one-to-one-onto restricted function. (Contributed by
|
|
Paul Chapman, 21-Apr-2008.) $)
|
|
f1orescnv $p |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) ->
|
|
( `' F |` P ) : P -1-1-onto-> R ) $=
|
|
( ccnv wfun cres wf1o wa f1ocnv adantl wceq wb cima funcnvres dfima3 dff1o5
|
|
crn wf1 simprbi syl5eq reseq2d sylan9eq f1oeq1 syl mpbid ) CDZEZBACBFZGZHZA
|
|
BUHDZGZABUFAFZGZUIULUGBAUHIJUJUKUMKULUNLUGUIUKUFCBMZFUMBCNUIUOAUFUIUOUHQZAC
|
|
BOUIBAUHRUPAKBAUHPSTUAUBABUKUMUCUDUE $.
|
|
|
|
$( Preimage of an image. (Contributed by set.mm contributors,
|
|
30-Sep-2004.) $)
|
|
f1imacnv $p |- ( ( F : A -1-1-> B /\ C C_ A )
|
|
-> ( `' F " ( F " C ) ) = C ) $=
|
|
( wf1 wss wa ccnv cima cres resima wfun wceq df-f1 simprbi adantr funcnvres
|
|
wf 3syl wf1o imaeq1 f1ores f1ocnv crn cdm imadmrn fdm imaeq2 syl5reqr f1ofo
|
|
f1of wfo forn syl eqtrd eqtr3d syl5eqr ) ABDEZCAFZGZDHZDCIZIVAVBJZVBIZCVAVB
|
|
KUTDCJZHZVBIZVDCUTVALZVFVCMVGVDMURVHUSURABDRVHABDNOPCDQVFVCVBUASUTCVBVETVBC
|
|
VFTZVGCMABCDUBCVBVEUCVIVGVFUDZCVIVJVFVFUEZIZVGVFUFVIVBCVFRVKVBMVLVGMVBCVFUK
|
|
VBCVFUGVKVBVFUHSUIVIVBCVFULVJCMVBCVFUJVBCVFUMUNUOSUPUQ $.
|
|
|
|
$( A reverse version of ~ f1imacnv . (Contributed by Jeffrey Hankins,
|
|
16-Jul-2009.) $)
|
|
foimacnv $p |- ( ( F : A -onto-> B /\ C C_ B )
|
|
-> ( F " ( `' F " C ) ) = C ) $=
|
|
( wfo wss wa ccnv cima cres resima wfun wceq fofun adantr syl crn cdm dfrn4
|
|
syl5eqr funcnvres2 imaeq1d wfn resss cnvss ax-mp cnvcnv sseqtri funss mpsyl
|
|
dfima3 eqtr2i jctir df-fn sylibr df-dm forn sseq2d biimpar syl6sseq ssdmres
|
|
sylib df-fo sylanbrc foima eqtr3d ) ABDEZCBFZGZDDHZCIZIDVKJZVKIZCDVKKVIVJCJ
|
|
ZHZVKIZVMCVIVOVLVKVIDLZVOVLMVGVQVHABDNZOCDUAPUBVIVKCVOEZVPCMVIVOVKUCZVOQZCM
|
|
VSVIVOLZVORZVKMZGVTVIWBWDVGWBVHVODFVGVQWBVOVJHZDVNVJFVOWEFVJCUDVNVJUEUFDUGU
|
|
HVRVODUIUJOVKVNQWCVJCUKVNSULUMVOVKUNUOVIWAVNRZCVNUPVICVJRZFWFCMVICDQZWGVGCW
|
|
HFVHVGWHBCABDUQURUSDSUTCVJVAVBTVKCVOVCVDVKCVOVEPVFT $.
|
|
|
|
$( The union of two one-to-one onto functions with disjoint domains and
|
|
ranges. (Contributed by set.mm contributors, 26-Mar-1998.) $)
|
|
f1oun $p |- ( ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D )
|
|
/\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) )
|
|
-> ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) ) $=
|
|
( wf1o wa cin c0 wceq cun wfn ccnv wi dff1o4 fnun ex cnvun fneq1i im2anan9
|
|
sylibr an4s syl2anb syl6ibr imp ) ABEGZCDFGZHZACIJKZBDIJKZHZACLZBDLZEFLZGZU
|
|
IULUOUMMZUONZUNMZHZUPUGEAMZENZBMZHFCMZFNZDMZHULUTOZUHABEPCDFPVAVDVCVFVGVAVD
|
|
HZUJUQVCVFHZUKUSVHUJUQACEFQRVIUKUSVIUKHVBVELZUNMUSBDVBVEQUNURVJEFSTUBRUAUCU
|
|
DUMUNUOPUEUF $.
|
|
|
|
${
|
|
$d A u v x z $. $d A u v y $. $d B u v y $. $d B u v z $.
|
|
$d C u v x z $. $d D u v $. $d S u v x $. $d y z $.
|
|
fun11iun.1 $e |- ( x = y -> B = C ) $.
|
|
fun11iun.2 $e |- B e. _V $.
|
|
$( The union of a chain (with respect to inclusion) of one-to-one functions
|
|
is a one-to-one function. (Contributed by Mario Carneiro,
|
|
20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.) $)
|
|
fun11iun $p |- ( A. x e. A ( B : D -1-1-> S /\
|
|
A. y e. A ( B C_ C \/ C C_ B ) ) ->
|
|
U_ x e. A B : U_ x e. A D -1-1-> S ) $=
|
|
( vz vu vv wss wa wfun wceq wrex wcel syl wex wf1 wo wral ciun wf wfn crn
|
|
ccnv cdm cab cuni vex weq eqeq1 rexbidv elab r19.29 nfre1 nfab nfral nfan
|
|
cv nfv wi f1eq1 biimparc f1fun df-f1 simprbi jca adantlr eqeq2d syl6bb wb
|
|
cbvrexv sseq12 orbi12d biimprcd expdimp rexlimivw imp sylan an32s sylan2b
|
|
ancoms ralrimiva adantll a1i rexlimi fun11uni simpld dfiun2 funeqi sylibr
|
|
cop nfra1 eldm2 eleq2d syl5bbr adantr rexbida eliun exbii rexcom4 3bitr4i
|
|
rsp f1dm 3bitr4g eqrdv df-fn sylanbrc elrn2 f1f frn sseld syl5bir rexlimd
|
|
syl6 syl5bi ssrdv df-f simprd cnveqi ) FGDUAZDEMZEDMZUBZBCUCZNZACUCZACFUD
|
|
ZGACDUDZUEZYLUHZOZYKGYLUAYJYLYKUFZYLUGZGMYMYJYLOZYLUIZYKPYPYJJVBZDPZACQZJ
|
|
UJZUKZOZYRYJUUEUUDUHZOZYJKVBZOZUUHUHOZNZUUHLVBZMZUULUUHMZUBZLUUCUCZNZKUUC
|
|
UCUUEUUGNYJUUQKUUCUUHUUCRYJUUHDPZACQZUUQUUBUUSJUUHKULJKUMUUAUURACYTUUHDUN
|
|
UOUPYJUUSNYIUURNZACQUUQYIUURACUQUUTUUQACUUKUUPAUUKAVCUUOALUUCUUBAJUUAACUR
|
|
USUUOAVCUTVAUUTUUQVDAVBCRZUUTUUKUUPYDUURUUKYHYDUURNFGUUHUAZUUKUURUVBYDFGU
|
|
UHDVEVFUVBUUIUUJFGUUHVGUVBFGUUHUEUUJFGUUHVHVIVJSVKYHUURUUPYDYHUURNZUUOLUU
|
|
CUULUUCRUVCUULEPZBCQZUUOUUBUVEJUULLULJLUMZUUBUULDPZACQUVEUVFUUAUVGACYTUUL
|
|
DUNUOUVGUVDABCABUMDEUULHVLVOVMUPYHUVEUURUUOYHUVENYGUVDNZBCQZUURUUOYGUVDBC
|
|
UQUVIUURUUOUVHUURUUOVDBCYGUVDUURUUOUVDUURNZUUOYGUVJUUMYEUUNYFUURUVDUUMYEV
|
|
NUUHDUULEVPWEUULEUUHDVPVQVRVSVTWAWBWCWDWFWGVJWHWISWDWFUUCKLWJSZWKYLUUDAJC
|
|
DIWLZWMWNYJKYSYKYJUUHUULWOZDRZLTZACQZUUHFRZACQUUHYSRZUUHYKRYJUVOUVQACYIAC
|
|
WPZYJUVANYIUVOUVQVNZYJUVAYIYIACXFZWAYDUVTYHUVOUUHDUIZRYDUVQLUUHDWQYDUWBFU
|
|
UHFGDXGWRWSWTSXAUVMYLRZLTUVNACQZLTUVRUVPUWCUWDLAUVMCDXBZXCLUUHYLWQUVNALCX
|
|
DXEAUUHCFXBXHXIYLYKXJXKYJLYQGUULYQRZUVNKTZACQZYJUULGRZUWCKTUWDKTUWFUWHUWC
|
|
UWDKUWEXCKUULYLXLUVNAKCXDXEYJUWGUWIACUVSUWIAVCYJUVAYIUWGUWIVDZUWAYDUWJYHU
|
|
WGUULDUGZRYDUWIKUULDXLYDUWKGUULYDFGDUEUWKGMFGDXMFGDXNSXOXPWTXRXQXSXTYKGYL
|
|
YAXKYJUUGYOYJUUEUUGUVKYBYNUUFYLUUDUVLYCWMWNYKGYLVHXK $.
|
|
$}
|
|
|
|
$( The restriction of a one-to-one onto function to a difference maps onto
|
|
the difference of the images. (Contributed by Paul Chapman,
|
|
11-Apr-2009.) $)
|
|
resdif $p |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\
|
|
( F |` B ) : B -onto-> D ) ->
|
|
( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) ) $=
|
|
( ccnv wfun cres wfo cdif cima wf1o wss wceq syl wb ax-mp crn dfima3 wa w3a
|
|
cdm fofun difss fof fdm syl5sseqr fores syl2anc resabs1 foeq1 rneqi 3eqtr4i
|
|
wf foeq3 bitri sylib funres11 dff1o3 biimpri syl2anr 3adant3 syl5eq anim12i
|
|
forn imadif difeq12 sylan9eq sylan2 3impb f1oeq3 mpbid ) EFGZACEAHZIZBDEBHZ
|
|
IZUAZABJZEVSKZEVSHZLZVSCDJZWALZVMVOWBVQVOVSVTWAIZWAFGZWBVMVOVSVNVSKZVNVSHZI
|
|
ZWEVOVNGVSVNUBZMWIACVNUCVOAVSWJABUDZVOACVNUNWJANACVNUEACVNUFOUGVSVNUHUIWIVS
|
|
WGWAIZWEWHWANZWIWLPVSAMWMWKEVSAUJQZVSWGWHWAUKQWGVTNWLWEPWHRWARWGVTWHWAWNULV
|
|
NVSSEVSSUMWGVTVSWAUOQUPUQVSEURWBWEWFTVSVTWAUSUTVAVBVRVTWCNZWBWDPVMVOVQWOVOV
|
|
QTVMEAKZCNZEBKZDNZTZWOVOWQVQWSVOWPVNRCEASACVNVEVCVQWRVPRDEBSBDVPVEVCVDVMWTV
|
|
TWPWRJWCABEVFWPCWRDVGVHVIVJVTWCVSWAVKOVL $.
|
|
|
|
$( The restriction of a one-to-one onto function to an intersection maps onto
|
|
the intersection of the images. (Contributed by Paul Chapman,
|
|
11-Apr-2009.) $)
|
|
resin $p |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\
|
|
( F |` B ) : B -onto-> D ) ->
|
|
( F |` ( A i^i B ) ) : ( A i^i B ) -1-1-onto-> ( C i^i D ) ) $=
|
|
( ccnv wfun cres wfo w3a cdif wf1o cin resdif f1ofo syl wceq wb dfin4 ax-mp
|
|
syld3an3 f1oeq3 f1oeq2 reseq2i f1oeq1 3bitrri sylib ) EFGZACEAHIZBDEBHIZJZA
|
|
ABKZKZCCDKZKZEUMHZLZABMZCDMZEURHZLZUHUIUJULUNEULHZIZUQUKULUNVBLVCABCDENULUN
|
|
VBOPAULCUNENUAVAURUOUTLZUMUOUTLZUQUSUOQVAVDRCDSUSUOURUTUBTURUMQVDVERABSZURU
|
|
MUOUTUCTUTUPQVEUQRURUMEVFUDUMUOUTUPUETUFUG $.
|
|
|
|
$( Composition of one-to-one onto functions. (Contributed by set.mm
|
|
contributors, 19-Mar-1998.) $)
|
|
f1oco $p |- ( ( F : B -1-1-onto-> C /\ G : A -1-1-onto-> B ) ->
|
|
( F o. G ) : A -1-1-onto-> C ) $=
|
|
( wf1 wfo wa ccom wf1o f1co foco anim12i an4s df-f1o anbi12i 3imtr4i ) BCDF
|
|
ZBCDGZHZABEFZABEGZHZHACDEIZFZACUDGZHZBCDJZABEJZHACUDJRUASUBUGRUAHUESUBHUFAB
|
|
CDEKABCDELMNUHTUIUCBCDOABEOPACUDOQ $.
|
|
|
|
$( The composition of a one-to-one onto function and its converse equals the
|
|
identity relation restricted to the function's range. (Contributed by
|
|
set.mm contributors, 13-Dec-2003.) $)
|
|
f1ococnv2 $p |- ( F : A -1-1-onto-> B -> ( F o. `' F ) = ( _I |` B ) ) $=
|
|
( wf1o ccnv ccom cid cdm cres wfun wceq f1ofun wss df-fun bitri sylib df-dm
|
|
iss crn dmcoeq ax-mp dfrn4 eqtr4i wfo f1ofo forn syl syl5eq reseq2d eqtrd )
|
|
ABCDZCCEZFZGUMHZIZGBIUKCJZUMUOKZABCLUPUMGMUQCNUMROPUKUNBGUKUNCSZBUNULHZURCH
|
|
ULSKUNUSKCQCULTUACUBUCUKABCUDURBKABCUEABCUFUGUHUIUJ $.
|
|
|
|
$( The composition of a one-to-one onto function's converse and itself equals
|
|
the identity relation restricted to the function's domain. (Contributed
|
|
by set.mm contributors, 13-Dec-2003.) $)
|
|
f1ococnv1 $p |- ( F : A -1-1-onto-> B -> ( `' F o. F ) = ( _I |` A ) ) $=
|
|
( wf1o ccnv ccom cid cres cnvcnv coeq2i wceq f1ocnv f1ococnv2 syl syl5eqr )
|
|
ABCDZCEZCFQQEZFZGAHZRCQCIJPBAQDSTKABCLBAQMNO $.
|
|
|
|
$( The converse of an injective function is bijective. (Contributed by FL,
|
|
11-Nov-2011.) $)
|
|
f1cnv $p |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A ) $=
|
|
( wf1 crn wf1o ccnv f1f1orn f1ocnv syl ) ABCDACEZCFKACGFABCHAKCIJ $.
|
|
|
|
$( Composition of an injective function with its converse. (Contributed by
|
|
FL, 11-Nov-2011.) $)
|
|
f1cocnv1 $p |- ( F : A -1-1-> B -> ( `' F o. F ) = ( _I |` A ) ) $=
|
|
( wf1 crn wf1o ccnv ccom cid cres wceq f1f1orn f1ococnv1 syl ) ABCDACEZCFCG
|
|
CHIAJKABCLAOCMN $.
|
|
|
|
$( Composition of an injective function with its converse. (Contributed by
|
|
FL, 11-Nov-2011.) $)
|
|
f1cocnv2 $p |- ( F : A -1-1-> B -> ( F o. `' F ) = ( _I |` ran F ) ) $=
|
|
( wf1 crn wf1o ccnv ccom cid cres wceq f1f1orn f1ococnv2 syl ) ABCDACEZCFCC
|
|
GHIOJKABCLAOCMN $.
|
|
|
|
${
|
|
$d x F $. $d x A $. $d x B $.
|
|
f11o.1 $e |- F e. _V $.
|
|
$( Relationship between a mapping and an onto mapping. Figure 38 of
|
|
[Enderton] p. 145. (Contributed by set.mm contributors,
|
|
10-May-1998.) $)
|
|
ffoss $p |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) ) $=
|
|
( wf cv wfo wss wa wex crn wfn df-f dffn4 anbi1i bitri rnex wceq foeq3
|
|
sseq1 anbi12d spcev sylbi fof fss sylan exlimiv impbii ) BCDFZBAGZDHZUKCI
|
|
ZJZAKZUJBDLZDHZUPCIZJZUOUJDBMZURJUSBCDNUTUQURBDOPQUNUSAUPDERUKUPSULUQUMUR
|
|
UKUPBDTUKUPCUAUBUCUDUNUJAULBUKDFUMUJBUKDUEBUKCDUFUGUHUI $.
|
|
|
|
$( Relationship between one-to-one and one-to-one onto function.
|
|
(Contributed by set.mm contributors, 4-Apr-1998.) $)
|
|
f11o $p |- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) ) $=
|
|
( wf ccnv wfun wa cv wfo wss wex wf1 wf1o ffoss anbi1i df-f1 dff1o3 bitri
|
|
an32 exbii 19.41v 3bitr4i ) BCDFZDGHZIBAJZDKZUGCLZIZAMZUFIZBCDNBUGDOZUIIZ
|
|
AMZUEUKUFABCDEPQBCDRUOUJUFIZAMULUNUPAUNUHUFIZUIIUPUMUQUIBUGDSQUHUFUIUATUB
|
|
UJUFAUCTUD $.
|
|
$}
|
|
|
|
$( The empty set maps one-to-one into any class. (Contributed by set.mm
|
|
contributors, 7-Apr-1998.) $)
|
|
f10 $p |- (/) : (/) -1-1-> A $=
|
|
( c0 wf1 wf ccnv wfun f0 fun0 cnv0 funeqi mpbir df-f1 mpbir2an ) BABCBABDBE
|
|
ZFZAGOBFHNBIJKBABLM $.
|
|
|
|
$( One-to-one onto mapping of the empty set. (Contributed by set.mm
|
|
contributors, 15-Apr-1998.) $)
|
|
f1o00 $p |- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) ) $=
|
|
( c0 wf1o wfn ccnv wceq dff1o4 fn0 biimpi adantr cdm dm0 cnveq syl6eq sylbi
|
|
wa cnv0 fneq1d jca biimpa fndm syl syl5reqr biimpri eqid mpbir fneq2 mpbiri
|
|
sylan9bb impbii bitri ) CABDBCEZBFZAEZQZBCGZACGZQZCABHUPUSUPUQURUMUQUOUMUQB
|
|
IZJKUPCCLZAMUPCAEZVAAGUMUOVBUMAUNCUMUQUNCGUTUQUNCFCBCNROZPSUAACUBUCUDTUSUMU
|
|
OUQUMURUMUQUTUEKUSUOCCEZVDCCGCUFCIUGUQUOVBURVDUQAUNCVCSACCUHUJUITUKUL $.
|
|
|
|
$( Onto mapping of the empty set. (Contributed by set.mm contributors,
|
|
22-Mar-2006.) $)
|
|
fo00 $p |- ( F : (/) -onto-> A <-> ( F = (/) /\ A = (/) ) ) $=
|
|
( c0 wfo wf1o wceq wf1 wfn fofn fn0 f10 f1eq1 mpbiri sylbi syl ancri df-f1o
|
|
wa sylibr f1ofo impbii f1o00 bitri ) CABDZCABEZBCFZACFRUDUEUDCABGZUDRUEUDUG
|
|
UDBCHZUGCABIUHUFUGBJUFUGCACGAKCABCLMNOPCABQSCABTUAABUBUC $.
|
|
|
|
$( One-to-one onto mapping of the empty set. (Contributed by set.mm
|
|
contributors, 10-Feb-2004.) (Revised by set.mm contributors,
|
|
16-Feb-2004.) $)
|
|
f1o0 $p |- (/) : (/) -1-1-onto-> (/) $=
|
|
( wf1o wf1 wfo f10 wfn crn wceq wfun cdm fun0 dm0 df-fn mpbir2an rn0 df-f1o
|
|
c0 df-fo ) PPPAPPPBPPPCZPDRPPEZPFPGSPHPIPGJKPPLMNPPPQMPPPOM $.
|
|
|
|
$( A restriction of the identity relation is a one-to-one onto function.
|
|
(The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by
|
|
set.mm contributors, 30-Apr-1998.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
f1oi $p |- ( _I |` A ) : A -1-1-onto-> A $=
|
|
( cid cres wf1o wfn ccnv fnresi cnvresid fneq1i mpbir dff1o4 mpbir2an ) AAB
|
|
ACZDMAEZMFZAEZAGZPNQAOMAHIJAAMKL $.
|
|
|
|
$( The identity relation is a one-to-one onto function on the universe.
|
|
(Contributed by set.mm contributors, 16-May-2004.) $)
|
|
f1ovi $p |- _I : _V -1-1-onto-> _V $=
|
|
( cvv cid wf1o wfn ccnv wfun cdm wceq funi df-fn mpbir2an cnvi fneq1i mpbir
|
|
dmi dff1o4 ) AABCBADZBEZADZQBFBGAHIOBAJKZSQTARBLMNAABPK $.
|
|
|
|
${
|
|
f1osn.1 $e |- A e. _V $.
|
|
f1osn.2 $e |- B e. _V $.
|
|
$( A singleton of an ordered pair is one-to-one onto function. (The proof
|
|
was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm
|
|
contributors, 18-May-1998.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
f1osn $p |- { <. A , B >. } : { A } -1-1-onto-> { B } $=
|
|
( csn cop wf1o wfn ccnv fnsn cnvsn fneq1i mpbir dff1o4 mpbir2an ) AEZBEZA
|
|
BFEZGRPHRIZQHZABCDJTBAFEZQHBADCJQSUAABCDKLMPQRNO $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B b $.
|
|
$( A singleton of an ordered pair is one-to-one onto function.
|
|
(Contributed by Mario Carneiro, 12-Jan-2013.) $)
|
|
f1osng $p |- ( ( A e. V /\ B e. W ) ->
|
|
{ <. A , B >. } : { A } -1-1-onto-> { B } ) $=
|
|
( va vb cv csn cop wf1o wceq sneq f1oeq2 syl opeq1 f1oeq1 3syl bitrd vex
|
|
wb f1oeq3 opeq2 f1osn vtocl2g ) EGZHZFGZHZUEUGIZHZJZAHZUHAUGIZHZJZULBHZAB
|
|
IZHZJZEFABCDUEAKZUKULUHUJJZUOUTUFULKUKVATUEALUFULUHUJMNUTUIUMKUJUNKVAUOTU
|
|
EAUGOUIUMLULUHUJUNPQRUGBKZUOULUPUNJZUSVBUHUPKUOVCTUGBLUHUPULUNUANVBUMUQKU
|
|
NURKVCUSTUGBAUBUMUQLULUPUNURPQRUEUGESFSUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( Alternate definition of function value. Definition 10.11 of [Quine]
|
|
p. 68. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)
|
|
(Contributed by set.mm contributors, 30-Apr-2004.) (Revised by set.mm
|
|
contributors, 18-Sep-2011.) $)
|
|
fv2 $p |- ( F ` A ) = U. { x | A. y ( A F y <-> y = x ) } $=
|
|
( cfv cv wbr cio weq wb wal cab cuni df-fv dfiota2 eqtri ) CDECBFDGZBHQBA
|
|
IJBKALMBCDNQBAOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( A function's value at a proper class is the empty set. (Contributed by
|
|
set.mm contributors, 20-May-1998.) $)
|
|
fvprc $p |- ( -. A e. _V -> ( F ` A ) = (/) ) $=
|
|
( vx cvv wcel wn cfv cv wbr cio c0 df-fv weu wceq wex euex simpld exlimiv
|
|
brex syl con3i iotanul syl5eq ) ADEZFZABGACHZBIZCJZKCABLUEUGCMZFUHKNUIUDU
|
|
IUGCOUDUGCPUGUDCUGUDUFDEAUFBSQRTUAUGCUBTUC $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x y F $.
|
|
$( Membership in a function value. (Contributed by set.mm contributors,
|
|
30-Apr-2004.) $)
|
|
elfv $p |- ( A e. ( F ` B ) <->
|
|
E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) ) $=
|
|
( cfv wcel cv wbr weq wb wal cab cuni wa wex fv2 eleq2i eluniab bitri ) C
|
|
DEFZGCDBHEIBAJKBLZAMNZGCAHGUBOAPUAUCCABDEQRUBACST $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $. $d x G $.
|
|
$( Equality theorem for function value. (Contributed by set.mm
|
|
contributors, 29-Dec-1996.) $)
|
|
fveq1 $p |- ( F = G -> ( F ` A ) = ( G ` A ) ) $=
|
|
( vx wceq cv wbr cio cfv breq iotabidv df-fv 3eqtr4g ) BCEZADFZBGZDHAOCGZ
|
|
DHABIACINPQDAOBCJKDABLDACLM $.
|
|
|
|
$( Equality theorem for function value. (Contributed by set.mm
|
|
contributors, 29-Dec-1996.) $)
|
|
fveq2 $p |- ( A = B -> ( F ` A ) = ( F ` B ) ) $=
|
|
( vx wceq cv wbr cio cfv breq1 iotabidv df-fv 3eqtr4g ) ABEZADFZCGZDHBOCG
|
|
ZDHACIBCINPQDABOCJKDACLDBCLM $.
|
|
$}
|
|
|
|
${
|
|
fveq1i.1 $e |- F = G $.
|
|
$( Equality inference for function value. (Contributed by set.mm
|
|
contributors, 2-Sep-2003.) $)
|
|
fveq1i $p |- ( F ` A ) = ( G ` A ) $=
|
|
( wceq cfv fveq1 ax-mp ) BCEABFACFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
fveq1d.1 $e |- ( ph -> F = G ) $.
|
|
$( Equality deduction for function value. (Contributed by set.mm
|
|
contributors, 2-Sep-2003.) $)
|
|
fveq1d $p |- ( ph -> ( F ` A ) = ( G ` A ) ) $=
|
|
( wceq cfv fveq1 syl ) ACDFBCGBDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
fveq2i.1 $e |- A = B $.
|
|
$( Equality inference for function value. (Contributed by set.mm
|
|
contributors, 28-Jul-1999.) $)
|
|
fveq2i $p |- ( F ` A ) = ( F ` B ) $=
|
|
( wceq cfv fveq2 ax-mp ) ABEACFBCFEDABCGH $.
|
|
$}
|
|
|
|
${
|
|
fveq2d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for function value. (Contributed by set.mm
|
|
contributors, 29-May-1999.) $)
|
|
fveq2d $p |- ( ph -> ( F ` A ) = ( F ` B ) ) $=
|
|
( wceq cfv fveq2 syl ) ABCFBDGCDGFEBCDHI $.
|
|
$}
|
|
|
|
${
|
|
fveq12d.1 $e |- ( ph -> F = G ) $.
|
|
fveq12d.2 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for function value. (Contributed by FL,
|
|
22-Dec-2008.) $)
|
|
fveq12d $p |- ( ph -> ( F ` A ) = ( G ` B ) ) $=
|
|
( cfv fveq1d fveq2d eqtrd ) ABDHBEHCEHABDEFIABCEGJK $.
|
|
$}
|
|
|
|
${
|
|
$d y F $. $d y A $. $d x y $.
|
|
nffv.1 $e |- F/_ x F $.
|
|
nffv.2 $e |- F/_ x A $.
|
|
$( Bound-variable hypothesis builder for function value. (Contributed by
|
|
NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) $)
|
|
nffv $p |- F/_ x ( F ` A ) $=
|
|
( vy cfv cv wbr cio df-fv nfcv nfbr nfiota nfcxfr ) ABCGBFHZCIZFJFBCKQAFA
|
|
BPCEDAPLMNO $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z F $. $d x z $.
|
|
nffvd.2 $e |- ( ph -> F/_ x F ) $.
|
|
nffvd.3 $e |- ( ph -> F/_ x A ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nffv .
|
|
(Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro,
|
|
15-Oct-2016.) $)
|
|
nffvd $p |- ( ph -> F/_ x ( F ` A ) ) $=
|
|
( vz cv wcel wal cab cfv wnfc nfaba1 nffv wb wa nfnfc1 wceq abidnf adantr
|
|
nfan adantl fveq12d nfceqdf syl2anc mpbii ) ABGHZCIZBJGKZUHDIZBJGKZLZMZBC
|
|
DLZMZBUJULUKBGNUIBGNOABDMZBCMZUNUPPEFUQURQZBUMUOUQURBBDRBCRUBUSUJCULDUQUL
|
|
DSURBGDTUAURUJCSUQBGCTUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d y F $. $d x y $.
|
|
$( Move class substitution in and out of a function value. (Contributed by
|
|
NM, 11-Nov-2005.) $)
|
|
csbfv12g $p |- ( A e. C ->
|
|
[_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) ) $=
|
|
( vy wcel wbr cio csb cfv csbiotag sbcbrg csbconstg breq2d bitrd iotabidv
|
|
cv wsbc df-fv eqtrd csbeq2i 3eqtr4g ) BDGZABCFRZEHZFIZJZABCJZUEABEJZHZFIZ
|
|
ABCEKZJUIUJKUDUHUFABSZFIULUFAFBDLUDUNUKFUDUNUIABUEJZUJHUKABCUEDEMUDUOUEUI
|
|
UJABUEDNOPQUAABUMUGFCETUBFUIUJTUC $.
|
|
$}
|
|
|
|
${
|
|
$d F x $.
|
|
$( Move class substitution in and out of a function value. (Contributed by
|
|
NM, 10-Nov-2005.) $)
|
|
csbfv2g $p |- ( A e. C -> [_ A / x ]_ ( F ` B ) =
|
|
( F ` [_ A / x ]_ B ) ) $=
|
|
( wcel cfv csb csbfv12g csbconstg fveq1d eqtrd ) BDFZABCEGHABCHZABEHZGNEG
|
|
ABCDEIMNOEABEDJKL $.
|
|
|
|
$( Substitution for a function value. (Contributed by NM, 1-Jan-2006.) $)
|
|
csbfvg $p |- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) ) $=
|
|
( wcel cv cfv csb csbfv2g csbvarg fveq2d eqtrd ) BCEZABAFZDGHABNHZDGBDGAB
|
|
NCDIMOBDABCJKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27.
|
|
(Contributed by set.mm contributors, 30-Dec-1996.) $)
|
|
fvex $p |- ( F ` A ) e. _V $=
|
|
( vx cfv cv wbr cio cvv df-fv iotaex eqeltri ) ABDACEBFZCGHCABILCJK $.
|
|
$}
|
|
|
|
$( Move a conditional outside of a function. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
fvif $p |- ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) $=
|
|
( cif cfv fveq2 ifsb ) ABCABCEZDFBDFCDFIBDGICDGH $.
|
|
|
|
${
|
|
$d x y z F $. $d x y z A $.
|
|
$( Alternate definition of the value of a function. Definition 6.11 of
|
|
[TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by
|
|
Mario Carneiro, 31-Aug-2015.) $)
|
|
fv3 $p |- ( F ` A ) = { x | ( E. y ( x e. y /\ A F y ) /\
|
|
E! y A F y ) } $=
|
|
( vz cv wcel wbr wa wex weu cfv wceq wb wal elfv wi bi2 breq2 sylib alimi
|
|
vex ceqsalv anim2i eximi elequ2 anbi12d cbvexv 19.40 simprd df-eu jca nfv
|
|
sylibr nfeu1 nfa1 nfan nfex nfim bi1 ax-14 syl6 com23 imp3a anc2ri eximdv
|
|
sps com12 syl5bi exlimi imp impbii bitri abbi2i ) AFZBFZGZCVPDHZIZBJZVRBK
|
|
ZIZACDLZVOWCGVOEFZGZVRVPWDMZNZBOZIZEJZWBEBVOCDPWJWBWJVTWAWJWECWDDHZIZEJVT
|
|
WIWLEWHWKWEWHWFVRQZBOWKWGWMBVRWFRUAVRWKBWDEUBVPWDCDSUCTUDUEWLVSEBWDVPMWEV
|
|
QWKVREBAUFWDVPCDSUGUHTWJWHEJZWAWJWEEJWNWEWHEUIUJVRBEUKZUNULVTWAWJVSWAWJQB
|
|
WAWJBVRBUOWIBEWEWHBWEBUMWGBUPUQURUSWAWNVSWJWOVSWHWIEWHVSWIWHVSWEWGVSWEQBW
|
|
GVQVRWEWGVRVQWEWGVRWFVQWEQVRWFUTBEAVAVBVCVDVGVEVHVFVIVJVKVLVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d x F $. $d x A $. $d x B $.
|
|
$( The value of a restricted function. (Contributed by set.mm
|
|
contributors, 2-Aug-1994.) (Revised by set.mm contributors,
|
|
16-Feb-2004.) $)
|
|
fvres $p |- ( A e. B -> ( ( F |` B ) ` A ) = ( F ` A ) ) $=
|
|
( vx wcel cv cres wbr cio cfv iba brres syl6rbbr iotabidv df-fv 3eqtr4g
|
|
wa ) ABEZADFZCBGZHZDIASCHZDIATJACJRUAUBDRUBUBRQUARUBKASCBLMNDATODACOP $.
|
|
$}
|
|
|
|
$( The value of a member of the domain of a subclass of a function.
|
|
(Contributed by set.mm contributors, 15-Aug-1994.) (Revised by set.mm
|
|
contributors, 29-May-2007.) $)
|
|
funssfv $p |- ( ( Fun F /\ G C_ F /\ A e. dom G ) ->
|
|
( F ` A ) = ( G ` A ) ) $=
|
|
( wfun wss cdm wcel cfv wceq wa cres fvres eqcomd funssres fveq1d sylan9eqr
|
|
3impa ) BDZCBEZACFZGZABHZACHZIUARSJZUBABTKZHZUCUAUFUBATBLMUDAUECBCNOPQ $.
|
|
|
|
${
|
|
$d A x y $. $d B x $. $d F x y $.
|
|
$( Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
|
|
by NM, 30-Apr-2004.) $)
|
|
tz6.12-1 $p |- ( ( A F B /\ E! y A F y ) -> ( F ` A ) = B ) $=
|
|
( vx cv wbr weu cfv wceq nfv breq2 cbveu wa cio cvv wcel brrelrnex adantr
|
|
df-fv wi iota2 biimpd ex com23 imp3a mpcom syl5eq sylan2b ) BAFZDGZAHBCDG
|
|
ZBEFZDGZEHZBDIZCJUKUNAEUKEKUNAKUJUMBDLMULUONZUPUNEOZCEBDTCPQZUQURCJZULUSU
|
|
OBCDRSUSULUOUTUSUOULUTUSUOULUTUAUSUONULUTUNULECPUMCBDLUBUCUDUEUFUGUHUI $.
|
|
|
|
$( Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
|
|
by NM, 10-Jul-1994.) $)
|
|
tz6.12 $p |- ( ( <. A , B >. e. F /\ E! y <. A , y >. e. F ) ->
|
|
( F ` A ) = B ) $=
|
|
( cop wcel cv weu wa wbr cfv wceq df-br eubii anbi12i tz6.12-1 sylbir ) B
|
|
CEDFZBAGZEDFZAHZIBCDJZBSDJZAHZIBDKCLUBRUDUABCDMUCTABSDMNOABCDPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z F $.
|
|
$( Function value when ` F ` is not a function. Theorem 6.12(2) of
|
|
[TakeutiZaring] p. 27. (Contributed by set.mm contributors,
|
|
30-Apr-2004.) $)
|
|
tz6.12-2 $p |- ( -. E! y A F y -> ( F ` A ) = (/) ) $=
|
|
( vx vz cv wbr weu wn cfv wel wa wex cab c0 fv3 wcel vex weq anbi1d con3i
|
|
elequ1 exbidv elab simprbi eq0rdv syl5eq ) BAFCGZAHZIZBCJDAKZUHLZAMZUILZD
|
|
NZODABCPUJEUOEFZUOQZUIUQEAKZUHLZAMZUIUNUTUILDUPERDESZUMUTUIVAULUSAVAUKURU
|
|
HDEAUBTUCTUDUEUAUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d y F $. $d y A $.
|
|
$( Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by
|
|
NM, 30-Apr-2004.) $)
|
|
tz6.12c $p |- ( E! y A F y -> ( ( F ` A ) = B <-> A F B ) ) $=
|
|
( cv wbr weu cfv wceq wex euex wi nfeu1 nfv tz6.12-1 expcom breq2 biimprd
|
|
nfim syli com12 exlimi mpcom syl5ibcom impbid ) BAEZDFZAGZBDHZCIZBCDFZUHB
|
|
UIDFZUJUKUGAJUHULUGAKUGUHULLAUHULAUGAMULANSUHUGULUGUHUIUFIZULUGUHUMABUFDO
|
|
PUMULUGUIUFBDQRTUAUBUCUICBDQUDUKUHUJABCDOPUE $.
|
|
$}
|
|
|
|
${
|
|
$d y F $. $d y A $. $d y B $.
|
|
$( Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by
|
|
set.mm contributors, 30-Apr-2004.) (Revised by set.mm contributors,
|
|
6-Apr-2007.) $)
|
|
tz6.12i $p |- ( B =/= (/) -> ( ( F ` A ) = B -> A F B ) ) $=
|
|
( vy cfv wceq c0 wne wbr wi cv tz6.12-2 necon1ai eqid tz6.12c mpbii neeq1
|
|
weu syl breq2 imbi12d com12 ) ACEZBFZBGHZABCIZUDUCGHZAUCCIZJUEUFJUGADKCID
|
|
RZUHUIUCGDACLMUIUCUCFUHUCNDAUCCOPSUDUGUEUHUFUCBGQUCBACTUAPUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( The value of a class outside its domain is the empty set. (Contributed
|
|
by set.mm contributors, 24-Aug-1995.) $)
|
|
ndmfv $p |- ( -. A e. dom F -> ( F ` A ) = (/) ) $=
|
|
( vx cdm wcel cv wbr wex cfv c0 wceq eldm wn weu euex tz6.12-2 syl sylnbi
|
|
con3i ) ABDEACFBGZCHZABIJKZCABLUAMTCNZMUBUCUATCOSCABPQR $.
|
|
$}
|
|
|
|
${
|
|
ndmfvrcl.1 $e |- dom F = S $.
|
|
ndmfvrcl.2 $e |- -. (/) e. S $.
|
|
$( Reverse closure law for function with the empty set not in its domain.
|
|
(Contributed by set.mm contributors, 26-Apr-1996.) $)
|
|
ndmfvrcl $p |- ( ( F ` A ) e. S -> A e. S ) $=
|
|
( cfv wcel cdm wn c0 ndmfv eleq1d mtbiri con4i syl6eleq ) ACFZBGZACHZBARG
|
|
ZQSIZQJBGETPJBACKLMNDO $.
|
|
$}
|
|
|
|
$( If a function value has a member, the argument belongs to the domain.
|
|
(Contributed by set.mm contributors, 12-Feb-2007.) $)
|
|
elfvdm $p |- ( A e. ( F ` B ) -> B e. dom F ) $=
|
|
( cfv wcel c0 wne cdm ne0i ndmfv necon1ai syl ) ABCDZEMFGBCHEZMAINMFBCJKL
|
|
$.
|
|
|
|
$( The value of a non-member of a restriction is the empty set. (Contributed
|
|
by set.mm contributors, 13-Nov-1995.) $)
|
|
nfvres $p |- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) ) $=
|
|
( wcel wn cres cdm cfv c0 wceq cin wa dmres eleq2i elin bitri simplbi con3i
|
|
ndmfv syl ) ABDZEACBFZGZDZEAUBHIJUDUAUDUAACGZDZUDABUEKZDUAUFLUCUGACBMNABUEO
|
|
PQRAUBST $.
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( If the restriction of a class to a singleton is not a function, its
|
|
value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof
|
|
shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
nfunsn $p |- ( -. Fun ( F |` { A } ) -> ( F ` A ) = (/) ) $=
|
|
( vx vy cfv c0 wceq csn cres wfun wn cv wbr wmo wal weu eumo wcel sylbi
|
|
wa brres wb elsn breq1 biimpac moimi tz6.12-2 nsyl4 alrimiv dffun6 sylibr
|
|
syl con1i ) ABEFGZBAHZIZJZUNKZCLZDLZUPMZDNZCOUQURVBCAUTBMZDPZVBUNVDVCDNVB
|
|
VCDQVAVCDVAUSUTBMZUSUORZTVCUSUTBUOUAVFVEVCVFUSAGVEVCUBCAUCUSAUTBUDSUESUFU
|
|
LDABUGUHUICDUPUJUKUM $.
|
|
$}
|
|
|
|
$( Function value of the empty set. (Contributed by Stefan O'Rear,
|
|
26-Nov-2014.) $)
|
|
fv01 $p |- ( (/) ` A ) = (/) $=
|
|
( c0 cdm wcel wn cfv wceq noel dm0 eleq2i mtbir ndmfv ax-mp ) ABCZDZEABFBGO
|
|
ABDAHNBAIJKABLM $.
|
|
|
|
$( Equal values imply equal values in a restriction. (Contributed by set.mm
|
|
contributors, 13-Nov-1995.) $)
|
|
fveqres $p |- ( ( F ` A ) = ( G ` A ) ->
|
|
( ( F |` B ) ` A ) = ( ( G |` B ) ` A ) ) $=
|
|
( wcel cfv wceq cres wi fvres eqeq12d biimprd wn nfvres eqtr4d a1d pm2.61i
|
|
c0 ) ABEZACFZADFZGZACBHFZADBHFZGZISUEUBSUCTUDUAABCJABDJKLSMZUEUBUFUCRUDABCN
|
|
ABDNOPQ $.
|
|
|
|
${
|
|
$d y A $. $d y F $. $d y B $.
|
|
$( The second argument of a binary relation on a function is the function's
|
|
value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro,
|
|
28-Apr-2015.) $)
|
|
funbrfv $p |- ( Fun F -> ( A F B -> ( F ` A ) = B ) ) $=
|
|
( vy wfun wbr cfv wceq wa cv weu funeu tz6.12-1 sylan2 anabss7 ex ) CEZAB
|
|
CFZACGBHZQRSQRIRADJCFDKSDABCLDABCMNOP $.
|
|
$}
|
|
|
|
$( The second element in an ordered pair member of a function is the
|
|
function's value. (Contributed by set.mm contributors, 19-Jul-1996.) $)
|
|
funopfv $p |- ( Fun F -> ( <. A , B >. e. F -> ( F ` A ) = B ) ) $=
|
|
( cop wcel wbr wfun cfv wceq df-br funbrfv syl5bir ) ABDCEABCFCGACHBIABCJAB
|
|
CKL $.
|
|
|
|
${
|
|
$d x F $. $d x A $. $d x B $. $d x C $.
|
|
$( Equivalence of function value and binary relation. (Contributed by NM,
|
|
19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) $)
|
|
fnbrfvb $p |- ( ( F Fn A /\ B e. A ) ->
|
|
( ( F ` B ) = C <-> B F C ) ) $=
|
|
( vx wfn wcel wa cv wbr weu cfv wceq wb fneu tz6.12c syl ) DAFBAGHBEIDJEK
|
|
BDLCMBCDJNEABDOEBCDPQ $.
|
|
|
|
$}
|
|
|
|
$( Equivalence of function value and ordered pair membership. (Contributed
|
|
by set.mm contributors, 9-Jan-2015.) $)
|
|
fnopfvb $p |- ( ( F Fn A /\ B e. A ) ->
|
|
( ( F ` B ) = C <-> <. B , C >. e. F ) ) $=
|
|
( wfn wcel wa cfv wceq wbr cop fnbrfvb df-br syl6bb ) DAEBAFGBDHCIBCDJBCKDF
|
|
ABCDLBCDMN $.
|
|
|
|
$( Equivalence of function value and binary relation. (Contributed by set.mm
|
|
contributors, 9-Jan-2015.) $)
|
|
funbrfvb $p |- ( ( Fun F /\ A e. dom F ) ->
|
|
( ( F ` A ) = B <-> A F B ) ) $=
|
|
( wfun cdm wfn wcel cfv wceq wbr wb funfn fnbrfvb sylanb ) CDCCEZFAOGACHBIA
|
|
BCJKCLOABCMN $.
|
|
|
|
$( Equivalence of function value and ordered pair membership. Theorem
|
|
4.3(ii) of [Monk1] p. 42. (Contributed by set.mm contributors,
|
|
9-Jan-2015.) $)
|
|
funopfvb $p |- ( ( Fun F /\ A e. dom F ) ->
|
|
( ( F ` A ) = B <-> <. A , B >. e. F ) ) $=
|
|
( wfun cdm wcel wa cfv wceq wbr cop funbrfvb df-br syl6bb ) CDACEFGACHBIABC
|
|
JABKCFABCLABCMN $.
|
|
|
|
${
|
|
$d x y z w A $. $d x y B $. $d x y z w F $.
|
|
|
|
$( Function value in terms of a binary relation. (Contributed by Mario
|
|
Carneiro, 19-Mar-2014.) $)
|
|
funbrfv2b $p |- ( Fun F ->
|
|
( A F B <-> ( A e. dom F /\ ( F ` A ) = B ) ) ) $=
|
|
( wfun wbr cdm wcel cfv wceq breldm a1i pm4.71rd funbrfvb pm5.32da bitr4d
|
|
wa wi ) CDZABCEZACFGZSPTACHBIZPRSTSTQRABCJKLRTUASABCMNO $.
|
|
|
|
$( Representation of a function in terms of its values. (Contributed by
|
|
set.mm contributors, 29-Jan-2004.) $)
|
|
dffn5 $p |- ( F Fn A <->
|
|
F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } ) $=
|
|
( vz vw wfn cv wcel cfv wceq wa copab cop fnop ex pm4.71rd eqcom vex weq
|
|
fnopfvb syl5bb pm5.32da bitr4d eleq1 fveq2 eqeq2d anbi12d anbi2d opelopab
|
|
eqeq1 syl6bbr eqrelrdv fvex eqid fnopab2 fneq1 mpbiri impbii ) DCGZDAHZCI
|
|
ZBHZVADJZKZLZABMZKZUTEFDVGUTEHZFHZNZDIZVICIZVJVIDJZKZLZVKVGIUTVLVMVLLVPUT
|
|
VLVMUTVLVMCVIVJDOPQUTVMVOVLVOVNVJKUTVMLVLVJVNRCVIVJDUAUBUCUDVFVMVCVNKZLVP
|
|
ABVIVJESFSAETZVBVMVEVQVAVICUEVRVDVNVCVAVIDUFUGUHBFTVQVOVMVCVJVNUKUIUJULUM
|
|
VHUTVGCGABCVDVGVADUNVGUOUPCDVGUQURUS $.
|
|
|
|
$( The range of a function expressed as a collection of the function's
|
|
values. (Contributed by set.mm contributors, 20-Oct-2005.) $)
|
|
fnrnfv $p |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } ) $=
|
|
( wfn crn cv cop wcel wex cab wceq wrex dfrn3 wa fnop ex pm4.71rd fnopfvb
|
|
cfv pm5.32da bitr4d exbidv eqcom rexbii df-rex syl6bbr abbidv syl5eq
|
|
bitri ) DCEZDFAGZBGZHDIZAJZBKUMULDTZLZACMZBKABDNUKUOURBUKUOULCIZUPUMLZOZA
|
|
JZURUKUNVAAUKUNUSUNOVAUKUNUSUKUNUSCULUMDPQRUKUSUTUNCULUMDSUAUBUCURUTACMVB
|
|
UQUTACUMUPUDUEUTACUFUJUGUHUI $.
|
|
|
|
$( A member of a function's range is a value of the function. (Contributed
|
|
by set.mm contributors, 31-Oct-1995.) $)
|
|
fvelrnb $p |- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) ) $=
|
|
( vy wfn crn wcel cv cfv wceq wrex cab fnrnfv eleq2d cvv fvex eleq1 mpbii
|
|
syl6bb rexlimivw eqeq1 eqcom rexbidv elab3 ) DBFZCDGZHCEIZAIZDJZKZABLZEMZ
|
|
HUJCKZABLZUFUGUMCAEBDNOULUOECUNCPHZABUNUJPHUPUIDQUJCPRSUAUHCKZUKUNABUQUKC
|
|
UJKUNUHCUJUBCUJUCTUDUET $.
|
|
|
|
$( Alternate definition of the image of a function. (Contributed by Raph
|
|
Levien, 20-Nov-2006.) $)
|
|
dfimafn $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( F " A ) = { y | E. x e. A ( F ` x ) = y } ) $=
|
|
( wfun cdm wss wa cv cfv wceq wrex cab cima wcel wb ssel2 funbrfvb sylan2
|
|
wbr anassrs rexbidva abbidv df-ima syl6reqr ) DEZCDFZGZHZAIZDJBIZKZACLZBM
|
|
UJUKDTZACLZBMDCNUIUMUOBUIULUNACUFUHUJCOZULUNPZUHUPHUFUJUGOUQCUGUJQUJUKDRS
|
|
UAUBUCBADCUDUE $.
|
|
|
|
$( Alternate definition of the image of a function as an indexed union of
|
|
singletons of function values. (Contributed by Raph Levien,
|
|
20-Nov-2006.) $)
|
|
dfimafn2 $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( F " A ) = U_ x e. A { ( F ` x ) } ) $=
|
|
( vy wfun cdm wss wa cima cv cfv wceq cab ciun wrex dfimafn iunab syl6eqr
|
|
csn wcel df-sn eqcom abbii eqtri a1i iuneq2i ) CEBCFGHZCBIZABAJZCKZDJZLZD
|
|
MZNZABUJSZNUGUHULABODMUNADBCPULADBQRABUOUMUOUMLUIBTUOUKUJLZDMUMDUJUAUPULD
|
|
UKUJUBUCUDUEUFR $.
|
|
|
|
$( Membership relation for the values of a function whose image is a
|
|
subclass. (Contributed by Raph Levien, 20-Nov-2006.) $)
|
|
funimass4 $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) ) $=
|
|
( vy cima wss cv wcel wi wal wfun cdm wa cfv wral dfss2 wceq wrex syl5bb
|
|
wbr wb ssel2 eqcom funbrfvb sylan2 anassrs rexbidva elima syl6rbbr imbi1d
|
|
r19.23v syl6bbr albidv ralcom4 fvex eleq1 ceqsalv ralbii bitr3i syl6bb )
|
|
DBFZCGEHZVBIZVCCIZJZEKZDLZBDMZGZNZAHZDOZCIZABPZEVBCQVKVGVCVMRZVEJZABPZEKZ
|
|
VOVKVFVREVKVFVPABSZVEJVRVKVDVTVEVKVTVLVCDUAZABSVDVKVPWAABVHVJVLBIZVPWAUBZ
|
|
VJWBNVHVLVIIZWCBVIVLUCVPVMVCRVHWDNWAVCVMUDVLVCDUETUFUGUHAVCDBUIUJUKVPVEAB
|
|
ULUMUNVSVQEKZABPVOVQAEBUOWEVNABVEVNEVMVLDUPVCVMCUQURUSUTVAT $.
|
|
|
|
$( Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42.
|
|
(The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed
|
|
by set.mm contributors, 29-Apr-2004.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
fvelima $p |- ( ( Fun F /\ A e. ( F " B ) ) ->
|
|
E. x e. B ( F ` x ) = A ) $=
|
|
( wfun cima wcel cv cfv wceq wrex wbr elima funbrfv reximdv syl5bi imp )
|
|
DEZBDCFGZAHZDIBJZACKZSTBDLZACKRUBABDCMRUCUAACTBDNOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d x y B $. $d x y C $. $d x y F $.
|
|
$( Function value in an image. (The proof was shortened by Andrew Salmon,
|
|
22-Oct-2011.) (An unnecessary distinct variable restriction was removed
|
|
by David Abernethy, 17-Dec-2011.) (Contributed by set.mm contributors,
|
|
20-Jan-2007.) (Revised by set.mm contributors, 25-Dec-2011.) $)
|
|
fvelimab $p |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <->
|
|
E. x e. B ( F ` x ) = C ) ) $=
|
|
( vy wfn wss wa cima wcel cv cfv wceq wrex cvv anim2i eleq1 wb wi rexbidv
|
|
elex fvex mpbii rexlimivw eqeq2 bibi12d imbi2d wfun cdm fnfun adantr fndm
|
|
cab sseq2d biimpar dfimafn syl2anc abeq2d vtoclg impcom pm5.21nd ) EBGZCB
|
|
HZIZDECJZKZALZEMZDNZACOZVEDPKZIVGVLVEDVFUBQVKVLVEVJVLACVJVIPKVLVHEUCVIDPR
|
|
UDUEQVLVEVGVKSZVEFLZVFKZVIVNNZACOZSZTVEVMTFDPVNDNZVRVMVEVSVOVGVQVKVNDVFRV
|
|
SVPVJACVNDVIUFUAUGUHVEVQFVFVEEUIZCEUJZHZVFVQFUNNVCVTVDBEUKULVCWBVDVCWABCB
|
|
EUMUOUPAFCEUQURUSUTVAVB $.
|
|
$}
|
|
|
|
$( Membership in the preimage of a singleton, under a function. (Contributed
|
|
by Mario Carneiro, 12-May-2014.) $)
|
|
fniniseg $p |- ( F Fn A ->
|
|
( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) ) $=
|
|
( ccnv csn cima wcel wbr wfn cfv wceq wa eliniseg breldm fndm eleq2d syl5ib
|
|
cdm pm4.71rd fnbrfvb pm5.32da bitr4d syl5bb ) CDEBFGHCBDIZDAJZCAHZCDKBLZMZD
|
|
BCNUFUEUGUEMUIUFUEUGUECDSZHUFUGCBDOUFUJACADPQRTUFUGUHUEACBDUAUBUCUD $.
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y F $.
|
|
$( The indexed intersection of a function's values is the intersection of
|
|
its range. (Contributed by set.mm contributors, 20-Oct-2005.) $)
|
|
fniinfv $p |- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F ) $=
|
|
( vy wfn crn cint cv cfv wceq wrex cab ciin fnrnfv inteqd dfiin2 syl6reqr
|
|
fvex ) CBEZCFZGDHAHZCIZJABKDLZGABUBMSTUCADBCNOADBUBUACRPQ $.
|
|
|
|
$( Singleton of function value. (Contributed by set.mm contributors,
|
|
22-May-1998.) $)
|
|
fnsnfv $p |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) ) $=
|
|
( vy wfn wcel wa cv cfv wceq cab wbr csn cima eqcom fnbrfvb syl5bb abbidv
|
|
df-sn imasn 3eqtr4g ) CAEBAFGZDHZBCIZJZDKBUCCLZDKUDMCBMNUBUEUFDUEUDUCJUBU
|
|
FUCUDOABUCCPQRDUDSDBCTUA $.
|
|
$}
|
|
|
|
$( The image of a pair under a funtion. (Contributed by Jeff Madsen,
|
|
6-Jan-2011.) $)
|
|
fnimapr $p |- ( ( F Fn A /\ B e. A /\ C e. A ) ->
|
|
( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } ) $=
|
|
( wfn wcel w3a csn cima cun cfv fnsnfv 3adant3 3adant2 uneq12d eqcomd df-pr
|
|
cpr wceq imaeq2i imaundi eqtri 3eqtr4g ) DAEZBAFZCAFZGZDBHZIZDCHZIZJZBDKZHZ
|
|
CDKZHZJZDBCRZIZUMUORUGUQULUGUNUIUPUKUDUEUNUISUFABDLMUDUFUPUKSUEACDLNOPUSDUH
|
|
UJJZIULURUTDBCQTDUHUJUAUBUMUOQUC $.
|
|
|
|
$( A simplified expression for the value of a function when we know it's a
|
|
function. (Contributed by NM, 22-May-1998.) $)
|
|
funfv $p |- ( Fun F -> ( F ` A ) = U. ( F " { A } ) ) $=
|
|
( wfun cdm wcel cfv csn cima cuni wceq fvex unisn wfn df-fn mpbiran2 fnsnfv
|
|
wa eqid unieqd c0 sylanbr syl5eqr ex wn ndmfv ndmima syl6eq eqtr4d pm2.61d1
|
|
uni0 ) BCZABDZEZABFZBAGHZIZJZUKUMUQUKUMQZUNUNGZIUPUNABKLURUSUOUKBULMZUMUSUO
|
|
JUTUKULULJULRBULNOULABPUASUBUCUMUDZUNTUPABUEVAUPTITVAUOTABUFSUJUGUHUI $.
|
|
|
|
${
|
|
$d y A $. $d y F $.
|
|
$( The value of a function. Definition of function value in [Enderton]
|
|
p. 43. (Contributed by set.mm contributors, 22-May-1998.) (Revised by
|
|
set.mm contributors, 11-May-2005.) $)
|
|
funfv2 $p |- ( Fun F -> ( F ` A ) = U. { y | A F y } ) $=
|
|
( wfun cfv csn cima cuni cv wbr cab funfv imasn unieqi syl6eq ) CDBCECBFG
|
|
ZHBAICJAKZHBCLPQABCMNO $.
|
|
$}
|
|
|
|
${
|
|
$d w A $. $d w F $. $d w y $.
|
|
funfv2f.1 $e |- F/_ y A $.
|
|
funfv2f.2 $e |- F/_ y F $.
|
|
$( The value of a function. Version of ~ funfv2 using a bound-variable
|
|
hypotheses instead of distinct variable conditions. (Contributed by NM,
|
|
19-Feb-2006.) $)
|
|
funfv2f $p |- ( Fun F -> ( F ` A ) = U. { y | A F y } ) $=
|
|
( vw wfun cfv wbr cab cuni funfv2 nfcv nfbr nfv breq2 cbvab unieqi syl6eq
|
|
cv ) CGBCHBFTZCIZFJZKBATZCIZAJZKFBCLUCUFUBUEFAABUACDEAUAMNUEFOUAUDBCPQRS
|
|
$.
|
|
$}
|
|
|
|
$( Value of the union of two functions when the domains are separate.
|
|
(Contributed by FL, 7-Nov-2011.) $)
|
|
fvun $p |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) ->
|
|
( ( F u. G ) ` A ) = ( ( F ` A ) u. ( G ` A ) ) ) $=
|
|
( wfun wa cdm cin c0 wceq cun cfv csn cima cuni funun funfv imaundir eqcomd
|
|
syl a1i unieqd uniun anim12i adantr uneq12 syl5eq 3eqtrd ) BDZCDZEZBFCFGHIZ
|
|
EZABCJZKZUMALZMZNZBUOMZCUOMZJZNZABKZACKZJZULUMDUNUQIBCOAUMPSULUPUTUPUTIULBC
|
|
UOQTUAULVAURNZUSNZJZVDURUSUBULVEVBIZVFVCIZEZVGVDIUJVJUKUHVHUIVIUHVBVEABPRUI
|
|
VCVFACPRUCUDVEVBVFVCUESUFUG $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d X x $.
|
|
$( The value of a union when the argument is in the first domain.
|
|
(Contributed by Scott Fenton, 29-Jun-2013.) $)
|
|
fvun1 $p |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) ->
|
|
( ( F u. G ) ` X ) = ( F ` X ) ) $=
|
|
( vx wfn cin c0 wceq wcel wa cun cfv wfun cdm fnfun 3ad2ant2 fndm wn fvun
|
|
w3a 3ad2ant1 ineq12 syl2an eqeq1d biimprd adantrd 3impia syl21anc cv wral
|
|
wi disj eleq1 notbid rspccv sylbi imp 3ad2ant3 eleq2d mtbird ndmfv uneq2d
|
|
syl eqtrd un0 syl6eq ) CAGZDBGZABHZIJZEAKZLZUBZECDMNZECNZIMZVQVOVPVQEDNZM
|
|
ZVRVOCOZDOZCPZDPZHZIJZVPVTJVIVJWAVNACQUCVJVIWBVNBDQRVIVJVNWFVIVJLZVLWFVMW
|
|
GWFVLWGWEVKIVIWCAJWDBJZWEVKJVJACSBDSZWCAWDBUDUEUFUGUHUIECDUAUJVOVSIVQVOEW
|
|
DKZTVSIJVOWJEBKZVNVIWKTZVJVLVMWLVLFUKZBKZTZFAULVMWLUMFABUNWOWLFEAWMEJWNWK
|
|
WMEBUOUPUQURUSUTVOWDBEVJVIWHVNWIRVAVBEDVCVEVDVFVQVGVH $.
|
|
$}
|
|
|
|
$( The value of a union when the argument is in the second domain.
|
|
(Contributed by Scott Fenton, 29-Jun-2013.) $)
|
|
fvun2 $p |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) ->
|
|
( ( F u. G ) ` X ) = ( G ` X ) ) $=
|
|
( wfn cin c0 wceq wcel w3a cun cfv uncom fveq1i incom eqeq1i anbi1i fvun1
|
|
wa syl3an3b 3com12 syl5eq ) CAFZDBFZABGZHIZEBJZTZKECDLZMEDCLZMZEDMZEUJUKCDN
|
|
OUEUDUIULUMIZUIUEUDBAGZHIZUHTUNUGUPUHUFUOHABPQRBADCESUAUBUC $.
|
|
|
|
${
|
|
$d y z A $. $d y z F $. $d y z G $.
|
|
$( Domains of a function composition. (Contributed by set.mm contributors,
|
|
27-Jan-1997.) $)
|
|
dmfco $p |- ( ( Fun G /\ A e. dom G ) ->
|
|
( A e. dom ( F o. G ) <-> ( G ` A ) e. dom F ) ) $=
|
|
( vy vz wfun cdm wcel wa cv wbr wex cfv ccom wceq fvex breq1 ceqsexv eldm
|
|
exbidv eqcom funbrfvb syl5bb anbi1d syl5rbbr brco exbii bitri 3bitr4g ) C
|
|
FACGHIZADJZCKZUKEJZBKZIZDLZELZACMZUMBKZELABCNZGHZURBGHUJUPUSEUSUKUROZUNIZ
|
|
DLUJUPUNUSDURACPUKURUMBQRUJVCUODUJVBULUNVBURUKOUJULUKURUAAUKCUBUCUDTUETVA
|
|
AUMUTKZELUQEAUTSVDUPEDAUMBCUFUGUHEURBSUI $.
|
|
|
|
$d C y z $.
|
|
$( Value of a function composition. Similar to second part of Theorem 3H
|
|
of [Enderton] p. 47. (The proof was shortened by Andrew Salmon,
|
|
22-Oct-2011.) (Contributed by set.mm contributors, 9-Oct-2004.)
|
|
(Revised by set.mm contributors, 22-Oct-2011.) $)
|
|
fvco2 $p |- ( ( G Fn A /\ C e. A ) ->
|
|
( ( F o. G ) ` C ) = ( F ` ( G ` C ) ) ) $=
|
|
( vy vz csn cima cv wceq cab cfv wbr cio df-iota df-fv imasn eqeq1i abbii
|
|
cuni wfn wcel wa ccom fnsnfv imaeq2d syl6reqr eqeq1d abbidv unieqd unieqi
|
|
imaco 3eqtr4i 3eqtr4g ) DAUABAUBUCZCDUDZBGZHZEIGZJZEKZTZCBDLZGZHZUSJZEKZT
|
|
ZBUPLZVCCLZUOVAVGUOUTVFEUOURVEUSUOVECDUQHZHURUOVDVKCABDUEUFCDUQULUGUHUIUJ
|
|
BFIZUPMZFNVMFKZUSJZEKZTVIVBVMFEOFBUPPVAVPUTVOEURVNUSFBUPQRSUKUMVCVLCMZFNV
|
|
QFKZUSJZEKZTVJVHVQFEOFVCCPVGVTVFVSEVEVRUSFVCCQRSUKUMUN $.
|
|
|
|
$}
|
|
|
|
$( Value of a function composition. Similar to Exercise 5 of [TakeutiZaring]
|
|
p. 28. (Contributed by set.mm contributors, 22-Apr-2006.) $)
|
|
fvco $p |- ( ( Fun G /\ A e. dom G ) ->
|
|
( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) ) $=
|
|
( wfun cdm wfn wcel ccom cfv wceq funfn fvco2 sylanb ) CDCCEZFANGABCHIACIBI
|
|
JCKNABCLM $.
|
|
|
|
$( Value of a function composition. (Contributed by set.mm contributors,
|
|
3-Jan-2004.) (Revised by set.mm contributors, 21-Aug-2006.) $)
|
|
fvco3 $p |- ( ( G : A --> B /\ C e. A ) ->
|
|
( ( F o. G ) ` C ) = ( F ` ( G ` C ) ) ) $=
|
|
( wf wfn wcel ccom cfv wceq ffn fvco2 sylan ) ABEFEAGCAHCDEIJCEJDJKABELACDE
|
|
MN $.
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y D $.
|
|
$( Closed theorem form of ~ fvopab4 . (Contributed by set.mm contributors,
|
|
21-Feb-2013.) $)
|
|
fvopab4t $p |- ( ( A. x A. y ( x = A -> B = C )
|
|
/\ A. x F = { <. x , y >. | ( x e. D /\ y = B ) }
|
|
/\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = C ) $=
|
|
( wcel wa cv wceq wi wal copab cvv cfv elex wfun wb anim2i funopab4 simp2
|
|
w3a cop 19.21bi funeqd mpbiri simp3l eqidd eleq1 eqeq2 bi2anan9 ex 2alimi
|
|
a2i 3ad2ant1 eqeq1 anbi2d gen2 simp3 opelopabt syl3anc mpbir2and eleqtrrd
|
|
a1i funopfv sylc syl3an3 ) CFIZEHIZJAKZCLZDELZMZBNANZGVLFIZBKZDLZJZABOZLZ
|
|
ANZVJEPIZJZCGQELZVKWDVJEHRUAVPWCWEUDZGSZCEUEZGIWFWGWHWASVQABDUBWGGWAWGWBA
|
|
VPWCWEUCUFZUGUHWGWIWAGWGWIWAIZVJEELZVPWCVJWDUIWGEUJWGVMVTVJVRELZJZTZMZBNA
|
|
NZWMWNVJWLJZTMZBNANZWEWKWRTVPWCWQWEVOWPABVMVNWOVMVNWOVMVQVJVNVSWMVLCFUKDE
|
|
VRULUMUNUPUOUQWTWGWSABWMWMWLVJVREEURUSUTVFVPWCWEVAVTWNWRABCEFPVBVCVDWJVEC
|
|
EGVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y ch $.
|
|
fvopab3g.1 $e |- B e. _V $.
|
|
fvopab3g.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
fvopab3g.3 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
fvopab3g.4 $e |- ( x e. C -> E! y ph ) $.
|
|
fvopab3g.5 $e |- F = { <. x , y >. | ( x e. C /\ ph ) } $.
|
|
$( Value of a function given by ordered-pair class abstraction.
|
|
(Contributed by set.mm contributors, 6-Mar-1996.) $)
|
|
fvopab3g $p |- ( A e. C -> ( ( F ` A ) = B <-> ch ) ) $=
|
|
( wcel cv wa wceq cvv wb cop copab cfv eleq1 anbi12d anbi2d opelopabg wfn
|
|
mpan2 fnopab fnopfvb mpan eleq2i syl6bb ibar 3bitr4d ) FHOZFGUAZDPZHOZAQZ
|
|
DEUBZOZUQCQZFIUCGRZCUQGSOVCVDTJVAUQBQVDDEFGHSUSFRUTUQABUSFHUDKUEEPGRBCUQL
|
|
UFUGUIUQVEURIOZVCIHUHUQVEVFTADEHIMNUJHFGIUKULIVBURNUMUNUQCUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y ch $.
|
|
fvopab3ig.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
fvopab3ig.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
fvopab3ig.3 $e |- ( x e. C -> E* y ph ) $.
|
|
fvopab3ig.4 $e |- F = { <. x , y >. | ( x e. C /\ ph ) } $.
|
|
$( Value of a function given by ordered-pair class abstraction.
|
|
(Contributed by set.mm contributors, 23-Oct-1999.) $)
|
|
fvopab3ig $p |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) ) $=
|
|
( wcel wa cv cfv wceq wmo copab cop funopab wi moanimv mpbir mpgbir simpl
|
|
wfun eleq1 anbi12d anbi2d opelopabg biimprd funopfv fveq1i eqeq1i syl6ibr
|
|
mpand ee02 ) FHOZGIOZPZCFDQZHOZAPZDEUAZRZGSZFJRZGSVGUIZVCCFGUBVGOZVIVKVFE
|
|
TZDVFDEUCVMVEAETUDMVEAEUEUFUGVCVACVLVAVBUHVCVLVACPZVFVABPVNDEFGHIVDFSVEVA
|
|
ABVDFHUJKUKEQGSBCVALULUMUNUSFGVGUOUTVJVHGFJVGNUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y D $.
|
|
fvopab4g.1 $e |- ( x = A -> B = C ) $.
|
|
fvopab4g.2 $e |- F = { <. x , y >. | ( x e. D /\ y = B ) } $.
|
|
$( Value of a function given by ordered-pair class abstraction.
|
|
(Contributed by set.mm contributors, 23-Oct-1999.) $)
|
|
fvopab4g $p |- ( ( A e. D /\ C e. R ) -> ( F ` A ) = C ) $=
|
|
( wcel wa wceq cfv eqid cv eqeq2d eqeq1 wmo moeq a1i fvopab3ig mpi ) CFKE
|
|
GKLEEMZCHNEMEOBPZDMZUEEMUDABCEFGHAPZCMDEUEIQUEEERUFBSUGFKBDTUAJUBUC $.
|
|
|
|
${
|
|
fvopab4.3 $e |- C e. _V $.
|
|
$( Value of a function given by ordered-pair class abstraction.
|
|
(Contributed by set.mm contributors, 23-Oct-1999.) $)
|
|
fvopab4 $p |- ( A e. D -> ( F ` A ) = C ) $=
|
|
( wcel cvv cfv wceq fvopab4g mpan2 ) CFKELKCGMENJABCDEFLGHIOP $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
fvopab4ndm.1 $e |- F = { <. x , y >. | ( x e. A /\ ph ) } $.
|
|
$( Value of a function given by an ordered-pair class abstraction, outside
|
|
of its domain. (Contributed by set.mm contributors, 28-Mar-2008.) $)
|
|
fvopab4ndm $p |- ( -. B e. A -> ( F ` B ) = (/) ) $=
|
|
( wcel wn cdm cfv c0 wceq cv wa copab dmeqi dmopabss eqsstri sseli con3i
|
|
ndmfv syl ) EDHZIEFJZHZIEFKLMUFUDUEDEUEBNDHAOBCPZJDFUGGQABCDRSTUAEFUBUC
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $.
|
|
fvopabg.1 $e |- ( x = A -> B = C ) $.
|
|
$( The value of a function given by ordered-pair class abstraction.
|
|
(Contributed by set.mm contributors, 2-Sep-2003.) $)
|
|
fvopabg $p |- ( ( A e. V /\ C e. W ) ->
|
|
( { <. x , y >. | y = B } ` A ) = C ) $=
|
|
( wcel cvv cv wceq copab cfv elex wa vex biantrur opabbii fvopab4g sylan
|
|
) CFICJIEGICBKDLZABMZNELCFOABCDEJGUCHUBAKJIZUBPABUDUBAQRSTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y F $. $d x y G $. $d x ph $.
|
|
$( Equality of functions is determined by their values. Special case of
|
|
Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
|
|
(The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed
|
|
by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors,
|
|
22-Oct-2011.) $)
|
|
eqfnfv $p |- ( ( F Fn A /\ G Fn A ) -> ( F = G <->
|
|
A. x e. A ( F ` x ) = ( G ` x ) ) ) $=
|
|
( vy wfn wa wceq cv cfv wcel wi wal wb adantl fnopfvb syl5ib cdm opeldm
|
|
wn wral fveq1 ralrimivw cop pm2.27 eqeq1 adantlr adantll syld expcom fndm
|
|
bibi12d eleq2d adantr con3d impcom 2falsed ex a1dd pm2.61i alrimdv alimdv
|
|
df-ral eqrel 3imtr4g impbid2 ) CBFZDBFZGZCDHZAIZCJZVKDJZHZABUAZVJVNABVKCD
|
|
UBUCVIVKBKZVNLZAMVKEIZUDZCKZVSDKZNZEMZAMVOVJVIVQWCAVIVQWBEVPVIVQWBLZLVIVP
|
|
WDVIVPGZVQVNWBVPVQVNLVIVPVNUEOVNVLVRHZVMVRHZNWEWBVLVMVRUFWEWFVTWGWAVGVPWF
|
|
VTNVHBVKVRCPUGVHVPWGWANVGBVKVRDPUHULQUIUJVPTZVIWBVQWHVIWBWHVIGVTWAVIWHVTT
|
|
VIVTVPVGVTVPLVHVTVKCRZKVGVPVKVRCSVGWIBVKBCUKUMQUNUOUPVIWHWATVIWAVPVHWAVPL
|
|
VGWAVKDRZKVHVPVKVRDSVHWJBVKBDUKUMQOUOUPUQURUSUTVAVBVNABVCAECDVDVEVF $.
|
|
|
|
$( Equality of functions is determined by their values. Exercise 4 of
|
|
[TakeutiZaring] p. 28. (Contributed by set.mm contributors,
|
|
3-Aug-1994.) (Revised by set.mm contributors, 5-Feb-2004.) $)
|
|
eqfnfv2 $p |- ( ( F Fn A /\ G Fn B ) -> ( F = G <->
|
|
( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) ) $=
|
|
( wfn wa wceq cv cfv wral dmeq fndm eqeqan12d syl5ib pm4.71rd wb biimparc
|
|
cdm fneq2 eqfnfv sylan2 anassrs pm5.32da bitrd ) DBFZECFZGZDEHZBCHZUIGUJA
|
|
IZDJUKEJHABKZGUHUIUJUIDSZESZHUHUJDELUFUGUMBUNCBDMCEMNOPUHUJUIULUFUGUJUIUL
|
|
QZUGUJGUFEBFZUOUJUPUGBCETRABDEUAUBUCUDUE $.
|
|
|
|
$d x B $.
|
|
$( Derive equality of functions from equality of their values.
|
|
(Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
eqfnfv3 $p |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A
|
|
( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) ) $=
|
|
( wfn wa wceq cv cfv wral wss wcel eqfnfv2 ancom bitri anbi1i anass dfss3
|
|
eqss r19.26 bitr4i anbi2i 3bitri syl6bb ) DBFECFGDEHBCHZAIZDJUGEJHZABKZGZ
|
|
CBLZUGCMZUHGABKZGZABCDENUJUKBCLZGZUIGUKUOUIGZGUNUFUPUIUFUOUKGUPBCTUOUKOPQ
|
|
UKUOUIRUQUMUKUQULABKZUIGUMUOURUIABCSQULUHABUAUBUCUDUE $.
|
|
eqfnfvd.1 $e |- ( ph -> F Fn A ) $.
|
|
eqfnfvd.2 $e |- ( ph -> G Fn A ) $.
|
|
eqfnfvd.3 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) ) $.
|
|
$( Deduction for equality of functions. (Contributed by Mario Carneiro,
|
|
24-Jul-2014.) $)
|
|
eqfnfvd $p |- ( ph -> F = G ) $=
|
|
( wceq cv cfv wral ralrimiva wfn wb eqfnfv syl2anc mpbird ) ADEIZBJZDKTEK
|
|
IZBCLZAUABCHMADCNECNSUBOFGBCDEPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d z F $. $d z G $.
|
|
eqfnfv2f.1 $e |- F/_ x F $.
|
|
eqfnfv2f.2 $e |- F/_ x G $.
|
|
$( Equality of functions is determined by their values. Special case of
|
|
Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
|
|
This version of ~ eqfnfv uses bound-variable hypotheses instead of
|
|
distinct variable conditions. (Contributed by NM, 29-Jan-2004.) $)
|
|
eqfnfv2f $p |- ( ( F Fn A /\ G Fn A ) -> ( F = G <->
|
|
A. x e. A ( F ` x ) = ( G ` x ) ) ) $=
|
|
( vz wfn wa wceq cv cfv wral eqfnfv nfcv nffv nfeq nfv fveq2 eqeq12d
|
|
cbvral syl6bb ) CBHDBHICDJGKZCLZUCDLZJZGBMAKZCLZUGDLZJZABMGBCDNUFUJGABAUD
|
|
UEAUCCEAUCOZPAUCDFUKPQUJGRUCUGJUDUHUEUIUCUGCSUCUGDSTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d G x $.
|
|
$( Equality of functions is determined by their values. (Contributed by
|
|
Scott Fenton, 19-Jun-2011.) $)
|
|
eqfunfv $p |- ( ( Fun F /\ Fun G ) -> ( F = G <->
|
|
( dom F = dom G /\
|
|
A. x e. dom F ( F ` x ) = ( G ` x ) ) ) ) $=
|
|
( wfun cdm wfn wceq cv cfv wral wa wb funfn eqfnfv2 syl2anb ) BDBBEZFCCEZ
|
|
FBCGPQGAHZBIRCIGAPJKLCDBMCMAPQBCNO $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x F $. $d x G $.
|
|
$( Equality of restricted functions is determined by their values.
|
|
(Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm
|
|
contributors, 6-Feb-2004.) $)
|
|
fvreseq $p |- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) ->
|
|
( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) ) $=
|
|
( wfn wa wss cres wceq cv cfv wral wb fnssres anim12i anandirs wcel fvres
|
|
eqfnfv eqeq12d ralbiia syl6bb syl ) DBFZEBFZGCBHZGDCIZCFZECIZCFZGZUHUJJZA
|
|
KZDLZUNELZJZACMZNUEUFUGULUEUGGUIUFUGGUKBCDOBCEOPQULUMUNUHLZUNUJLZJZACMURA
|
|
CUHUJTVAUQACUNCRUSUOUTUPUNCDSUNCESUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( The range of a choice function (a function that chooses an element from
|
|
each member of its domain) is included in the union of its domain.
|
|
(Contributed by set.mm contributors, 31-Aug-1999.) $)
|
|
chfnrn $p |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A ) $=
|
|
( vy wfn cv cfv wcel wral crn cuni wel wrex wceq fvelrnb biimpd nfra1 rsp
|
|
wa wi eleq1 biimpcd syl6 reximdai sylan9 eluni2 syl6ibr ssrdv ) CBEZAFZCG
|
|
ZUJHZABIZSZDCJZBKZUNDFZUOHZDALZABMZUQUPHUIURUKUQNZABMZUMUTUIURVBABUQCOPUM
|
|
VAUSABULABQUMUJBHULVAUSTULABRVAULUSUKUQUJUAUBUCUDUEAUQBUFUGUH $.
|
|
$}
|
|
|
|
$( Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1]
|
|
p. 41. (Contributed by set.mm contributors, 14-Oct-1996.) $)
|
|
funfvop $p |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F ) $=
|
|
( wfun cdm wcel wa cfv wceq cop eqid funopfvb mpbii ) BCABDEFABGZMHAMIBEMJA
|
|
MBKL $.
|
|
|
|
$( Two ways to say that ` A ` is in the domain of ` F ` . (Contributed by
|
|
Mario Carneiro, 1-May-2014.) $)
|
|
funfvbrb $p |- ( Fun F -> ( A e. dom F <-> A F ( F ` A ) ) ) $=
|
|
( wfun cdm wcel cfv wbr wa cop funfvop df-br sylibr breldm adantl impbida )
|
|
BCZABDEZAABFZBGZPQHARIBESABJARBKLSQPARBMNO $.
|
|
|
|
$( A member of a preimage is a function value argument. (Contributed by
|
|
set.mm contributors, 4-May-2007.) $)
|
|
fvimacnvi $p |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B ) $=
|
|
( wfun ccnv cima wcel cfv csn wss snssi funimass2 sylan2 fvex snss cdm wceq
|
|
wa cnvimass sseli wfn funfn fnsnfv sylanb sseq1d syl5bb mpbird ) CDZACEBFZG
|
|
ZRZACHZBGZCAIZFZBJZUJUHUNUIJUPAUIKUNBCLMUMULIZBJUKUPULBACNOUKUQUOBUJUHACPZG
|
|
ZUQUOQZUIURACBSTUHCURUAUSUTCUBURACUCUDMUEUFUG $.
|
|
|
|
$( The argument of a function value belongs to the preimage of any class
|
|
containing the function value. (Contributed by Raph Levien,
|
|
20-Nov-2006.) He remarks: "This proof is unsatisfying, because it seems
|
|
to me that ~ funimass2 could probably be strengthened to a
|
|
biconditional." $)
|
|
fvimacnv $p |- ( ( Fun F /\ A e. dom F ) ->
|
|
( ( F ` A ) e. B <-> A e. ( `' F " B ) ) ) $=
|
|
( wfun cdm wcel wa cfv ccnv cima csn cop funfvop opelcnv sylibr elimasn wss
|
|
fvex snss imass2 sylbi sseld syl5com wi fvimacnvi ex adantr impbid ) CDZACE
|
|
FZGZACHZBFZACIZBJZFZUKAUNULKZJZFZUMUPUKULALUNFZUSUKAULLCFUTACMULACNOUNULAPO
|
|
UMURUOAUMUQBQURUOQULBACRSUQBUNTUAUBUCUIUPUMUDUJUIUPUMABCUEUFUGUH $.
|
|
|
|
${
|
|
$d F x $. $d A x $. $d B x $.
|
|
$( A kind of contraposition law that infers an image subclass from a
|
|
subclass of a preimage. (Contributed by Raph Levien, 20-Nov-2006.) He
|
|
remarks: "Likely this could be proved directly, and ~ fvimacnv would be
|
|
the special case of ` A ` being a singleton, but it works this way round
|
|
too." $)
|
|
funimass3 $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) ) $=
|
|
( vx wfun cdm wss wa cima cv ccnv wcel wral funimass4 wb ssel fvimacnv ex
|
|
cfv syl9r imp31 ralbidva bitrd dfss3 syl6bbr ) CEZACFZGZHZCAIBGZDJZCKBIZL
|
|
ZDAMZAULGUIUJUKCSBLZDAMUNDABCNUIUOUMDAUFUHUKALZUOUMOZUHUPUKUGLZUFUQAUGUKP
|
|
UFURUQUKBCQRTUAUBUCDAULUDUE $.
|
|
|
|
$( A subclass of a preimage in terms of function values. (Contributed by
|
|
set.mm contributors, 15-May-2007.) $)
|
|
funimass5 $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( A C_ ( `' F " B ) <-> A. x e. A ( F ` x ) e. B ) ) $=
|
|
( wfun cdm wss wa cima ccnv cv cfv wcel wral funimass3 funimass4 bitr3d )
|
|
DEBDFGHDBICGBDJCIGAKDLCMABNBCDOABCDPQ $.
|
|
|
|
$( Two ways of specifying that a function is constant on a subdomain.
|
|
(Contributed by set.mm contributors, 8-Mar-2007.) $)
|
|
funconstss $p |- ( ( Fun F /\ A C_ dom F ) ->
|
|
( A. x e. A ( F ` x ) = B <-> A C_ ( `' F " { B } ) ) ) $=
|
|
( wfun cdm wss wa cfv wceq wral cima csn ccnv wcel funimass4 elsnc ralbii
|
|
cv fvex syl6rbb funimass3 bitrd ) DEBDFGHZASZDIZCJZABKZDBLCMZGZBDNUILGUDU
|
|
JUFUIOZABKUHABUIDPUKUGABUFCUEDTQRUABUIDUBUC $.
|
|
$}
|
|
|
|
$( Membership in the preimage of a set under a function. (Contributed by
|
|
Jeff Madsen, 2-Sep-2009.) $)
|
|
elpreima $p |- ( F Fn A -> ( B e. ( `' F " C )
|
|
<-> ( B e. A /\ ( F ` B ) e. C ) ) ) $=
|
|
( wfn ccnv cima wcel cfv wa cdm cnvimass sseli fndm eleq2d syl5ib fvimacnvi
|
|
wfun fnfun sylan ex jcad wb fvimacnv funfni biimpd expimpd impbid ) DAEZBDF
|
|
CGZHZBAHZBDICHZJUIUKULUMUKBDKZHUIULUJUNBDCLMUIUNABADNOPUIUKUMUIDRUKUMADSBCD
|
|
QTUAUBUIULUMUKUIULJUMUKUMUKUCABDBCDUDUEUFUGUH $.
|
|
|
|
${
|
|
$d x F $. $d x A $. $d x B $.
|
|
$( Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.) $)
|
|
unpreima $p |- ( Fun F -> ( `' F " ( A u. B ) )
|
|
= ( ( `' F " A ) u. ( `' F " B ) ) ) $=
|
|
( vx wfun cdm wfn ccnv cun cima wceq funfn cv wcel wa wo wb elun elpreima
|
|
cfv anbi2i andi bitri a1i orbi12d syl5bb 3bitr4d eqrdv sylbi ) CECCFZGZCH
|
|
ZABIZJZULAJZULBJZIZKCLUKDUNUQUKDMZUJNZURCTZUMNZOZUSUTANZOZUSUTBNZOZPZURUN
|
|
NURUQNZVBVGQUKVBUSVCVEPZOVGVAVIUSUTABRUAUSVCVEUBUCUDUJURUMCSVHURUONZURUPN
|
|
ZPUKVGURUOUPRUKVJVDVKVFUJURACSUJURBCSUEUFUGUHUI $.
|
|
|
|
$( Preimage of an intersection. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
inpreima $p |- ( Fun F -> ( `' F " ( A i^i B ) )
|
|
= ( ( `' F " A ) i^i ( `' F " B ) ) ) $=
|
|
( vx wfun cdm wfn ccnv cin cima wceq funfn cv wcel wa cfv anbi2i elpreima
|
|
wb elin a1i anandi syl6bb anbi12d 3bitr4d syl6bbr eqrdv sylbi ) CECCFZGZC
|
|
HZABIZJZUKAJZUKBJZIZKCLUJDUMUPUJDMZUMNZUQUNNZUQUONZOZUQUPNUJUQUINZUQCPZUL
|
|
NZOZVBVCANZOZVBVCBNZOZOZURVAUJVEVBVFVHOZOZVJVEVLSUJVDVKVBVCABTQUAVBVFVHUB
|
|
UCUIUQULCRUJUSVGUTVIUIUQACRUIUQBCRUDUEUQUNUOTUFUGUH $.
|
|
|
|
$( The preimage of a restricted function. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
respreima $p |- ( Fun F -> ( `' ( F |` B ) " A )
|
|
= ( ( `' F " A ) i^i B ) ) $=
|
|
( vx wfun cres ccnv cima cin cdm wfn cv wcel wb cfv wa bitri syl elpreima
|
|
elin funfn ancom anbi1i fvres eleq1d adantl pm5.32i a1i an32 syl6bb fnfun
|
|
wceq funres dmres jctir df-fn sylibr anbi1d syl5bb 3bitr4d sylbi eqrdv )
|
|
CEZDCBFZGAHZCGAHZBIZVCCCJZKZDLZVEMZVJVGMZNCUAVIVJBVHIZMZVJVDOZAMZPZVJVHMZ
|
|
VJCOZAMZPZVJBMZPZVKVLVIVQVRWBPZVTPZWCVQWENVIVQWDVPPWEVNWDVPVNWBVRPWDVJBVH
|
|
TWBVRUBQUCWDVPVTWBVPVTNVRWBVOVSAVJBCUDUEUFUGQUHVRWBVTUIUJVIVDVMKZVKVQNVIV
|
|
DEZVDJVMULZPWFVIWGWHVIVCWGVHCUKBCUMRCBUNUOVDVMUPUQVMVJAVDSRVLVJVFMZWBPVIW
|
|
CVJVFBTVIWIWAWBVHVJACSURUSUTVAVB $.
|
|
$}
|
|
|
|
$( The preimage of the codomain of a mapping is the mapping's domain.
|
|
(Contributed by FL, 25-Jan-2007.) $)
|
|
fimacnv $p |- ( F : A --> B -> ( `' F " B ) = A ) $=
|
|
( ccnv cima crn imassrn cdm df-dm fdm wss ssid a1i eqsstrd syl5eqssr syl5ss
|
|
wf frn wfun wb ffun syl5sseqr funimass3 syl2anc mpbid eqssd ) ABCQZCDZBEZAU
|
|
GUIUHFZAUHBGUGUJCHZACIUGUKAAABCJZAAKUGALZMNOPUGCAEZBKZAUIKZUGUNCFBCAGABCRPU
|
|
GCSAUKKUOUPTABCUAUGAAUKUMULUBABCUCUDUEUF $.
|
|
|
|
$( Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1]
|
|
p. 41. (Contributed by set.mm contributors, 30-Sep-2004.) $)
|
|
fnopfv $p |- ( ( F Fn A /\ B e. A ) -> <. B , ( F ` B ) >. e. F ) $=
|
|
( cfv cop wcel funfvop funfni ) BBCDECFABCBCGH $.
|
|
|
|
${
|
|
$d x F $. $d x A $.
|
|
$( A function's value belongs to its range. (Contributed by set.mm
|
|
contributors, 14-Oct-1996.) $)
|
|
fvelrn $p |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F ) $=
|
|
( vx wfun cdm wcel wa cv cfv cop wex crn simpr funfvop wceq eleq1d spcegv
|
|
opeq1 sylc elrn2 sylibr ) BDZABEZFZGZCHZABIZJZBFZCKZUGBLFUEUDAUGJZBFZUJUB
|
|
UDMABNUIULCAUCUFAOUHUKBUFAUGRPQSCUGBTUA $.
|
|
$}
|
|
|
|
$( A function's value belongs to its range. (Contributed by set.mm
|
|
contributors, 15-Oct-1996.) $)
|
|
fnfvelrn $p |- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F ) $=
|
|
( cfv crn wcel fvelrn funfni ) BCDCEFABCBCGH $.
|
|
|
|
$( A function's value belongs to its codomain. (Contributed by set.mm
|
|
contributors, 12-Aug-1999.) $)
|
|
ffvelrn $p |- ( ( F : A --> B /\ C e. A ) -> ( F ` C ) e. B ) $=
|
|
( wf wcel wa cfv crn wfn ffn fnfvelrn sylan wi frn sseld adantr mpd ) ABDEZ
|
|
CAFZGCDHZDIZFZUABFZSDAJTUCABDKACDLMSUCUDNTSUBBUAABDOPQR $.
|
|
|
|
${
|
|
ffvrni.1 $e |- F : A --> B $.
|
|
$( A function's value belongs to its codomain. (Contributed by set.mm
|
|
contributors, 6-Apr-2005.) $)
|
|
ffvelrni $p |- ( C e. A -> ( F ` C ) e. B ) $=
|
|
( wf wcel cfv ffvelrn mpan ) ABDFCAGCDHBGEABCDIJ $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z F $.
|
|
$( A function expressed as the range of another function. (Contributed by
|
|
Mario Carneiro, 22-Jun-2013.) $)
|
|
fnasrn $p |- ( F Fn A -> F = ran { <. x , y >. |
|
|
( x e. A /\ y = <. x , ( F ` x ) >. ) } ) $=
|
|
( vz vw wfn cv cfv cop wceq wrex cab wcel wa copab crn wi wb vex weq fndm
|
|
cdm opeldm eleq2 syl5ib syl eleq1 biimpcd adantrd rexlimiv fveq2 ceqsrexv
|
|
a1i eqeq1d adantl fnopfvb bitr2d ex pm5.21ndd opex eqeq1 eqcom opth bitri
|
|
syl6bb rexbidv elab syl6bbr eqrelrdv rnopab2 syl6eqr ) DCGZDBHZAHZVODIZJZ
|
|
KZACLZBMZVOCNZVROABPQVMEFDVTVMEHZFHZJZDNZAEUAZVPWCKZOZACLZWDVTNVMWBCNZWEW
|
|
IVMDUCZCKZWEWJRCDUBWEWBWKNWLWJWBWCDUDWKCWBUEUFUGWIWJRVMWHWJACWAWFWJWGWFWA
|
|
WJVOWBCUHUIUJUKUNVMWJWEWISVMWJOWIWBDIZWCKZWEWJWIWNSVMWGWNAWBCWFVPWMWCVOWB
|
|
DULUOUMUPCWBWCDUQURUSUTVSWIBWDWBWCETFTVAVNWDKZVRWHACWOVRWDVQKZWHVNWDVQVBW
|
|
PVQWDKWHWDVQVCVOVPWBWCVDVEVFVGVHVIVJABCVQVKVL $.
|
|
$}
|
|
|
|
${
|
|
f0cl.1 $e |- F : A --> B $.
|
|
f0cl.2 $e |- (/) e. B $.
|
|
$( Unconditional closure of a function when the range includes the empty
|
|
set. (Contributed by Mario Carneiro, 12-Sep-2013.) $)
|
|
f0cli $p |- ( F ` C ) e. B $=
|
|
( wcel cfv ffvelrni cdm fdmi eleq2i wn c0 ndmfv syl6eqel sylnbir pm2.61i
|
|
) CAGZCDHZBGZABCDEISCDJZGZUAUBACABDEKLUCMTNBCDOFPQR $.
|
|
$}
|
|
|
|
$( Alternate definition of a mapping. (Contributed by set.mm contributors,
|
|
14-Nov-2007.) $)
|
|
dff2 $p |- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) ) $=
|
|
( wf wfn cxp wss wa ffn fssxp jca crn rnss rnxpss syl6ss anim2i df-f sylibr
|
|
impbii ) ABCDZCAEZCABFZGZHZTUAUCABCIABCJKUDUACLZBGZHTUCUFUAUCUEUBLBCUBMABNO
|
|
PABCQRS $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( Alternate definition of a mapping. (Contributed by set.mm contributors,
|
|
20-Mar-2007.) $)
|
|
dff3 $p |- ( F : A --> B <->
|
|
( F C_ ( A X. B ) /\ A. x e. A E! y x F y ) ) $=
|
|
( wss cv wral wa wcel cdm adantr sylanbrc crn wal wi syl6ss sylibr syl6
|
|
wn wf cxp wbr weu fssxp wex wmo eleq2d biimpar eldm sylib wfun ffun funmo
|
|
fdm syl eu5 ralrimiva jca wfn wceq df-ral dmss dmxpss sseld syl5bir con3d
|
|
pm2.21 df-mo a1dd pm2.27 eumo pm2.61d2 alimdv syl5bi dffun6 ralimi adantl
|
|
imp euex dfss3 eqssd df-fn rnss rnxpss df-f impbii ) CDEUAZECDUBZFZAGZBGE
|
|
UCZBUDZACHZIZWHWJWNCDEUEWHWMACWHWKCJZIZWLBUFZWLBUGZWMWQWKEKZJZWRWHXAWPWHW
|
|
TCWKCDEUOUHUIBWKEUJZUKWQEULZWSWHXCWPCDEUMLBWKEUNUPWLBUQMURUSWOECUTZENZDFZ
|
|
WHWOXCWTCVAXDWOWSAOZXCWJWNXGWNWPWMPZAOWJXGWMACVBWJXHWSAWJWPXHWSPWJWPTZWSX
|
|
HWJXIWRTZWSWJWRWPWRXAWJWPXBWJWTCWKWJWTWIKCEWIVCCDVDQZVEVFVGXJWRWMPWSWRWMV
|
|
HWLBVIRSVJWPXHWMWSWPWMVKWLBVLSVMVNVOVSABEVPRWOWTCWJWTCFWNXKLWNCWTFZWJWNXA
|
|
ACHXLWMXAACWMWRXAWLBVTXBRVQACWTWARVRWBECWCMWJXFWNWJXEWINDEWIWDCDWEQLCDEWF
|
|
MWG $.
|
|
|
|
$( Alternate definition of a mapping. (Contributed by set.mm contributors,
|
|
20-Mar-2007.) $)
|
|
dff4 $p |- ( F : A --> B <->
|
|
( F C_ ( A X. B ) /\ A. x e. A E! y e. B x F y ) ) $=
|
|
( wf cxp wss cv wbr weu wral wa wreu dff3 wcel crn brelrn rnss rnxpss
|
|
syl6ss sseld syl5 pm4.71rd eubidv df-reu syl6bbr ralbidv pm5.32i bitri )
|
|
CDEFECDGZHZAIZBIZEJZBKZACLZMULUOBDNZACLZMABCDEOULUQUSULUPURACULUPUNDPZUOM
|
|
ZBKURULUOVABULUOUTUOUNEQZPULUTUMUNERULVBDUNULVBUKQDEUKSCDTUAUBUCUDUEUOBDU
|
|
FUGUHUIUJ $.
|
|
|
|
$( An onto mapping expressed in terms of function values. (Contributed by
|
|
set.mm contributors, 29-Oct-2006.) $)
|
|
dffo3 $p |- ( F : A -onto-> B <-> ( F : A --> B /\
|
|
A. y e. B E. x e. A y = ( F ` x ) ) ) $=
|
|
( wfo wf crn wceq wa cv cfv wrex wral dffo2 cab wb wcel wal wi wfn fnrnfv
|
|
ffn eqeq1d simpr ffvelrn adantr eqeltrd exp31 rexlimdv biantrurd syl6rbbr
|
|
syl dfbi2 albidv abeq1 df-ral 3bitr4g bitrd pm5.32i bitri ) CDEFCDEGZEHZD
|
|
IZJVBBKZAKZELZIZACMZBDNZJCDEOVBVDVJVBVDVIBPZDIZVJVBECUAZVDVLQCDEUCVMVCVKD
|
|
ABCEUBUDUMVBVIVEDRZQZBSVNVITZBSVLVJVBVOVPBVBVPVIVNTZVPJVOVBVQVPVBVHVNACVB
|
|
VFCRZVHVNVBVRJZVHJVEVGDVSVHUEVSVGDRVHCDVFEUFUGUHUIUJUKVIVNUNULUOVIBDUPVIB
|
|
DUQURUSUTVA $.
|
|
|
|
$( Alternate definition of an onto mapping. (Contributed by set.mm
|
|
contributors, 20-Mar-2007.) $)
|
|
dffo4 $p |- ( F : A -onto-> B <->
|
|
( F : A --> B /\ A. y e. B E. x e. A x F y ) ) $=
|
|
( wfo wf cv wbr wrex wral wa fof wcel wex crn eleq2d wi wceq sylibr ancrd
|
|
elrn syl5bbr biimpar cdm breldm fdm syl5ib eximdv adantr df-rex ralrimiva
|
|
forn syl mpd jca cfv wfn ffn eqcom fnbrfvb biimprd sylan reximdva ralimdv
|
|
syl5bb imdistani dffo3 impbii ) CDEFZCDEGZAHZBHZEIZACJZBDKZLZVJVKVPCDEMZV
|
|
JVOBDVJVMDNZLZVLCNZVNLZAOZVOVTVNAOZWCVJWDVSWDVMEPZNVJVSAVMEUBVJWEDVMCDEUM
|
|
QUCUDVJWDWCRVSVJVNWBAVJVNWAVNVLEUEZNVJWAVLVMEUFVJWFCVLVJVKWFCSVRCDEUGUNQU
|
|
HUAUIUJUOVNACUKTULUPVQVKVMVLEUQZSZACJZBDKZLVJVKVPWJVKVOWIBDVKVNWHACVKECUR
|
|
ZWAVNWHRCDEUSWKWALZWHVNWHWGVMSWLVNVMWGUTCVLVMEVAVFVBVCVDVEVGABCDEVHTVI $.
|
|
|
|
$( Alternate definition of an onto mapping. (Contributed by set.mm
|
|
contributors, 20-Mar-2007.) $)
|
|
dffo5 $p |- ( F : A -onto-> B <->
|
|
( F : A --> B /\ A. y e. B E. x x F y ) ) $=
|
|
( wfo wf cv wbr wrex wral wa wex dffo4 rexex ralimi anim2i wcel wfn wi ex
|
|
ffn fnbr syl ancrd eximdv df-rex syl6ibr ralimdv imdistani impbii bitri )
|
|
CDEFCDEGZAHZBHZEIZACJZBDKZLZUMUPAMZBDKZLZABCDENUSVBURVAUMUQUTBDUPACOPQUMV
|
|
AURUMUTUQBDUMUTUNCRZUPLZAMUQUMUPVDAUMUPVCUMECSZUPVCTCDEUBVEUPVCCUNUOEUCUA
|
|
UDUEUFUPACUGUHUIUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d F x y $. $d A x y $. $d B x y $. $d C x y $.
|
|
$( Property of a surjective function. (Contributed by Jeff Madsen,
|
|
4-Jan-2011.) $)
|
|
foelrn $p |- ( ( F : A -onto-> B /\ C e. B )
|
|
-> E. x e. A C = ( F ` x ) ) $=
|
|
( vy wfo cv wceq wrex wral wcel dffo3 simprbi eqeq1 rexbidv rspccva sylan
|
|
cfv wf ) BCEGZFHZAHESZIZABJZFCKZDCLDUCIZABJZUABCETUFAFBCEMNUEUHFDCUBDIUDU
|
|
GABUBDUCOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d F x y z $. $d G x y z $. $d A y z $. $d B x y z $. $d C x y z $.
|
|
$( If a composition of two functions is surjective, then the function on
|
|
the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) $)
|
|
foco2 $p |- ( ( F : B --> C /\ G : A --> B /\
|
|
( F o. G ) : A -onto-> C ) -> F : B -onto-> C ) $=
|
|
( vy vx vz wf ccom wfo w3a cv cfv wceq wrex wral wa wcel adantll sylanbrc
|
|
simp1 foelrn ffvelrn fvco3 fveq2 eqeq2d rspcev syl2anc rexbidv syl5ibrcom
|
|
eqeq1 rexlimdva syl5 impl ralrimiva 3impa dffo3 ) BCDIZABEIZACDEJZKZLUSFM
|
|
ZGMZDNZOZGBPZFCQZBCDKUSUTVBUBUSUTVBVHUSUTRZVBRVGFCVIVBVCCSZVGVBVJRVCHMZVA
|
|
NZOZHAPVIVGHACVCVAUCVIVMVGHAVIVKASZRZVGVMVLVEOZGBPZVOVKENZBSZVLVRDNZOZVQU
|
|
TVNVSUSABVKEUDTUTVNWAUSABVKDEUETVPWAGVRBVDVROVEVTVLVDVRDUFUGUHUIVMVFVPGBV
|
|
CVLVEULUJUKUMUNUOUPUQGFBCDURUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
$( A function maps to a class to which all values belong. (Contributed by
|
|
NM, 3-Dec-2003.) $)
|
|
ffnfv $p |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) $=
|
|
( vy wf wfn cv cfv wcel wral ffn ffvelrn ralrimiva jca crn wss simpl wceq
|
|
wa wrex fvelrnb biimpd nfra1 nfv wi rsp eleq1 biimpcd syl6 rexlimd sylan9
|
|
ssrdv df-f sylanbrc impbii ) BCDFZDBGZAHZDIZCJZABKZTZUQURVBBCDLUQVAABBCUS
|
|
DMNOVCURDPZCQUQURVBRVCEVDCUREHZVDJZUTVESZABUAZVBVECJZURVFVHABVEDUBUCVBVGV
|
|
IABVAABUDVIAUEVBUSBJVAVGVIUFVAABUGVGVAVIUTVECUHUIUJUKULUMBCDUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d z F $. $d x z $.
|
|
ffnfvf.1 $e |- F/_ x A $.
|
|
ffnfvf.2 $e |- F/_ x B $.
|
|
ffnfvf.3 $e |- F/_ x F $.
|
|
$( A function maps to a class to which all values belong. This version of
|
|
~ ffnfv uses bound-variable hypotheses instead of distinct variable
|
|
conditions. (Contributed by NM, 28-Sep-2006.) $)
|
|
ffnfvf $p |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) $=
|
|
( vz wf wfn cv cfv wcel wral wa ffnfv nfcv nffv nfel nfv wceq fveq2 bitri
|
|
eleq1d cbvralf anbi2i ) BCDIDBJZHKZDLZCMZHBNZOUGAKZDLZCMZABNZOHBCDPUKUOUG
|
|
UJUNHABHBQEAUICAUHDGAUHQRFSUNHTUHULUAUIUMCUHULDUBUDUEUFUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x B $. $d x y F $.
|
|
$( An upper bound for range determined by function values. (Contributed by
|
|
set.mm contributors, 8-Oct-2004.) $)
|
|
fnfvrnss $p |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B ) $=
|
|
( wfn cv cfv wcel wral wa wf crn wss ffnfv frn sylbir ) DBEAFDGCHABIJBCDK
|
|
DLCMABCDNBCDOP $.
|
|
|
|
$( Representation of a mapping in terms of its values. (Contributed by
|
|
set.mm contributors, 21-Feb-2004.) $)
|
|
fopabfv $p |- ( F : A --> B <->
|
|
( F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
|
|
/\ A. x e. A ( F ` x ) e. B ) ) $=
|
|
( wf wfn cv cfv wcel wral wa wceq copab ffnfv dffn5 anbi1i bitri ) CDEFEC
|
|
GZAHZEIZDJACKZLETCJBHUAMLABNMZUBLACDEOSUCUBABCEPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
$( A necessary and sufficient condition for a restricted function.
|
|
(Contributed by Mario Carneiro, 14-Nov-2013.) $)
|
|
ffvresb $p |- ( Fun F -> ( ( F |` A ) : A --> B <->
|
|
A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) ) $=
|
|
( wfun cres wf cv cdm wcel cfv wa wral fdm wss cin dmres inss2 adantl wfn
|
|
eqsstri a1i eqsstr3d sselda wceq fvres ffvelrn eqeltrrd jca ralrimiva crn
|
|
simpl ralimi sylibr fnssres sylanb sylan2 eleq1d syl5ibr ralimia fnfvrnss
|
|
dfss3 funfn simpr syl2anc df-f sylanbrc ex impbid2 ) DEZBCDBFZGZAHZDIZJZV
|
|
MDKZCJZLZABMZVLVRABVLVMBJZLZVOVQVLBVNVMVLBVKIZVNBCVKNWBVNOVLWBBVNPVNDBQBV
|
|
NRUAUBUCUDWAVMVKKZVPCVTWCVPUEVLVMBDUFZSBCVMVKUGUHUIUJVJVSVLVJVSLZVKBTZVKU
|
|
KCOZVLVSVJBVNOZWFVSVOABMWHVRVOABVOVQULUMABVNVBUNVJDVNTWHWFDVCVNBDUOUPUQZW
|
|
EWFWCCJZABMZWGWIVSWKVJVRWJABVRWJVTVQVOVQVDVTWCVPCWDURUSUTSABCVKVAVEBCVKVF
|
|
VGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y F $.
|
|
fsn.1 $e |- A e. _V $.
|
|
fsn.2 $e |- B e. _V $.
|
|
$( A function maps a singleton to a singleton iff it is the singleton of an
|
|
ordered pair. (Contributed by NM, 10-Dec-2003.) $)
|
|
fsn $p |- ( F : { A } --> { B } <-> F = { <. A , B >. } ) $=
|
|
( vx vy csn cop wceq cv wcel wa elsn weu eleq1d bitr4i bitr2i vex wf1o wf
|
|
opelf anbi12i sylib ex wreu snid feu mpan2 anbi1i opeq2 ancom eubii eueq1
|
|
pm5.32i biantru euanv df-reu 3bitr4i sylibr opeq12 syl5ibrcom impbid opex
|
|
elsnc opth syl6bb eqrelrdv f1osn f1oeq1 mpbiri f1of syl impbii ) AHZBHZCU
|
|
AZCABIZHZJZVQFGCVSVQFKZGKZIZCLZWAAJZWBBJZMZWCVSLZVQWDWGVQWDWGVQWDMWAVOLZW
|
|
BVPLZMWGVOVPWAWBCUBWIWEWJWFFANGBNZUCUDUEVQWDWGVRCLZVQAWBIZCLZGVPUFZWLVQAV
|
|
OLWOADUGGVOVPACUHUIWLWFMZGOZWJWNMZGOWLWOWPWRGWRWFWNMZWPWJWFWNWKUJWSWFWLMW
|
|
PWFWNWLWFWMVRCWBBAUKPUOWLWFULQRUMWLWLWFGOZMWQWTWLGBEUNUPWLWFGUQQWNGVPURUS
|
|
UTWGWCVRCWAAWBBVAPVBVCWHWCVRJWGWCVRWAWBFSGSVDVEWAWBABVFRVGVHVTVOVPCTZVQVT
|
|
XAVOVPVSTABDEVIVOVPCVSVJVKVOVPCVLVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B b $. $d F a b $.
|
|
$( A function maps a singleton to a singleton iff it is the singleton of an
|
|
ordered pair. (Contributed by set.mm contributors, 26-Oct-2012.) $)
|
|
fsng $p |- ( ( A e. C /\ B e. D ) ->
|
|
( F : { A } --> { B } <-> F = { <. A , B >. } ) ) $=
|
|
( va vb cv csn wf cop wceq wb sneq feq2d opeq1 sneqd eqeq2d bibi12d vex
|
|
feq3 syl opeq2 fsn vtocl2g ) FHZIZGHZIZEJZEUFUHKZIZLZMAIZUIEJZEAUHKZIZLZM
|
|
UNBIZEJZEABKZIZLZMFGABCDUFALZUJUOUMURVDUGUNUIEUFANOVDULUQEVDUKUPUFAUHPQRS
|
|
UHBLZUOUTURVCVEUIUSLUOUTMUHBNUIUSUNEUAUBVEUQVBEVEUPVAUHBAUCQRSUFUHEFTGTUD
|
|
UE $.
|
|
$}
|
|
|
|
${
|
|
fsn2.1 $e |- A e. _V $.
|
|
$( A function that maps a singleton to a class is the singleton of an
|
|
ordered pair. (Contributed by set.mm contributors, 19-May-2004.) $)
|
|
fsn2 $p |- ( F : { A } --> B <->
|
|
( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) $=
|
|
( csn wf cfv wcel cop wceq snid ffvelrn mpan2 wfn ffn crn dffn3 cima syl
|
|
wa biimpi wb cdm imadmrn fndm imaeq2d syl5eqr fnsnfv eqtr4d mpbid jca wss
|
|
feq3 snssi fss ancoms sylan impbii fvex fsn anbi2i bitri ) AEZBCFZACGZBHZ
|
|
VCVEEZCFZTZVFCAVEIEJZTVDVIVDVFVHVDAVCHZVFADKZVCBACLMVDCVCNZVHVCBCOVMVCCPZ
|
|
CFZVHVMVOVCCQUAVMVNVGJVOVHUBVMVNCVCRZVGVMVNCCUCZRVPCUDVMVQVCCVCCUEUFUGVMV
|
|
KVGVPJVLVCACUHMUIVNVGVCCUMSUJSUKVFVGBULZVHVDVEBUNVHVRVDVCVGBCUOUPUQURVHVJ
|
|
VFAVECDACUSUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
xpsn.1 $e |- A e. _V $.
|
|
xpsn.2 $e |- B e. _V $.
|
|
$( The cross product of two singletons. (Contributed by set.mm
|
|
contributors, 4-Nov-2006.) $)
|
|
xpsn $p |- ( { A } X. { B } ) = { <. A , B >. } $=
|
|
( csn cxp wf cop wceq fconst fsn mpbi ) AEZBEZMNFZGOABHEIMBDJABOCDKL $.
|
|
$}
|
|
|
|
$( If ` A ` is not in ` C ` , then the restriction of a singleton of
|
|
` <. A , B >. ` to ` C ` is null. (Contributed by Scott Fenton,
|
|
15-Apr-2011.) $)
|
|
ressnop0 $p |- ( -. A e. C -> ( { <. A , B >. } |` C ) = (/) ) $=
|
|
( wcel wn cop cvv cxp csn cres c0 wceq opelxp simplbi con3i cin incom eqtri
|
|
df-res disjsn biimpri syl5eq syl ) ACDZEABFZCGHZDZEZUEIZCJZKLUGUDUGUDBGDABC
|
|
GMNOUHUJUFUIPZKUJUIUFPUKUICSUIUFQRUKKLUHUFUETUAUBUC $.
|
|
|
|
${
|
|
fpr.1 $e |- A e. _V $.
|
|
fpr.2 $e |- B e. _V $.
|
|
fpr.3 $e |- C e. _V $.
|
|
fpr.4 $e |- D e. _V $.
|
|
$( A function with a domain of two elements. (Contributed by Jeff Madsen,
|
|
20-Jun-2010.) (The proof was shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
fpr $p |- ( A =/= B
|
|
-> { <. A , C >. , <. B , D >. } : { A , B } --> { C , D } ) $=
|
|
( wne cop cpr wfn crn wa jctir sylibr csn cun rnsnop df-pr wss wfun funpr
|
|
wf cdm wceq dmprop df-fn uneq12i rneqi rnun eqtri 3eqtr4i eqimssi df-f )
|
|
ABIZACJZBDJZKZABKZLZUSMZCDKZUAZNUTVCUSUDUPVAVDUPUSUBZUSUEUTUFZNVAUPVEVFAB
|
|
CDGHUCACBDGHUGOUSUTUHPVBVCUQQZMZURQZMZRZCQZDQZRVBVCVHVLVJVMACESBDFSUIVBVG
|
|
VIRZMVKUSVNUQURTUJVGVIUKULCDTUMUNOUTVCUSUOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x F $.
|
|
$( A function restricted to a singleton. (Contributed by set.mm
|
|
contributors, 9-Oct-2004.) $)
|
|
fnressn $p |- ( ( F Fn A /\ B e. A ) ->
|
|
( F |` { B } ) = { <. B , ( F ` B ) >. } ) $=
|
|
( vx wcel wfn csn cres cfv cop wceq cv wi sneq reseq2d fveq2 opeq12 mpdan
|
|
wa cvv sneqd eqeq12d imbi2d wss vex snss fnssres sylan2b wf fsn2 biantrur
|
|
dffn2 fvres ax-mp opeq2i sneqi eqeq2i bitr3i 3bitri expcom vtoclga impcom
|
|
fvex snid sylib ) BAECAFZCBGZHZBBCIZJZGZKZVFCDLZGZHZVMVMCIZJZGZKZMVFVLMDB
|
|
AVMBKZVSVLVFVTVOVHVRVKVTVNVGCVMBNOVTVQVJVTVPVIKVQVJKVMBCPVMBVPVIQRUAUBUCV
|
|
FVMAEZVSVFWASVOVNFZVSWAVFVNAUDWBVMADUEZUFAVNCUGUHWBVNTVOUIVMVOIZTEZVOVMWD
|
|
JZGZKZSZVSVNVOULVMTVOWCUJWIWHVSWEWHVMVOVCUKWGVRVOWFVQWDVPVMVMVNEWDVPKVMWC
|
|
VDVMVNCUMUNUOUPUQURUSVEUTVAVB $.
|
|
|
|
$( The value of a function restricted to a singleton. (Contributed by
|
|
set.mm contributors, 9-Oct-2004.) $)
|
|
fressnfv $p |- ( ( F Fn A /\ B e. A ) ->
|
|
( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) ) $=
|
|
( vx wcel wfn csn cres wf cfv wb cv wi wceq sneq reseq2 syl wa cop eleq1d
|
|
feq1d feq2 bitrd fveq2 bibi12d imbi2d fnressn vex snid fvres ax-mp opeq2i
|
|
sneqi eqeq2i fsn2 eleq1i iba syl5rbbr syl5bb sylbir expcom vtoclga impcom
|
|
) BAFDAGZBHZCDVFIZJZBDKZCFZLZVEEMZHZCDVMIZJZVLDKZCFZLZNVEVKNEBAVLBOZVRVKV
|
|
EVSVOVHVQVJVSVMVFOZVOVHLVLBPVTVOVMCVGJVHVTVMCVNVGVMVFDQUBVMVFCVGUCUDRVSVP
|
|
VICVLBDUEUAUFUGVEVLAFZVRVEWASVNVLVPTZHZOZVRAVLDUHWDVNVLVLVNKZTZHZOZVRWGWC
|
|
VNWFWBWEVPVLVLVMFWEVPOVLEUIZUJVLVMDUKULZUMUNUOVOWECFZWHSZWHVQVLCVNWIUPVQW
|
|
KWHWLWEVPCWJUQWHWKURUSUTVARVBVCVD $.
|
|
$}
|
|
|
|
$( The value of a constant function. (Contributed by set.mm contributors,
|
|
30-May-1999.) $)
|
|
fvconst $p |- ( ( F : A --> { B } /\ C e. A ) -> ( F ` C ) = B ) $=
|
|
( csn wf wcel wa cfv wceq ffvelrn elsni syl ) ABEZDFCAGHCDIZNGOBJANCDKOBLM
|
|
$.
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
fopabsn.1 $e |- A e. _V $.
|
|
fopabsn.2 $e |- B e. _V $.
|
|
$( The singleton of an ordered pair expressed as an ordered pair class
|
|
abstraction. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|
|
(Contributed by set.mm contributors, 6-Jun-2006.) (Revised by set.mm
|
|
contributors, 22-Oct-2011.) $)
|
|
fopabsn $p |- { <. A , B >. } =
|
|
{ <. x , y >. | ( x e. { A } /\ y = B ) } $=
|
|
( csn cxp cop cv wcel wceq wa copab xpsn fconstopab eqtr3i ) CGZDGHCDIGAJ
|
|
RKBJDLMABNCDEFOABRDPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The value of the identity function. (Contributed by set.mm
|
|
contributors, 1-May-2004.) $)
|
|
fvi $p |- ( A e. V -> ( _I ` A ) = A ) $=
|
|
( vx cv cid cfv wceq fveq2 id eqeq12d cvv wfn wcel wfun cdm funi mpbir2an
|
|
dmi df-fn vex wa cop weq equid wbr ideq df-br bitr3i fnopfvb mpbiri mp2an
|
|
mpbi vtoclg ) CDZEFZUNGZAEFZAGCABUNAGZUOUQUNAUNAEHURIJEKLZUNKMZUPUSENEOKG
|
|
PREKSQCTZUSUTUAUPUNUNUBEMZCCUCZVBCUDVCUNUNEUEVBUNUNVAUFUNUNEUGUHULKUNUNEU
|
|
IUJUKUM $.
|
|
$}
|
|
|
|
$( The value of a restricted identity function. (Contributed by set.mm
|
|
contributors, 19-May-2004.) $)
|
|
fvresi $p |- ( B e. A -> ( ( _I |` A ) ` B ) = B ) $=
|
|
( wcel cid cres cfv fvres fvi eqtrd ) BACBDAEFBDFBBADGBAHI $.
|
|
|
|
$( Remove an ordered pair not participating in a function value.
|
|
(Contributed by set.mm contributors, 1-Oct-2013.) (Revised by Mario
|
|
Carneiro, 28-May-2014.) $)
|
|
fvunsn $p |- ( B =/= D
|
|
-> ( ( A u. { <. B , C >. } ) ` D ) = ( A ` D ) ) $=
|
|
( wne cop csn cun cres cfv c0 wcel wceq syl cvv fvres fvprc eqtr4d pm2.61i
|
|
wn resundir elsni necon3ai ressnop0 uneq2d un0 syl6eq syl5eq fveq1d 3eqtr3g
|
|
snidg ) BDEZDABCFGZHZDGZIZJZDAUOIZJZDUNJZDAJZULDUPURULUPURUMUOIZHZURAUMUOUA
|
|
ULVCURKHURULVBKURULBUOLZTVBKMVDBDBDUBUCBCUOUDNUEURUFUGUHUIDOLZUQUTMZVEDUOLZ
|
|
VFDOUKZDUOUNPNVETZUQKUTDUPQDUNQRSVEUSVAMZVEVGVJVHDUOAPNVIUSKVADURQDAQRSUJ
|
|
$.
|
|
|
|
${
|
|
fvsn.1 $e |- A e. _V $.
|
|
fvsn.2 $e |- B e. _V $.
|
|
$( The value of a singleton of an ordered pair is the second member.
|
|
(Contributed by set.mm contributors, 12-Aug-1994.) $)
|
|
fvsn $p |- ( { <. A , B >. } ` A ) = B $=
|
|
( cop csn wfun wcel cfv wceq funsn opex snid funopfv mp2 ) ABEZFZGPQHAQIB
|
|
JABKPABCDLMABQNO $.
|
|
$}
|
|
|
|
${
|
|
$d A a b $. $d B b $.
|
|
$( The value of a singleton of an ordered pair is the second member.
|
|
(Contributed by set.mm contributors, 26-Oct-2012.) $)
|
|
fvsng $p |- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B ) $=
|
|
( va vb cv cop csn cfv opeq1 sneqd id fveq12d eqeq1d opeq2 fveq1d eqeq12d
|
|
wceq vex fvsn vtocl2g ) EGZUCFGZHZIZJZUDSAAUDHZIZJZUDSAABHZIZJZBSEFABCDUC
|
|
ASZUGUJUDUNUCAUFUIUNUEUHUCAUDKLUNMNOUDBSZUJUMUDBUOAUIULUOUHUKUDBAPLQUOMRU
|
|
CUDETFTUAUB $.
|
|
$}
|
|
|
|
${
|
|
fvsnun.1 $e |- A e. _V $.
|
|
fvsnun.2 $e |- B e. _V $.
|
|
fvsnun.3 $e |- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) $.
|
|
$( The value of a function with one of its ordered pairs replaced, at the
|
|
replaced ordered pair. See also ~ fvsnun2 . (Contributed by set.mm
|
|
contributors, 23-Sep-2007.) $)
|
|
fvsnun1 $p |- ( G ` A ) = B $=
|
|
( csn cres cfv wcel wceq fvres ax-mp cun c0 cin eqtri 3eqtri snid reseq1i
|
|
cop cdif resundir incom disjdif resdisj uneq2i un0 fveq1i fvsn eqtr3i ) A
|
|
EAIZJZKZAEKZBAUNLZUPUQMAFUAZAUNENOUPAABUCIZUNJZKZAUTKZBAUOVAUOUTDCUNUDZJZ
|
|
PZUNJVAVEUNJZPZVAEVFUNHUBUTVEUNUEVHVAQPVAVGQVAVDUNRZQMVGQMVIUNVDRQVDUNUFU
|
|
NCUGSVDUNDUHOUIVAUJSTUKURVBVCMUSAUNUTNOABFGULTUM $.
|
|
|
|
$( The value of a function with one of its ordered pairs replaced, at
|
|
arguments other than the replaced one. See also ~ fvsnun1 .
|
|
(Contributed by set.mm contributors, 23-Sep-2007.) $)
|
|
fvsnun2 $p |- ( D e. ( C \ { A } ) -> ( G ` D ) = ( F ` D ) ) $=
|
|
( csn cdif wcel cres cfv fvres cop cun c0 wceq 3eqtri reseq1i cin disjdif
|
|
resundir wfn wb fnsn fnresdisj ax-mp mpbi residm uneq12i uncom un0 fveq1i
|
|
syl5eq eqtr3d ) DCAJZKZLZDFUSMZNZDFNDENZDUSFOUTVBDEUSMZNVCDVAVDVAABPJZVDQ
|
|
ZUSMVEUSMZVDUSMZQZVDFVFUSIUAVEVDUSUDVIRVDQVDRQVDVGRVHVDURUSUBRSZVGRSZURCU
|
|
CVEURUEVJVKUFABGHUGURUSVEUHUIUJEUSUKULRVDUMVDUNTTUODUSEOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
fvpr1.1 $e |- A e. _V $.
|
|
fvpr1.2 $e |- C e. _V $.
|
|
$( The value of a function with a domain of two elements. (Contributed by
|
|
Jeff Madsen, 20-Jun-2010.) $)
|
|
fvpr1 $p |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = C ) $=
|
|
( wne cop cpr cfv csn df-pr fveq1i wceq necom fvunsn sylbi syl5eq fvsn
|
|
cun syl6eq ) ABGZAACHZBDHZIZJZAUCKZJZCUBUFAUGUDKTZJZUHAUEUIUCUDLMUBBAGUJU
|
|
HNABOUGBDAPQRACEFSUA $.
|
|
$}
|
|
|
|
${
|
|
fvpr2.1 $e |- B e. _V $.
|
|
fvpr2.2 $e |- D e. _V $.
|
|
$( The value of a function with a domain of two elements. (Contributed by
|
|
Jeff Madsen, 20-Jun-2010.) $)
|
|
fvpr2 $p |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` B ) = D ) $=
|
|
( wne cop cpr cfv wceq fvpr1 necom prcom fveq1i eqeq1i 3imtr4i ) BAGBBDHZ
|
|
ACHZIZJZDKABGBSRIZJZDKBADCEFLABMUCUADBUBTSRNOPQ $.
|
|
$}
|
|
|
|
$( The value of a constant function. (Contributed by set.mm contributors,
|
|
20-Aug-2005.) $)
|
|
fvconst2g $p |- ( ( B e. D /\ C e. A ) -> ( ( A X. { B } ) ` C ) = B ) $=
|
|
( wcel csn cxp wf cfv wceq fconstg fvconst sylan ) BDEABFZANGZHCAECOIBJABDK
|
|
ABCOLM $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x F $.
|
|
$( A constant function expressed as a cross product. (Contributed by
|
|
set.mm contributors, 27-Nov-2007.) $)
|
|
fconst2g $p |- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) ) $=
|
|
( vx wcel csn wf cxp wceq wa cv cfv wral fvconst adantlr fvconst2g eqtr4d
|
|
adantll wfn ralrimiva wb ffn fnconstg eqfnfv syl2an mpbird expcom fconstg
|
|
feq1 syl5ibrcom impbid ) BCFZABGZDHZDAUNIZJZUOUMUQUOUMKZUQELZDMZUSUPMZJZE
|
|
ANZURVBEAURUSAFZKUTBVAUOVDUTBJUMABUSDOPUMVDVABJUOABUSCQSRUAUODATUPATUQVCU
|
|
BUMAUNDUCABCUDEADUPUEUFUGUHUMUOUQAUNUPHABCUIAUNDUPUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
fvconst2.1 $e |- B e. _V $.
|
|
$( The value of a constant function. (Contributed by set.mm contributors,
|
|
16-Apr-2005.) $)
|
|
fvconst2 $p |- ( C e. A -> ( ( A X. { B } ) ` C ) = B ) $=
|
|
( cvv wcel csn cxp cfv wceq fvconst2g mpan ) BEFCAFCABGHIBJDABCEKL $.
|
|
|
|
$( A constant function expressed as a cross product. (Contributed by
|
|
set.mm contributors, 20-Aug-1999.) $)
|
|
fconst2 $p |- ( F : A --> { B } <-> F = ( A X. { B } ) ) $=
|
|
( cvv wcel csn wf cxp wceq wb fconst2g ax-mp ) BEFABGZCHCANIJKDABECLM $.
|
|
$}
|
|
|
|
$( Two ways to express that a function is constant. (Contributed by set.mm
|
|
contributors, 27-Nov-2007.) $)
|
|
fconst5 $p |- ( ( F Fn A /\ A =/= (/) ) -> ( F = ( A X. { B } ) <->
|
|
ran F = { B } ) ) $=
|
|
( wfn c0 wne wa csn cxp wceq crn wi rneq rnxp eqeq2d syl5ib adantl cvv wcel
|
|
wf wfo df-fo fof sylbir fconst2g exp3a adantrd wn wb rneq0 a1i snprc biimpi
|
|
xpeq2d xp0 syl6eq 3bitr4d biimprd a1d pm2.61i impbid ) CADZAEFZGZCABHZIZJZC
|
|
KZVEJZVCVGVILVBVGVHVFKZJVCVICVFMVCVJVEVHAVENOPQBRSZVDVIVGLZLVKVBVLVCVKVBVIV
|
|
GVBVIGZAVECTZVKVGVMAVECUAVNAVECUBAVECUCUDABRCUEPUFUGVKUHZVLVDVOVGVIVOCEJZVH
|
|
EJZVGVIVPVQUIVOCUJUKVOVFECVOVFAEIEVOVEEAVOVEEJBULUMZUNAUOUPOVOVEEVHVROUQURU
|
|
SUTVA $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z F $.
|
|
$( A constant function expressed in terms of its functionality, domain, and
|
|
value. See also ~ fconst2 . (Contributed by NM, 27-Aug-2004.) $)
|
|
fconstfv $p |- ( F : A --> { B } <->
|
|
( F Fn A /\ A. x e. A ( F ` x ) = B ) ) $=
|
|
( vy vz csn wf wfn cv cfv wceq wral wa syl6bb wcel wrex eqeq1d sylan9bbr
|
|
c0 ffn fvconst ralrimiva jca wi fneq2 fn0 feq1 mpbiri syl6bi feq2 sylibrd
|
|
adantrd wne crn fvelrnb fveq2 rspccva rexbidva r19.9rzv bicomd elsn eqcom
|
|
f0 bitr2i eqrdv an32s exp31 imdistand wfo df-fo fof sylbir syl6 pm2.61ine
|
|
impbii ) BCGZDHZDBIZAJZDKZCLZABMZNZVRVSWCBVQDUAVRWBABBCVTDUBUCUDWDVRUEBTB
|
|
TLZVSVRWCWEVSTVQDHZVRWEVSDTLZWFWEVSDTIWGBTDUFDUGOWGWFTVQTHVQVDTVQDTUHUIUJ
|
|
BTVQDUKULUMBTUNZWDVSDUOZVQLZNZVRWHVSWCWJWHVSWCWJWHWCVSWJWHWCNZVSNZEWIVQWM
|
|
EJZWIPZCWNLZWNVQPZVSWOFJZDKZWNLZFBQZWLWPFBWNDUPWCXAWPFBQZWHWPWCWTWPFBWCWR
|
|
BPNWSCWNWBWSCLAWRBVTWRLWAWSCVTWRDUQRURRUSWHWPXBWPFBUTVASSWQWNCLWPECVBWNCV
|
|
CVEOVFVGVHVIWKBVQDVJVRBVQDVKBVQDVLVMVNVOVP $.
|
|
|
|
$( Two ways to express a constant function. (Contributed by set.mm
|
|
contributors, 15-Mar-2007.) $)
|
|
fconst3 $p |- ( F : A --> { B } <->
|
|
( F Fn A /\ A C_ ( `' F " { B } ) ) ) $=
|
|
( vx csn wf wfn cv cfv wceq wral wa ccnv cima wss fconstfv wfun cdm fnfun
|
|
wb fndm eqimss2 syl funconstss syl2anc pm5.32i bitri ) ABEZCFCAGZDHCIBJDA
|
|
KZLUIACMUHNOZLDABCPUIUJUKUICQACRZOZUJUKTACSUIULAJUMACUAAULUBUCDABCUDUEUFU
|
|
G $.
|
|
$}
|
|
|
|
$( Two ways to express a constant function. (Contributed by set.mm
|
|
contributors, 8-Mar-2007.) $)
|
|
fconst4 $p |- ( F : A --> { B } <->
|
|
( F Fn A /\ ( `' F " { B } ) = A ) ) $=
|
|
( csn wf wfn ccnv cima wss wa wceq fconst3 cnvimass fndm syl5sseq biantrurd
|
|
cdm eqss syl6bbr pm5.32i bitri ) ABDZCECAFZACGUBHZIZJUCUDAKZJABCLUCUEUFUCUE
|
|
UDAIZUEJUFUCUGUEUCCQUDACUBMACNOPUDARSTUA $.
|
|
|
|
$( A function's value in a preimage belongs to the image. (Contributed by
|
|
set.mm contributors, 23-Sep-2003.) $)
|
|
funfvima $p |- ( ( Fun F /\ B e. dom F ) -> ( B e. A ->
|
|
( F ` B ) e. ( F " A ) ) ) $=
|
|
( wfun cdm wcel wa cfv cima wi cres cin dmres eleq2i elin crn funres fvelrn
|
|
bitri sylan fvres eleq1d syl6rbbr syl5ibrcom ex syl5bir exp3a com12 pm2.43b
|
|
dfima3 imp3a ) CDZBCEZFZGBAFZBCHZCAIZFZUOULUNUOURJZULUOUNUSJULUOUNUSUOUNGZB
|
|
CAKZEZFZULUSVCBAUMLZFUTVBVDBCAMNBAUMOSULVCUSULVCGURUOBVAHZVAPZFZULVADVCVGAC
|
|
QBVARTUOVGUPVFFURUOVEUPVFBACUAUBUQVFUPCAUJNUCUDUEUFUGUHUKUI $.
|
|
|
|
$( A function's value in an included preimage belongs to the image.
|
|
(Contributed by set.mm contributors, 3-Feb-1997.) $)
|
|
funfvima2 $p |- ( ( Fun F /\ A C_ dom F ) -> ( B e. A ->
|
|
( F ` B ) e. ( F " A ) ) ) $=
|
|
( wfun cdm wss wcel cfv cima wi ssel funfvima ex com23 a2d syl5 imp ) CDZAC
|
|
EZFZBAGZBCHCAIGZJZTUABSGZJRUCASBKRUAUDUBRUDUAUBRUDUCABCLMNOPQ $.
|
|
|
|
$( A class including a function contains the function's value in the image of
|
|
the singleton of the argument. (Contributed by set.mm contributors,
|
|
23-Mar-2004.) $)
|
|
funfvima3 $p |- ( ( Fun F /\ F C_ G ) -> ( A e. dom F ->
|
|
( F ` A ) e. ( G " { A } ) ) ) $=
|
|
( wss wfun cdm wcel cfv csn cima wi cop funfvop ssel2 sylan2 elimasn sylibr
|
|
wa exp32 impcom ) BCDZBEZABFGZABHZCAIJGZKUAUBUCUEUAUBUCRZRAUDLZCGZUEUFUAUGB
|
|
GUHABMBCUGNOCAUDPQST $.
|
|
|
|
${
|
|
$d x y F $.
|
|
$( Upper bound for the class of values of a class. (Contributed by NM,
|
|
9-Nov-1995.) $)
|
|
fvclss $p |- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } ) $=
|
|
( cv cfv wceq wex crn c0 csn cun wcel wo wn wne wbr eqcom tz6.12i syl6ibr
|
|
syl5bi eximdv com12 elrn necon1bd vex elsnc orrd elun sylibr abssi ) BDZA
|
|
DZCEZFZAGZBCHZIJZKZUOUKUPLZUKUQLZMUKURLUOUSUTUOUSNUKIFUTUOUSUKIUOUKIOZULU
|
|
KCPZAGZUSVAUOVCVAUNVBAUNUMUKFVAVBUKUMQULUKCRTUAUBAUKCUCSUDUKIBUEUFSUGUKUP
|
|
UQUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d B y z $. $d C w $. $d D y $. $d w x y $. $d w z y $.
|
|
abrexco.1 $e |- B e. _V $.
|
|
abrexco.2 $e |- ( y = B -> C = D ) $.
|
|
$( Composition of two image maps ` C ( y ) ` and ` B ( w ) ` .
|
|
(Contributed by set.mm contributors, 27-May-2013.) $)
|
|
abrexco $p |- { x | E. y e. { z | E. w e. A z = B } x = C } =
|
|
{ x | E. w e. A x = D } $=
|
|
( cv wceq wrex cab wa wex wcel df-rex vex eqeq1 rexbidv elab anbi1i exbii
|
|
r19.41v bitr4i bitri rexcom4 eqeq2d ceqsexv rexbii 3bitr2i abbii ) AKZGLZ
|
|
BCKZFLZDEMZCNZMZUNHLZDEMZAUTBKZFLZUOOZDEMZBPZVEBPZDEMVBUTVCUSQZUOOZBPVGUO
|
|
BUSRVJVFBVJVDDEMZUOOVFVIVKUOURVKCVCBSUPVCLUQVDDEUPVCFTUAUBUCVDUODEUEUFUDU
|
|
GVEDBEUHVHVADEUOVABFIVDGHUNJUIUJUKULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d y z C $.
|
|
$( The image of an indexed union is the indexed union of the images.
|
|
(Contributed by Mario Carneiro, 18-Jun-2014.) $)
|
|
imaiun $p |- ( A " U_ x e. B C ) = U_ x e. B ( A " C ) $=
|
|
( vy vz ciun cima cv wcel cop wex wrex rexcom4 elima3 rexbii eliun anbi1i
|
|
wa r19.41v bitr4i exbii 3bitr4ri 3bitr4i eqriv ) EBACDGZHZACBDHZGZFIZUFJZ
|
|
UJEIZKBJZSZFLZULUHJZACMZULUGJULUIJUJDJZUMSZFLZACMUSACMZFLUQUOUSAFCNUPUTAC
|
|
FULBDOPUNVAFUNURACMZUMSVAUKVBUMAUJCDQRURUMACTUAUBUCFULBUFOAULCUHQUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( The image of a union is the indexed union of the images. Theorem 3K(a)
|
|
of [Enderton] p. 50. (The proof was shortened by Mario Carneiro,
|
|
18-Jun-2014.) (Contributed by set.mm contributors, 9-Aug-2004.)
|
|
(Revised by set.mm contributors, 18-Jun-2014.) $)
|
|
imauni $p |- ( A " U. B ) = U_ x e. B ( A " x ) $=
|
|
( cuni cima cv ciun uniiun imaeq2i imaiun eqtri ) BCDZEBACAFZGZEACBMEGLNB
|
|
ACHIABCMJK $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z F $.
|
|
$( The indexed union of a function's values is the union of its range.
|
|
Compare Definition 5.4 of [Monk1] p. 50. (Contributed by set.mm
|
|
contributors, 27-Sep-2004.) $)
|
|
fniunfv $p |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F ) $=
|
|
( vy wfn crn cuni cv cfv wceq wrex cab ciun fnrnfv unieqd dfiun2 syl6reqr
|
|
fvex ) CBEZCFZGDHAHZCIZJABKDLZGABUBMSTUCADBCNOADBUBUACRPQ $.
|
|
|
|
$( The indexed union of a function's values is the union of its image under
|
|
the index class.
|
|
|
|
Note: This theorem depends on the fact that our function value is the
|
|
empty set outside of its domain. If the antecedent is changed to
|
|
` F Fn A ` , the theorem can be proved without this dependency.
|
|
(Contributed by set.mm contributors, 26-Mar-2006.) $)
|
|
funiunfv $p |- ( Fun F -> U_ x e. A ( F ` x ) = U. ( F " A ) ) $=
|
|
( vy vz vw wfun cv cfv ciun wcel wceq wa cuni fvex wrex wex c0 wn syl5eq
|
|
copab crn cima fveq2 fvopab4 iuneq2i wfn fnopab2 fniunfv ax-mp eqtr3i cab
|
|
rnopab2 unieqi wel wbr eqcom idd wi funbrfv adantr wb cdm n0i ndmfv eqeq1
|
|
eqid syl5ib con1d mpan9 funbrfvb sylan2 pm5.21ndd syl5bb rexbidv pm5.32da
|
|
expr exbidv eluniab eluni elima anbi2i exbii bitri 3bitr4g eqrdv ) CGZABA
|
|
HZCIZJZDHZBKEHZWKCIZLZMDEUAZUBZNZCBUCZNZABWHWOIZJZWJWQABWTWIDEWHWMWIBWOWK
|
|
WHCUDWOVGZWHCOUEUFWOBUGXAWQLDEBWMWOWKCOXBUHABWOUIUJUKWGWQWNDBPZEULZNZWSWP
|
|
XDDEBWMUMUNWGFXEWSWGFEUOZXCMZEQXFWKWLCUPZDBPZMZEQZFHZXEKXLWSKZWGXGXJEWGXF
|
|
XCXIWGXFMZWNXHDBWNWMWLLZXNXHWLWMUQXNXOXOXHXNXOURWGXHXOUSXFWKWLCUTVAWGXFXO
|
|
XOXHVBZXFXOMWGWKCVCKZXPXFWLRLZSXOXQWLXLVDXOXQXRXQSWMRLXOXRWKCVEWMWLRVFVHV
|
|
IVJWKWLCVKVLVQVMVNVOVPVRXCEXLVSXMXFWLWRKZMZEQXKEXLWRVTXTXJEXSXIXFDWLCBWAW
|
|
BWCWDWEWFTT $.
|
|
$}
|
|
|
|
${
|
|
$d x z A $. $d z F $.
|
|
funiunfvf.1 $e |- F/_ x F $.
|
|
$( The indexed union of a function's values is the union of its image under
|
|
the index class. This version of ~ funiunfv uses a bound-variable
|
|
hypothesis in place of a distinct variable condition. (Contributed by
|
|
NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.) $)
|
|
funiunfvf $p |- ( Fun F -> U_ x e. A ( F ` x ) = U. ( F " A ) ) $=
|
|
( vz wfun cv cfv ciun cima cuni nfcv nffv fveq2 cbviun funiunfv syl5eqr )
|
|
CFABAGZCHZIEBEGZCHZICBJKEABUASATCDATLMESLTRCNOEBCPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $.
|
|
$( Membership in the union of an image of a function. (Contributed by
|
|
set.mm contributors, 28-Sep-2006.) $)
|
|
eluniima $p |- ( Fun F ->
|
|
( B e. U. ( F " A ) <-> E. x e. A B e. ( F ` x ) ) ) $=
|
|
( cv cfv wcel wrex ciun wfun cima cuni eliun funiunfv eleq2d syl5rbbr ) C
|
|
AEDFZGABHCABQIZGDJZCDBKLZGACBQMSRTCABDNOP $.
|
|
|
|
$( Membership in the union of the range of a function. (Contributed by
|
|
set.mm contributors, 24-Sep-2006.) $)
|
|
elunirn $p |- ( Fun F ->
|
|
( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) ) $=
|
|
( crn cuni wcel cdm cima wfun cv cfv wrex imadmrn unieqi eluniima syl5bbr
|
|
eleq2i ) BCDZEZFBCCGZHZEZFCIBAJCKFATLUBSBUARCMNQATBCOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d z B $. $d x y z F $.
|
|
$( A one-to-one function in terms of function values. Compare Theorem
|
|
4.8(iv) of [Monk1] p. 43. (Contributed by set.mm contributors,
|
|
29-Oct-1996.) $)
|
|
dff13 $p |- ( F : A -1-1-> B <-> ( F : A --> B /\
|
|
A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
|
|
( vz cv wbr wal wa cfv wceq wi wral wb wcel breldm eleq2d syl5ib eqcom wf
|
|
wf1 wmo weq wfn ffn fndm anim12d pm4.71rd fnbrfvb syl5bb bi2anan9 anandis
|
|
cdm pm5.32da bitr4d imbi1d impexp syl6bb albidv 19.21v 19.23v fvex eqvinc
|
|
dff12 wex imbi1i bitr4i imbi2i bitri 2albidv breq1 mo4 albii alcom 3bitri
|
|
r2al 3bitr4g syl pm5.32i ) CDEUBCDEUAZAGZFGZEHZAUCZFIZJWAWBEKZBGZEKZLZABU
|
|
DZMZBCNACNZJAFCDEVEWAWFWMWAECUEZWFWMOCDEUFWNWDWHWCEHZJZWKMZFIZBIZAIZWBCPZ
|
|
WHCPZJZWLMZBIAIWFWMWNWRXDABWNWRXCWCWGLZWCWILZJZWKMZMZFIZXDWNWQXIFWNWQXCXG
|
|
JZWKMXIWNWPXKWKWNWPXCWPJXKWNWPXCWNWDXAWOXBWDWBEUNZPWNXAWBWCEQWNXLCWBCEUGZ
|
|
RSWOWHXLPWNXBWHWCEQWNXLCWHXMRSUHUIWNXCXGWPWNXAXBXGWPOWNXAJZXEWDWNXBJZXFWO
|
|
XEWGWCLXNWDWCWGTCWBWCEUJUKXFWIWCLXOWOWCWITCWHWCEUJUKULUMUOUPUQXCXGWKURUSU
|
|
TXJXCXHFIZMXDXCXHFVAXPWLXCXPXGFVFZWKMWLXGWKFVBWJXQWKFWGWIWBEVCVDVGVHVIVJU
|
|
SVKWFWQBIZAIZFIXRFIZAIWTWEXSFWDWOABWBWHWCEVLVMVNXRFAVOXTWSAWQFBVOVNVPWLAB
|
|
CCVQVRVSVTVJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y w v A $. $d w v B $. $d w v F $.
|
|
dff13f.1 $e |- F/_ x F $.
|
|
dff13f.2 $e |- F/_ y F $.
|
|
$( A one-to-one function in terms of function values. Compare Theorem
|
|
4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.) $)
|
|
dff13f $p |- ( F : A -1-1-> B <-> ( F : A --> B /\
|
|
A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
|
|
( vw vv cv cfv wceq wi wral wa nfcv nffv nfeq nfv nfim wf1 wf dff13 fveq2
|
|
eqeq2d eqeq2 imbi12d cbvral ralbii nfral eqeq1d eqeq1 ralbidv anbi2i
|
|
bitri ) CDEUACDEUBZHJZEKZIJZEKZLZUQUSLZMZICNZHCNZOUPAJZEKZBJZEKZLZVFVHLZM
|
|
ZBCNZACNZOHICDEUCVEVNUPVEURVILZUQVHLZMZBCNZHCNVNVDVRHCVCVQIBCVAVBBBURUTBU
|
|
QEGBUQPQBUSEGBUSPQRVBBSTVQISUSVHLZVAVOVBVPVSUTVIURUSVHEUDUEUSVHUQUFUGUHUI
|
|
VRVMHACVQABCACPVOVPAAURVIAUQEFAUQPQAVHEFAVHPQRVPASTUJVMHSUQVFLZVQVLBCVTVO
|
|
VJVPVKVTURVGVIUQVFEUDUKUQVFVHULUGUMUHUOUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y F $.
|
|
$( Equality of function values for a one-to-one function. (Contributed by
|
|
set.mm contributors, 11-Feb-1997.) $)
|
|
f1fveq $p |- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) ->
|
|
( ( F ` C ) = ( F ` D ) <-> C = D ) ) $=
|
|
( vx vy wf1 wcel wa cfv wceq wi cv fveq2 eqeq1d eqeq1 imbi12d imbi2d wral
|
|
eqeq2d eqeq2 wf dff13 simprbi rsp2 syl com12 vtocl2ga impcom impbid1 ) AB
|
|
EHZCAIDAIJZJCEKZDEKZLZCDLZUMULUPUQMZULFNZEKZGNZEKZLZUSVALZMZMULUNVBLZCVAL
|
|
ZMZMULURMFGCDAAUSCLZVEVHULVIVCVFVDVGVIUTUNVBUSCEOPUSCVAQRSVADLZVHURULVJVF
|
|
UPVGUQVJVBUOUNVADEOUAVADCUBRSULUSAIVAAIJZVEULVEGATFATZVKVEMULABEUCVLFGABE
|
|
UDUEVEFGAAUFUGUHUIUJCDEOUK $.
|
|
$}
|
|
|
|
${
|
|
$d F z $. $d A z $. $d Y z $. $d X z $. $d B z $.
|
|
$( Membership in the image of a 1-1 map. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
f1elima $p |- ( ( F : A -1-1-> B /\ X e. A /\ Y C_ A )
|
|
-> ( ( F ` X ) e. ( F " Y ) <-> X e. Y ) ) $=
|
|
( vz wf1 wcel wss w3a cfv cima cv wceq wrex wb wfn wi wa anassrs fvelimab
|
|
f1fn sylan 3adant2 ssel impac f1fveq ancom2s biimpd biimpcd sylan9 anasss
|
|
eleq1 sylan2 rexlimdva 3impa eqid fveq2 eqeq1d rspcev mpan2 impbid1 bitrd
|
|
) ABCGZDAHZEAIZJZDCKZCELHZFMZCKZVHNZFEOZDEHZVDVFVIVMPZVEVDCAQVFVOABCUBFAE
|
|
VHCUAUCUDVGVMVNVDVEVFVMVNRVDVESZVFSVLVNFEVPVFVJEHZVLVNRZVFVQSVPVJAHZVQSVR
|
|
VFVQVSEAVJUEUFVPVSVQVRVPVSSVLVJDNZVQVNVDVEVSVLVTRVDVEVSSSVLVTVDVSVEVLVTPA
|
|
BVJDCUGUHUITVTVQVNVJDEUMUJUKULUNTUOUPVNVHVHNZVMVHUQVLWAFDEVTVKVHVHVJDCURU
|
|
SUTVAVBVC $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( A one-to-one onto function in terms of function values. (Contributed by
|
|
set.mm contributors, 29-Mar-2008.) $)
|
|
dff1o6 $p |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ ran F = B /\
|
|
A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) $=
|
|
( wf1o wf1 wfo wa wf cv cfv wceq wi wral wfn crn w3a df-f1o dff13 anbi12i
|
|
df-fo df-3an wss eqimss anim2i df-f sylibr pm4.71ri anbi1i 3bitrri 3bitri
|
|
an32 ) CDEFCDEGZCDEHZICDEJZAKZELBKZELMUQURMNBCOACOZIZECPZEQZDMZIZIZVAVCUS
|
|
RZCDESUNUTUOVDABCDETCDEUBUAVFVDUSIUPVDIZUSIVEVAVCUSUCVDVGUSVDUPVDVAVBDUDZ
|
|
IUPVCVHVAVBDUEUFCDEUGUHUIUJUPVDUSUMUKUL $.
|
|
$}
|
|
|
|
$( The converse value of the value of a one-to-one onto function.
|
|
(Contributed by set.mm contributors, 20-May-2004.) $)
|
|
f1ocnvfv1 $p |- ( ( F : A -1-1-onto-> B /\ C e. A ) ->
|
|
( `' F ` ( F ` C ) ) = C ) $=
|
|
( wf1o wcel wa ccnv ccom cfv cid cres wceq f1ococnv1 fveq1d adantr wf fvco3
|
|
f1of sylan fvresi adantl 3eqtr3d ) ABDEZCAFZGCDHZDIZJZCKALZJZCDJUFJZCUDUHUJ
|
|
MUEUDCUGUIABDNOPUDABDQUEUHUKMABDSABCUFDRTUEUJCMUDACUAUBUC $.
|
|
|
|
$( The value of the converse value of a one-to-one onto function.
|
|
(Contributed by set.mm contributors, 20-May-2004.) $)
|
|
f1ocnvfv2 $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
|
|
( F ` ( `' F ` C ) ) = C ) $=
|
|
( wf1o wcel wa ccnv cfv cnvcnv fveq1i wceq f1ocnv f1ocnvfv1 sylan syl5eqr )
|
|
ABDEZCBFZGCDHZIZDITSHZIZCTUADDJKQBASERUBCLABDMBACSNOP $.
|
|
|
|
$( Relationship between the value of a one-to-one onto function and the value
|
|
of its converse. (Contributed by Raph Levien, 10-Apr-2004.) $)
|
|
f1ocnvfv $p |- ( ( F : A -1-1-onto-> B /\ C e. A ) ->
|
|
( ( F ` C ) = D -> ( `' F ` D ) = C ) ) $=
|
|
( cfv wceq ccnv wf1o wcel wa fveq2 eqcoms f1ocnvfv1 eqeq2d syl5ib ) CEFZDGD
|
|
EHZFZQRFZGZABEICAJKZSCGUADQDQRLMUBTCSABCENOP $.
|
|
|
|
$( Relationship between the value of a one-to-one onto function and the value
|
|
of its converse. (Contributed by set.mm contributors, 20-May-2004.)
|
|
(Revised by set.mm contributors, 9-Aug-2006.) $)
|
|
f1ocnvfvb $p |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) ->
|
|
( ( F ` C ) = D <-> ( `' F ` D ) = C ) ) $=
|
|
( wf1o wcel w3a cfv wceq ccnv wi f1ocnvfv 3adant3 wa fveq2 eqcoms f1ocnvfv2
|
|
eqeq2d syl5ib 3adant2 impbid ) ABEFZCAGZDBGZHCEIZDJZDEKIZCJZUCUDUGUILUEABCD
|
|
EMNUCUEUIUGLUDUIUFUHEIZJZUCUEOZUGUKCUHCUHEPQULUJDUFABDERSTUAUB $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x C $. $d x F $.
|
|
$( There is one domain element for each value of a one-to-one onto
|
|
function. (Contributed by set.mm contributors, 26-May-2006.) $)
|
|
f1ofveu $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
|
|
E! x e. A ( F ` x ) = C ) $=
|
|
( wf1o wcel wa cv cfv wceq wreu cop ccnv wf f1ocnv f1of syl feu sylan wfn
|
|
wb f1ofn fnopfvb opelcnv syl6bbr reubidva adantr mpbird ) BCEFZDCGZHAIZEJ
|
|
DKZABLZDULMENZGZABLZUJCBUOOZUKUQUJCBUOFURBCEPCBUOQRACBDUOSTUJUNUQUBZUKUJE
|
|
BUAZUSBCEUCUTUMUPABUTULBGHUMULDMEGUPBULDEUDDULEUEUFUGRUHUI $.
|
|
$}
|
|
|
|
$( The value of the converse of a one-to-one onto function belongs to its
|
|
domain. (Contributed by set.mm contributors, 26-May-2006.) $)
|
|
f1ocnvdm $p |- ( ( F : A -1-1-onto-> B /\ C e. B ) ->
|
|
( `' F ` C ) e. A ) $=
|
|
( wf1o ccnv wf wcel cfv f1ocnv f1of syl ffvelrn sylan ) ABDEZBADFZGZCBHCPIA
|
|
HOBAPEQABDJBAPKLBACPMN $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y H $. $d x y G $.
|
|
$d x y R $. $d x y S $. $d x y T $.
|
|
$( Equality theorem for isomorphisms. (Contributed by set.mm contributors,
|
|
17-May-2004.) $)
|
|
isoeq1 $p |- ( H = G ->
|
|
( H Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq1 fveq1 df-iso 2ralbidv
|
|
breq12d bibi2d anbi12d 3bitr4g ) FEIZABFJZGKZHKZCLZUHFMZUIFMZDLZNZHAOGAOZ
|
|
PABEJZUJUHEMZUIEMZDLZNZHAOGAOZPABCDFQABCDEQUFUGUPUOVAABFERUFUNUTGHAAUFUMU
|
|
SUJUFUKUQULURDUHFESUIFESUBUCUAUDGHABCDFTGHABCDETUE $.
|
|
|
|
$( Equality theorem for isomorphisms. (Contributed by set.mm contributors,
|
|
17-May-2004.) $)
|
|
isoeq2 $p |- ( R = T ->
|
|
( H Isom R , S ( A , B ) <-> H Isom T , S ( A , B ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso breq bibi1d df-iso 2ralbidv
|
|
anbi2d 3bitr4g ) CEIZABFJZGKZHKZCLZUFFMUGFMDLZNZHAOGAOZPUEUFUGELZUINZHAOG
|
|
AOZPABCDFQABEDFQUDUKUNUEUDUJUMGHAAUDUHULUIUFUGCERSUAUBGHABCDFTGHABEDFTUC
|
|
$.
|
|
|
|
$( Equality theorem for isomorphisms. (Contributed by set.mm contributors,
|
|
17-May-2004.) $)
|
|
isoeq3 $p |- ( S = T ->
|
|
( H Isom R , S ( A , B ) <-> H Isom R , T ( A , B ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso breq bibi2d df-iso 2ralbidv
|
|
anbi2d 3bitr4g ) DEIZABFJZGKZHKZCLZUFFMZUGFMZDLZNZHAOGAOZPUEUHUIUJELZNZHA
|
|
OGAOZPABCDFQABCEFQUDUMUPUEUDULUOGHAAUDUKUNUHUIUJDERSUAUBGHABCDFTGHABCEFTU
|
|
C $.
|
|
|
|
$( Equality theorem for isomorphisms. (Contributed by set.mm contributors,
|
|
17-May-2004.) $)
|
|
isoeq4 $p |- ( A = C ->
|
|
( H Isom R , S ( A , B ) <-> H Isom R , S ( C , B ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq2 raleq df-iso anbi12d
|
|
raleqbi1dv 3bitr4g ) ACIZABFJZGKZHKZDLUFFMUGFMELNZHAOZGAOZPCBFJZUHHCOZGCO
|
|
ZPABDEFQCBDEFQUDUEUKUJUMACBFRUIULGACUHHACSUBUAGHABDEFTGHCBDEFTUC $.
|
|
|
|
$( Equality theorem for isomorphisms. (Contributed by set.mm contributors,
|
|
17-May-2004.) $)
|
|
isoeq5 $p |- ( B = C ->
|
|
( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) ) $=
|
|
( vx vy wceq wf1o cv wbr cfv wb wral wa wiso f1oeq3 anbi1d df-iso 3bitr4g
|
|
) BCIZABFJZGKZHKZDLUDFMUEFMELNHAOGAOZPACFJZUFPABDEFQACDEFQUBUCUGUFBCAFRSG
|
|
HABDEFTGHACDEFTUA $.
|
|
$}
|
|
|
|
${
|
|
$d y z H $. $d y z R $. $d y z S $. $d y z A $. $d y z B $.
|
|
$d x y z $.
|
|
nfiso.1 $e |- F/_ x H $.
|
|
nfiso.2 $e |- F/_ x R $.
|
|
nfiso.3 $e |- F/_ x S $.
|
|
nfiso.4 $e |- F/_ x A $.
|
|
nfiso.5 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for an isomorphism. (Contributed by
|
|
NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
nfiso $p |- F/ x H Isom R , S ( A , B ) $=
|
|
( vy vz cv wbr cfv wral nfcv nfbr nffv wiso wf1o wb wa df-iso nff1o nfral
|
|
nfbi nfan nfxfr ) BCDEFUABCFUBZLNZMNZDOZULFPZUMFPZEOZUCZMBQZLBQZUDALMBCDE
|
|
FUEUKUTAABCFGJKUFUSALBJURAMBJUNUQAAULUMDAULRZHAUMRZSAUOUPEAULFGVATIAUMFGV
|
|
BTSUHUGUGUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$( An isomorphism is a one-to-one onto function. (Contributed by set.mm
|
|
contributors, 27-Apr-2004.) $)
|
|
isof1o $p |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B ) $=
|
|
( vx vy wiso wf1o cv wbr cfv wb wral df-iso simplbi ) ABCDEHABEIFJZGJZCKQ
|
|
ELRELDKMGANFANFGABCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$d x y C $. $d x y D $.
|
|
$( An isomorphism connects binary relations via its function values.
|
|
(Contributed by set.mm contributors, 27-Apr-2004.) $)
|
|
isorel $p |- ( ( H Isom R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) ->
|
|
( C R D <-> ( H ` C ) S ( H ` D ) ) ) $=
|
|
( vx vy wiso cv wbr cfv wb wral wcel wa wceq fveq2 bibi12d df-iso simprbi
|
|
wf1o breq1 breq1d breq2 breq2d rspc2v mpan9 ) ABEFGJZHKZIKZELZUKGMZULGMZF
|
|
LZNZIAOHAOZCAPDAPQCDELZCGMZDGMZFLZNZUJABGUCURHIABEFGUAUBUQVCCULELZUTUOFLZ
|
|
NHICDAAUKCRZUMVDUPVEUKCULEUDVFUNUTUOFUKCGSUETULDRZVDUSVEVBULDCEUFVGUOVAUT
|
|
FULDGSUGTUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring]
|
|
p. 33. (Contributed by set.mm contributors, 27-Apr-2004.) $)
|
|
isoid $p |- ( _I |` A ) Isom R , R ( A , A ) $=
|
|
( vx vy cid cres wiso wf1o wbr cfv wral f1oi wcel fvresi breqan12d bicomd
|
|
cv wb wa rgen2a df-iso mpbir2an ) AABBEAFZGAAUCHCQZDQZBIZUDUCJZUEUCJZBIZR
|
|
ZDAKCAKALUJCDAUDAMZUEAMZSUIUFUKULUGUDUHUEBAUDNAUENOPTCDAABBUCUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w R $. $d x y z w S $.
|
|
$d x y z w H $.
|
|
$( Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring]
|
|
p. 33. (Contributed by set.mm contributors, 27-Apr-2004.) $)
|
|
isocnv $p |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) ) $=
|
|
( vx vy vz vw wf1o cv wbr cfv wb wral wa wiso wcel wceq f1ocnvfv2 adantrr
|
|
ccnv f1ocnv adantr adantrl breq12d adantlr wf syl ffvelrn anim12dan breq1
|
|
f1of fveq2 breq1d bibi12d bicom syl6bb breq2d breq2 rspc2va sylan sylanl1
|
|
an32s bitr3d ralrimivva jca df-iso 3imtr4i ) ABEJZFKZGKZCLZVKEMZVLEMZDLZN
|
|
ZGAOFAOZPZBAEUBZJZHKZIKZDLZWBVTMZWCVTMZCLZNZIBOHBOZPABCDEQBADCVTQVSWAWIVJ
|
|
WAVRABEUCZUDVSWHHIBBVSWBBRZWCBRZPZPWEEMZWFEMZDLZWDWGVJWMWPWDNVRVJWMPWNWBW
|
|
OWCDVJWKWNWBSWLABWBETUAVJWLWOWCSWKABWCETUEUFUGVJBAVTUHZVRWMWPWGNZVJWAWQWJ
|
|
BAVTUMUIWQWMVRWRWQWMPWEARZWFARZPVRWRWQWKWSWLWTBAWBVTUJBAWCVTUJUKVQWRWNVOD
|
|
LZWEVLCLZNZFGWEWFAAVKWESZVQXBXANXCXDVMXBVPXAVKWEVLCULXDVNWNVODVKWEEUNUOUP
|
|
XBXAUQURVLWFSZXAWPXBWGXEVOWOWNDVLWFEUNUSVLWFWECUTUPVAVBVDVCVEVFVGFGABCDEV
|
|
HHIBADCVTVHVI $.
|
|
$( Converse law for isomorphism. (Contributed by Mario Carneiro,
|
|
30-Jan-2014.) $)
|
|
isocnv2 $p |- ( H Isom R , S ( A , B ) <->
|
|
H Isom `' R , `' S ( A , B ) ) $=
|
|
( vx vy wf1o cv ccnv wbr cfv wb wral wa wiso brcnv bibi12i 2ralbii df-iso
|
|
ralcom bitri anbi2i 3bitr4ri ) ABEHZFIZGIZCJZKZUFELZUGELZDJZKZMZGANFANZOU
|
|
EUGUFCKZUKUJDKZMZFANGANZOABUHULEPABCDEPUOUSUEUOURGANFANUSUNURFGAAUIUPUMUQ
|
|
UFUGCQUJUKDQRSURFGAAUAUBUCFGABUHULETGFABCDETUD $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d H x y $. $d R x y $. $d S x y $.
|
|
$( An isomorphism from one well-order to another can be restricted on
|
|
either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) $)
|
|
isores2 $p |- ( H Isom R , S ( A , B ) <->
|
|
H Isom R , ( S i^i ( B X. B ) ) ( A , B ) ) $=
|
|
( vx vy wf1o cv wbr cfv wb wral wa cxp wiso wcel ffvelrn ralbidva df-iso
|
|
cin wf f1of adantrr adantrl brinxp syl2anc anassrs bibi2d pm5.32i 3bitr4i
|
|
sylan ) ABEHZFIZGIZCJZUNEKZUOEKZDJZLZGAMZFAMZNUMUPUQURDBBOUAZJZLZGAMZFAMZ
|
|
NABCDEPABCVCEPUMVBVGUMVAVFFAUMUNAQZNZUTVEGAVIUOAQZNUSVDUPUMVHVJUSVDLZUMAB
|
|
EUBZVHVJNZVKABEUCVLVMNUQBQZURBQZVKVLVHVNVJABUNERUDVLVJVOVHABUOERUEUQURBBD
|
|
UFUGULUHUISSUJFGABCDETFGABCVCETUK $.
|
|
$}
|
|
|
|
$( An isomorphism from one well-order to another can be restricted on either
|
|
well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) $)
|
|
isores1 $p |- ( H Isom R , S ( A , B ) <->
|
|
H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) $=
|
|
( wiso cxp cin ccnv isocnv isores2 sylib wceq wb cnvcnv isoeq1 ax-mp sylibr
|
|
syl impbii ) ABCDEFZABCAAGHZDEFZUAABUBDEIZIZFZUCUABADUBUDFZUFUABADCUDFZUGAB
|
|
CDEJBADCUDKZLBADUBUDJSUEEMZUFUCNEOZABUBDEUEPQLUCABCDUEFZUAUCUHULUCUGUHABUBD
|
|
EJUIRBADCUDJSUJULUANUKABCDEUEPQLT $.
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y z w v u B $. $d x y z w v u C $.
|
|
$d x y z w v u R $. $d x y z w v u S $. $d x y z w v u T $.
|
|
$d x y z w v u G $. $d x y z w v u H $.
|
|
$( Composition (transitive) law for isomorphism. Proposition 6.30(3) of
|
|
[TakeutiZaring] p. 33. (Contributed by set.mm contributors,
|
|
27-Apr-2004.) $)
|
|
isotr $p |- ( ( H Isom R , S ( A , B ) /\ G Isom S , T ( B , C ) ) ->
|
|
( G o. H ) Isom R , T ( A , C ) ) $=
|
|
( vz vw vu vv vx vy cv wbr cfv wb wral wa wf1o ccom f1oco ad2ant2r ancoms
|
|
wiso wcel wi wf f1of ffvelrn anim12d syl adantr wceq breq1 breq1d bibi12d
|
|
ex fveq2 breq2 breq2d rspc2v adantl sylan9 imp weq impcom adantll adantlr
|
|
com12 fvco3 breqan12d anandis sylan 3bitr4d ralrimivva jca df-iso anbi12i
|
|
ad2antrr 3imtr4i ) ABHUAZIOZJOZDPZWDHQZWEHQZEPZRZJASIASZTZBCGUAZKOZLOZEPZ
|
|
WNGQZWOGQZFPZRZLBSKBSZTZTZACGHUBZUAZMOZNOZDPZXFXDQZXGXDQZFPZRZNASMASZTABD
|
|
EHUFZBCEFGUFZTACDFXDUFXCXEXMXBWLXEWMWCXEXAWKABCGHUCUDUEXCXLMNAAXCXFAUGZXG
|
|
AUGZTZTXFHQZXGHQZEPZXSGQZXTGQZFPZXHXKXCXRYAYDRZWLXRXSBUGZXTBUGZTZXBYEWCXR
|
|
YHUHZWKWCABHUIZYIABHUJZYJXPYFXQYGYJXPYFABXFHUKUSYJXQYGABXGHUKUSULUMUNXAYH
|
|
YEUHWMYHXAYEWTYEXSWOEPZYBWRFPZRKLXSXTBBWNXSUOZWPYLWSYMWNXSWOEUPYNWQYBWRFW
|
|
NXSGUTUQURWOXTUOZYLYAYMYDWOXTXSEVAYOWRYCYBFWOXTGUTVBURVCVKVDVEVFWLXRXHYAR
|
|
ZXBWKXRYPWCXRWKYPWJYPXFWEDPZXSWHEPZRIJXFXGAAIMVGZWFYQWIYRWDXFWEDUPYSWGXSW
|
|
HEWDXFHUTUQURJNVGZYQXHYRYAWEXGXFDVAYTWHXTXSEWEXGHUTVBURVCVHVIVJXCYJXRXKYD
|
|
RZWCYJWKXBYKWAYJXPXQUUAYJXPTYJXQTXIYBXJYCFABXFGHVLABXGGHVLVMVNVOVPVQVRXNW
|
|
LXOXBIJABDEHVSKLBCEFGVSVTMNACDFXDVSWB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$d x y C $. $d x y D $.
|
|
$( Isomorphisms preserve minimal elements. Note that ` ( ``' R " { D } ) `
|
|
is Takeuti and Zaring's idiom for the initial segment
|
|
` { x | x R D } ` . Proposition 6.31(1) of [TakeutiZaring] p. 33.
|
|
(Contributed by set.mm contributors, 19-Apr-2004.) $)
|
|
isomin $p |- ( ( H Isom R , S ( A , B ) /\ ( C C_ A /\ D e. A ) ) ->
|
|
( ( C i^i ( `' R " { D } ) ) = (/) <->
|
|
( ( H " C ) i^i ( `' S " { ( H ` D ) } ) ) = (/) ) ) $=
|
|
( vy vx wcel wa ccnv csn cima cin wceq wbr wrex wex 3bitr4i wss c0 cfv cv
|
|
wiso wn wb ssel2 anim1i an32s isorel fvex breq1 ceqsexv eqcom wf1o isof1o
|
|
wfn f1ofn simpl fnbrfvb syl2an syl5bb anbi1d exbidv syl5bbr bitrd anassrs
|
|
syl sylan2 rexbidva elin eliniseg anbi2i bitri exbii df-rex elima anbi12i
|
|
neq0 r19.41v rexcom4 3bitr4g con4bid ) ABEFGUEZCAUAZDAJZKZKZCELDMNZOZUBPZ
|
|
GCNZFLDGUCZMNZOZUBPZWIHUDZDEQZHCRZWRIUDZGQZXAWNFQZKZISZHCRZWLUFZWQUFZWIWS
|
|
XEHCWEWHWRCJZWSXEUGZWHXIKWEWRAJZWGKZXJWFXIWGXLWFXIKXKWGCAWRUHUIUJWEXLKZWS
|
|
WRGUCZWNFQZXEABWRDEFGUKXOXAXNPZXCKZISXMXEXCXOIXNWRGULXAXNWNFUMUNXMXQXDIXM
|
|
XPXBXCXPXNXAPZXMXBXAXNUOWEGAURZXKXRXBUGXLWEABGUPXSABEFGUQABGUSVIXKWGUTAWR
|
|
XAGVAVBVCVDVEVFVGVJVHVKWRWKJZHSXIWSKZHSXGWTXTYAHXTXIWRWJJZKYAWRCWJVLYBWSX
|
|
IEDWRVMVNVOVPHWKVTWSHCVQTXAWPJZISXDHCRZISXHXFYCYDIXAWMJZXAWOJZKXBHCRZXCKY
|
|
CYDYEYGYFXCHXAGCVRFWNXAVMVSXAWMWOVLXBXCHCWATVPIWPVTXDHICWBTWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $. $d x y S $. $d x y H $.
|
|
$d x y D $.
|
|
$( Isomorphisms preserve initial segments. Proposition 6.31(2) of
|
|
[TakeutiZaring] p. 33. (Contributed by set.mm contributors,
|
|
20-Apr-2004.) $)
|
|
isoini $p |- ( ( H Isom R , S ( A , B ) /\ D e. A ) ->
|
|
( H " ( A i^i ( `' R " { D } ) ) ) =
|
|
( B i^i ( `' S " { ( H ` D ) } ) ) ) $=
|
|
( vx vy wcel wa ccnv cfv csn cima cin wbr wrex bitri wb syl wiso cab elin
|
|
cv eliniseg anbi2i wceq crn wfo wf1o isof1o f1ofo forn eleq2d wfn fvelrnb
|
|
f1ofn bitr3d anbi1d adantr anbi1i anass wi fnbrfvb adantrr bicomd anbi12d
|
|
sylan isorel ancom breq1 pm5.32i 3bitr3g exp32 com23 imp pm5.32d rexbidv2
|
|
syl5bb r19.41v syl6bb bitr4d abbi2dv df-ima syl6reqr ) ABDEFUAZCAIZJZBEKC
|
|
FLZMNZOZGUDZHUDZFPZGADKCMNZOZQZHUBFWPNWHWQHWKWMWKIZWMBIZWMWIEPZJZWHWQWRWS
|
|
WMWJIZJXAWMBWJUCXBWTWSEWIWMUEUFRWHXAWLFLZWMUGZGAQZWTJZWQWFXAXFSWGWFWSXEWT
|
|
WFWMFUHZIZWSXEWFXGBWMWFABFUIZXGBUGWFABFUJZXIABDEFUKZABFULTABFUMTUNWFFAUOZ
|
|
XHXESWFXJXLXKABFUQTZGAWMFUPTURUSUTWHWQXDWTJZGAQXFWHWNXNGWPAWLWPIZWNJZWLAI
|
|
ZWLCDPZWNJZJZWHXQXNJXPXQXRJZWNJXTXOYAWNXOXQWLWOIZJYAWLAWOUCYBXRXQDCWLUEUF
|
|
RVAXQXRWNVBRWHXQXSXNWFWGXQXSXNSZVCWFXQWGYCWFXQWGYCWFXQWGJJZWNXRJXDXCWIEPZ
|
|
JXSXNYDWNXDXRYEYDXDWNWFXQXDWNSZWGWFXLXQYFXMAWLWMFVDVHVEVFABWLCDEFVIVGWNXR
|
|
VJXDYEWTXCWMWIEVKVLVMVNVOVPVQVSVRXDWTGAVTWAWBVSWCHGFWPWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $. $d x y H $.
|
|
$d x y R $. $d x y S $.
|
|
isoini2.1 $e |- C = ( A i^i ( `' R " { X } ) ) $.
|
|
isoini2.2 $e |- D = ( B i^i ( `' S " { ( H ` X ) } ) ) $.
|
|
$( Isomorphisms are isomorphisms on their initial segments. (Contributed
|
|
by Mario Carneiro, 29-Mar-2014.) $)
|
|
isoini2 $p |- ( ( H Isom R , S ( A , B ) /\ X e. A ) ->
|
|
( H |` C ) Isom R , S ( C , D ) ) $=
|
|
( vx vy wiso wcel wf1o wbr cfv wb wral cima wa cres cv wf1 wss isof1o syl
|
|
f1of1 adantr ccnv csn inss1 eqsstri f1ores sylancl isoini imaeq2i 3eqtr4g
|
|
cin wceq f1oeq3 mpbid df-iso simprbi ssralv ralimdv mpsyl fvres breqan12d
|
|
bibi2d ralbidva ralbiia sylibr sylanbrc ) ABEFGMZHANZUAZCDGCUBZOZKUCZLUCZ
|
|
EPZVTVRQZWAVRQZFPZRZLCSZKCSZCDEFVRMVQCGCTZVROZVSVQABGUDZCAUEZWJVOWKVPVOAB
|
|
GOZWKABEFGUFABGUHUGUICAEUJHUKTZUSZAIAWNULUMZABCGUNUOVQWIDUTWJVSRVQGWOTBFU
|
|
JHGQUKTUSWIDABHEFGUPCWOGIUQJURWIDCVRVAUGVBVQWBVTGQZWAGQZFPZRZLCSZKCSZWHWL
|
|
VQXAKASZXBWPWLVQWTLASZKASZXCWPVOXEVPVOWMXEKLABEFGVCVDUIWLXDXAKAWTLCAVEVFV
|
|
GXAKCAVEVGWGXAKCVTCNZWFWTLCXFWACNZUAWEWSWBXFXGWCWQWDWRFVTCGVHWACGVHVIVJVK
|
|
VLVMKLCDEFVRVCVN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d x y v u B $. $d x y z w v u H $.
|
|
$d x y z w v u R $. $d v u S $.
|
|
$( Any one-to-one onto function determines an isomorphism with an induced
|
|
relation ` S ` . Proposition 6.33 of [TakeutiZaring] p. 34.
|
|
(Contributed by set.mm contributors, 30-Apr-2004.) $)
|
|
f1oiso $p |- ( ( H : A -1-1-onto-> B /\ S = { <. z , w >. |
|
|
E. x e. A E. y e. A ( ( z = ( H ` x ) /\ w = ( H ` y ) ) /\ x R y ) } ) ->
|
|
H Isom R , S ( A , B ) ) $=
|
|
( vv vu cv cfv wceq wa wbr wrex wb wcel anbi1d wf1o copab wral wiso simpl
|
|
wf1 f1of1 cop df-br eleq2 fvex eqeq1 2rexbidv anbi2d opelopab anass eqcom
|
|
f1fveq syl6bb anassrs syl5bb rexbidv rexbidva breq1 ceqsrexv adantl bitrd
|
|
r19.42v breq2 sylan9bb anandis sylan9bbr an32s ralrimivva df-iso sylanbrc
|
|
syl5rbb sylan ) EFIUAZHCLZALZIMZNZDLZBLZIMZNZOZWAWEGPZOZBEQAEQZCDUBZNZOVS
|
|
JLZKLZGPZWNIMZWOIMZHPZRZKEUCJEUCZEFGHIUDVSWMUEVSEFIUFZWMXAEFIUGXBWMOZWTJK
|
|
EEWSWQWRUHZHSZXCWNESZWOESZOZOWPWQWRHUIXBXHWMXEWPRWMXEXDWLSZXBXHOZWPHWLXDU
|
|
JXIWQWBNZWRWFNZOZWIOZBEQZAEQZXJWPWKXKWGOZWIOZBEQAEQXPCDWQWRWNIUKWOIUKVTWQ
|
|
NZWJXRABEEXSWHXQWIXSWCXKWGVTWQWBULTTUMWDWRNZXRXNABEEXTXQXMWIXTWGXLXKWDWRW
|
|
FULUNTUMUOXBXFXGXPWPRXBXFOZXPXLWNWEGPZOZBEQZXBXGOZWPYAXPWAWNNZXLWIOZBEQZO
|
|
ZAEQZYDYAXOYIAEYAWAESZOZXOYFYGOZBEQYIYLXNYMBEXNXKYGOYLYMXKXLWIUPYLXKYFYGX
|
|
BXFYKXKYFRXBXFYKOOXKWNWANYFEFWNWAIURWNWAUQUSUTTVAVBYFYGBEVHUSVCXFYJYDRXBY
|
|
HYDAWNEYFYGYCBEYFWIYBXLWAWNWEGVDUNVBVEVFVGYEYDWEWONZYBOZBEQZWPYEYCYOBEYEW
|
|
EESZOXLYNYBXBXGYQXLYNRXBXGYQOOXLWOWENYNEFWOWEIURWOWEUQUSUTTVCXGYPWPRXBYBW
|
|
PBWOEWEWOWNGVIVEVFVGVJVKVAVLVMVQVNVRJKEFGHIVOVP $.
|
|
$}
|
|
|
|
${
|
|
$d A w x y z $. $d B w x y z $. $d H w x y z $. $d R w x y z $.
|
|
f1oiso2.1 $e |- S = { <. x , y >. |
|
|
( ( x e. B /\ y e. B ) /\ ( `' H ` x ) R ( `' H ` y ) ) } $.
|
|
$( Any one-to-one onto function determines an isomorphism with an induced
|
|
relation ` S ` . (Contributed by Mario Carneiro, 9-Mar-2013.) $)
|
|
f1oiso2 $p |- ( H : A -1-1-onto-> B -> H Isom R , S ( A , B ) ) $=
|
|
( vz vw cv cfv wceq wa wbr wrex wcel 3adant3 eqcomd syl2anc wf1o wiso w3a
|
|
copab ccnv f1ocnvdm adantrr f1ocnvfv2 anim12dan simp3 fveq2 eqeq2d anbi2d
|
|
adantrl breq2 anbi12d rspcev syl12anc anbi1d breq1 rexbidv 3expib simp3ll
|
|
simp1 simp2l wf f1of ffvelrn sylan eqeltrd simp3lr simp2r simp3r f1ocnvfv
|
|
wi mpd 3brtr4d jca31 3exp rexlimdvv impbid opabbidv syl5eq f1oiso mpdan )
|
|
CDGUAZFAKZIKZGLZMZBKZJKZGLZMZNZWHWLEOZNZJCPZICPZABUDZMCDEFGUBWFFWGDQZWKDQ
|
|
ZNZWGGUEZLZWKXDLZEOZNZABUDWTHWFXHWSABWFXHWSWFXCXGWSWFXCXGUCZXECQZWGXEGLZM
|
|
ZWNNZXEWLEOZNZJCPZWSWFXCXJXGWFXAXJXBCDWGGUFUGRXIXFCQZXLWKXFGLZMZNZXGXPWFX
|
|
CXQXGWFXBXQXACDWKGUFUNRWFXCXTXGWFXAXLXBXSWFXANXKWGCDWGGUHSWFXBNXRWKCDWKGU
|
|
HSUIRWFXCXGUJXOXTXGNJXFCWLXFMZXMXTXNXGYAWNXSXLYAWMXRWKWLXFGUKULUMWLXFXEEU
|
|
OUPUQURWRXPIXECWHXEMZWQXOJCYBWOXMWPXNYBWJXLWNYBWIXKWGWHXEGUKULUSWHXEWLEUT
|
|
UPVAUQTVBWFWQXHIJCCWFWHCQZWLCQZNZWQXHWFYEWQUCZXAXBXGYFWGWIDWJWNWPWFYEVCZY
|
|
FWFYCWIDQZWFYEWQVDZWFYCYDWQVEZWFCDGVFZYCYHCDGVGZCDWHGVHVITVJYFWKWMDWJWNWP
|
|
WFYEVKZYFWFYDWMDQZYIWFYCYDWQVLZWFYKYDYNYLCDWLGVHVITVJYFWHWLXEXFEWFYEWOWPV
|
|
MYFWIWGMZXEWHMZYFWGWIYGSYFWFYCYPYQVOYIYJCDWHWGGVNTVPYFWMWKMZXFWLMZYFWKWMY
|
|
MSYFWFYDYRYSVOYIYOCDWLWKGVNTVPVQVRVSVTWAWBWCIJABCDEFGWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $.
|
|
opbr1st.1 $e |- A e. _V $.
|
|
opbr1st.2 $e |- B e. _V $.
|
|
$( Binary relationship of an ordered pair over ` 1st ` . (Contributed by
|
|
SF, 6-Feb-2015.) $)
|
|
opbr1st $p |- ( <. A , B >. 1st C <-> A = C ) $=
|
|
( vx vy cop c1st wbr cvv wcel wceq brex simprd cv wex wa eqcom bitri opth
|
|
eleq1 mpbii breq2 eqeq2 br1st biidd ceqsexv ancom exbii 3bitr4i pm5.21nii
|
|
vex vtoclbg ) ABHZCIJZCKLZACMZUPUOKLUQUOCINOURAKLUQDACKUBUCUOFPZIJZAUSMZU
|
|
PURFCKUSCUOIUDUSCAUEUTUOUSGPZHZMZGQZVAGUOUSFUMUFVBBMZUSAMZRZGQVGVEVAVGVGG
|
|
BEVFVGUGUHVDVHGVDVCUOMZVHUOVCSVIVGVFRVHUSVBABUAVGVFUITTUJAUSSUKTUNUL $.
|
|
|
|
$( Binary relationship of an ordered pair over ` 2nd ` . (Contributed by
|
|
SF, 6-Feb-2015.) $)
|
|
opbr2nd $p |- ( <. A , B >. 2nd C <-> B = C ) $=
|
|
( vx vy cop c2nd wbr cvv wcel wceq brex simprd eleq1 cv wex eqcom bitri
|
|
mpbii breq2 eqeq2 vex br2nd wa biidd ceqsexv opth exbii 3bitr4i pm5.21nii
|
|
vtoclbg ) ABHZCIJZCKLZBCMZUOUNKLUPUNCINOUQBKLUPEBCKPUAUNFQZIJZBURMZUOUQFC
|
|
KURCUNIUBURCBUCUSUNGQZURHZMZGRZUTGUNURFUDUEVAAMZURBMZUFZGRVFVDUTVFVFGADVE
|
|
VFUGUHVCVGGVCVBUNMVGUNVBSVAURABUITUJBURSUKTUMUL $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Alternate definition of the identity relationship. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
dfid4 $p |- _I = ( _S i^i `' _S ) $=
|
|
( vx vy cid csset ccnv cin weq cv wss wbr eqss vex ideq brin brsset brcnv
|
|
wa bitri anbi12i 3bitr4i eqbrriv ) ABCDDEZFZABGAHZBHZIZUEUDIZQZUDUECJUDUE
|
|
UCJZUDUEKUDUEBLZMUIUDUEDJZUDUEUBJZQUHUDUEDUBNUKUFULUGUDUEALZUJOULUEUDDJUG
|
|
UDUEDPUEUDUJUMORSRTUA $.
|
|
$}
|
|
|
|
$( The identity relationship is a set. (Contributed by SF, 11-Feb-2015.) $)
|
|
idex $p |- _I e. _V $=
|
|
( cid csset ccnv cin cvv dfid4 ssetex cnvex inex eqeltri ) ABBCZDEFBKGBGHIJ
|
|
$.
|
|
|
|
${
|
|
$d x y z w t $.
|
|
$( ` 1st ` is a mapping from the universe onto the universe.
|
|
(Contributed by SF, 12-Feb-2015.) (Revised by Scott Fenton,
|
|
17-Apr-2021.) $)
|
|
1stfo $p |- 1st : _V -onto-> _V $=
|
|
( vx vy vz vw vt cvv c1st wceq cv wbr wa weq wal cop wex vex br1st mpgbir
|
|
wcel eqv wfo wfn crn wfun cdm wi dffun2 anbi12i eeanv bitr4i opth simplbi
|
|
eqtr2 exlimivv sylbi gen2 cproj1 cproj2 opeq eqid proj1ex proj2ex opbr1st
|
|
syl mpbir breldm ax-mp eqeltri df-fn mpbir2an brelrn df-fo ) FFGUAGFUBZGU
|
|
CZFHZVMGUDZGUEZFHZVPAIZBIZGJZVSCIZGJZKZBCLZUFZCMBMAABCGUGWFBCWDVSVTDIZNZH
|
|
ZVSWBEIZNZHZKZEODOZWEWDWIDOZWLEOZKWNWAWOWCWPDVSVTBPQEVSWBCPQUHWIWLDEUIUJW
|
|
MWEDEWMWHWKHZWEVSWHWKUMWQWEDELVTWGWBWJUKULVDUNUOUPRVRVSVQSAAVQTVSVSUQZVSU
|
|
RZNZVQVSUSWTWRGJZWTVQSXAWRWRHWRUTWRWSWRVSAPZVAVSXBVBVCVEWTWRGVFVGVHRGFVIV
|
|
JVOVSVNSZAAVNTVSVSNZVSGJZXCXEAALVSUTVSVSVSXBXBVCVEXDVSGVKVGRFFGVLVJ $.
|
|
|
|
$( ` 2nd ` is a mapping from the universe onto the universe.
|
|
(Contributed by SF, 12-Feb-2015.) (Revised by Scott Fenton,
|
|
17-Apr-2021.) $)
|
|
2ndfo $p |- 2nd : _V -onto-> _V $=
|
|
( vx vy vz vw vt cvv c2nd wceq cv wbr wa weq wal cop wex vex br2nd mpgbir
|
|
wcel eqv wfo wfn crn wfun cdm wi dffun2 anbi12i eeanv bitr4i opth simprbi
|
|
eqtr2 exlimivv sylbi gen2 cproj1 cproj2 opeq eqid proj1ex proj2ex opbr2nd
|
|
syl mpbir breldm ax-mp eqeltri df-fn mpbir2an equid brelrn df-fo ) FFGUAG
|
|
FUBZGUCZFHZVNGUDZGUEZFHZVQAIZBIZGJZVTCIZGJZKZBCLZUFZCMBMAABCGUGWGBCWEVTDI
|
|
ZWANZHZVTEIZWCNZHZKZEODOZWFWEWJDOZWMEOZKWOWBWPWDWQDVTWABPQEVTWCCPQUHWJWMD
|
|
EUIUJWNWFDEWNWIWLHZWFVTWIWLUMWRDELWFWHWAWKWCUKULVDUNUOUPRVSVTVRSAAVRTVTVT
|
|
UQZVTURZNZVRVTUSXAWTGJZXAVRSXBWTWTHWTUTWSWTWTVTAPZVAVTXCVBVCVEXAWTGVFVGVH
|
|
RGFVIVJVPVTVOSZAAVOTVTVTNZVTGJZXDXFAALAVKVTVTVTXCXCVCVEXEVTGVLVGRFFGVMVJ
|
|
$.
|
|
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Alternate definition of domain. (Contributed by SF, 23-Feb-2015.) $)
|
|
dfdm4 $p |- dom A = ( 1st " A ) $=
|
|
( vx vy vz cdm c1st cima cv cop wcel wex wbr wrex wceq rexcom4 vex rexbii
|
|
br1st risset exbii 3bitr4ri eldm2 elima 3bitr4i eqriv ) BAEZFAGZBHZCHIZAJ
|
|
ZCKZDHZUHFLZDAMZUHUFJUHUGJULUINZCKZDAMUODAMZCKUNUKUODCAOUMUPDACULUHBPRQUJ
|
|
UQCDUIASTUACUHAUBDUHFAUCUDUE $.
|
|
|
|
$( Alternate definition of range. (Contributed by SF, 23-Feb-2015.) $)
|
|
dfrn5 $p |- ran A = ( 2nd " A ) $=
|
|
( vy vx vz crn c2nd cima cv cop wcel wex wbr wrex wceq rexcom4 vex rexbii
|
|
br2nd risset exbii 3bitr4ri elrn2 elima 3bitr4i eqriv ) BAEZFAGZCHBHZIZAJ
|
|
ZCKZDHZUHFLZDAMZUHUFJUHUGJULUINZCKZDAMUODAMZCKUNUKUODCAOUMUPDACULUHBPRQUJ
|
|
UQCDUIASTUACUHAUBDUHFAUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d B a $. $d B b $. $d b x $. $d B x $. $d b y $. $d B y $. $d x y $.
|
|
|
|
$( Binary relationship of ` Swap ` . (Contributed by SF, 23-Feb-2015.) $)
|
|
brswap $p |- ( A Swap B <-> E. x E. y ( A = <. x , y >. /\ B = <. y , x >.
|
|
) ) $=
|
|
( va vb cswap cvv wcel wa cv cop wceq wex opex eleq1 mpbiri eqeq1 2exbidv
|
|
vex wbr brex anim12i exlimivv anbi1d anbi2d df-swap brabg pm5.21nii ) CDG
|
|
UACHIZDHIZJZCAKZBKZLZMZDUNUMLZMZJZBNANZCDGUBUSULABUPUJURUKUPUJUOHIUMUNATZ
|
|
BTZOCUOHPQURUKUQHIUNUMVBVAODUQHPQUCUDEKZUOMZFKZUQMZJZBNANUPVFJZBNANUTEFCD
|
|
HHGVCCMZVGVHABVIVDUPVFVCCUORUESVEDMZVHUSABVJVFURUPVEDUQRUFSEFABUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d a b x y $.
|
|
$( The converse of ` Swap ` is ` Swap ` . (Contributed by SF,
|
|
23-Feb-2015.) $)
|
|
cnvswap $p |- `' Swap = Swap $=
|
|
( va vb vy vx cswap ccnv cv cop wceq wa wex wbr ancom 2exbii brcnv brswap
|
|
excom 3bitri 3bitr4i eqbrriv ) ABEFZEBGZCGZDGZHIZAGZUDUCHIZJZCKDKZUGUEJZC
|
|
KDKUFUBUALZUFUBELUHUJDCUEUGMNUKUBUFELUHDKCKUIUFUBEOCDUBUFPUHCDQRDCUFUBPST
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( ` Swap ` is a bijection over the universe. (Contributed by SF,
|
|
23-Feb-2015.) (Revised by Scott Fenton, 17-Apr-2021.) $)
|
|
swapf1o $p |- Swap : _V -1-1-onto-> _V $=
|
|
( vx vy vz cvv cswap wfn wceq cv wbr wa wal cproj2 cproj1 cop vex proj1ex
|
|
opeq proj2ex brswap2 bitri wf1o ccnv cdm weq wi dffun2 breq2i eqtr2 ancom
|
|
wfun opth eqeq12i 3bitr4i sylib syl2anb gen2 mpgbir wcel eqv mpbir breldm
|
|
eqid ax-mp eqeltri df-fn mpbir2an cnvswap fneq1i dff1o4 ) DDEUAEDFZEUBZDF
|
|
ZVJEUJZEUCZDGZVMAHZBHZEIZVPCHZEIZJBCUDZUEZCKBKAABCEUFWBBCVRVPVQLZVQMZNZGZ
|
|
VPVSLZVSMZNZGZWAVTVRVPWDWCNZEIWFVQWKVPEVQQZUGVPWDWCVQBOZPVQWMRSTVTVPWHWGN
|
|
ZEIWJVSWNVPEVSQZUGVPWHWGVSCOZPVSWPRSTWFWJJWEWIGZWAVPWEWIUHWCWGGZWDWHGZJWS
|
|
WRJZWQWAWRWSUIWCWDWGWHUKWAWKWNGWTVQWKVSWNWLWOULWDWCWHWGUKTUMUNUOUPUQVOVPV
|
|
NURAAVNUSVPVPMZVPLZNZVNVPQXCXBXANZEIZXCVNURXEXCXCGXCVBXCXBXAVPAOZRVPXFPSU
|
|
TXCXDEVAVCVDUQEDVEVFZVLVJXGDVKEVGVHUTDDEVIVF $.
|
|
|
|
$}
|
|
|
|
$( Bijection law for restrictions of ` Swap ` . (Contributed by SF,
|
|
23-Feb-2015.) (Modified by Scott Fenton, 17-Apr-2021.) $)
|
|
swapres $p |- ( Swap |` A ) : A -1-1-onto-> `' A $=
|
|
( ccnv cswap cres wf1o cima cvv wf1 wss swapf1o f1of1 ax-mp ssv f1ores wceq
|
|
mp2an wb dfcnv2 f1oeq3 mpbir ) AABZCADZEZACAFZUBEZGGCHZAGIUEGGCEUFJGGCKLAMG
|
|
GACNPUAUDOUCUEQARUAUDAUBSLT $.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d A z $. $d B x $. $d B y $. $d C x $. $d C y $.
|
|
$d C z $. $d D x $. $d D y $. $d x y $. $d x z $. $d y z $.
|
|
xpnedisj.1 $e |- C e. _V $.
|
|
xpnedisj.2 $e |- C =/= D $.
|
|
$( Cross products with non-equal singletons are disjoint. (Contributed by
|
|
SF, 23-Feb-2015.) $)
|
|
xpnedisj $p |- ( ( A X. { C } ) i^i ( B X. { D } ) ) = (/) $=
|
|
( vx vy vz csn cxp cin c0 wceq cv wcel wn disj cop wrex elxp2 opeq2 rexsn
|
|
eqeq2d rexbii bitri wa df-ne mpbi elsni intnan eleq1 opelxp syl6bb mtbiri
|
|
wne mto rexlimivw sylbi mprgbir ) ACJZKZBDJZKZLMNGOZVDPZQZGVBGVBVDRVEVBPZ
|
|
VEHOZCSZNZHATZVGVHVEVIIOZSZNZIVATZHATVLHIVEAVAUAVPVKHAVOVKICEVMCNVNVJVEVM
|
|
CVIUBUDUCUEUFVKVGHAVKVFVIBPZCVCPZUGZVRVQVRCDNZCDUPVTQFCDUHUICDUJUQUKVKVFV
|
|
JVDPVSVEVJVDULVICBVCUMUNUOURUSUT $.
|
|
$}
|
|
|
|
${
|
|
opfv1st.1 $e |- A e. _V $.
|
|
opfv1st.2 $e |- B e. _V $.
|
|
$( The value of the ` 1st ` function on an ordered pair. (Contributed by
|
|
SF, 23-Feb-2015.) $)
|
|
opfv1st $p |- ( 1st ` <. A , B >. ) = A $=
|
|
( cop c1st cfv wceq wbr eqid opbr1st mpbir cvv wfn wcel wb wfo 1stfo fofn
|
|
ax-mp opex fnbrfvb mp2an ) ABEZFGAHZUDAFIZUFAAHAJABACDKLFMNZUDMOUEUFPMMFQ
|
|
UGRMMFSTABCDUAMUDAFUBUCL $.
|
|
|
|
$( The value of the ` 2nd ` function on an ordered pair. (Contributed by
|
|
SF, 23-Feb-2015.) $)
|
|
opfv2nd $p |- ( 2nd ` <. A , B >. ) = B $=
|
|
( cop c2nd cfv wceq wbr eqid opbr2nd mpbir cvv wfn wcel wb wfo 2ndfo fofn
|
|
ax-mp opex fnbrfvb mp2an ) ABEZFGBHZUDBFIZUFBBHBJABBCDKLFMNZUDMOUEUFPMMFQ
|
|
UGRMMFSTABCDUAMUDBFUBUCL $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C x y $.
|
|
$( Reconstruction of a member of a cross product in terms of its ordered
|
|
pair components. (Contributed by SF, 20-Oct-2013.) $)
|
|
1st2nd2 $p |- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) $=
|
|
( vx vy cxp wcel cv cop wceq wrex c1st cfv c2nd elxp2 vex opfv1st opfv2nd
|
|
fveq2 rexlimivw opeq12i eqcomi id opeq12d 3eqtr4a sylbi ) ABCFGADHZEHZIZJ
|
|
ZECKZDBKAALMZANMZIZJZDEABCOUKUODBUJUOECUJUIUILMZUINMZIZAUNURUIUPUGUQUHUGU
|
|
HDPZEPZQUGUHUSUTRUAUBUJUCUJULUPUMUQAUILSAUINSUDUETTUF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d A z $. $d B x $. $d B y $. $d B z $. $d C x $.
|
|
$d C y $. $d C z $. $d F x $. $d F y $. $d F z $. $d x y $. $d x z $.
|
|
$d y z $.
|
|
$( Implicational form of part of the definition of a function.
|
|
(Contributed by SF, 24-Feb-2015.) $)
|
|
fununiq $p |- ( ( Fun F /\ A F B /\ A F C ) -> B = C ) $=
|
|
( vx vy vz cvv wcel wa wbr w3a wceq brex 3adant1 wi cv wal wb breq12 wfun
|
|
anim12i anandi sylibr dffun2 3adant3 3adant2 anbi12d eqeq12 spc3gv syl5bi
|
|
weq imbi12d exp4a 3impd 3expb mpcom ) AHIZBHIZCHIZJJZDUAZABDKZACDKZLZBCMZ
|
|
VCVDVAVBVCVDJZURUSJZURUTJZJVAVCVHVDVIABDNACDNUBURUSUTUCUDOURUSUTVEVFPURUS
|
|
UTLZVBVCVDVFVJVBVCVDVFVBEQZFQZDKZVKGQZDKZJZFGULZPZGRFRERVJVGVFPZEFGDUEVRV
|
|
SEFGABCHHHVKAMZVLBMZVNCMZLZVPVGVQVFWCVMVCVOVDVTWAVMVCSWBVKAVLBDTUFVTWBVOV
|
|
DSWAVKAVNCDTUGUHWAWBVQVFSVTVLBVNCUIOUMUJUKUNUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a x $. $d a y $. $d b x $. $d b y $. $d R a $. $d R b $.
|
|
$d R x $. $d R y $. $d x y $.
|
|
$( Calculate the converse of a singleton image. (Contributed by SF,
|
|
26-Feb-2015.) $)
|
|
cnvsi $p |- `' SI R = SI `' R $=
|
|
( vx vy vb va csi ccnv csn wceq wbr w3a wex 3ancoma 3anbi3i bitr4i 2exbii
|
|
cv brcnv brsi excom 3bitri 3bitr4i eqbrriv ) BCAFZGZAGZFZCQZDQZHIZBQZEQZH
|
|
IZUIULAJZKZDLELZUMUJULUIUFJZKZDLELUKUHUEJZUKUHUGJUOUREDUOUMUJUNKURUJUMUNM
|
|
UQUNUMUJULUIARNOPUSUHUKUDJUOELDLUPUKUHUDRDEUHUKASUODETUAEDUKUHUFSUBUC $.
|
|
|
|
$( Calculate the domain of a singleton image. Theorem X.4.29.I of [Rosser]
|
|
p. 301. (Contributed by SF, 26-Feb-2015.) $)
|
|
dmsi $p |- dom SI R = ~P1 dom R $=
|
|
( vx va vy vb csi cdm cpw1 cv csn wceq wbr wcel wa bitri exbii excom eldm
|
|
wex 3bitr4i wrex 3anass 2exbii 19.42vv isseti 19.41v mpbiran anbi2i ancom
|
|
w3a snex df-rex brsi elpw1 eqriv ) BAFZGZAGZHZBIZCIZJKZDIZEIZJZKZVAVDALZU
|
|
JZESZCSZDSZVBCURUAZUTUQMZUTUSMVIDSZCSVAURMZVBNZCSVKVLVNVPCVNVBVFVGNZESDSZ
|
|
NZVPVNVBVQNZESDSVSVHVTDEVBVFVGUBUCVBVQDEUDOVSVBVONVPVRVOVBVQDSZESVGESVRVO
|
|
WAVGEWAVFDSVGDVEVDUKUEVFVGDUFUGPVQDEQEVAARTUHVBVOUIOOPVIDCQVBCURULTVMUTVC
|
|
UPLZDSVKDUTUPRWBVJDCEUTVCAUMPOCUTURUNTUO $.
|
|
|
|
$d a c $. $d a d $. $d a z $. $d b c $. $d b d $. $d b z $. $d c d $.
|
|
$d c x $. $d c y $. $d c z $. $d d x $. $d d y $. $d d z $. $d F a $.
|
|
$d F b $. $d F c $. $d F d $. $d F x $. $d F y $. $d F z $. $d x z $.
|
|
$d y z $.
|
|
$( The singleton image of a function is a function. (Contributed by SF,
|
|
26-Feb-2015.) $)
|
|
funsi $p |- ( Fun F -> Fun SI F ) $=
|
|
( vx vy vz va vb vc vd wfun cv wbr wa weq wi wal csn wceq w3a wex wb brsi
|
|
anbi12i ee4anv bitr4i fununiq 3exp breq1 bicomd adantr eqeq2 sneqb syl6bb
|
|
csi vex adantl imbi12d biimprcd exp3a 3impd syl6 eqeq1 3anbi1d imbi2d syl
|
|
imp3a exlimdvv syl5bi alrimiv alrimivv dffun2 sylibr ) AIZBJZCJZAUMZKZVMD
|
|
JZVOKZLZCDMZNZDOZCOBOVOIVLWBBCVLWADVSVMEJZPZQZVNFJZPZQZWCWFAKZRZVMGJZPZQZ
|
|
VQHJZPZQZWKWNAKZRZLZHSGSZFSESZVLVTVSWJFSESZWRHSGSZLXAVPXBVRXCEFVMVNAUAGHV
|
|
MVQAUAUBWJWREFGHUCUDVLWTVTEFVLWSVTGHVLWJWRVTVLWIEGMZWPWQRZWGVQQZNZNZWJWRV
|
|
TNZNVLWIWCWNAKZFHMZNZXGVLWIXJXKWCWFWNAUEUFXLXDWPWQXFXLXDWPWQXFNZXDWPLZXMX
|
|
LXNWQXJXFXKXDWQXJTWPXDXJWQWCWKWNAUGUHUIWPXFXKTXDWPXFWGWOQXKVQWOWGUJWFWNFU
|
|
NUKULUOUPUQURUSUTXHWEWHWIXIXHWEWHWIXINZWEWHLZXOXHXPXIXGWIXPWRXEVTXFWEWRXE
|
|
TWHWEWMXDWPWQWEWMWDWLQXDVMWDWLVAWCWKEUNUKULVBUIWHVTXFTWEVNWGVQVAUOUPVCUQU
|
|
RUSVDVEVFVFVGVHVIBCDVOVJVK $.
|
|
$}
|
|
|
|
$( Calculate the range of a singleton image. Theorem X.4.29.II of [Rosser]
|
|
p. 302. (Contributed by SF, 26-Feb-2015.) $)
|
|
rnsi $p |- ran SI R = ~P1 ran R $=
|
|
( csi ccnv cdm cpw1 cnvsi dmeqi dmsi eqtri dfrn4 wceq pw1eq ax-mp 3eqtr4i
|
|
crn ) ABZCZDZACZDZEZPOAOZEZRSBZDUAQUDAFGSHIPJUBTKUCUAKAJUBTLMN $.
|
|
|
|
${
|
|
op1st.1 $e |- A e. _V $.
|
|
op1st.2 $e |- B e. _V $.
|
|
$( Extract the first member of an ordered pair. (Contributed by Mario
|
|
Carneiro, 31-Aug-2015.) $)
|
|
op1std $p |- ( C = <. A , B >. -> ( 1st ` C ) = A ) $=
|
|
( cop wceq c1st cfv fveq2 opfv1st syl6eq ) CABFZGCHIMHIACMHJABDEKL $.
|
|
|
|
$( Extract the second member of an ordered pair. (Contributed by Mario
|
|
Carneiro, 31-Aug-2015.) $)
|
|
op2ndd $p |- ( C = <. A , B >. -> ( 2nd ` C ) = B ) $=
|
|
( cop wceq c2nd cfv fveq2 opfv2nd syl6eq ) CABFZGCHIMHIBCMHJABDEKL $.
|
|
$}
|
|
|
|
$( The domain of the epsilon relationship. (Contributed by Scott Fenton,
|
|
20-Apr-2021.) $)
|
|
dmep $p |- dom _E = _V $=
|
|
( vx cep cdm cvv wceq cv wcel eqv csn wbr vex snid epelc mpbir breldm ax-mp
|
|
snex mpgbir ) BCZDEAFZSGZAASHTTIZBJZUAUCTUBGTAKLTUBTQMNTUBBOPR $.
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Operations
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Extend class notation to include the value of an operation ` F ` for two
|
|
arguments ` A ` and ` B ` . Note that the syntax is simply three class
|
|
symbols in a row surrounded by parentheses. Since operation values are
|
|
the only possible class expressions consisting of three class expressions
|
|
in a row surrounded by parentheses, the syntax is unambiguous. $)
|
|
co $a class ( A F B ) $.
|
|
|
|
$( Define the value of an operation. Definition of operation value in
|
|
[Enderton] p. 79. Note that the syntax is simply three class expressions
|
|
in a row bracketed by parentheses. There are no restrictions of any kind
|
|
on what those class expressions may be, although only certain kinds of
|
|
class expressions - a binary operation ` F ` and its arguments ` A ` and
|
|
` B ` - will be useful for proving meaningful theorems. This definition
|
|
is well-defined, although not very meaningful, when classes ` A ` and/or
|
|
` B ` are proper classes (i.e. are not sets). On the other hand, we often
|
|
find uses for this definition when ` F ` is a proper class. ` F ` is
|
|
normally equal to a class of nested ordered pairs of the form defined by
|
|
~ df-oprab . (Contributed by SF, 5-Jan-2015.) $)
|
|
df-ov $a |- ( A F B ) = ( F ` <. A , B >. ) $.
|
|
|
|
$( Extend class notation to include class abstraction (class builder) of
|
|
nested ordered pairs. $)
|
|
coprab $a class { <. <. x , y >. , z >. | ph } $.
|
|
|
|
${
|
|
$d x w $. $d y w $. $d z w $. $d w ph $.
|
|
$( Define the class abstraction (class builder) of a collection of nested
|
|
ordered pairs (for use in defining operations). This is a special case
|
|
of Definition 4.16 of [TakeutiZaring] p. 14. Normally ` x ` , ` y ` ,
|
|
and ` z ` are distinct, although the definition doesn't strictly require
|
|
it. See ~ df-ov for the value of an operation. The brace notation is
|
|
called "class abstraction" by Quine; it is also called a "class builder"
|
|
in the literature. The value of the most common operation class builder
|
|
is given by ov2 in set.mm. (Contributed by SF, 5-Jan-2015.) $)
|
|
df-oprab $a |- { <. <. x , y >. , z >. | ph } =
|
|
{ w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } $.
|
|
$}
|
|
|
|
$( Equality theorem for operation value. (Contributed by set.mm
|
|
contributors, 28-Feb-1995.) $)
|
|
oveq $p |- ( F = G -> ( A F B ) = ( A G B ) ) $=
|
|
( wceq cop cfv co fveq1 df-ov 3eqtr4g ) CDEABFZCGLDGABCHABDHLCDIABCJABDJK
|
|
$.
|
|
|
|
$( Equality theorem for operation value. (Contributed by set.mm
|
|
contributors, 28-Feb-1995.) $)
|
|
oveq1 $p |- ( A = B -> ( A F C ) = ( B F C ) ) $=
|
|
( wceq cop cfv co opeq1 fveq2d df-ov 3eqtr4g ) ABEZACFZDGBCFZDGACDHBCDHMNOD
|
|
ABCIJACDKBCDKL $.
|
|
|
|
$( Equality theorem for operation value. (Contributed by set.mm
|
|
contributors, 28-Feb-1995.) $)
|
|
oveq2 $p |- ( A = B -> ( C F A ) = ( C F B ) ) $=
|
|
( wceq cop cfv co opeq2 fveq2d df-ov 3eqtr4g ) ABEZCAFZDGCBFZDGCADHCBDHMNOD
|
|
ABCIJCADKCBDKL $.
|
|
|
|
$( Equality theorem for operation value. (Contributed by set.mm
|
|
contributors, 16-Jul-1995.) $)
|
|
oveq12 $p |- ( ( A = B /\ C = D ) -> ( A F C ) = ( B F D ) ) $=
|
|
( wceq co oveq1 oveq2 sylan9eq ) ABFCDFACEGBCEGBDEGABCEHCDBEIJ $.
|
|
|
|
${
|
|
oveq1i.1 $e |- A = B $.
|
|
$( Equality inference for operation value. (Contributed by set.mm
|
|
contributors, 28-Feb-1995.) $)
|
|
oveq1i $p |- ( A F C ) = ( B F C ) $=
|
|
( wceq co oveq1 ax-mp ) ABFACDGBCDGFEABCDHI $.
|
|
|
|
$( Equality inference for operation value. (Contributed by set.mm
|
|
contributors, 28-Feb-1995.) $)
|
|
oveq2i $p |- ( C F A ) = ( C F B ) $=
|
|
( wceq co oveq2 ax-mp ) ABFCADGCBDGFEABCDHI $.
|
|
|
|
${
|
|
oveq12i.2 $e |- C = D $.
|
|
$( Equality inference for operation value. (The proof was shortened by
|
|
Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors,
|
|
28-Feb-1995.) (Revised by set.mm contributors, 22-Oct-2011.) $)
|
|
oveq12i $p |- ( A F C ) = ( B F D ) $=
|
|
( wceq co oveq12 mp2an ) ABHCDHACEIBDEIHFGABCDEJK $.
|
|
$}
|
|
|
|
$( Equality inference for operation value. (Contributed by set.mm
|
|
contributors, 24-Nov-2007.) $)
|
|
oveqi $p |- ( C A D ) = ( C B D ) $=
|
|
( wceq co oveq ax-mp ) ABFCDAGCDBGFECDABHI $.
|
|
$}
|
|
|
|
${
|
|
oveq1d.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for operation value. (Contributed by set.mm
|
|
contributors, 13-Mar-1995.) $)
|
|
oveq1d $p |- ( ph -> ( A F C ) = ( B F C ) ) $=
|
|
( wceq co oveq1 syl ) ABCGBDEHCDEHGFBCDEIJ $.
|
|
|
|
$( Equality deduction for operation value. (Contributed by set.mm
|
|
contributors, 13-Mar-1995.) $)
|
|
oveq2d $p |- ( ph -> ( C F A ) = ( C F B ) ) $=
|
|
( wceq co oveq2 syl ) ABCGDBEHDCEHGFBCDEIJ $.
|
|
|
|
$( Equality deduction for operation value. (Contributed by set.mm
|
|
contributors, 9-Sep-2006.) $)
|
|
oveqd $p |- ( ph -> ( C A D ) = ( C B D ) ) $=
|
|
( wceq co oveq syl ) ABCGDEBHDECHGFDEBCIJ $.
|
|
|
|
${
|
|
oveq12d.2 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for operation value. (The proof was shortened by
|
|
Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors,
|
|
13-Mar-1995.) (Revised by set.mm contributors, 22-Oct-2011.) $)
|
|
oveq12d $p |- ( ph -> ( A F C ) = ( B F D ) ) $=
|
|
( wceq co oveq12 syl2anc ) ABCIDEIBDFJCEFJIGHBCDEFKL $.
|
|
$}
|
|
|
|
${
|
|
opreqan12i.2 $e |- ( ps -> C = D ) $.
|
|
$( Equality deduction for operation value. (Contributed by set.mm
|
|
contributors, 10-Aug-1995.) $)
|
|
oveqan12d $p |- ( ( ph /\ ps ) -> ( A F C ) = ( B F D ) ) $=
|
|
( wceq co oveq12 syl2an ) ACDJEFJCEGKDFGKJBHICDEFGLM $.
|
|
|
|
$( Equality deduction for operation value. (Contributed by set.mm
|
|
contributors, 10-Aug-1995.) $)
|
|
oveqan12rd $p |- ( ( ps /\ ph ) -> ( A F C ) = ( B F D ) ) $=
|
|
( co wceq oveqan12d ancoms ) ABCEGJDFGJKABCDEFGHILM $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
oveq123d.1 $e |- ( ph -> F = G ) $.
|
|
oveq123d.2 $e |- ( ph -> A = B ) $.
|
|
oveq123d.3 $e |- ( ph -> C = D ) $.
|
|
$( Equality deduction for operation value. (Contributed by FL,
|
|
22-Dec-2008.) $)
|
|
oveq123d $p |- ( ph -> ( A F C ) = ( B G D ) ) $=
|
|
( co oveqd oveq1d oveq2d 3eqtrd ) ABDFKBDGKCDGKCEGKAFGBDHLABCDGIMADECGJNO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nfovd.2 $e |- ( ph -> F/_ x A ) $.
|
|
nfovd.3 $e |- ( ph -> F/_ x F ) $.
|
|
nfovd.4 $e |- ( ph -> F/_ x B ) $.
|
|
$( Deduction version of bound-variable hypothesis builder ~ nfov .
|
|
(Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon,
|
|
22-Oct-2011.) $)
|
|
nfovd $p |- ( ph -> F/_ x ( A F B ) ) $=
|
|
( co cop cfv df-ov nfopd nffvd nfcxfrd ) ABCDEICDJZEKCDELABPEGABCDFHMNO
|
|
$.
|
|
$}
|
|
|
|
${
|
|
nfov.1 $e |- F/_ x A $.
|
|
nfov.2 $e |- F/_ x F $.
|
|
nfov.3 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for operation value. (Contributed by
|
|
NM, 4-May-2004.) $)
|
|
nfov $p |- F/_ x ( A F B ) $=
|
|
( co wnfc wtru a1i nfovd trud ) ABCDHIJABCDABIJEKADIJFKACIJGKLM $.
|
|
$}
|
|
|
|
${
|
|
$d w x $. $d w y $. $d w z $. $d w ph $.
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy,
|
|
19-Jun-2012.) $)
|
|
nfoprab1 $p |- F/_ x { <. <. x , y >. , z >. | ph } $=
|
|
( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfab nfcxfr ) BABCDFEGBG
|
|
CGHDGHIAJDKCKZBKZELABCDEMRBEQBNOP $.
|
|
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy,
|
|
30-Jul-2012.) $)
|
|
nfoprab2 $p |- F/_ y { <. <. x , y >. , z >. | ph } $=
|
|
( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfex nfab nfcxfr ) CABCD
|
|
FEGBGCGHDGHIAJDKZCKZBKZELABCDEMTCESCBRCNOPQ $.
|
|
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Contributed by NM, 22-Aug-2013.) $)
|
|
nfoprab3 $p |- F/_ z { <. <. x , y >. , z >. | ph } $=
|
|
( vw coprab cv cop wceq wa wex cab df-oprab nfe1 nfex nfab nfcxfr ) DABCD
|
|
FEGBGCGHDGHIAJZDKZCKZBKZELABCDEMUADETDBSDCRDNOOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d v w x $. $d v w y $. $d v w z $. $d v ph $.
|
|
nfoprab.1 $e |- F/ w ph $.
|
|
$( Bound-variable hypothesis builder for an operation class abstraction.
|
|
(Contributed by NM, 22-Aug-2013.) $)
|
|
nfoprab $p |- F/_ w { <. <. x , y >. , z >. | ph } $=
|
|
( vv coprab cv cop wceq wa wex cab df-oprab nfv nfan nfex nfab nfcxfr ) E
|
|
ABCDHGIBICIJDIJKZALZDMZCMZBMZGNABCDGOUEEGUDEBUCECUBEDUAAEUAEPFQRRRST $.
|
|
$}
|
|
|
|
${
|
|
$d a ph r s t w $. $d a r s t w x $. $d a r s t w y $. $d a r s t w z $.
|
|
$( The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61.
|
|
(Contributed by Mario Carneiro, 20-Mar-2013.) $)
|
|
oprabid $p |- ( <. <. x , y >. , z >. e.
|
|
{ <. <. x , y >. , z >. | ph } <-> ph ) $=
|
|
( vw va vt vr vs cv cop wceq wa wex vex wi weq opth wal wn coprab eqvinop
|
|
biimpi eqeq1 simplbi syl6bi opeq1 eqeq2d anbi1i bitri anass 3bitri 3exbii
|
|
opex nfcvf2 nfcvd nfeqd exdistrf eximi excom anim2i 3syl sylbi w3a df-3an
|
|
3imtr4i bitr4i weu euequ1 eupick mpan syl6 3impd syl5bi com12 syl5 syl6bb
|
|
eqcom anbi1d 3exbidv imbi1d mpbiri adantr exlimivv com3l mpdd mpcom 19.8a
|
|
imbi12d ex impbid df-oprab elab2 ) EJZBJZCJZKZDJZKZLZAMZDNZCNZBNZAEWSABCD
|
|
UAWQWRWOWPBOZCOZUNZDOZUNWTXDAWNFJZGJZKZLZXKWSLZMZGNFNZWTXDAPZWTXOFGWNWQWR
|
|
XGXHUBUCXNWTXPPZFGXLXQXMXLWTXIWQLZXPXLWTXMXRWNXKWSUDXMXRGDQXIXJWQWRRUEUFX
|
|
RXLWTXPXRXIHJZIJZKZLZYAWQLZMZINHNXLXQPZHIXIWOWPXEXFUBYDYEHIYBYEYCYBXLWNYA
|
|
XJKZLZXQYBXKYFWNXIYAXJUGUHYGXQWSYFLZYHAMZDNCNBNZAPZPYJBHQZCIQZDGQZAMZDNZM
|
|
CNZMZBNZYHAYJYLYMYOMZMZDNZCNBNZYSYIUUABCDYIYLYMMZYNMZAMUUDYOMUUAYHUUEAYHW
|
|
QYALZYNMUUEWQWRYAXJRUUFUUDYNWOWPXSXTRUIUJZUIUUDYNAUKYLYMYOUKULUMUUCYLYTDN
|
|
ZMZCNBNZYLUUHCNZMZBNYSUUBBNZCNUUIBNZCNUUCUUJUUMUUNCYLYTBDBDQBSTZDWOXSBDUO
|
|
UUODXSUPUQURUSUUBBCUTUUIBCUTVFYLUUHBCBCQBSTZCWOXSBCUOUUPCXSUPUQURUULYRBUU
|
|
KYQYLYMYOCDCDQCSTZDWPXTCDUOUUQDXTUPUQURVAUSVBVCYSYHAYHYLYMYNVDZYSAYHUUEUU
|
|
RUUGYLYMYNVEVGYSYLYMYNAYSYLYQYMYNAPZPYLBVHYSYLYQPBHVIYLYQBVJVKYQYMYPUUSYM
|
|
CVHYQYMYPPCIVIYMYPCVJVKYNDVHYPUUSDGVIYNADVJVKVLVLVMVNVOVPYGWTYHXPYKYGWTYF
|
|
WSLYHWNYFWSUDYFWSVRVQZYGXDYJAYGXAYIBCDYGWTYHAUUTVSVTWAWIWBUFWCWDVCWEWFWCW
|
|
DWGWTAXDXAXBXCXDXADWHXBCWHXCBWHVBWJWKABCDEWLWM $.
|
|
$}
|
|
|
|
$( The result of an operation is a set. (Contributed by set.mm contributors,
|
|
13-Mar-1995.) $)
|
|
ovex $p |- ( A F B ) e. _V $=
|
|
( co cop cfv cvv df-ov fvex eqeltri ) ABCDABEZCFGABCHKCIJ $.
|
|
|
|
${
|
|
$d y A $. $d y B $. $d y C $. $d y D $. $d y F $. $d x y $.
|
|
$( Move class substitution in and out of an operation. (Contributed by NM,
|
|
12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) $)
|
|
csbovg $p |- ( A e. D -> [_ A / x ]_ ( B F C ) =
|
|
( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) ) $=
|
|
( vy cv co csb wceq csbeq1 oveq123d eqeq12d nfcsb1v csbeq1a csbief vtoclg
|
|
vex nfov ) AGHZCDFIZJZAUACJZAUADJZAUAFJZIZKABUBJZABCJZABDJZABFJZIZKGBEUAB
|
|
KZUCUHUGULAUABUBLUMUDUIUEUJUFUKAUABFLAUABCLAUABDLMNAUAUBUGGSAUDUEUFAUACOA
|
|
UAFOAUADOTAHUAKCUDDUEFUFAUAFPAUACPAUADPMQR $.
|
|
$}
|
|
|
|
${
|
|
$d x F $.
|
|
$( Move class substitution in and out of an operation. (Contributed by NM,
|
|
12-Nov-2005.) $)
|
|
csbov12g $p |- ( A e. D ->
|
|
[_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) ) $=
|
|
( wcel co csb csbovg csbconstg oveqd eqtrd ) BEGZABCDFHIABCIZABDIZABFIZHO
|
|
PFHABCDEFJNQFOPABFEKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x C $. $d x F $.
|
|
$( Move class substitution in and out of an operation. (Contributed by NM,
|
|
12-Nov-2005.) $)
|
|
csbov1g $p |- ( A e. D ->
|
|
[_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) ) $=
|
|
( wcel co csb csbov12g csbconstg oveq2d eqtrd ) BEGZABCDFHIABCIZABDIZFHOD
|
|
FHABCDEFJNPDOFABDEKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x B $. $d x F $.
|
|
$( Move class substitution in and out of an operation. (Contributed by NM,
|
|
12-Nov-2005.) $)
|
|
csbov2g $p |- ( A e. D ->
|
|
[_ A / x ]_ ( B F C ) = ( B F [_ A / x ]_ C ) ) $=
|
|
( wcel co csb csbov12g csbconstg oveq1d eqtrd ) BEGZABCDFHIABCIZABDIZFHCP
|
|
FHABCDEFJNOCPFABCEKLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x y B $. $d x y C $. $d y D $. $d x y F $. $d x y S $.
|
|
$( A frequently used special case of ~ rspc2ev for operation values.
|
|
(Contributed by set.mm contributors, 21-Mar-2007.) $)
|
|
rspceov $p |- ( ( C e. A /\ D e. B /\ S = ( C F D ) ) ->
|
|
E. x e. A E. y e. B S = ( x F y ) ) $=
|
|
( cv co wceq oveq1 eqeq2d oveq2 rspc2ev ) GAIZBIZHJZKGEFHJZKGEQHJZKABEFCD
|
|
PEKRTGPEQHLMQFKTSGQFEHNMO $.
|
|
$}
|
|
|
|
$( Equivalence of operation value and ordered triple membership, analogous to
|
|
~ fnopfvb . (Contributed by set.mm contributors, 17-Dec-2008.) $)
|
|
fnopovb $p |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) ->
|
|
( ( C F D ) = R <-> <. <. C , D >. , R >. e. F ) ) $=
|
|
( cxp wfn wcel co wceq cop wb wa opelxp cfv df-ov eqeq1i fnopfvb syl5bb
|
|
sylan2br 3impb ) FABGZHZCAIZDBIZCDFJZEKZCDLZELFIZMZUEUFNUDUIUCIZUKCDABOUHUI
|
|
FPZEKUDULNUJUGUMECDFQRUCUIEFSTUAUB $.
|
|
|
|
${
|
|
$d x z w v $. $d y z w v $. $d w ph v $.
|
|
$( Class abstraction for operations in terms of class abstraction of
|
|
ordered pairs. (Contributed by set.mm contributors, 12-Mar-1995.) $)
|
|
dfoprab2 $p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } $=
|
|
( vv cv cop wceq wex cab coprab copab excom exrot4 an12 exbii vex bitri
|
|
wa opex opeq1 eqeq2d anbi1d ceqsexv 3exbii 19.42vv 3bitr3i abbii df-oprab
|
|
2exbii df-opab 3eqtr4i ) FGZBGZCGZHZDGZHZIZATZDJCJBJZFKUNEGZURHZIZVCUQIZA
|
|
TZCJBJZTZDJEJZFKABCDLVHEDMVBVJFVEVGTZCJBJZEJDJZVLDJEJVBVJVLDENVMVKEJZDJCJ
|
|
BJVBVKDEBCOVNVABCDVNVFVEATZTZEJVAVKVPEVEVFAPQVOVAEUQUOUPBRCRUAVFVEUTAVFVD
|
|
USUNVCUQURUBUCUDUESUFSVLVIEDVEVGBCUGUKUHUIABCDFUJVHEDFULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x w v $. $d y v $. $d z v $. $d v ph $.
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Unnecessary distinct variable restrictions were removed by David
|
|
Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors,
|
|
25-Apr-1995.) (Revised by set.mm contributors, 24-Jul-2012.) $)
|
|
hboprab1 $p |- ( w e. { <. <. x , y >. , z >. | ph } ->
|
|
A. x w e. { <. <. x , y >. , z >. | ph } ) $=
|
|
( vv coprab cv cop wceq wa wex cab df-oprab hbe1 hbab hbxfreq ) BEABCDGFH
|
|
BHCHIDHIJAKDLCLZBLZFMABCDFNSBFERBOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x v $. $d y w v $. $d z v $. $d v ph $.
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Unnecessary distinct variable restrictions were removed by David
|
|
Abernethy, 30-Jul-2012.) (Contributed by set.mm contributors,
|
|
25-Apr-1995.) (Revised by set.mm contributors, 31-Jul-2012.) $)
|
|
hboprab2 $p |- ( w e. { <. <. x , y >. , z >. | ph } ->
|
|
A. y w e. { <. <. x , y >. , z >. | ph } ) $=
|
|
( vv coprab cv cop wceq wa wex cab df-oprab hbe1 hbex hbab hbxfreq ) CEAB
|
|
CDGFHBHCHIDHIJAKDLZCLZBLZFMABCDFNUACFETCBSCOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d v x $. $d v y $. $d v z $. $d w z $. $d v ph $.
|
|
$( The abstraction variables in an operation class abstraction are not
|
|
free. (Contributed by set.mm contributors, 22-Aug-2013.) $)
|
|
hboprab3 $p |- ( w e. { <. <. x , y >. , z >. | ph } ->
|
|
A. z w e. { <. <. x , y >. , z >. | ph } ) $=
|
|
( vv coprab cv cop wceq wa wex cab df-oprab hbe1 hbex hbab hbxfreq ) DEAB
|
|
CDGFHBHCHIDHIJAKZDLZCLZBLZFMABCDFNUBDFEUADBTDCSDOPPQR $.
|
|
$}
|
|
|
|
${
|
|
$d u w $. $d v w x $. $d v w y $. $d v w z $. $d v ph $.
|
|
hboprab.1 $e |- ( ph -> A. w ph ) $.
|
|
$( Bound-variable hypothesis builder for an operation class abstraction.
|
|
(Contributed by set.mm contributors, 22-Aug-2013.) $)
|
|
hboprab $p |- ( u e. { <. <. x , y >. , z >. | ph } ->
|
|
A. w u e. { <. <. x , y >. , z >. | ph } ) $=
|
|
( vv coprab cv cop wceq wa wex cab df-oprab ax-17 hban hbex hbab hbxfreq
|
|
) EFABCDIHJBJCJKDJKLZAMZDNZCNZBNZHOABCDHPUFEHFUEEBUDECUCEDUBAEUBEQGRSSSTU
|
|
A $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $. $d w ps $. $d w ch $.
|
|
oprabbid.1 $e |- F/ x ph $.
|
|
oprabbid.2 $e |- F/ y ph $.
|
|
oprabbid.3 $e |- F/ z ph $.
|
|
oprabbid.4 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal operation class abstractions (deduction
|
|
rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro,
|
|
24-Jun-2014.) $)
|
|
oprabbid $p |- ( ph ->
|
|
{ <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } ) $=
|
|
( vw cv cop wceq wa wex cab coprab exbid df-oprab anbi2d abbidv 3eqtr4g )
|
|
AKLDLELMFLMNZBOZFPZEPZDPZKQUDCOZFPZEPZDPZKQBDEFRCDEFRAUHULKAUGUKDGAUFUJEH
|
|
AUEUIFIABCUDJUASSSUBBDEFKTCDEFKTUC $.
|
|
$}
|
|
|
|
${
|
|
$d x z ph $. $d y z ph $.
|
|
oprabbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
|
|
$( Equivalent wff's yield equal operation class abstractions (deduction
|
|
rule). (Contributed by NM, 21-Feb-2004.) $)
|
|
oprabbidv $p |- ( ph ->
|
|
{ <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } ) $=
|
|
( nfv oprabbid ) ABCDEFADHAEHAFHGI $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $. $d w ps $.
|
|
oprabbii.1 $e |- ( ph <-> ps ) $.
|
|
$( Equivalent wff's yield equal operation class abstractions. (Unnecessary
|
|
distinct variable restrictions were removed by David Abernethy,
|
|
19-Jun-2012.) (Contributed by set.mm contributors, 28-May-1995.)
|
|
(Revised by set.mm contributors, 24-Jul-2012.) $)
|
|
oprabbii $p |- { <. <. x , y >. , z >. | ph }
|
|
= { <. <. x , y >. , z >. | ps } $=
|
|
( vw cv wceq coprab eqid wb a1i oprabbidv ax-mp ) GHZPIZACDEJBCDEJIPKQABC
|
|
DEABLQFMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Two ways to state the domain of an operation. (Contributed by FL,
|
|
24-Jan-2010.) $)
|
|
oprab4 $p |-
|
|
{ <. <. x , y >. , z >. | ( <. x , y >. e. ( A X. B ) /\ ph ) } =
|
|
{ <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } $=
|
|
( cv cop cxp wcel wa opelxp anbi1i oprabbii ) BGZCGZHEFIJZAKOEJPFJKZAKBCD
|
|
QRAOPEFLMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v $. $d v ph $. $d v ps $.
|
|
cbvoprab1.1 $e |- F/ w ph $.
|
|
cbvoprab1.2 $e |- F/ x ps $.
|
|
cbvoprab1.3 $e |- ( x = w -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change first bound variable in an operation abstraction,
|
|
using implicit substitution. (Contributed by NM, 20-Dec-2008.)
|
|
(Revised by Mario Carneiro, 5-Dec-2016.) $)
|
|
cbvoprab1 $p |- { <. <. x , y >. , z >. | ph }
|
|
= { <. <. w , y >. , z >. | ps } $=
|
|
( vv cv cop wceq wa wex copab coprab nfv nfan nfex eqeq2d anbi12d opabbii
|
|
opeq1 exbidv cbvex dfoprab2 3eqtr4i ) JKZCKZDKZLZMZANZDOZCOZJEPUIFKZUKLZM
|
|
ZBNZDOZFOZJEPACDEQBFDEQUPVBJEUOVACFUNFDUMAFUMFRGSTUTCDUSBCUSCRHSTUJUQMZUN
|
|
UTDVCUMUSABVCULURUIUJUQUKUDUAIUBUEUFUCACDEJUGBFDEJUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d v w x y z $. $d ph v $. $d ps v $.
|
|
cbvoprab2.1 $e |- F/ w ph $.
|
|
cbvoprab2.2 $e |- F/ y ps $.
|
|
cbvoprab2.3 $e |- ( y = w -> ( ph <-> ps ) ) $.
|
|
$( Change the second bound variable in an operation abstraction.
|
|
(Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro,
|
|
11-Dec-2016.) $)
|
|
cbvoprab2 $p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. <. x , w >. , z >. | ps } $=
|
|
( vv cv cop wceq wa wex cab coprab nfv nfan nfex opeq2 opeq1d cbvex exbii
|
|
eqeq2d anbi12d exbidv abbii df-oprab 3eqtr4i ) JKZCKZDKZLZEKZLZMZANZEOZDO
|
|
ZCOZJPUKULFKZLZUOLZMZBNZEOZFOZCOZJPACDEQBCFEQVAVIJUTVHCUSVGDFURFEUQAFUQFR
|
|
GSTVFDEVEBDVEDRHSTUMVBMZURVFEVJUQVEABVJUPVDUKVJUNVCUOUMVBULUAUBUEIUFUGUCU
|
|
DUHACDEJUIBCFEJUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u $. $d u ph $. $d u ps $.
|
|
cbvoprab12.1 $e |- F/ w ph $.
|
|
cbvoprab12.2 $e |- F/ v ph $.
|
|
cbvoprab12.3 $e |- F/ x ps $.
|
|
cbvoprab12.4 $e |- F/ y ps $.
|
|
cbvoprab12.5 $e |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change first two bound variables in an operation
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
|
|
cbvoprab12 $p |- { <. <. x , y >. , z >. | ph }
|
|
= { <. <. w , v >. , z >. | ps } $=
|
|
( vu cv cop wceq wa wex nfv nfan coprab weq opeq12 eqeq2d anbi12d opabbii
|
|
copab cbvex2 dfoprab2 3eqtr4i ) MNZCNZDNZOZPZAQZDRCRZMEUGUKFNZGNZOZPZBQZG
|
|
RFRZMEUGACDEUABFGEUAUQVCMEUPVBCDFGUOAFUOFSHTUOAGUOGSITVABCVACSJTVABDVADSK
|
|
TCFUBDGUBQZUOVAABVDUNUTUKULURUMUSUCUDLUEUHUFACDEMUIBFGEMUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v $. $d w v ph $. $d x y ps $.
|
|
cbvoprab12v.1 $e |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change first two bound variables in an operation
|
|
abstraction, using implicit substitution. (Contributed by set.mm
|
|
contributors, 8-Oct-2004.) $)
|
|
cbvoprab12v $p |- { <. <. x , y >. , z >. | ph }
|
|
= { <. <. w , v >. , z >. | ps } $=
|
|
( nfv cbvoprab12 ) ABCDEFGAFIAGIBCIBDIHJ $.
|
|
$}
|
|
|
|
${
|
|
$d x z w v $. $d y z w v $. $d v ph $. $d v ps $.
|
|
cbvoprab3.1 $e |- F/ w ph $.
|
|
cbvoprab3.2 $e |- F/ z ps $.
|
|
cbvoprab3.3 $e |- ( z = w -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change the third bound variable in an operation
|
|
abstraction, using implicit substitution. (Contributed by NM,
|
|
22-Aug-2013.) $)
|
|
cbvoprab3 $p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. <. x , y >. , w >. | ps } $=
|
|
( vv cv wceq wa wex copab coprab nfv nfan nfex dfoprab2 2exbidv cbvopab2
|
|
cop anbi2d 3eqtr4i ) JKCKDKUCLZAMZDNZCNZJEOUFBMZDNZCNZJFOACDEPBCDFPUIULJE
|
|
FUHFCUGFDUFAFUFFQGRSSUKECUJEDUFBEUFEQHRSSEKFKLZUGUJCDUMABUFIUDUAUBACDEJTB
|
|
CDFJTUE $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $. $d z ps $.
|
|
cbvoprab3v.1 $e |- ( z = w -> ( ph <-> ps ) ) $.
|
|
$( Rule used to change the third bound variable in an operation
|
|
abstraction, using implicit substitution. (Unnecessary distinct
|
|
variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|
|
(Contributed by set.mm contributors, 8-Oct-2004.) (Revised by set.mm
|
|
contributors, 24-Jul-2012.) $)
|
|
cbvoprab3v $p |- { <. <. x , y >. , z >. | ph } =
|
|
{ <. <. x , y >. , w >. | ps } $=
|
|
( nfv cbvoprab3 ) ABCDEFAFHBEHGI $.
|
|
$}
|
|
|
|
${
|
|
elimdelov.1 $e |- ( ph -> C e. ( A F B ) ) $.
|
|
elimdelov.2 $e |- Z e. ( X F Y ) $.
|
|
$( Eliminate a hypothesis which is a predicate expressing membership in the
|
|
result of an operator (deduction version). (Contributed by Paul
|
|
Chapman, 25-Mar-2008.) $)
|
|
elimdelov $p |- if ( ph , C , Z ) e.
|
|
( if ( ph , A , X ) F if ( ph , B , Y ) ) $=
|
|
( cif co wcel iftrue eqeltrd oveq12d eleqtrrd wn iffalse syl6eqel pm2.61i
|
|
) AADHKZABFKZACGKZELZMAUBBCELZUEAUBDUFADHNIOAUCBUDCEABFNACGNPQARZUBFGELZU
|
|
EUGUBHUHADHSJTUGUCFUDGEABFSACGSPQUA $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $.
|
|
$( The domain of an operation class abstraction. (Unnecessary distinct
|
|
variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|
|
(Contributed by set.mm contributors, 17-Mar-1995.) (Revised by set.mm
|
|
contributors, 24-Jul-2012.) $)
|
|
dmoprab $p |- dom { <. <. x , y >. , z >. | ph } =
|
|
{ <. x , y >. | E. z ph } $=
|
|
( vw coprab cdm cv cop wceq wa wex copab cab dfoprab2 dmeqi dmopab exrot3
|
|
19.42v 2exbii bitri abbii df-opab eqtr4i 3eqtri ) ABCDFZGEHBHCHIJZAKZCLBL
|
|
ZEDMZGUIDLZENZADLZBCMZUFUJABCDEOPUIEDQULUGUMKZCLBLZENUNUKUPEUKUHDLZCLBLUP
|
|
UHDBCRUQUOBCUGADSTUAUBUMBCEUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $.
|
|
$( The domain of an operation class abstraction. (Contributed by set.mm
|
|
contributors, 24-Aug-1995.) $)
|
|
dmoprabss $p |- dom { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ ph ) } C_ ( A X. B ) $=
|
|
( cv wcel wa coprab cdm wex copab dmoprab 19.42v opabbii opabssxp eqsstri
|
|
cxp ) BGEHCGFHIZAIZBCDJKUADLZBCMZEFSZUABCDNUCTADLZIZBCMUDUBUFBCTADOPUEBCE
|
|
FQRR $.
|
|
$}
|
|
|
|
${
|
|
$d x z w $. $d y z w $. $d w ph $.
|
|
$( The range of an operation class abstraction. (Unnecessary distinct
|
|
variable restrictions were removed by David Abernethy, 19-Apr-2013.)
|
|
(Contributed by set.mm contributors, 30-Aug-2004.) (Revised by set.mm
|
|
contributors, 19-Apr-2013.) $)
|
|
rnoprab $p |- ran { <. <. x , y >. , z >. | ph } =
|
|
{ z | E. x E. y ph } $=
|
|
( vw coprab crn cv cop wceq wa wex copab cab dfoprab2 rneqi rnopab exrot3
|
|
vex bitri 19.41v opex isseti biantrur bicomi 2exbii abbii 3eqtri ) ABCDFZ
|
|
GEHBHZCHZIZJZAKZCLBLZEDMZGUOELZDNACLBLZDNUIUPABCDEOPUOEDQUQURDUQUNELZCLBL
|
|
URUNEBCRUSABCUSUMELZAKZAUMAEUAAVAUTAEULUJUKBSCSUBUCUDUETUFTUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d x y z $.
|
|
$( The range of a restricted operation class abstraction. (Contributed by
|
|
Scott Fenton, 21-Mar-2012.) $)
|
|
rnoprab2 $p |- ran { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ ph ) } =
|
|
{ z | E. x e. A E. y e. B ph } $=
|
|
( cv wcel wa coprab crn wex cab wrex rnoprab r2ex abbii eqtr4i ) BGEHCGFH
|
|
IAIZBCDJKSCLBLZDMACFNBENZDMSBCDOUATDABCEFPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w C $. $d w ph $.
|
|
$d x y z w ps $.
|
|
eloprabga.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
$( The law of concretion for operation class abstraction. Compare
|
|
~ elopab . (Contributed by set.mm contributors, 17-Dec-2013.) (Revised
|
|
by David Abernethy, 18-Jun-2012.) Removed unnecessary distinct variable
|
|
requirements. (Revised by Mario Carneiro, 19-Dec-2013.) $)
|
|
eloprabga $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) ) $=
|
|
( vw wcel cvv cop wb wa wceq wex coprab w3a opexg sylan 3impa wi cv eqeq1
|
|
elex eqcom opth anbi1i df-3an 3bitr4i bitri syl6bb anbi1d pm5.32i 3exbidv
|
|
adantl cab df-oprab eleq2i bitr2i eleq1 syl5bb isset eeeanv bitr4i biimpi
|
|
3anbi123i biantrurd 19.41vvv syl6rbbr adantr 3bitr3d expcom vtocleg mpcom
|
|
abid syl3an ) FINFONZGJNGONZHKNHONZFGPZHPZACDEUAZNZBQZFIUIGJUIHKUIWFONZWB
|
|
WCWDUBZWIWBWCWDWJWBWCRWEONWDWJFGOOUCWEHOOUCUDUEWKWIUFMWFOWKMUGZWFSZWIWKWM
|
|
RWLCUGZDUGZPZEUGZPZSZARZETDTCTZWNFSZWOGSZWQHSZUBZBRZETDTCTZWHBWMXAXGQWKWM
|
|
WTXFCDEWMWTXEARXFWMWSXEAWMWSWFWRSZXEWLWFWRUHXHWRWFSZXEWFWRUJWPWESZXDRXBXC
|
|
RZXDRXIXEXJXKXDWNWOFGUKULWPWQWEHUKXBXCXDUMUNUOUPUQXEABLURUPUSUTWMXAWHQWKX
|
|
AWLWGNZWMWHXLWLXAMVAZNXAWGXMWLACDEMVBVCXAMVTVDWLWFWGVEVFUTWKXGBQWMWKBXEET
|
|
DTCTZBRXGWKXNBWKXNWKXBCTZXCDTZXDETZUBXNWBXOWCXPWDXQCFVGDGVGEHVGVKXBXCXDCD
|
|
EVHVIVJVLXEBCDEVMVNVOVPVQVRVSWA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z th $.
|
|
eloprabg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
eloprabg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
eloprabg.3 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
$( The law of concretion for operation class abstraction. Compare
|
|
~ elopab . (Contributed by set.mm contributors, 14-Sep-1999.) (Revised
|
|
by David Abernethy, 19-Jun-2012.) Removed unnecessary distinct variable
|
|
requirements. (Revised by set.mm contributors, 19-Dec-2013.) $)
|
|
eloprabg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) ) $=
|
|
( cv wceq syl3an9b eloprabga ) ADEFGHIJKLMEQHRABFQIRCGQJRDNOPST $.
|
|
$}
|
|
|
|
${
|
|
$d ph w $. $d ps w $. $d x z w $. $d y z w $.
|
|
ssoprab2i.1 $e |- ( ph -> ps ) $.
|
|
$( Inference of operation class abstraction subclass from implication.
|
|
(Unnecessary distinct variable restrictions were removed by David
|
|
Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors,
|
|
11-Nov-1995.) (Revised by set.mm contributors, 24-Jul-2012.) $)
|
|
ssoprab2i $p |- { <. <. x , y >. , z >. | ph } C_
|
|
{ <. <. x , y >. , z >. | ps } $=
|
|
( vw cv cop wceq wex copab coprab anim2i 2eximi ssopab2i dfoprab2 3sstr4i
|
|
wa ) GHCHDHIJZASZDKCKZGELTBSZDKCKZGELACDEMBCDEMUBUDGEUAUCCDABTFNOPACDEGQB
|
|
CDEGQR $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w ph $.
|
|
$( Restriction of an operation class abstraction. (Contributed by set.mm
|
|
contributors, 10-Feb-2007.) $)
|
|
resoprab $p |- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) =
|
|
{ <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } $=
|
|
( vw cv cop wceq wex copab cxp cres wcel coprab resopab 19.42vv dfoprab2
|
|
wa eleq1 opelxp syl6bb anbi1d pm5.32i bitri 2exbii bitr3i opabbii reseq1i
|
|
an12 eqtri 3eqtr4i ) GHZBHZCHZIZJZATZCKBKZGDLZEFMZNZURUOEOUPFOTZATZTZCKBK
|
|
ZGDLZABCDPZVBNVEBCDPVCUNVBOZUTTZGDLVHUTGDVBQVKVGGDVKVJUSTZCKBKVGVJUSBCRVL
|
|
VFBCVLURVJATZTVFVJURAUKURVMVEURVJVDAURVJUQVBOVDUNUQVBUAUOUPEFUBUCUDUEUFUG
|
|
UHUIULVIVAVBABCDGSUJVEBCDGSUM $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $.
|
|
$( Restriction of an operator abstraction. (Contributed by Jeff Madsen,
|
|
2-Sep-2009.) $)
|
|
resoprab2 $p |- ( ( C C_ A /\ D C_ B ) -> ( { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ ph ) } |` ( C X. D ) ) =
|
|
{ <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ph ) } ) $=
|
|
( wss wa cv wcel coprab cxp wi wb ssel pm4.71 sylib bicomd resoprab anass
|
|
cres an4 bi2anan9 syl5bb anbi1d syl5bbr oprabbidv syl5eq ) GEIZHFIZJZBKZE
|
|
LZCKZFLZJZAJZBCDMGHNUCUNGLZUPHLZJZUSJZBCDMVBAJZBCDMUSBCDGHUAUMVCVDBCDVCVB
|
|
URJZAJUMVDVBURAUBUMVEVBAVEUTUOJZVAUQJZJUMVBUTVAUOUQUDUKVFUTULVGVAUKUTVFUK
|
|
UTUOOUTVFPGEUNQUTUORSTULVAVGULVAUQOVAVGPHFUPQVAUQRSTUEUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $. $d w ph $.
|
|
$( "At most one" is a sufficient condition for an operation class
|
|
abstraction to be a function. (Contributed by set.mm contributors,
|
|
28-Aug-2007.) $)
|
|
funoprabg $p |- ( A. x A. y E* z ph ->
|
|
Fun { <. <. x , y >. , z >. | ph } ) $=
|
|
( vw wmo wal cv cop wceq wa wex coprab wfun mosubopt alrimiv copab funeqi
|
|
dfoprab2 funopab bitr2i sylib ) ADFCGBGZEHZBHCHIJAKCLBLZDFZEGZABCDMZNZUCU
|
|
FEADBCUDOPUIUEEDQZNUGUHUJABCDESRUEEDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
funoprab.1 $e |- E* z ph $.
|
|
$( "At most one" is a sufficient condition for an operation class
|
|
abstraction to be a function. (Contributed by set.mm contributors,
|
|
17-Mar-1995.) $)
|
|
funoprab $p |- Fun { <. <. x , y >. , z >. | ph } $=
|
|
( wmo wal coprab wfun gen2 funoprabg ax-mp ) ADFZCGBGABCDHIMBCEJABCDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
$( Functionality and domain of an operation class abstraction.
|
|
(Contributed by set.mm contributors, 28-Aug-2007.) $)
|
|
fnoprabg $p |- ( A. x A. y ( ph -> E! z ps ) ->
|
|
{ <. <. x , y >. , z >. | ( ph /\ ps ) } Fn { <. x , y >. | ph } ) $=
|
|
( weu wi wal wa coprab wfun cdm copab wceq wfn wmo eumo imim2i wex sps wb
|
|
moanimv sylibr 2alimi funoprabg syl dmoprab nfa1 nfa2 simpl exlimiv ancld
|
|
euex 19.42v syl6ibr impbid2 opabbid syl5eq df-fn sylanbrc ) ABEFZGZDHZCHZ
|
|
ABIZCDEJZKZVFLZACDMZNVFVIOVDVEEPZDHCHVGVBVJCDVBABEPZGVJVAVKABEQRABEUBUCUD
|
|
VECDEUEUFVDVHVEESZCDMVIVECDEUGVDVLACDVCCUHVBDCUIVCVLAUAZCVBVMDVBVLAVEAEAB
|
|
UJUKVBAABESZIVLVBAVNVAVNABEUMRULABEUNUOUPTTUQURVFVIUSUT $.
|
|
$}
|
|
|
|
|
|
${
|
|
$d x y z $. $d z ph $.
|
|
fnoprab.1 $e |- ( ph -> E! z ps ) $.
|
|
$( Functionality and domain of an operation class abstraction.
|
|
(Contributed by set.mm contributors, 15-May-1995.) $)
|
|
fnoprab $p |- { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn
|
|
{ <. x , y >. | ph } $=
|
|
( weu wi wal wa coprab copab wfn gen2 fnoprabg ax-mp ) ABEGHZDICIABJCDEKA
|
|
CDLMQCDFNABCDEOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y w A $. $d x y w B $. $d x y w C $. $d x y w F $.
|
|
$( An operation maps to a class to which all values belong. (Contributed
|
|
by set.mm contributors, 7-Feb-2004.) $)
|
|
ffnov $p |- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\
|
|
A. x e. A A. y e. B ( x F y ) e. C ) ) $=
|
|
( vw cxp wf wfn cv cfv wcel wral wa co ffnfv cop wceq fveq2 df-ov syl6eqr
|
|
eleq1d ralxp anbi2i bitri ) CDHZEFIFUGJZGKZFLZEMZGUGNZOUHAKZBKZFPZEMZBDNA
|
|
CNZOGUGEFQULUQUHUKUPGABCDUIUMUNRZSZUJUOEUSUJURFLUOUIURFTUMUNFUAUBUCUDUEUF
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y F $. $d x y R $. $d x y S $.
|
|
fovcl.1 $e |- F : ( R X. S ) --> C $.
|
|
$( Closure law for an operation. (Contributed by set.mm contributors,
|
|
19-Apr-2007.) $)
|
|
fovcl $p |- ( ( A e. R /\ B e. S ) -> ( A F B ) e. C ) $=
|
|
( vx vy wcel wa cv co wral cxp wf wfn ffnov wceq eleq1d ax-mp oveq1 oveq2
|
|
simprbi rspc2v mpi ) ADJBEJKHLZILZFMZCJZIENHDNZABFMZCJZDEOZCFPZUKGUOFUNQU
|
|
KHIDECFRUDUAUJUMAUHFMZCJHIABDEUGASUIUPCUGAUHFUBTUHBSUPULCUHBAFUCTUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d z C $. $d z D $. $d x y z F $.
|
|
$d x y z G $.
|
|
$( Equality of two operations is determined by their values. (Contributed
|
|
by set.mm contributors, 1-Sep-2005.) $)
|
|
eqfnov $p |- ( ( F Fn ( A X. B ) /\ G Fn ( C X. D ) ) -> ( F = G <->
|
|
( ( A X. B ) = ( C X. D ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) ) $=
|
|
( vz cxp wfn wa wceq cv cfv wral co eqfnfv2 fveq2 df-ov cop eqeq12d ralxp
|
|
eqeq12i syl6bbr anbi2i syl6bb ) GCDJZKHEFJZKLGHMUHUIMZINZGOZUKHOZMZIUHPZL
|
|
UJANZBNZGQZUPUQHQZMZBDPACPZLIUHUIGHRUOVAUJUNUTIABCDUKUPUQUAZMZUNVBGOZVBHO
|
|
ZMUTVCULVDUMVEUKVBGSUKVBHSUBURVDUSVEUPUQGTUPUQHTUDUEUCUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d F x y $. $d G x y $.
|
|
$( Two operators with the same domain are equal iff their values at each
|
|
point in the domain are equal. (Contributed by Jeff Madsen,
|
|
7-Jun-2010.) $)
|
|
eqfnov2 $p |- ( ( F Fn ( A X. B ) /\ G Fn ( A X. B ) ) ->
|
|
( F = G <-> A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) $=
|
|
( cxp wfn wa wceq cv co wral eqfnov simpr eqidd ancri impbii syl6bb ) ECD
|
|
GZHFTHIEFJTTJZAKZBKZELUBUCFLJBDMACMZIZUDABCDCDEFNUEUDUAUDOUDUAUDTPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d x y z w B $. $d x y z w F $.
|
|
$( Representation of an operation class abstraction in terms of its
|
|
values. (Contributed by set.mm contributors, 7-Feb-2004.) $)
|
|
fnov $p |- ( F Fn ( A X. B ) <-> F = { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) } ) $=
|
|
( vw cxp wfn cv wcel cfv wceq wa copab co coprab dffn5 wex bitri cop elxp
|
|
anbi1i 19.41vv anass syl6eqr eqeq2d anbi2d pm5.32i 2exbii 3bitr2i opabbii
|
|
fveq2 df-ov dfoprab2 eqtr4i eqeq2i ) FDEHZIFGJZURKZCJZUSFLZMZNZGCOZMFAJZD
|
|
KBJZEKNZVAVFVGFPZMZNZABCQZMGCURFRVEVLFVEUSVFVGUAZMZVKNZBSASZGCOVLVDVPGCVD
|
|
VNVHNZBSASZVCNVQVCNZBSASVPUTVRVCABUSDEUBUCVQVCABUDVSVOABVSVNVHVCNZNVOVNVH
|
|
VCUEVNVTVKVNVCVJVHVNVBVIVAVNVBVMFLVIUSVMFUMVFVGFUNUFUGUHUITUJUKULVKABCGUO
|
|
UPUQT $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z F $.
|
|
$( Representation of an operation class abstraction in terms of its
|
|
values. (Contributed by set.mm contributors, 7-Feb-2004.) $)
|
|
fov $p |- ( F : ( A X. B ) --> C <-> ( F = { <. <. x , y >. , z >. |
|
|
( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) }
|
|
/\ A. x e. A A. y e. B ( x F y ) e. C ) ) $=
|
|
( cxp wf wfn cv co wcel wral wa wceq coprab ffnov fnov anbi1i bitri ) DEH
|
|
ZFGIGUBJZAKZBKZGLZFMBENADNZOGUDDMUEEMOCKUFPOABCQPZUGOABDEFGRUCUHUGABCDEGS
|
|
TUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
ovidig.1 $e |- E* z ph $.
|
|
ovidig.2 $e |- F = { <. <. x , y >. , z >. | ph } $.
|
|
$( The value of an operation class abstraction. Compare ~ ovidi . The
|
|
condition ` ( x e. R /\ y e. S ) ` is been removed. (Contributed by
|
|
Mario Carneiro, 29-Dec-2014.) $)
|
|
ovidig $p |- ( ph -> ( x F y ) = z ) $=
|
|
( cv co cop cfv df-ov wcel wceq coprab eleq2i oprabid bitri wfun wi mpbir
|
|
funoprab funeqi funopfv ax-mp sylbir syl5eq ) ABHZCHZEIUHUIJZEKZDHZUHUIEL
|
|
AUJULJZEMZUKULNZUNUMABCDOZMAEUPUMGPABCDQRESZUNUOTUQUPSABCDFUBEUPGUCUAUJUL
|
|
EUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z R $. $d z S $.
|
|
ovidi.2 $e |- ( ( x e. R /\ y e. S ) -> E* z ph ) $.
|
|
ovidi.3 $e |- F =
|
|
{ <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
|
|
$( The value of an operation class abstraction (weak version).
|
|
(Contributed by Mario Carneiro, 29-Dec-2014.) $)
|
|
ovidi $p |- ( ( x e. R /\ y e. S ) -> ( ph -> ( x F y ) = z ) ) $=
|
|
( cv wcel wa co wceq wmo wi moanimv mpbir ovidig ex ) BJZEKCJZFKLZAUAUBGM
|
|
DJNUCALZBCDGUDDOUCADOPHUCADQRIST $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z R $. $d x y z S $.
|
|
$d x y z th $.
|
|
ov.1 $e |- C e. _V $.
|
|
ov.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ov.3 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ov.4 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ov.5 $e |- ( ( x e. R /\ y e. S ) -> E! z ph ) $.
|
|
ov.6 $e |- F =
|
|
{ <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
|
|
$( The value of an operation class abstraction. (Unnecessary distinct
|
|
variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|
|
(Contributed by set.mm contributors, 16-May-1995.) (Revised by set.mm
|
|
contributors, 24-Jul-2012.) $)
|
|
ov $p |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) ) $=
|
|
( wcel wa co wceq cop cv coprab cfv df-ov fveq1i eqtri eqeq1i wfn fnoprab
|
|
copab eleq1 anbi1d anbi2d opelopabg ibir fnopfvb sylancr anbi12d eloprabg
|
|
wb cvv mp3an3 bitrd syl5bb bianabs ) HKTZILTZUAZHIMUBZJUCZDVNHIUDZEUEZKTZ
|
|
FUEZLTZUAZAUAZEFGUFZUGZJUCZVLVLDUAZVMWCJVMVOMUGWCHIMUHVOMWBSUIUJUKVLWDVOJ
|
|
UDWBTZWEVLWBVTEFUNZULVOWGTZWDWFVDVTAEFGRUMVLWHVTVJVSUAZVLEFHIKLVPHUCZVQVJ
|
|
VSVPHKUOUPZVRIUCZVSVKVJVRILUOUQZURUSWGVOJWBUTVAVJVKJVETWFWEVDNWAWIBUAVLCU
|
|
AWEEFGHIJKLVEWJVTWIABWKOVBWLWIVLBCWMPVBGUEJUCCDVLQUQVCVFVGVHVI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z ps $.
|
|
ovigg.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
ovigg.4 $e |- E* z ph $.
|
|
ovigg.5 $e |- F = { <. <. x , y >. , z >. | ph } $.
|
|
$( The value of an operation class abstraction. Compare ~ ovig . The
|
|
condition ` ( x e. R /\ y e. S ) ` is been removed. (Contributed by FL,
|
|
24-Mar-2007.) $)
|
|
ovigg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( ps -> ( A F B ) = C ) ) $=
|
|
( wcel w3a cop cfv wceq coprab co eloprabga wfun funoprab funopfv syl6bir
|
|
wi ax-mp df-ov fveq1i eqtri eqeq1i syl6ibr ) FJPGKPHLPQZBFGRZACDEUAZSZHTZ
|
|
FGIUBZHTUOBUPHRUQPZUSABCDEFGHJKLMUCUQUDVAUSUHACDENUEUPHUQUFUIUGUTURHUTUPI
|
|
SURFGIUJUPIUQOUKULUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z R $. $d x y z S $.
|
|
$d x y z ps $.
|
|
ovig.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $.
|
|
ovig.2 $e |- ( ( x e. R /\ y e. S ) -> E* z ph ) $.
|
|
ovig.3 $e |- F =
|
|
{ <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $.
|
|
$( The value of an operation class abstraction (weak version).
|
|
(Contributed by set.mm contributors, 14-Sep-1999.) (Unnecessary
|
|
distinct variable restrictions were removed by David Abernethy,
|
|
19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) $)
|
|
ovig $p |- ( ( A e. R /\ B e. S /\ C e. D ) ->
|
|
( ps -> ( A F B ) = C ) ) $=
|
|
( wcel w3a wa wceq cv co 3simpa wb eleq1 bi2anan9 3adant3 anbi12d moanimv
|
|
wmo wi mpbir ovigg mpand ) FJPZGKPZHIPZQUNUORZBFGLUAHSUNUOUPUBCTZJPZDTZKP
|
|
ZRZARZUQBRCDEFGHLJKIURFSZUTGSZETHSZQVBUQABVDVEVBUQUCVFVDUSUNVEVAUOURFJUDU
|
|
TGKUDUEUFMUGVCEUIVBAEUIUJNVBAEUHUKOULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $. $d z R $.
|
|
$d x y z S $.
|
|
ov2ag.1 $e |- ( ( x = A /\ y = B ) -> R = S ) $.
|
|
ov2ag.3 $e |- F = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D )
|
|
/\ z = R ) } $.
|
|
$( The value of an operation class abstraction. Special case.
|
|
(Contributed by Mario Carneiro, 19-Dec-2013.) $)
|
|
ov2ag $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
|
|
( wcel w3a wceq co eqid cv simp3 3adant3 eqeq12d wmo wa moeq a1i ovig mpi
|
|
) DFNEGNIKNOIIPZDEJQIPIRCSZHPZUIABCDEIKFGJASZDPZBSZEPZUJIPZOUJIHIUMUOUPTU
|
|
MUOHIPUPLUAUBUKCUCULFNUNGNUDCHUEUFMUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d f u v w x y z A $. $d f u v w x y z B $. $d x y z R $.
|
|
$d f u v w y z C $. $d f u v w y z D $. $d f u v w x y z H $.
|
|
$d f u v w z S $.
|
|
ov3.1 $e |- S e. _V $.
|
|
ov3.2 $e |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) ->
|
|
R = S ) $.
|
|
ov3.3 $e |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\
|
|
y e. ( H X. H ) ) /\
|
|
E. w E. v E. u E. f ( ( x = <. w , v >. /\
|
|
y = <. u , f >. ) /\ z = R ) ) } $.
|
|
$( The value of an operation class abstraction. Special case.
|
|
(Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro,
|
|
29-Dec-2014.) $)
|
|
ov3 $p |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) ->
|
|
( <. A , B >. F <. C , D >. ) = S ) $=
|
|
( wa wex wcel cv wceq cop isseti nfv nfcv cxp coprab nfoprab3 nfcxfr nfov
|
|
co nfeq1 eqeq2d copsex4g wi opelxp nfoprab1 nfoprab2 eqeq1 anbi1d 4exbidv
|
|
nfim oveq1 eqeq1d imbi12d anbi2d oveq2 mosubop anass 2exbii 19.42vv bitri
|
|
wmo moeq mobii mpbir ovidi vtocl2gaf syl2anbr sylbird eqeq2 mpbidi exlimd
|
|
a1i mpi ) GOUAHOUASZIOUAJOUASZSZCUBZLUCZCTGHUDZIJUDZNUMZLUCZCLPUEWJWLWPCW
|
|
JCUFCWOLCWMWNNCWMUGCNAUBZOOUHZUABUBZWRUASZWQDUBZEUBZUDZUCZWSFUBZMUBZUDZUC
|
|
ZSZWKKUCZSZMTFTZETDTZSZABCUIZRXNABCUJUKCWNUGULUNWLWOWKUCZWPWJWJWLWMXCUCZW
|
|
NXGUCZSZXJSZMTFTETDTZXPXJWLDEFMGHIJOOXAGUCXBHUCSXEIUCXFJUCSSKLWKQUOUPWHWM
|
|
WRUAWNWRUAYAXPUQZWIGHOOURIJOOURXMWQWSNUMZWKUCZUQXQXHSZXJSZMTFTETDTZWMWSNU
|
|
MZWKUCZUQYBABWMWNWRWRAWMUGZBWMUGZBWNUGZYGYIAYGAUFAYHWKAWMWSNYJANXORXNABCU
|
|
SUKAWSUGULUNVDYAXPBYABUFBWOWKBWMWNNYKBNXORXNABCUTUKYLULUNVDWQWMUCZXMYGYDY
|
|
IYMXKYFDEFMYMXIYEXJYMXDXQXHWQWMXCVAVBVBVCYMYCYHWKWQWMWSNVEVFVGWSWNUCZYGYA
|
|
YIXPYNYFXTDEFMYNYEXSXJYNXHXRXQWSWNXGVAVHVBVCYNYHWOWKWSWNWMNVIVFVGXMABCWRW
|
|
RNXMCVOZWTYOXDXHXJSZMTFTZSZETDTZCVOYQCDEWQXJCFMWSCKVPVJVJXMYSCXLYRDEXLXDY
|
|
PSZMTFTYRXKYTFMXDXHXJVKVLXDYPFMVMVNVLVQVRWFRVSVTWAWBWKLWOWCWDWEWG $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w x y z C $. $d w z R $.
|
|
$d w x y z S $.
|
|
ov6g.1 $e |- ( <. x , y >. = <. A , B >. -> R = S ) $.
|
|
ov6g.2 $e |- F = { <. <. x , y >. , z >. | ( <. x , y >. e. C
|
|
/\ z = R ) } $.
|
|
$( The value of an operation class abstraction. Special case.
|
|
(Contributed by set.mm contributors, 13-Nov-2006.) $)
|
|
ov6g $p |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J )
|
|
-> ( A F B ) = S ) $=
|
|
( vw wcel wa cv wceq wex cop w3a co cfv df-ov eqid biidd copsex2g 3adant3
|
|
mpbiri adantr wi eqeq1 anbi1d eqeq2d eqcoms pm5.32i syl6bb 2exbidv anbi2d
|
|
wb wmo moeq mosubop a1i coprab copab dfoprab2 eleq1 bitr3i 2exbii 19.42vv
|
|
an12 bitri opabbii 3eqtri fvopab3ig 3ad2antl3 mpd syl5eq ) DJPZEKPZDEUAZF
|
|
PZUBZHLPZQZDEIUCWCIUDZHDEIUEWGWCARZBRZUAZSZHHSZQZBTATZWHHSZWEWOWFWAWBWOWD
|
|
WAWBQWOWMHUFWMWMABDEJKWIDSWJESQWMUGUHUJUIUKWDWAWFWOWPULWBORZWKSZCRZGSZQZB
|
|
TATZWLWSHSZQZBTATWOOCWCHFLIWQWCSZXAXDABXEXAWLWTQXDXEWRWLWTWQWCWKUMUNWLWTX
|
|
CWTXCVAWKWCWKWCSGHWSMUOUPUQURUSXCXDWNABXCXCWMWLWSHHUMUTUSXBCVBWQFPZWTCABW
|
|
QCGVCVDVEIWKFPZWTQZABCVFWRXHQZBTATZOCVGXFXBQZOCVGNXHABCOVHXJXKOCXJXFXAQZB
|
|
TATXKXIXLABXIWRXFWTQZQXLWRXMXHWRXFXGWTWQWKFVIUNUQWRXFWTVMVJVKXFXAABVLVNVO
|
|
VPVQVRVSVT $.
|
|
$}
|
|
|
|
${
|
|
$d ps x $. $d ch x y $. $d th x y z $. $d ta x y $. $d R x y z $.
|
|
$d S x y z $. $d A x y z $. $d B x y z $. $d C x y z $.
|
|
ovg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
|
|
ovg.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
|
|
ovg.3 $e |- ( z = C -> ( ch <-> th ) ) $.
|
|
ovg.4 $e |- ( ( ta /\ ( x e. R /\ y e. S ) ) -> E! z ph ) $.
|
|
ovg.5 $e |- F = { <. <. x , y >. , z >. |
|
|
( ( x e. R /\ y e. S ) /\ ph ) } $.
|
|
$( The value of an operation class abstraction. (Contributed by Jeff
|
|
Madsen, 10-Jun-2010.) $)
|
|
ovg $p |- ( ( ta /\ ( A e. R /\ B e. S /\ C e. D ) )
|
|
-> ( ( A F B ) = C <-> th ) ) $=
|
|
( wcel w3a wa co wceq cop cv coprab cfv df-ov fveq1i eqtri eqeq1i cxp wfn
|
|
wb copab weu wi wal ex alrimivv fnoprabg syl fneq2i sylibr opelxp biimpri
|
|
df-xp 3adant3 fnopfvb syl2an anbi1d anbi12d anbi2d eloprabg adantl syl5bb
|
|
eleq1 bitrd biidd bianabs ) EIMUAZJNUAZKLUAZUBZUCZIJOUDZKUEZWCWDUCZDUCZDW
|
|
IIJUFZFUGZMUAZGUGZNUAZUCZAUCZFGHUHZUIZKUEZWGWKWHWTKWHWLOUIWTIJOUJWLOWSTUK
|
|
ULUMWGXAWLKUFWSUAZWKEWSMNUNZUOZWLXCUAZXAXBUPWFEWSWQFGUQZUOZXDEWQAHURZUSZG
|
|
UTFUTXGEXIFGEWQXHSVAVBWQAFGHVCVDXCXFWSFGMNVIVEVFWCWDXEWEXEWJIJMNVGVHVJXCW
|
|
LKWSVKVLWFXBWKUPEWRWCWPUCZBUCWJCUCWKFGHIJKMNLWMIUEZWQXJABXKWNWCWPWMIMVSVM
|
|
PVNWOJUEZXJWJBCXLWPWDWCWOJNVSVOQVNHUGKUECDWJRVOVPVQVTVRWFWKDUPZEWCWDXMWEW
|
|
JWKDWJWKWAWBVJVQVT $.
|
|
$}
|
|
|
|
$( The value of a restricted operation. (Contributed by FL, 10-Nov-2006.) $)
|
|
ovres $p |- ( ( A e. C /\ B e. D ) -> ( A ( F |` ( C X. D ) ) B )
|
|
= ( A F B ) ) $=
|
|
( wcel wa cop cxp cres co wceq opelxp cfv fvres df-ov 3eqtr4g sylbir ) ACFB
|
|
DFGABHZCDIZFZABETJZKZABEKZLABCDMUASUBNSENUCUDSTEOABUBPABEPQR $.
|
|
|
|
$( The value of a member of the domain of a subclass of an operation.
|
|
(Contributed by set.mm contributors, 23-Aug-2007.) $)
|
|
oprssov $p |- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\
|
|
( A e. C /\ B e. D ) ) -> ( A F B ) = ( A G B ) ) $=
|
|
( wfun cxp wfn wss w3a wcel wa cres co wceq ovres adantl cdm eqtr3d reseq2d
|
|
fndm 3ad2ant2 funssres 3adant2 oveqd adantr ) EGZFCDHZIZFEJZKZACLBDLMZMABEU
|
|
INZOZABEOZABFOZUMUOUPPULABCDEQRULUOUQPUMULUNFABULEFSZNZUNFUJUHUSUNPUKUJURUI
|
|
EUIFUBUAUCUHUKUSFPUJEFUDUETUFUGT $.
|
|
|
|
$( A operations's value belongs to its codomain. (Contributed by set.mm
|
|
contributors, 27-Aug-2006.) $)
|
|
fovrn $p |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) ->
|
|
( A F B ) e. C ) $=
|
|
( cxp wf wcel co wa cop opelxp cfv df-ov ffvelrn syl5eqel sylan2br 3impb )
|
|
DEGZCFHZADIZBEIZABFJZCIZUBUCKUAABLZTIZUEABDEMUAUGKUDUFFNCABFOTCUFFPQRS $.
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w z C $. $d w x y z F $.
|
|
$( The range of an operation expressed as a collection of the operation's
|
|
values. (Contributed by set.mm contributors, 29-Oct-2006.) $)
|
|
fnrnov $p |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B
|
|
z = ( x F y ) } ) $=
|
|
( vw cxp wfn crn cv cfv wceq wrex cab co fnrnfv cop fveq2 df-ov syl6eqr
|
|
eqeq2d rexxp abbii syl6eq ) FDEHZIFJCKZGKZFLZMZGUFNZCOUGAKZBKZFPZMZBENADN
|
|
ZCOGCUFFQUKUPCUJUOGABDEUHULUMRZMZUIUNUGURUIUQFLUNUHUQFSULUMFTUAUBUCUDUE
|
|
$.
|
|
|
|
$( An onto mapping of an operation expressed in terms of operation values.
|
|
(Contributed by set.mm contributors, 29-Oct-2006.) $)
|
|
foov $p |- ( F : ( A X. B ) -onto-> C <-> ( F : ( A X. B ) --> C /\
|
|
A. z e. C E. x e. A E. y e. B z = ( x F y ) ) ) $=
|
|
( vw cxp wfo wf cv cfv wceq wrex wral wa co dffo3 cop fveq2 df-ov syl6eqr
|
|
eqeq2d rexxp ralbii anbi2i bitri ) DEIZFGJUIFGKZCLZHLZGMZNZHUIOZCFPZQUJUK
|
|
ALZBLZGRZNZBEOADOZCFPZQHCUIFGSUPVBUJUOVACFUNUTHABDEULUQURTZNZUMUSUKVDUMVC
|
|
GMUSULVCGUAUQURGUBUCUDUEUFUGUH $.
|
|
$}
|
|
|
|
$( An operation's value belongs to its range. (Contributed by set.mm
|
|
contributors, 10-Feb-2007.) $)
|
|
fnovrn $p |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) ->
|
|
( C F D ) e. ran F ) $=
|
|
( cxp wfn wcel co crn cop opelxp cfv df-ov fnfvelrn syl5eqel sylan2br 3impb
|
|
wa ) EABFZGZCAHZDBHZCDEIZEJZHZUBUCSUACDKZTHZUFCDABLUAUHSUDUGEMUECDENTUGEOPQ
|
|
R $.
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $. $d x y z F $.
|
|
$( A member of an operation's range is a value of the operation.
|
|
(Contributed by set.mm contributors, 7-Feb-2007.) (Revised by Mario
|
|
Carneiro, 30-Jan-2014.) $)
|
|
ovelrn $p |- ( F Fn ( A X. B ) -> ( C e. ran F <->
|
|
E. x e. A E. y e. B C = ( x F y ) ) ) $=
|
|
( vz cxp wfn crn wcel cv co wceq wrex cab fnrnov eleq2d cvv rexlimivw
|
|
ovex eleq1 mpbiri eqeq1 2rexbidv elab3 syl6bb ) FCDHIZEFJZKEGLZALZBLZFMZN
|
|
ZBDOACOZGPZKEUMNZBDOZACOZUHUIUPEABGCDFQRUOUSGEURESKZACUQUTBDUQUTUMSKUKULF
|
|
UAEUMSUBUCTTUJENUNUQABCDUJEUMUDUEUFUG $.
|
|
|
|
$( Membership relation for the values of a function whose image is a
|
|
subclass. (Contributed by Mario Carneiro, 23-Dec-2013.) $)
|
|
funimassov $p |- ( ( Fun F /\ ( A X. B ) C_ dom F ) ->
|
|
( ( F " ( A X. B ) ) C_ C <-> A. x e. A A. y e. B ( x F y ) e. C ) ) $=
|
|
( vz wfun cxp cdm wss wa cima cv cfv wcel wral co funimass4 cop syl6eqr
|
|
wceq fveq2 df-ov eleq1d ralxp syl6bb ) FHCDIZFJKLFUHMEKGNZFOZEPZGUHQANZBN
|
|
ZFRZEPZBDQACQGUHEFSUKUOGABCDUIULUMTZUBZUJUNEUQUJUPFOUNUIUPFUCULUMFUDUAUEU
|
|
FUG $.
|
|
|
|
$( Operation value in an image. (Contributed by Mario Carneiro,
|
|
23-Dec-2013.) $)
|
|
ovelimab $p |- ( ( F Fn A /\ ( B X. C ) C_ A ) ->
|
|
( D e. ( F " ( B X. C ) ) <-> E. x e. B E. y e. C D = ( x F y ) ) ) $=
|
|
( vz wfn cxp wss wa cima wcel cv cfv wceq wrex co syl6bb fvelimab syl6eqr
|
|
cop fveq2 df-ov eqeq1d eqcom rexxp ) GCIDEJZCKLFGUIMNHOZGPZFQZHUIRFAOZBOZ
|
|
GSZQZBERADRHCUIFGUAULUPHABDEUJUMUNUCZQZULUOFQUPURUKUOFURUKUQGPUOUJUQGUDUM
|
|
UNGUEUBUFUOFUGTUHT $.
|
|
$}
|
|
|
|
${
|
|
oprvalconst2.1 $e |- C e. _V $.
|
|
$( The value of a constant operation. (Contributed by set.mm contributors,
|
|
5-Nov-2006.) $)
|
|
ovconst2 $p |- ( ( R e. A /\ S e. B ) ->
|
|
( R ( ( A X. B ) X. { C } ) S ) = C ) $=
|
|
( wcel wa cxp csn co cop cfv df-ov wceq opelxp fvconst2 sylbir syl5eq ) D
|
|
AGEBGHZDEABIZCJIZKDELZUBMZCDEUBNTUCUAGUDCODEABPUACUCFQRS $.
|
|
$}
|
|
|
|
${
|
|
$d x y S $. $d x y F $.
|
|
oprssdm.1 $e |- -. (/) e. S $.
|
|
oprssdm.2 $e |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S ) $.
|
|
$( Domain of closure of an operation. (Contributed by set.mm contributors,
|
|
24-Aug-1995.) $)
|
|
oprssdm $p |- ( S X. S ) C_ dom F $=
|
|
( cxp cdm cv cop wcel wa opelxp wn cfv c0 wceq ndmfv co df-ov eqeq1i nsyl
|
|
eleq1 mtbiri sylbir syl con4i sylbi relssi ) ABCCGZDHZAIZBIZJZUJKULCKUMCK
|
|
LZUNUKKZULUMCCMUPUOUPNUNDOZPQZUONUNDRURULUMDSZCKZUOURUSPQZUTNUSUQPULUMDTU
|
|
AVAUTPCKEUSPCUCUDUEFUBUFUGUHUI $.
|
|
$}
|
|
|
|
$( The value of an operation outside its domain. (Contributed by set.mm
|
|
contributors, 28-Mar-2008.) $)
|
|
ndmovg $p |- ( ( dom F = ( R X. S ) /\ -. ( A e. R /\ B e. S ) )
|
|
-> ( A F B ) = (/) ) $=
|
|
( cdm cxp wceq wcel wa wn co cop cfv c0 df-ov eleq2 opelxp syl6bb biimpd
|
|
con3d imp ndmfv syl syl5eq ) EFZCDGZHZACIBDIJZKZJZABELABMZENZOABEPUKULUFIZK
|
|
ZUMOHUHUJUOUHUNUIUHUNUIUHUNULUGIUIUFUGULQABCDRSTUAUBULEUCUDUE $.
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x F $. $d x S $.
|
|
ndmovcl.1 $e |- dom F = ( S X. S ) $.
|
|
ndmovcl.2 $e |- ( ( A e. S /\ x e. S ) -> ( A F x ) e. S ) $.
|
|
ndmovcl.3 $e |- (/) e. S $.
|
|
$( The closure of an operation outside its domain, when the domain includes
|
|
the empty set. This technical lemma can make the operation more
|
|
convenient to work in some cases. It is is dependent on our particular
|
|
definitions of operation value, function value, and ordered pair.
|
|
(Contributed by set.mm contributors, 24-Sep-2004.) $)
|
|
ndmovcl $p |- ( A F B ) e. S $=
|
|
( wcel wa co cv wi wceq oveq2 eleq1d imbi2d expcom wn c0 impcom cop df-ov
|
|
vtoclga cfv cdm eleq2i opelxp bitri notbii sylbir syl5eq syl6eqel pm2.61i
|
|
cxp ndmfv ) BDIZCDIZJZBCEKZDIZURUQVAUQBALZEKZDIZMUQVAMACDVBCNZVDVAUQVEVCU
|
|
TDVBCBEOPQUQVBDIVDGRUDUAUSSZUTTDVFUTBCUBZEUEZTBCEUCVFVGEUFZIZSVHTNVJUSVJV
|
|
GDDUOZIUSVIVKVGFUGBCDDUHUIUJVGEUPUKULHUMUN $.
|
|
$}
|
|
|
|
${
|
|
ndmov.1 $e |- B e. _V $.
|
|
ndmov.2 $e |- dom F = ( S X. S ) $.
|
|
$( The value of an operation outside its domain. (Contributed by set.mm
|
|
contributors, 24-Aug-1995.) $)
|
|
ndmov $p |- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = (/) ) $=
|
|
( wcel wa cop cdm co c0 wceq cxp eleq2i opelxp bitri wn cfv df-ov sylnbir
|
|
ndmfv syl5eq ) ACGBCGHZABIZDJZGZABDKZLMUGUECCNZGUDUFUIUEFOABCCPQUGRUHUEDS
|
|
LABDTUEDUBUCUA $.
|
|
|
|
${
|
|
ndmovrcl.3 $e |- -. (/) e. S $.
|
|
$( Reverse closure law, when an operation's domain doesn't contain the
|
|
empty set. (Contributed by set.mm contributors, 3-Feb-1996.) $)
|
|
ndmovrcl $p |- ( ( A F B ) e. S -> ( A e. S /\ B e. S ) ) $=
|
|
( wcel wa co wn c0 ndmov eleq1d mtbiri con4i ) ACHBCHIZABDJZCHZQKZSLCHG
|
|
TRLCABCDEFMNOP $.
|
|
$}
|
|
|
|
${
|
|
ndmov.3 $e |- A e. _V $.
|
|
$( Any operation is commutative outside its domain. (Contributed by
|
|
set.mm contributors, 24-Aug-1995.) $)
|
|
ndmovcom $p |- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = ( B F A ) ) $=
|
|
( wcel wa wn co c0 ndmov wceq ancom sylnbi eqtr4d ) ACHZBCHZIZJABDKLBAD
|
|
KZABCDEFMTSRIUALNRSOBACDGFMPQ $.
|
|
$}
|
|
|
|
${
|
|
ndmov.4 $e |- C e. _V $.
|
|
ndmov.5 $e |- -. (/) e. S $.
|
|
$( Any operation is associative outside its domain, if the domain doesn't
|
|
contain the empty set. (Contributed by set.mm contributors,
|
|
24-Aug-1995.) $)
|
|
ndmovass $p |- ( -. ( A e. S /\ B e. S /\ C e. S ) ->
|
|
( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
|
|
( wcel wn co c0 wa wceq ndmovrcl sylibr con3i ndmov syl w3a anim1i ovex
|
|
df-3an anim2i 3anass eqtr4d ) ADJZBDJZCDJZUAZKZABELZCELZMABCELZELZULUMD
|
|
JZUJNZKUNMOURUKURUHUINZUJNUKUQUSUJABDEFGIPUBUHUIUJUDQRUMCDEHGSTULUHUODJ
|
|
ZNZKUPMOVAUKVAUHUIUJNZNUKUTVBUHBCDEHGIPUEUHUIUJUFQRAUODEBCEUCGSTUG $.
|
|
|
|
${
|
|
ndmov.6 $e |- dom G = ( S X. S ) $.
|
|
$( Any operation is distributive outside its domain, if the domain
|
|
doesn't contain the empty set. (Contributed by set.mm contributors,
|
|
24-Aug-1995.) $)
|
|
ndmovdistr $p |- ( -. ( A e. S /\ B e. S /\ C e. S ) ->
|
|
( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) ) $=
|
|
( wcel wn co c0 wa wceq ndmovrcl sylibr con3i w3a anim2i 3anass ndmov
|
|
ovex syl anim12i anandi bitri eqtr4d ) ADLZBDLZCDLZUAZMZABCENZFNZOABF
|
|
NZACFNZENZUOUKUPDLZPZMUQOQVBUNVBUKULUMPZPZUNVAVCUKBCDEIHJRUBUKULUMUCZ
|
|
STAUPDFBCEUEKUDUFUOURDLZUSDLZPZMUTOQVHUNVHUKULPZUKUMPZPZUNVFVIVGVJABD
|
|
FGKJRACDFIKJRUGUNVDVKVEUKULUMUHUISTURUSDEACFUEHUDUFUJ $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ndmovord.3 $e |- A e. _V $.
|
|
ndmovord.4 $e |- R C_ ( S X. S ) $.
|
|
ndmovord.5 $e |- -. (/) e. S $.
|
|
ndmovord.6 $e |- ( ( A e. S /\ B e. S /\ C e. S ) ->
|
|
( A R B <-> ( C F A ) R ( C F B ) ) ) $.
|
|
$( Elimination of redundant antecedents in an ordering law. (Contributed
|
|
by set.mm contributors, 7-Mar-1996.) $)
|
|
ndmovord $p |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $=
|
|
( wcel wa wbr co wb brel ndmovrcl simprd wi 3expia anim12i syl pm5.21ni
|
|
wn a1d pm2.61i ) AEMZBEMZNZCEMZABDOZCAFPZCBFPZDOZQZUAUIUJULUQLUBUKUFUQU
|
|
LUMUKUPABEEDJRUPUNEMZUOEMZNUKUNUOEEDJRURUIUSUJURULUICAEFIHKSTUSULUJCBEF
|
|
GHKSTUCUDUEUGUH $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
ndmovordi.3 $e |- A e. _V $.
|
|
ndmovordi.2 $e |- dom F = ( S X. S ) $.
|
|
ndmovordi.4 $e |- R C_ ( S X. S ) $.
|
|
ndmovordi.5 $e |- -. (/) e. S $.
|
|
ndmovordi.6 $e |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $.
|
|
$( Elimination of redundant antecedent in an ordering law. (Contributed by
|
|
set.mm contributors, 25-Jun-1998.) $)
|
|
ndmovordi $p |- ( ( C F A ) R ( C F B ) -> A R B ) $=
|
|
( wcel co wbr brel simpld ndmovrcl syl biimprd mpcom ) CELZCAFMZCBFMZDNZA
|
|
BDNZUDUBELZUAUDUFUCELUBUCEEDIOPUFUAAELCAEFGHJQPRUAUEUDKST $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y C $. $d x y D $. $d x y E $. $d x y ph $.
|
|
$d x y F $.
|
|
caovcld.1 $e |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E ) $.
|
|
$( Convert an operation closure law to class notation. (Contributed by
|
|
Mario Carneiro, 26-May-2014.) $)
|
|
caovcld $p |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E ) $=
|
|
( cv co wcel wral wa ralrimivva wceq oveq1 eleq1d oveq2 rspc2v mpan9 ) AB
|
|
KZCKZILZHMZCGNBFNDFMEGMODEILZHMZAUFBCFGJPUFUHDUDILZHMBCDEFGUCDQUEUIHUCDUD
|
|
IRSUDEQUIUGHUDEDITSUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x y F $. $d x y S $.
|
|
caovcl.1 $e |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S ) $.
|
|
$( Convert an operation closure law to class notation. (Contributed by
|
|
set.mm contributors, 4-Aug-1995.) (Revised by set.mm contributors,
|
|
26-May-2014.) $)
|
|
caovcl $p |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S ) $=
|
|
( wtru wcel wa co tru cv adantl caovcld mpan ) HCEIDEIJCDFKEILHABCDEEEFAM
|
|
ZEIBMZEIJQRFKEIHGNOP $.
|
|
$}
|
|
|
|
${
|
|
$( General laws for commutative, associative, distributive operations. $)
|
|
$d x y z F $. $d x y z S $. $d x y z A $. $d x y z B $. $d x y z C $.
|
|
$d x y z D $. $d x y z G $. $d x y z R $. $d x y z ph $.
|
|
${
|
|
caovcomg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
|
|
( x F y ) = ( y F x ) ) $.
|
|
$( Convert an operation commutative law to class notation. (Contributed
|
|
by set.mm contributors, 1-Jun-2013.) (Revised by Mario Carneiro,
|
|
2-Jun-2013.) $)
|
|
caovcomg $p |- ( ( ph /\ ( A e. S /\ B e. S ) ) ->
|
|
( A F B ) = ( B F A ) ) $=
|
|
( cv co wceq wral wcel wa ralrimivva oveq1 oveq2 eqeq12d rspc2v mpan9 )
|
|
ABIZCIZGJZUBUAGJZKZCFLBFLDFMEFMNDEGJZEDGJZKZAUEBCFFHOUEUHDUBGJZUBDGJZKB
|
|
CDEFFUADKUCUIUDUJUADUBGPUADUBGQRUBEKUIUFUJUGUBEDGQUBEDGPRST $.
|
|
$}
|
|
|
|
${
|
|
caovcom.1 $e |- A e. _V $.
|
|
caovcom.2 $e |- B e. _V $.
|
|
caovcom.3 $e |- ( x F y ) = ( y F x ) $.
|
|
$( Convert an operation commutative law to class notation. (Contributed
|
|
by set.mm contributors, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
1-Jun-2013.) $)
|
|
caovcom $p |- ( A F B ) = ( B F A ) $=
|
|
( cvv wcel wa co wceq pm3.2i cv a1i caovcomg mp2an ) CIJZSDIJZKCDELDCEL
|
|
MFSTFGNSABCDIEAOZBOZELUBUAELMSUAIJUBIJKKHPQR $.
|
|
$}
|
|
|
|
${
|
|
caovassg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x F y ) F z ) = ( x F ( y F z ) ) ) $.
|
|
$( Convert an operation associative law to class notation. (Contributed
|
|
by set.mm contributors, 1-Jun-2013.) (Revised by Mario Carneiro,
|
|
2-Jun-2013.) $)
|
|
caovassg $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
|
|
( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
|
|
( cv co wceq wral wcel oveq1 oveq1d eqeq12d oveq2 oveq2d rspc3v mpan9
|
|
w3a ralrimivvva ) ABKZCKZILZDKZILZUEUFUHILZILZMZDHNCHNBHNEHOFHOGHOUCEFI
|
|
LZGILZEFGILZILZMZAULBCDHHHJUDULUQEUFILZUHILZEUJILZMUMUHILZEFUHILZILZMBC
|
|
DEFGHHHUEEMZUIUSUKUTVDUGURUHIUEEUFIPQUEEUJIPRUFFMZUSVAUTVCVEURUMUHIUFFE
|
|
ISQVEUJVBEIUFFUHIPTRUHGMZVAUNVCUPUHGUMISVFVBUOEIUHGFISTRUAUB $.
|
|
$}
|
|
|
|
${
|
|
caovass.1 $e |- A e. _V $.
|
|
caovass.2 $e |- B e. _V $.
|
|
caovass.3 $e |- C e. _V $.
|
|
caovass.4 $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
|
|
$( Convert an operation associative law to class notation. (Contributed
|
|
by set.mm contributors, 26-Aug-1995.) (Revised by Mario Carneiro,
|
|
1-Jun-2013.) $)
|
|
caovass $p |- ( ( A F B ) F C ) = ( A F ( B F C ) ) $=
|
|
( cvv wcel co wceq wtru w3a tru cv wa a1i caovassg mpan mp3an ) DLMZELM
|
|
ZFLMZDEGNFGNDEFGNGNOZHIJPUEUFUGQUHRPABCDEFLGASZBSZGNCSZGNUIUJUKGNGNOPUI
|
|
LMUJLMUKLMQTKUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
caovcan.1 $e |- C e. _V $.
|
|
caovcan.2 $e |- ( ( x e. S /\ y e. S ) ->
|
|
( ( x F y ) = ( x F z ) -> y = z ) ) $.
|
|
$( Convert an operation cancellation law to class notation. (Contributed
|
|
by set.mm contributors, 20-Aug-1995.) $)
|
|
caovcan $p |- ( ( A e. S /\ B e. S ) ->
|
|
( ( A F B ) = ( A F C ) -> B = C ) ) $=
|
|
( cv co wceq wi oveq1 eqeq12d imbi1d oveq2 imbi12d wcel eqeq1d eqeq1 wa
|
|
eqeq2d eqeq2 imbi2d vtocl vtocl2ga ) AKZBKZHLZUIFHLZMZUJFMZNZDUJHLZDFHL
|
|
ZMZUNNDEHLZUQMZEFMZNABDEGGUIDMZUMURUNVBUKUPULUQUIDUJHOUIDFHOPQUJEMZURUT
|
|
UNVAVCUPUSUQUJEDHRUAUJEFUBSUIGTUJGTUCZUKUICKZHLZMZUJVEMZNZNVDUONCFIVEFM
|
|
ZVIUOVDVJVGUMVHUNVJVFULUKVEFUIHRUDVEFUJUESUFJUGUH $.
|
|
$}
|
|
|
|
${
|
|
caovord.1 $e |- A e. _V $.
|
|
caovord.2 $e |- B e. _V $.
|
|
caovord.3 $e |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) ) $.
|
|
$( Convert an operation ordering law to class notation. (Contributed by
|
|
set.mm contributors, 19-Feb-1996.) $)
|
|
caovord $p |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $=
|
|
( wbr cv co wb wceq oveq1 wi oveq2 breq12d bibi2d wcel wa breq1 bibi12d
|
|
breq1d breq2 breq2d sylan9bb imbi2d vtocl2 vtoclga ) DEGMZCNZDIOZUOEIOZ
|
|
GMZPZUNFDIOZFEIOZGMZPCFHUOFQZURVBUNVCUPUTUQVAGUOFDIRUOFEIRUAUBUOHUCZANZ
|
|
BNZGMZUOVEIOZUOVFIOZGMZPZSVDUSSABDEJKVEDQZVFEQZUDVKUSVDVLVKDVFGMZUPVIGM
|
|
ZPVMUSVLVGVNVJVOVEDVFGUEVLVHUPVIGVEDUOITUGUFVMVNUNVOURVFEDGUHVMVIUQUPGV
|
|
FEUOITUIUFUJUKLULUM $.
|
|
|
|
$( (We don't bother to eliminate this redundant hypothesis.) $)
|
|
caovord2.3 $e |- C e. _V $.
|
|
caovord2.com $e |- ( x F y ) = ( y F x ) $.
|
|
$( Operation ordering law with commuted arguments. (Contributed by
|
|
set.mm contributors, 27-Feb-1996.) $)
|
|
caovord2 $p |- ( C e. S -> ( A R B <-> ( A F C ) R ( B F C ) ) ) $=
|
|
( wcel wbr co caovord caovcom breq12i syl6bb ) FHODEGPFDIQZFEIQZGPDFIQZ
|
|
EFIQZGPABCDEFGHIJKLRUBUDUCUEGABFDIMJNSABFEIMKNSTUA $.
|
|
|
|
$( (We don't bother to eliminate redundant hypotheses.) $)
|
|
caovord3.4 $e |- D e. _V $.
|
|
$( Ordering law. (Contributed by set.mm contributors, 29-Feb-1996.) $)
|
|
caovord3 $p |- ( ( ( B e. S /\ C e. S ) /\
|
|
( A F B ) = ( C F D ) ) -> ( A R C <-> D R B ) ) $=
|
|
( wcel wa co wbr wceq wb caovord2 adantr breq1 sylan9bb ad2antlr bitr4d
|
|
caovord ) EIQZFIQZRZDEJSZFGJSZUAZRDFHTZUNFEJSZHTZGEHTZULUPUMUQHTZUOURUJ
|
|
UPUTUBUKABCDFEHIJKNMLOUCUDUMUNUQHUEUFUKUSURUBUJUOABCGEFHIJPLMUIUGUH $.
|
|
$}
|
|
|
|
${
|
|
caovdig.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) ) $.
|
|
$( Convert an operation distributive law to class notation. (Contributed
|
|
by set.mm contributors, 25-Aug-1995.) (Revised by Mario Carneiro,
|
|
26-Jul-2014.) $)
|
|
caovdig $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
|
|
( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) ) $=
|
|
( cv co wceq wral wcel oveq1 eqeq12d oveq2d oveq2 oveq12d oveq1d rspc3v
|
|
w3a ralrimivvva mpan9 ) ABLZCLZDLZIMZJMZUGUHJMZUGUIJMZIMZNZDHOCHOBHOEHP
|
|
FHPGHPUDEFGIMZJMZEFJMZEGJMZIMZNZAUOBCDHHHKUEUOVAEUJJMZEUHJMZEUIJMZIMZNE
|
|
FUIIMZJMZURVDIMZNBCDEFGHHHUGENZUKVBUNVEUGEUJJQVIULVCUMVDIUGEUHJQUGEUIJQ
|
|
UARUHFNZVBVGVEVHVJUJVFEJUHFUIIQSVJVCURVDIUHFEJTUBRUIGNZVGUQVHUTVKVFUPEJ
|
|
UIGFITSVKVDUSURIUIGEJTSRUCUF $.
|
|
$}
|
|
|
|
${
|
|
caovdirg.1 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
|
|
( ( x F y ) G z ) = ( ( x G z ) F ( y G z ) ) ) $.
|
|
$( Convert an operation reverse distributive law to class notation.
|
|
(Contributed by set.mm contributors, 19-Oct-2014.) $)
|
|
caovdirg $p |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) ->
|
|
( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) ) $=
|
|
( cv co wceq wral wcel oveq1 oveq1d eqeq12d oveq2 oveq2d oveq12d rspc3v
|
|
w3a ralrimivvva mpan9 ) ABLZCLZIMZDLZJMZUGUJJMZUHUJJMZIMZNZDHOCHOBHOEHP
|
|
FHPGHPUDEFIMZGJMZEGJMZFGJMZIMZNZAUOBCDHHHKUEUOVAEUHIMZUJJMZEUJJMZUMIMZN
|
|
UPUJJMZVDFUJJMZIMZNBCDEFGHHHUGENZUKVCUNVEVIUIVBUJJUGEUHIQRVIULVDUMIUGEU
|
|
JJQRSUHFNZVCVFVEVHVJVBUPUJJUHFEITRVJUMVGVDIUHFUJJQUASUJGNZVFUQVHUTUJGUP
|
|
JTVKVDURVGUSIUJGEJTUJGFJTUBSUCUF $.
|
|
$}
|
|
|
|
${
|
|
caovdi.1 $e |- A e. _V $.
|
|
caovdi.2 $e |- B e. _V $.
|
|
caovdi.3 $e |- C e. _V $.
|
|
caovdi.4 $e |- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) $.
|
|
$( Convert an operation distributive law to class notation. (Contributed
|
|
by set.mm contributors, 25-Aug-1995.) (Revised by Mario Carneiro,
|
|
28-Jun-2013.) $)
|
|
caovdi $p |- ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) $=
|
|
( cvv wcel co wceq wtru w3a tru cv wa a1i caovdig mpan mp3an ) DMNZEMNZ
|
|
FMNZDEFGOHODEHODFHOGOPZIJKQUFUGUHRUISQABCDEFMGHATZBTZCTZGOHOUJUKHOUJULH
|
|
OGOPQUJMNUKMNULMNRUALUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
caopr.1 $e |- A e. _V $.
|
|
caopr.2 $e |- B e. _V $.
|
|
caopr.3 $e |- C e. _V $.
|
|
caopr.com $e |- ( x F y ) = ( y F x ) $.
|
|
caopr.ass $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by set.mm contributors, 26-Aug-1995.) $)
|
|
caov32 $p |- ( ( A F B ) F C ) = ( ( A F C ) F B ) $=
|
|
( co caovcom oveq2i caovass 3eqtr4i ) DEFGMZGMDFEGMZGMDEGMFGMDFGMEGMRSD
|
|
GABEFGIJKNOABCDEFGHIJLPABCDFEGHJILPQ $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by set.mm contributors, 26-Aug-1995.) $)
|
|
caov12 $p |- ( A F ( B F C ) ) = ( B F ( A F C ) ) $=
|
|
( co caovcom oveq1i caovass 3eqtr3i ) DEGMZFGMEDGMZFGMDEFGMGMEDFGMGMRSF
|
|
GABDEGHIKNOABCDEFGHIJLPABCEDFGIHJLPQ $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by set.mm contributors, 26-Aug-1995.) $)
|
|
caov31 $p |- ( ( A F B ) F C ) = ( ( C F B ) F A ) $=
|
|
( co caovass caov12 eqtri caov32 eqtr3i 3eqtr4i ) DFGMEGMZFDEGMZGMZUAFG
|
|
MFEGMZDGMZTDUCGMUBABCDFEGHJILNABCDFEGHJIKLOPABCDEFGHIJKLQFDGMEGMUDUBABC
|
|
FDEGJHIKLQABCFDEGJHILNRS $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by set.mm contributors, 26-Aug-1995.) $)
|
|
caov13 $p |- ( A F ( B F C ) ) = ( C F ( B F A ) ) $=
|
|
( co caov31 caovass 3eqtr3i ) DEGMFGMFEGMDGMDEFGMGMFEDGMGMABCDEFGHIJKLN
|
|
ABCDEFGHIJLOABCFEDGJIHLOP $.
|
|
|
|
${
|
|
caopr.4 $e |- D e. _V $.
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by set.mm contributors, 26-Aug-1995.) $)
|
|
caov4 $p |- ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) $=
|
|
( co caov12 oveq2i ovex caovass 3eqtr4i ) DEFGHOZHOZHODFEGHOZHOZHODEH
|
|
OUAHODFHOUCHOUBUDDHABCEFGHJKNLMPQABCDEUAHIJFGHRMSABCDFUCHIKEGHRMST $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by set.mm contributors, 26-Aug-1995.) $)
|
|
caov411 $p |- ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) $=
|
|
( co caov31 oveq1i ovex caovass 3eqtr3i ) DEHOZFHOZGHOFEHOZDHOZGHOUAF
|
|
GHOHOUCDGHOHOUBUDGHABCDEFHIJKLMPQABCUAFGHDEHRKNMSABCUCDGHFEHRINMST $.
|
|
|
|
$( Rearrange arguments in a commutative, associative operation.
|
|
(Contributed by set.mm contributors, 26-Aug-1995.) $)
|
|
caov42 $p |- ( ( A F B ) F ( C F D ) ) =
|
|
( ( A F C ) F ( D F B ) ) $=
|
|
( co caov4 caovcom oveq2i eqtri ) DEHOFGHOHODFHOZEGHOZHOTGEHOZHOABCDE
|
|
FGHIJKLMNPUAUBTHABEGHJNLQRS $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
caoprd.1 $e |- A e. _V $.
|
|
caoprd.2 $e |- B e. _V $.
|
|
caoprd.3 $e |- C e. _V $.
|
|
caoprd.com $e |- ( x G y ) = ( y G x ) $.
|
|
caoprd.distr $e |- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) $.
|
|
$( Reverse distributive law. (Contributed by set.mm contributors,
|
|
26-Aug-1995.) $)
|
|
caovdir $p |- ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) $=
|
|
( co caovdi ovex caovcom oveq12i 3eqtr3i ) FDEGNZHNFDHNZFEHNZGNTFHNDFHN
|
|
ZEFHNZGNABCFDEGHKIJMOABFTHKDEGPLQUAUCUBUDGABFDHKILQABFEHKJLQRS $.
|
|
|
|
${
|
|
$d x y z H $. $d x y z R $.
|
|
caoprdl.4 $e |- D e. _V $.
|
|
caoprdl.5 $e |- H e. _V $.
|
|
caoprdl.ass $e |- ( ( x G y ) G z ) = ( x G ( y G z ) ) $.
|
|
$( Lemma used by real number construction. (Contributed by set.mm
|
|
contributors, 26-Aug-1995.) $)
|
|
caovdilem $p |- ( ( ( A G C ) F ( B G D ) ) G H ) =
|
|
( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) $=
|
|
( co ovex caovdir caovass oveq12i eqtri ) DFISZEGISZHSJISUEJISZUFJISZ
|
|
HSDFJISISZEGJISISZHSABCUEUFJHIDFITEGITQNOUAUGUIUHUJHABCDFJIKMQRUBABCE
|
|
GJILPQRUBUCUD $.
|
|
|
|
${
|
|
caoprdl2.6 $e |- R e. _V $.
|
|
caoprdl2.com $e |- ( x F y ) = ( y F x ) $.
|
|
caoprdl2.ass $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
|
|
$( Lemma used in real number construction. (Contributed by set.mm
|
|
contributors, 26-Aug-1995.) $)
|
|
caovlem2 $p |- ( ( ( ( A G C ) F ( B G D ) ) G H ) F
|
|
( ( ( A G D ) F ( B G C ) ) G R ) ) =
|
|
( ( A G ( ( C G H ) F ( D G R ) ) ) F ( B G ( ( C G R ) F ( D G H ) ) ) ) $=
|
|
( co ovex caov42 caovdilem oveq12i caovdi 3eqtr4i ) DFKJUCZJUCZEGKJ
|
|
UCZJUCZIUCZDGHJUCZJUCZEFHJUCZJUCZIUCZIUCUKUPIUCZURUMIUCZIUCDFJUCEGJ
|
|
UCIUCKJUCZDGJUCEFJUCIUCHJUCZIUCDUJUOIUCJUCZEUQULIUCJUCZIUCABCUKUMUP
|
|
URIDUJJUDEULJUDDUOJUDUAUBEUQJUDUEVBUNVCUSIABCDEFGIJKLMNOPQRSUFABCDE
|
|
GFIJHLMQOPNTSUFUGVDUTVEVAIABCDUJUOIJLFKJUDGHJUDPUHABCEUQULIJMFHJUDG
|
|
KJUDPUHUGUI $.
|
|
$}
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y z w v $. $d w v S $. $d w v A $. $d w v B $. $d w v F $.
|
|
caovmo.1 $e |- A e. _V $.
|
|
$( Identity element. $)
|
|
caovmo.2 $e |- B e. S $.
|
|
caovmo.dom $e |- dom F = ( S X. S ) $.
|
|
caovmo.3 $e |- -. (/) e. S $.
|
|
caovmo.com $e |- ( x F y ) = ( y F x ) $.
|
|
caovmo.ass $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
|
|
caovmo.id $e |- ( x e. S -> ( x F B ) = x ) $.
|
|
$( Uniqueness of inverse element in commutative, associative operation
|
|
with identity. Remark in proof of Proposition 9-2.4 of [Gleason]
|
|
p. 119. (Contributed by set.mm contributors, 4-Mar-1996.) $)
|
|
caovmo $p |- E* w ( A F w ) = B $=
|
|
( vv wcel co wceq wa cv wmo wi wal eleq1 eqeq1d anbi12d mo4 vex caovass
|
|
oveq2 caov12 eqtri oveq1 id eqeq12d vtoclga sylan9eqr ad2ant2rl caovcom
|
|
syl5eq elexi sylan9eq eqtr3d ax-gen mpgbir mpbiri ndmovrcl simprd ancri
|
|
ad2ant2lr syl moimi ax-mp ) DUAZGQZEVOHRZFSZTZDUBZVRDUBVTVSPUAZGQZEWAHR
|
|
ZFSZTZTZVOWASZUCZPUDDVSWEDPWGVPWBVRWDVOWAGUEWGVQWCFVOWAEHUKUFUGUHWHPWFV
|
|
QWAHRZVOWAVPWDWIVOSVRWBVPWDTWIVOWCHRZVOWIEVOWAHRHRWJABCEVOWAHIDUIZPUIZN
|
|
UJABCEVOWAHIWKWLMNULUMWDVPWJVOFHRZVOWCFVOHUKAUAZFHRZWNSZWMVOSAVOGWNVOSZ
|
|
WOWMWNVOWNVOFHUNWQUOUPOUQURVAUSVRWBWIWASVPWDVRWBWIFWAHRZWAVQFWAHUNWBWRW
|
|
AFHRZWAABFWAHFGJVBWLMUTWPWSWASAWAGWNWASZWOWSWNWAWNWAFHUNWTUOUPOUQVAVCVK
|
|
VDVEVFVRVSDVRVPVRVQGQZVPVRXAFGQJVQFGUEVGXAEGQVPEVOGHWKKLVHVIVLVJVMVN $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y z w t u A $.
|
|
$( Identity law for operator abstractions. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
oprabid2 $p |- { <. <. x , y >. , z >. | <. <. x , y >. , z >. e. A }
|
|
= A $=
|
|
( vw vt vu cv cop wcel coprab cvv wb vex weq opeq1 opeq1d opeq2 eloprabg
|
|
eleq1d mp3an eqoprriv ) EFGAHZBHZIZCHZIZDJZABCKZDEHZLJFHZLJGHZLJUJUKIZUL
|
|
IZUIJUNDJZMENFNGNUHUJUDIZUFIZDJUMUFIZDJUOABCUJUKULLLLAEOZUGUQDUSUEUPUFUC
|
|
UJUDPQTBFOZUQURDUTUPUMUFUDUKUJRQTCGOURUNDUFULUMRTSUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $.
|
|
oprabbi2i.1 $e |- ( <. <. x , y >. , z >. e. A <-> ph ) $.
|
|
$( Biconditional for operators. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
oprabbi2i $p |- A = { <. <. x , y >. , z >. | ph } $=
|
|
( cv cop wcel coprab oprabid2 oprabbii eqtr3i ) BGCGHDGHEIZBCDJEABCDJBCDE
|
|
KNABCDFLM $.
|
|
$}
|
|
|
|
|
|
$( Eliminate antecedent for operator values: domain and range can be taken to
|
|
be a set. (Contributed by set.mm contributors, 25-Feb-2015.) $)
|
|
elovex12 $p |- ( A e. ( B F C ) -> ( B e. _V /\ C e. _V ) ) $=
|
|
( co wcel c0 wne cvv wa ne0i cop wceq opexb cfv df-ov fvprc syl5eq sylnbir
|
|
wn necon1ai syl ) ABCDEZFUCGHBIFCIFJZUCAKUDUCGUDBCLZIFZUCGMBCNUFTUCUEDOGBCD
|
|
PUEDQRSUAUB $.
|
|
|
|
$( Eliminate antecedent for operator values: domain can be taken to be a
|
|
set. (Contributed by set.mm contributors, 25-Feb-2015.) $)
|
|
elovex1 $p |- ( A e. ( B F C ) -> B e. _V ) $=
|
|
( co wcel cvv elovex12 simpld ) ABCDEFBGFCGFABCDHI $.
|
|
|
|
$( Eliminate antecedent for operator values: range can be taken to be a set.
|
|
(Contributed by set.mm contributors, 25-Feb-2015.) $)
|
|
elovex2 $p |- ( A e. ( B F C ) -> C e. _V ) $=
|
|
( co wcel cvv elovex12 simprd ) ABCDEFBGFCGFABCDHI $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
"Maps to" notation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c |-> $. $( Maps-to symbol $)
|
|
|
|
$( Extend the definition of a class to include maps-to notation for defining
|
|
a function via a rule. $)
|
|
cmpt $a class ( x e. A |-> B ) $.
|
|
|
|
${
|
|
$d x y $. $d y A $. $d y B $.
|
|
$( Define maps-to notation for defining a function via a rule. Read as
|
|
"the function defined by the map from ` x ` (in ` A ` ) to
|
|
` B ( x ) ` ." The class expression ` B ` is the value of the function
|
|
at ` x ` and normally contains the variable ` x ` . Similar to the
|
|
definition of mapping in [ChoquetDD] p. 2. (Contributed by SF,
|
|
5-Jan-2015.) $)
|
|
df-mpt $a |- ( x e. A |-> B ) =
|
|
{ <. x , y >. | ( x e. A /\ y = B ) } $.
|
|
$}
|
|
|
|
$( Extend the definition of a class to include maps-to notation for defining
|
|
an operation via a rule. $)
|
|
cmpt2 $a class ( x e. A , y e. B |-> C ) $.
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z A $. $d z B $. $d z C $.
|
|
$( Define maps-to notation for defining an operation via a rule. Read as
|
|
"the operation defined by the map from ` x , y ` (in ` A X. B ` ) to
|
|
` B ( x , y ) ` ." An extension of ~ df-mpt for two arguments.
|
|
(Contributed by SF, 5-Jan-2015.) $)
|
|
df-mpt2 $a |- ( x e. A , y e. B |-> C ) =
|
|
{ <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } $.
|
|
$}
|
|
|
|
${
|
|
$d x y ph $. $d y A $. $d y B $. $d y C $. $d y D $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpteq12f $p |- ( ( A. x A = C /\ A. x e. A B = D ) ->
|
|
( x e. A |-> B ) = ( x e. C |-> D ) ) $=
|
|
( vy wceq wal wral wa cv wcel copab cmpt nfa1 nfra1 nfan nfv rsp df-mpt
|
|
imp eqeq2d pm5.32da sp eleq2d anbi1d sylan9bbr opabbid 3eqtr4g ) BDGZAHZC
|
|
EGZABIZJZAKZBLZFKZCGZJZAFMUODLZUQEGZJZAFMABCNADENUNUSVBAFUKUMAUJAOULABPQU
|
|
NFRUMUSUPVAJUKVBUMUPURVAUMUPJCEUQUMUPULULABSUAUBUCUKUPUTVAUKBDUOUJAUDUEUF
|
|
UGUHAFBCTAFDETUI $.
|
|
|
|
mpteq12dv.1 $e |- ( ph -> A = C ) $.
|
|
mpteq12dv.2 $e |- ( ph -> B = D ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by set.mm
|
|
contributors, 24-Aug-2011.) (Revised by set.mm contributors,
|
|
16-Dec-2013.) $)
|
|
mpteq12dv $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
|
|
( wceq wal wral cmpt alrimiv ralrimivw mpteq12f syl2anc ) ACEIZBJDFIZBCKB
|
|
CDLBEFLIAQBGMARBCHNBCDEFOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $.
|
|
$( An equality theorem for the maps to notation. (Contributed by set.mm
|
|
contributors, 16-Dec-2013.) $)
|
|
mpteq12 $p |- ( ( A = C /\ A. x e. A B = D ) ->
|
|
( x e. A |-> B ) = ( x e. C |-> D ) ) $=
|
|
( wceq wal wral cmpt ax-17 mpteq12f sylan ) BDFZMAGCEFABHABCIADEIFMAJABCD
|
|
EKL $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpteq1 $p |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) ) $=
|
|
( wceq wral cmpt cv wcel eqidd rgen mpteq12 mpan2 ) BCEDDEZABFABDGACDGENA
|
|
BAHBIDJKABDCDLM $.
|
|
$}
|
|
|
|
${
|
|
mpteq2ia.1 $e |- ( x e. A -> B = C ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpteq2ia $p |- ( x e. A |-> B ) = ( x e. A |-> C ) $=
|
|
( wceq wal wral cmpt eqid ax-gen rgen mpteq12f mp2an ) BBFZAGCDFZABHABCIA
|
|
BDIFOABJKPABELABCBDMN $.
|
|
$}
|
|
|
|
${
|
|
mpteq2i.1 $e |- B = C $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpteq2i $p |- ( x e. A |-> B ) = ( x e. A |-> C ) $=
|
|
( wceq cv wcel a1i mpteq2ia ) ABCDCDFAGBHEIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d y z B $. $d x y z D $. $d y z E $. $d z C $.
|
|
$d z F $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.) $)
|
|
mpt2eq123 $p |- ( ( A = D /\ A. x e. A ( B = E /\ A. y e. B C = F ) ) ->
|
|
( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) ) $=
|
|
( vz wceq wral wa cv wcel coprab cmpt2 nfv nfra1 nfan wb nfcv nfral eqeq2
|
|
rsp pm5.32d eleq2 anbi1d sylan9bbr anass 3bitr4g oprabbid df-mpt2 3eqtr4g
|
|
syl6 ) CFJZDGJZEHJZBDKZLZACKZLZAMZCNZBMZDNZLIMZEJZLZABIOVBFNZVDGNZLVFHJZL
|
|
ZABIOABCDEPABFGHPVAVHVLABIUOUTAUOAQUSACRSUOUTBUOBQUSBACBCUAUPURBUPBQUQBDR
|
|
SUBSVAIQVAVCVEVGLZLZVIVJVKLZLZVHVLUTVNVCVOLUOVPUTVCVMVOUTVCUSVMVOTUSACUDU
|
|
RVMVEVKLUPVOURVEVGVKURVEUQVGVKTUQBDUDEHVFUCUNUEUPVEVJVKDGVDUFUGUHUNUEUOVC
|
|
VIVOCFVBUFUGUHVCVEVGUIVIVJVKUIUJUKABICDEULABIFGHULUM $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $. $d x y D $.
|
|
$( An equality theorem for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpt2eq12 $p |- ( ( A = C /\ B = D ) ->
|
|
( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) ) $=
|
|
( wceq wral wa cmpt2 eqid rgenw jctr ralrimivw mpt2eq123 sylan2 ) DFHZCEH
|
|
RGGHZBDIZJZACIABCDGKABEFGKHRUAACRTSBDGLMNOABCDGEFGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d z C $. $d z D $. $d z E $. $d z F $.
|
|
$d x z ph $. $d y z ph $.
|
|
mpt2eq123dv.1 $e |- ( ph -> A = D ) $.
|
|
mpt2eq123dv.2 $e |- ( ph -> B = E ) $.
|
|
mpt2eq123dv.3 $e |- ( ph -> C = F ) $.
|
|
$( An equality deduction for the maps to notation. (Contributed by set.mm
|
|
contributors, 12-Sep-2011.) $)
|
|
mpt2eq123dv $p |- ( ph
|
|
-> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) ) $=
|
|
( vz cv wcel wa wceq coprab cmpt2 eleq2d anbi12d eqeq2d oprabbidv df-mpt2
|
|
3eqtr4g ) ABNZDOZCNZEOZPZMNZFQZPZBCMRUFGOZUHHOZPZUKIQZPZBCMRBCDEFSBCGHISA
|
|
UMURBCMAUJUPULUQAUGUNUIUOADGUFJTAEHUHKTUAAFIUKLUBUAUCBCMDEFUDBCMGHIUDUE
|
|
$.
|
|
$}
|
|
|
|
${
|
|
mpt2eq123i.1 $e |- A = D $.
|
|
mpt2eq123i.2 $e |- B = E $.
|
|
mpt2eq123i.3 $e |- C = F $.
|
|
$( An equality inference for the maps to notation. (Contributed by set.mm
|
|
contributors, 15-Jul-2013.) $)
|
|
mpt2eq123i $p |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) $=
|
|
( cmpt2 wceq wtru a1i mpt2eq123dv trud ) ABCDELABFGHLMNABCDEFGHCFMNIODGMN
|
|
JOEHMNKOPQ $.
|
|
$}
|
|
|
|
${
|
|
mpteq12i.1 $e |- A = C $.
|
|
mpteq12i.2 $e |- B = D $.
|
|
$( An equality inference for the maps to notation. (Contributed by Scott
|
|
Fenton, 27-Oct-2010.) $)
|
|
mpteq12i $p |- ( x e. A |-> B ) = ( x e. C |-> D ) $=
|
|
( cmpt wceq wtru a1i mpteq12dv trud ) ABCHADEHIJABCDEBDIJFKCEIJGKLM $.
|
|
$}
|
|
|
|
${
|
|
mpteq2da.1 $e |- F/ x ph $.
|
|
mpteq2da.2 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Slightly more general equality inference for the maps to notation.
|
|
(Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro,
|
|
16-Dec-2013.) $)
|
|
mpteq2da $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
|
|
( wceq wal wral cmpt eqid ax-gen cv wcel ex ralrimi mpteq12f sylancr ) AC
|
|
CHZBIDEHZBCJBCDKBCEKHTBCLMAUABCFABNCOUAGPQBCDCERS $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
mpteq2dva.1 $e |- ( ( ph /\ x e. A ) -> B = C ) $.
|
|
$( Slightly more general equality inference for the maps to notation.
|
|
(Contributed by Scott Fenton, 25-Apr-2012.) $)
|
|
mpteq2dva $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
|
|
( nfv mpteq2da ) ABCDEABGFH $.
|
|
$}
|
|
|
|
${
|
|
$d x ph $.
|
|
mpteq2dv.1 $e |- ( ph -> B = C ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 23-Aug-2014.) $)
|
|
mpteq2dv $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
|
|
( wceq cv wcel adantr mpteq2dva ) ABCDEADEGBHCIFJK $.
|
|
$}
|
|
|
|
${
|
|
$d x z ph $. $d y z ph $. $d z A $. $d z B $. $d z C $. $d z D $.
|
|
mpt2eq3dva.1 $e |- ( ( ph /\ x e. A /\ y e. B ) -> C = D ) $.
|
|
$( Slightly more general equality inference for the maps to notation.
|
|
(Contributed by set.mm contributors, 17-Oct-2013.) (Revised by set.mm
|
|
contributors, 16-Dec-2013.) $)
|
|
mpt2eq3dva $p |- ( ph -> ( x e. A , y e. B |-> C )
|
|
= ( x e. A , y e. B |-> D ) ) $=
|
|
( vz cv wcel wa wceq coprab cmpt2 3expb eqeq2d pm5.32da oprabbidv df-mpt2
|
|
3eqtr4g ) ABJDKZCJEKZLZIJZFMZLZBCINUDUEGMZLZBCINBCDEFOBCDEGOAUGUIBCIAUDUF
|
|
UHAUDLFGUEAUBUCFGMHPQRSBCIDEFTBCIDEGTUA $.
|
|
$}
|
|
|
|
${
|
|
mpt2eq3ia.1 $e |- ( ( x e. A /\ y e. B ) -> C = D ) $.
|
|
$( An equality inference for the maps to notation. (Contributed by Mario
|
|
Carneiro, 16-Dec-2013.) $)
|
|
mpt2eq3ia $p |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) $=
|
|
( cmpt2 wceq wtru cv wcel 3adant1 mpt2eq3dva trud ) ABCDEHABCDFHIJABCDEFA
|
|
KCLBKDLEFIJGMNO $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d x y z $.
|
|
nfmpt.1 $e |- F/_ x A $.
|
|
nfmpt.2 $e |- F/_ x B $.
|
|
$( Bound-variable hypothesis builder for the maps-to notation.
|
|
(Contributed by NM, 20-Feb-2013.) $)
|
|
nfmpt $p |- F/_ x ( y e. A |-> B ) $=
|
|
( vz cmpt cv wcel wceq wa copab df-mpt nfcri nfeq2 nfan nfopab nfcxfr ) A
|
|
BCDHBICJZGIZDKZLZBGMBGCDNUCBGATUBAABCEOAUADFPQRS $.
|
|
$}
|
|
|
|
${
|
|
$d A z $. $d B z $. $d x z $.
|
|
$( Bound-variable hypothesis builder for the maps-to notation.
|
|
(Contributed by FL, 17-Feb-2008.) $)
|
|
nfmpt1 $p |- F/_ x ( x e. A |-> B ) $=
|
|
( vz cmpt cv wcel wceq wa copab df-mpt nfopab1 nfcxfr ) AABCEAFBGDFCHIZAD
|
|
JADBCKNADLM $.
|
|
$}
|
|
|
|
${
|
|
$d z A $. $d z B $. $d z C $. $d z x $. $d z y $.
|
|
$( Bound-variable hypothesis builder for an operation in maps-to notation.
|
|
(Contributed by NM, 27-Aug-2013.) $)
|
|
nfmpt21 $p |- F/_ x ( x e. A , y e. B |-> C ) $=
|
|
( vz cmpt2 cv wcel wa wceq coprab df-mpt2 nfoprab1 nfcxfr ) AABCDEGAHCIBH
|
|
DIJFHEKJZABFLABFCDEMPABFNO $.
|
|
|
|
$( Bound-variable hypothesis builder for an operation in maps-to notation.
|
|
(Contributed by NM, 27-Aug-2013.) $)
|
|
nfmpt22 $p |- F/_ y ( x e. A , y e. B |-> C ) $=
|
|
( vz cmpt2 cv wcel wa wceq coprab df-mpt2 nfoprab2 nfcxfr ) BABCDEGAHCIBH
|
|
DIJFHEKJZABFLABFCDEMPABFNO $.
|
|
$}
|
|
|
|
${
|
|
$d w x z $. $d w y z $. $d w A $. $d w B $. $d w C $.
|
|
nfmpt2.1 $e |- F/_ z A $.
|
|
nfmpt2.2 $e |- F/_ z B $.
|
|
nfmpt2.3 $e |- F/_ z C $.
|
|
$( Bound-variable hypothesis builder for the maps-to notation.
|
|
(Contributed by NM, 20-Feb-2013.) $)
|
|
nfmpt2 $p |- F/_ z ( x e. A , y e. B |-> C ) $=
|
|
( vw cmpt2 cv wcel wa wceq coprab df-mpt2 nfcri nfan nfeq2 nfoprab nfcxfr
|
|
) CABDEFKALDMZBLEMZNZJLZFOZNZABJPABJDEFQUHABJCUEUGCUCUDCCADGRCBEHRSCUFFIT
|
|
SUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d w z x A $. $d w z y A $. $d w z B $. $d w z C $.
|
|
cbvmpt.1 $e |- F/_ y B $.
|
|
cbvmpt.2 $e |- F/_ x C $.
|
|
cbvmpt.3 $e |- ( x = y -> B = C ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. This version has bound-variable hypotheses in place of
|
|
distinct variable conditions. (Contributed by NM, 11-Sep-2011.) $)
|
|
cbvmpt $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
|
|
( vz vw cv wcel wceq wa copab cmpt wsb nfv nfan eleq1 nfs1v sbequ12 nfeq2
|
|
anbi12d cbvopab1 nfsb sbequ eqeq2d sbie syl6bb eqtri df-mpt 3eqtr4i ) AKZ
|
|
CLZIKZDMZNZAIOZBKZCLZUPEMZNZBIOZACDPBCEPUSJKZCLZUQAJQZNZJIOVDURVHAIJURJRV
|
|
FVGAVFARUQAJUASUNVEMUOVFUQVGUNVECTUQAJUBUDUEVHVCJIBVFVGBVFBRUQAJBBUPDFUCU
|
|
FSVCJRVEUTMZVFVAVGVBVEUTCTVIVGUQABQVBUQJBAUGUQVBABAUPEGUCUNUTMDEUPHUHUIUJ
|
|
UDUEUKAICDULBICEULUM $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B y $. $d C x $.
|
|
cbvmptv.1 $e |- ( x = y -> B = C ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) $)
|
|
cbvmptv $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
|
|
( nfcv cbvmpt ) ABCDEBDGAEGFH $.
|
|
$}
|
|
|
|
${
|
|
$d u w x y z $. $d u w x y z A $. $d u w B $. $d u C $. $d u y D $.
|
|
$d u E $.
|
|
cbvmpt2x.1 $e |- F/_ z B $.
|
|
cbvmpt2x.2 $e |- F/_ x D $.
|
|
cbvmpt2x.3 $e |- F/_ z C $.
|
|
cbvmpt2x.4 $e |- F/_ w C $.
|
|
cbvmpt2x.5 $e |- F/_ x E $.
|
|
cbvmpt2x.6 $e |- F/_ y E $.
|
|
cbvmpt2x.7 $e |- ( x = z -> B = D ) $.
|
|
cbvmpt2x.8 $e |- ( ( x = z /\ y = w ) -> C = E ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. This version of ~ cbvmpt2 allows ` B ` to be a function
|
|
of ` x ` . (Contributed by NM, 29-Dec-2014.) $)
|
|
cbvmpt2x $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. D |-> E ) $=
|
|
( vu cv nfan wcel wa wceq coprab cmpt2 nfv nfcri nfeq2 nfcv adantr eleq2d
|
|
wb eleq1 sylan9bb anbi12d eqeq2d cbvoprab12 df-mpt2 3eqtr4i ) ASZEUAZBSZF
|
|
UAZUBZRSZGUCZUBZABRUDCSZEUAZDSZHUAZUBZVEIUCZUBZCDRUDABEFGUECDEHIUEVGVNABR
|
|
CDVDVFCVAVCCVACUFCBFJUGTCVEGLUHTVDVFDVAVCDVADUFDBFDFUIUGTDVEGMUHTVLVMAVIV
|
|
KAVIAUFADHKUGTAVEINUHTVLVMBVLBUFBVEIOUHTUTVHUCZVBVJUCZUBZVDVLVFVMVQVAVIVC
|
|
VKVOVAVIULVPUTVHEUMUJVOVCVBHUAVPVKVOFHVBPUKVBVJHUMUNUOVQGIVEQUPUOUQABREFG
|
|
URCDREHIURUS $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $.
|
|
cbvmpt2.1 $e |- F/_ z C $.
|
|
cbvmpt2.2 $e |- F/_ w C $.
|
|
cbvmpt2.3 $e |- F/_ x D $.
|
|
cbvmpt2.4 $e |- F/_ y D $.
|
|
cbvmpt2.5 $e |- ( ( x = z /\ y = w ) -> C = D ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. (Contributed by NM, 17-Dec-2013.) $)
|
|
cbvmpt2 $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D ) $=
|
|
( nfcv cv wceq eqidd cbvmpt2x ) ABCDEFGFHCFNAFNIJKLAOCOPFQMR $.
|
|
$}
|
|
|
|
${
|
|
$d w x y z A $. $d w x y z B $. $d w z C $. $d x y D $.
|
|
cbvmpt2v.1 $e |- ( x = z -> C = E ) $.
|
|
cbvmpt2v.2 $e |- ( y = w -> E = D ) $.
|
|
$( Rule to change the bound variable in a maps-to function, using implicit
|
|
substitution. With a longer proof analogous to ~ cbvmpt , some distinct
|
|
variable requirements could be eliminated. (Contributed by NM,
|
|
11-Jun-2013.) $)
|
|
cbvmpt2v $p |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D ) $=
|
|
( nfcv cv wceq sylan9eq cbvmpt2 ) ABCDEFGHCGLDGLAHLBHLAMCMNBMDMNGIHJKOP
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( Representation of a constant function using the mapping operation.
|
|
(Note that ` x ` cannot appear free in ` B ` .) (Contributed by set.mm
|
|
contributors, 16-Nov-2013.) $)
|
|
fconstmpt $p |- ( A X. { B } ) = ( x e. A |-> B ) $=
|
|
( vy csn cxp cv wcel wceq wa copab cmpt fconstopab df-mpt eqtr4i ) BCEFAG
|
|
BHDGCIJADKABCLADBCMADBCNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d y A $. $d y B $.
|
|
dmmpt2.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( The preimage of a function in maps-to notation. (Contributed by Stefan
|
|
O'Rear, 25-Jan-2015.) $)
|
|
mptpreima $p |- ( `' F " C ) = { x e. A | B e. C } $=
|
|
( vy ccnv cima cv wcel wceq wa copab crab eqtri crn wex cab bitri cnvopab
|
|
cmpt df-mpt cnveqi imaeq1i dfima3 resopab rneqi ancom anass exbii df-clel
|
|
cres 19.42v bicomi anbi2i abbii rnopab df-rab 3eqtr4i ) EHZDIAJBKZGJZCLZM
|
|
ZGANZDIZCDKZABOZVAVFDVAVEAGNZHVFEVJEABCUBVJFAGBCUCPUDVEAGUAPUEVGVFDUMZQZV
|
|
IVFDUFVLVCDKZVEMZGANZQZVIVKVOVEGADUGUHVNGRZASVBVHMZASVPVIVQVRAVQVBVDVMMZM
|
|
ZGRZVRVNVTGVNVEVMMVTVMVEUIVBVDVMUJTUKWAVBVSGRZMVRVBVSGUNWBVHVBVHWBGCDULUO
|
|
UPTTUQVNGAURVHABUSUTPPP $.
|
|
|
|
$( The domain of the mapping operation in general. (Contributed by Mario
|
|
Carneiro, 13-Sep-2013.) $)
|
|
dmmpt $p |- dom F = { x e. A | B e. _V } $=
|
|
( vy cdm cv wcel wceq wa copab wex cab cvv crab cmpt df-mpt eqtri dmeqi
|
|
dmopab 19.42v isset anbi2i bitr4i abbii df-rab eqtr4i 3eqtri ) DGAHBIZFHC
|
|
JZKZAFLZGULFMZANZCOIZABPZDUMDABCQUMEAFBCRSTULAFUAUOUJUPKZANUQUNURAUNUJUKF
|
|
MZKURUJUKFUBUPUSUJFCUCUDUEUFUPABUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( The domain of the mapping operation is the stated domain, if the
|
|
function value is always a set. (Contributed by Mario Carneiro,
|
|
9-Feb-2013.) $)
|
|
dmmptg $p |- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A ) $=
|
|
( wcel wral cvv crab cmpt cdm wceq elex ralimi rabid2 eqid dmmpt syl6reqr
|
|
sylibr ) CDEZABFZBCGEZABHZABCIZJTUAABFBUBKSUAABCDLMUAABNRABCUCUCOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B y $.
|
|
dmmptss.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( The domain of a mapping is a subset of its base class. (Contributed by
|
|
Scott Fenton, 17-Jun-2013.) $)
|
|
dmmptss $p |- dom F C_ A $=
|
|
( vy cdm cv wcel wceq wa copab cmpt df-mpt eqtri dmeqi dmopabss eqsstri )
|
|
DGAHBIFHCJZKAFLZGBDTDABCMTEAFBCNOPSAFBQR $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d x y $.
|
|
rnmpt.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( The range of a function in maps-to notation. (Contributed by Scott
|
|
Fenton, 21-Mar-2011.) $)
|
|
rnmpt $p |- ran F = { y | E. x e. A y = B } $=
|
|
( crn cv wcel wceq wa copab wrex cab cmpt df-mpt eqtri rneqi rnopab2 ) EG
|
|
AHCIBHDJZKABLZGTACMBNEUAEACDOUAFABCDPQRABCDSQ $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $. $d x y $.
|
|
$( A function in maps-to notation is a function. (Contributed by Mario
|
|
Carneiro, 13-Jan-2013.) $)
|
|
funmpt $p |- Fun ( x e. A |-> B ) $=
|
|
( vy cmpt wfun cv wcel wceq wa copab funopab4 df-mpt funeqi mpbir ) ABCEZ
|
|
FAGBHZDGCIJADKZFQADCLPRADBCMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
mptfng.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( The maps-to notation defines a function with domain. (Contributed by
|
|
Scott Fenton, 21-Mar-2011.) $)
|
|
mptfng $p |- ( A. x e. A B e. _V <-> F Fn A ) $=
|
|
( vy cmpt cv wcel wceq wa copab df-mpt eqtri fnopab2g ) AFBCDDABCGAHBIFHC
|
|
JKAFLEAFBCMNO $.
|
|
|
|
$( The maps-to notation defines a function with domain. (Contributed by
|
|
set.mm contributors, 9-Apr-2013.) $)
|
|
fnmpt $p |- ( A. x e. A B e. V -> F Fn A ) $=
|
|
( wcel wral cvv wfn elex ralimi mptfng sylib ) CEGZABHCIGZABHDBJOPABCEKLA
|
|
BCDFMN $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
fnmpti.1 $e |- B e. _V $.
|
|
fnmpti.2 $e |- F = ( x e. A |-> B ) $.
|
|
$( Functionality and domain of an ordered-pair class abstraction.
|
|
(Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
fnmpti $p |- F Fn A $=
|
|
( cvv wcel wral wfn rgenw mptfng mpbi ) CGHZABIDBJNABEKABCDFLM $.
|
|
|
|
$( Domain of an ordered-pair class abstraction that specifies a function.
|
|
(Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro,
|
|
31-Aug-2015.) $)
|
|
dmmpti $p |- dom F = A $=
|
|
( wfn cdm wceq fnmpti fndm ax-mp ) DBGDHBIABCDEFJBDKL $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d y F $.
|
|
fmpt.1 $e |- F = ( x e. A |-> C ) $.
|
|
$( Functionality of the mapping operation. (Contributed by Mario Carneiro,
|
|
26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
fmpt $p |- ( A. x e. A C e. B <-> F : A --> B ) $=
|
|
( vy wcel wral wf wfn crn wss fnmpt cv wceq wrex cab rnmpt wa biimparc ex
|
|
r19.29 rexlimivw syl abssdv syl5eqss df-f sylanbrc crab ccnv cima fimacnv
|
|
eleq1 mptpreima syl5reqr rabid2 sylib impbii ) DCHZABIZBCEJZVAEBKELZCMVBA
|
|
BDECFNVAVCGOZDPZABQZGRCAGBDEFSVAVFGCVAVFVDCHZVAVFTUTVETZABQVGUTVEABUCVHVG
|
|
ABVEVGUTVDDCUNUAUDUEUBUFUGBCEUHUIVBBUTABUJZPVAVBVIEUKCULBABDCEFUOBCEUMUPU
|
|
TABUQURUS $.
|
|
|
|
fmpti.2 $e |- ( x e. A -> C e. B ) $.
|
|
$( Functionality of the mapping operation. (Contributed by NM,
|
|
19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
fmpti $p |- F : A --> B $=
|
|
( wcel wral wf rgen fmpt mpbi ) DCHZABIBCEJNABGKABCDEFLM $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x ph $.
|
|
fmptd.1 $e |- ( ( ph /\ x e. A ) -> B e. C ) $.
|
|
fmptd.2 $e |- F = ( x e. A |-> B ) $.
|
|
$( Domain and co-domain of the mapping operation; deduction form.
|
|
(Contributed by Mario Carneiro, 13-Jan-2013.) $)
|
|
fmptd $p |- ( ph -> F : A --> C ) $=
|
|
( wcel wral wf ralrimiva fmpt sylib ) ADEIZBCJCEFKAOBCGLBCEDFHMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y C $. $d y F $. $d x y ph $.
|
|
fmpt2d.1 $e |- ( ph -> ( x e. A -> B e. V ) ) $.
|
|
fmpt2d.2 $e |- F = ( x e. A |-> B ) $.
|
|
fmpt2d.3 $e |- ( ph -> ( y e. A -> ( F ` y ) e. C ) ) $.
|
|
$( Domain and co-domain of the mapping operation; deduction form.
|
|
(Contributed by set.mm contributors, 9-Apr-2013.) $)
|
|
fmpt2d $p |- ( ph -> F : A --> C ) $=
|
|
( wfn crn wss wf wcel wral ralrimiv fnmpt syl cfv fnfvrnss df-f sylanbrc
|
|
cv syl2anc ) AGDLZGMFNZDFGOAEHPZBDQUGAUIBDIRBDEGHJSTZAUGCUEGUAFPZCDQUHUJA
|
|
UKCDKRCDFGUBUFDFGUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d C y $.
|
|
$( Restriction of the mapping operation. (Contributed by Mario Carneiro,
|
|
15-Jul-2013.) $)
|
|
resmpt $p |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) ) $=
|
|
( vy wss cv wcel wceq wa copab cres cmpt resopab2 df-mpt reseq1i 3eqtr4g
|
|
) CBFAGZBHEGDIZJAEKZCLRCHSJAEKABDMZCLACDMSAECBNUATCAEBDOPAECDOQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $. $d E z $.
|
|
$( Restriction of the mapping operation. (Contributed by Mario Carneiro,
|
|
17-Dec-2013.) $)
|
|
resmpt2 $p |- ( ( C C_ A /\ D C_ B ) ->
|
|
( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) =
|
|
( x e. C , y e. D |-> E ) ) $=
|
|
( vz wss wa cv wcel wceq coprab cxp cres cmpt2 resoprab2 df-mpt2 reseq1i
|
|
3eqtr4g ) ECIFDIJAKZCLBKZDLJHKGMZJABHNZEFOZPUBELUCFLJUDJABHNABCDGQZUFPABE
|
|
FGQUDABHCDEFRUGUEUFABHCDGSTABHEFGSUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d x C y $. $d x D y $.
|
|
fvmptg.1 $e |- ( x = A -> B = C ) $.
|
|
fvmptg.2 $e |- F = ( x e. D |-> B ) $.
|
|
$( Value of a function given in maps-to notation. Analogous to
|
|
~ fvopab4g . (Contributed by set.mm contributors, 2-Oct-2007.)
|
|
(Revised by set.mm contributors, 4-Aug-2008.) $)
|
|
fvmptg $p |- ( ( A e. D /\ C e. R ) -> ( F ` A ) = C ) $=
|
|
( vy cmpt cv wcel wceq wa copab df-mpt eqtri fvopab4g ) AJBCDEFGHGAECKALE
|
|
MJLCNOAJPIAJECQRS $.
|
|
|
|
$( Value of a function given in maps-to notation. (Contributed by Mario
|
|
Carneiro, 23-Apr-2014.) $)
|
|
fvmpti $p |- ( A e. D -> ( F ` A ) = ( _I ` C ) ) $=
|
|
( wcel cvv cfv cid wceq wa fvmptg fvi adantl eqtr4d wn c0 cv eleq1d dmmpt
|
|
cdm elrab2 baib notbid ndmfv syl6bir imp fvprc pm2.61dan ) BEIZDJIZBFKZDL
|
|
KZMUMUNNUODUPABCDEJFGHOUNUPDMUMDJPQRUMUNSZNUOTUPUMUQUOTMZUMUQBFUDZIZSURUM
|
|
UTUNUTUMUNCJIUNABEUSAUABMCDJGUBAECFHUCUEUFUGBFUHUIUJUQUPTMUMDLUKQRUL $.
|
|
|
|
${
|
|
fvmpt.3 $e |- C e. _V $.
|
|
$( Value of a function given in maps-to notation. (Contributed by set.mm
|
|
contributors, 17-Aug-2011.) $)
|
|
fvmpt $p |- ( A e. D -> ( F ` A ) = C ) $=
|
|
( wcel cvv cfv wceq fvmptg mpan2 ) BEJDKJBFLDMIABCDEKFGHNO $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d y B $. $d x y C $.
|
|
fvmpts.1 $e |- F = ( x e. C |-> B ) $.
|
|
$( Value of a function given in maps-to notation, using explicit class
|
|
substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by
|
|
Mario Carneiro, 31-Aug-2015.) $)
|
|
fvmpts $p |- ( ( A e. C /\ [_ A / x ]_ B e. V ) ->
|
|
( F ` A ) = [_ A / x ]_ B ) $=
|
|
( vy cv csb csbeq1 cmpt nfcv nfcsb1v csbeq1a cbvmpt eqtri fvmptg ) HBAHIZ
|
|
CJZABCJDFEASBCKEADCLHDTLGAHDCTHCMASCNASCOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $. $d x ph $.
|
|
fvmptd.1 $e |- ( ph -> F = ( x e. D |-> B ) ) $.
|
|
fvmptd.2 $e |- ( ( ph /\ x = A ) -> B = C ) $.
|
|
fvmptd.3 $e |- ( ph -> A e. D ) $.
|
|
fvmptd.4 $e |- ( ph -> C e. V ) $.
|
|
$( Deduction version of ~ fvmpt . (Contributed by Scott Fenton,
|
|
18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) $)
|
|
fvmptd $p |- ( ph -> ( F ` A ) = C ) $=
|
|
( cfv cmpt csb fveq1d wcel wceq csbied eqeltrd eqid fvmpts syl2anc 3eqtrd
|
|
) ACGMCBFDNZMZBCDOZEACGUEIPACFQUGHQUFUGRKAUGEHABCDEFKJSZLTBCDFUEHUEUAUBUC
|
|
UHUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
fvmpt2.1 $e |- F = ( x e. A |-> B ) $.
|
|
$( Value of a function given by the "maps to" notation. (Contributed by
|
|
Mario Carneiro, 23-Apr-2014.) $)
|
|
fvmpt2i $p |- ( x e. A -> ( F ` x ) = ( _I ` B ) ) $=
|
|
( vy csb wceq csbeq1 csbid syl6eq cmpt nfcv nfcsb1v csbeq1a cbvmpt fvmpti
|
|
cv eqtri ) FARZAFRZCGZCBDUATHUBATCGCAUATCIACJKDABCLFBUBLEAFBCUBFCMAUACNAU
|
|
ACOPSQ $.
|
|
|
|
$( Value of a function given by the "maps to" notation. (Contributed by
|
|
FL, 21-Jun-2010.) $)
|
|
fvmpt2 $p |- ( ( x e. A /\ B e. C ) -> ( F ` x ) = B ) $=
|
|
( cv wcel cfv cid fvmpt2i fvi sylan9eq ) AGZBHCDHNEICJICABCEFKCDLM $.
|
|
|
|
$d x y C $. $d y D $. $d y F $.
|
|
$( If all the values of the mapping are subsets of a class ` C ` , then so
|
|
is any evaluation of the mapping, even if ` D ` is not in the base set
|
|
` A ` . (Contributed by Mario Carneiro, 13-Feb-2015.) $)
|
|
fvmptss $p |- ( A. x e. A B C_ C -> ( F ` D ) C_ C ) $=
|
|
( vy wss wcel cfv cv wi wceq fveq2 sseq1d imbi2d nfcv wa c0 dmmptss sseli
|
|
wral cdm nfra1 cmpt nfmpt1 nfcxfr nffv nfss nfim cvv dmmpt rabeq2i fvmpt2
|
|
eqimss syl sylbi ndmfv 0ss a1i eqsstrd pm2.61i rsp impcom syl5ss vtoclgaf
|
|
wn ex vtoclga sylan2 adantl pm2.61dan ) CDIZABUCZEFUDZJZEFKZDIZVQVOEBJZVS
|
|
VPBEABCFGUAUBVTVOVSVOHLZFKZDIZMZVOVSMHEBWAENZWCVSVOWEWBVRDWAEFOPQVOALZFKZ
|
|
DIZMWDAWABAWARZVOWCAVNABUEAWBDAWAFAFABCUFGABCUGUHWIUIADRUJUKWFWANZWHWCVOW
|
|
JWGWBDWFWAFOPQWFBJZVOWHWKVOSWGCDWFVPJZWGCIZWLWKCULJZSZWMWNAVPBABCFGUMUNWO
|
|
WGCNWMABCULFGUOWGCUPUQURWLVHZWGTCWFFUSTCIWPCUTVAVBVCVOWKVNVNABVDVEVFVIVGV
|
|
JVEVKVOVQVHZSZVRTDWQVRTNVOEFUSVLTDIWRDUTVAVBVM $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y F $.
|
|
$( Representation of a function in terms of its values. (Contributed by
|
|
FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) $)
|
|
dffn5v $p |- ( F Fn A <-> F = ( x e. A |-> ( F ` x ) ) ) $=
|
|
( vy wfn cv wcel cfv wceq wa copab cmpt dffn5 df-mpt eqeq2i bitr4i ) CBEC
|
|
AFZBGDFQCHZIJADKZICABRLZIADBCMTSCADBRNOP $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z F $.
|
|
$( Representation of a function in terms of its values. (Contributed by
|
|
Mario Carneiro, 23-Dec-2013.) $)
|
|
fnov2 $p |- ( F Fn ( A X. B ) <->
|
|
F = ( x e. A , y e. B |-> ( x F y ) ) ) $=
|
|
( vz cxp wfn cv wcel wa co wceq coprab cmpt2 fnov df-mpt2 eqeq2i bitr4i )
|
|
ECDGHEAIZCJBIZDJKFITUAELZMKABFNZMEABCDUBOZMABFCDEPUDUCEABFCDUBQRS $.
|
|
|
|
$}
|
|
${
|
|
$d w x y z A $. $d w y z B $. $d w x y C $. $d w z D $.
|
|
mpt2mpt.1 $e |- ( z = <. x , y >. -> C = D ) $.
|
|
$( Express a two-argument function as a one-argument function, or
|
|
vice-versa. In this version ` B ( x ) ` is not assumed to be constant
|
|
w.r.t ` x ` . (Contributed by Mario Carneiro, 29-Dec-2014.) $)
|
|
mpt2mptx $p |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) =
|
|
( x e. A , y e. B |-> D ) $=
|
|
( vw cv csn cxp ciun cmpt wcel wceq wa copab wex eqtr4i df-mpt coprab cop
|
|
cmpt2 df-mpt2 eliunxp anbi1i 19.41vv eqeq2d anbi2d pm5.32i 2exbii 3bitr2i
|
|
anass bitri opabbii dfoprab2 ) CADAJZKELMZFNCJZUSOZIJZFPZQZCIRZABDEGUDZCI
|
|
USFUAVFURDOBJZEOQZVBGPZQZABIUBZVEABIDEGUEVEUTURVGUCPZVJQZBSASZCIRVKVDVNCI
|
|
VDVLVHQZBSASZVCQVOVCQZBSASVNVAVPVCABDEUTUFUGVOVCABUHVQVMABVQVLVHVCQZQVMVL
|
|
VHVCUNVLVRVJVLVCVIVHVLFGVBHUIUJUKUOULUMUPVJABICUQTTT $.
|
|
|
|
$d x B $.
|
|
$( Express a two-argument function as a one-argument function, or
|
|
vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by
|
|
Mario Carneiro, 29-Dec-2014.) $)
|
|
mpt2mpt $p |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D ) $=
|
|
( cv csn cxp ciun cmpt cmpt2 wceq iunxpconst mpteq1 ax-mp mpt2mptx eqtr3i
|
|
) CADAIJEKLZFMZCDEKZFMZABDEGNUAUCOUBUDOADEPCUAUCFQRABCDEFGHST $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $. $d z A $. $d z B $. $d z C $. $d z F $.
|
|
ovmpt4g.3 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( Value of a function given by the "maps to" notation. (This is the
|
|
operation analog of ~ fvmpt2 .) (Contributed by NM, 21-Feb-2004.)
|
|
(Revised by Mario Carneiro, 1-Sep-2015.) $)
|
|
ovmpt4g $p |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C ) $=
|
|
( vz cv wcel co wceq wex wa elisset wmo moeq a1i cmpt2 coprab eqtri ovidi
|
|
df-mpt2 eqeq2 mpbidi exlimdv syl5 3impia ) AJZCKZBJZDKZEGKZUJULFLZEMZUNIJ
|
|
ZEMZINUKUMOZUPIEGPUSURUPIURUOUQMUPUSURABICDFURIQUSIERSFABCDETUSUROABIUAHA
|
|
BICDEUDUBUCUQEUOUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y C $. $d x y D $.
|
|
ov2gf.a $e |- F/_ x A $.
|
|
ov2gf.c $e |- F/_ y A $.
|
|
ov2gf.d $e |- F/_ y B $.
|
|
ov2gf.1 $e |- F/_ x G $.
|
|
ov2gf.2 $e |- F/_ y S $.
|
|
ov2gf.3 $e |- ( x = A -> R = G ) $.
|
|
ov2gf.4 $e |- ( y = B -> G = S ) $.
|
|
ov2gf.5 $e |- F = ( x e. C , y e. D |-> R ) $.
|
|
$( The value of an operation class abstraction. A version of ~ ovmpt2g
|
|
using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.)
|
|
(Revised by Mario Carneiro, 19-Dec-2013.) $)
|
|
ov2gf $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
|
|
( wcel co wceq cvv wa elex cv wi nfel1 nfmpt21 nfcxfr nfcv nfov nfeq nfim
|
|
cmpt2 nfmpt22 eleq1d oveq1 eqeq12d imbi12d oveq2 ovmpt4g 3expia vtocl2gaf
|
|
syl5 3impia ) CETZDFTZHKTZCDIUAZHUBZVIHUCTZVGVHUDVKHKUEGUCTZAUFZBUFZIUAZG
|
|
UBZUGJUCTZCVOIUAZJUBZUGVLVKUGABCDEFLMNVRVTAAJUCOUHAVSJACVOILAIABEFGUOZSAB
|
|
EFGUIUJAVOUKULOUMUNVLVKBBHUCPUHBVJHBCDIMBIWASABEFGUPUJNULPUMUNVNCUBZVMVRV
|
|
QVTWBGJUCQUQWBVPVSGJVNCVOIURQUSUTVODUBZVRVLVTVKWCJHUCRUQWCVSVJJHVODCIVARU
|
|
SUTVNETVOFTVMVQABEFGIUCSVBVCVDVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $. $d L x y z $. $d D z $.
|
|
$d R z $. $d S x y z $.
|
|
ovmpt2x.1 $e |- ( ( x = A /\ y = B ) -> R = S ) $.
|
|
ovmpt2x.2 $e |- ( x = A -> D = L ) $.
|
|
ovmpt2x.3 $e |- F = ( x e. C , y e. D |-> R ) $.
|
|
$( The value of an operation class abstraction. Variant of ~ ovmpt2ga
|
|
which does not require ` D ` and ` x ` to be distinct. (Contributed by
|
|
Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.) $)
|
|
ovmpt2x $p |- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S ) $=
|
|
( vz wcel w3a wa wceq cv co 3simpa eqeq2d biimp3ar biantrud eleq1d eleq2d
|
|
wb simpl eleq1 sylan9bb anbi12d 3adant3 bitr3d moani cmpt2 coprab df-mpt2
|
|
moeq eqtri ovigg mpd ) CEPZDKPZHJPZQVCVDRZCDIUAHSVCVDVEUBATZEPZBTZFPZRZOT
|
|
ZGSZRZVFABOCDHIEKJVGCSZVIDSZVLHSZQZVKVNVFVRVMVKVOVPVMVQVOVPRZGHVLLUCUDUEV
|
|
OVPVKVFUHVQVSVHVCVJVDVSVGCEVOVPUIUFVOVJVIKPVPVDVOFKVIMUGVIDKUJUKULUMUNVMV
|
|
KOOGUSUOIABEFGUPVNABOUQNABOEFGURUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z D $. $d z R $.
|
|
$d x y z S $.
|
|
ovmpt2ga.1 $e |- ( ( x = A /\ y = B ) -> R = S ) $.
|
|
ovmpt2ga.2 $e |- F = ( x e. C , y e. D |-> R ) $.
|
|
$( Value of an operation given by a maps-to rule. Equivalent to ~ ov2ag .
|
|
(Contributed by Mario Carneiro, 19-Dec-2013.) $)
|
|
ovmpt2ga $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
|
|
( vz cmpt2 cv wcel wa wceq coprab df-mpt2 eqtri ov2ag ) ABMCDEFGHIJKIABEF
|
|
GNAOEPBOFPQMOGRQABMSLABMEFGTUAUB $.
|
|
|
|
${
|
|
ovmpt2a.4 $e |- S e. _V $.
|
|
$( Value of an operation given by a maps-to rule. Equivalent to
|
|
~ ov2ag . (Contributed by set.mm contributors, 19-Dec-2013.) $)
|
|
ovmpt2a $p |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S ) $=
|
|
( wcel cvv co wceq ovmpt2ga mp3an3 ) CEMDFMHNMCDIOHPLABCDEFGHINJKQR $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d B y $. $d x C y $. $d x D y $. $d x G $. $d y S $.
|
|
$d x S $. $d B x $.
|
|
ovmpt2g.1 $e |- ( x = A -> R = G ) $.
|
|
ovmpt2g.2 $e |- ( y = B -> G = S ) $.
|
|
ovmpt2g.3 $e |- F = ( x e. C , y e. D |-> R ) $.
|
|
$( Value of an operation given by a maps-to rule. (Unnecessary distinct
|
|
variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|
|
(Contributed by set.mm contributors, 2-Oct-2007.) (Revised by set.mm
|
|
contributors, 24-Jul-2012.) $)
|
|
ovmpt2g $p |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S ) $=
|
|
( cv wceq sylan9eq ovmpt2ga ) ABCDEFGHIKAOCPBODPGJHLMQNR $.
|
|
|
|
${
|
|
ovmpt2.4 $e |- S e. _V $.
|
|
$( Value of an operation given by a maps-to rule. Equivalent to ov2 in
|
|
set.mm. (Contributed by set.mm contributors, 12-Sep-2011.) $)
|
|
ovmpt2 $p |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S ) $=
|
|
( wcel cvv co wceq ovmpt2g mp3an3 ) CEODFOHPOCDIQHRNABCDEFGHIJPKLMST $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A y z $. $d B z $. $d C z $. $d x y z $.
|
|
rngop.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( The range of an operation given by the "maps to" notation. (Contributed
|
|
by FL, 20-Jun-2011.) $)
|
|
rnmpt2 $p |- ran F = { z | E. x e. A E. y e. B z = C } $=
|
|
( crn cv wcel wa wceq coprab wex cab wrex cmpt2 df-mpt2 eqtri r2ex bicomi
|
|
rneqi rnoprab abbii 3eqtri ) GIAJDKBJEKLCJFMZLZABCNZIUHBOAOZCPUGBEQADQZCP
|
|
GUIGABDEFRUIHABCDEFSTUCUHABCUDUJUKCUKUJUGABDEUAUBUEUF $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d x y $. $d y B $.
|
|
$( Function with universal domain in maps-to notation. (Contributed by
|
|
set.mm contributors, 16-Aug-2013.) $)
|
|
mptv $p |- ( x e. _V |-> B ) = { <. x , y >. | y = B } $=
|
|
( cvv cmpt cv wcel wceq wa copab df-mpt vex biantrur opabbii eqtr4i ) ADC
|
|
EAFDGZBFCHZIZABJQABJABDCKQRABPQALMNO $.
|
|
$}
|
|
|
|
${
|
|
$d x z $. $d y z $. $d z C $.
|
|
$( Operation with universal domain in maps-to notation. (Contributed by
|
|
set.mm contributors, 16-Aug-2013.) $)
|
|
mpt2v $p |- ( x e. _V , y e. _V |-> C )
|
|
= { <. <. x , y >. , z >. | z = C } $=
|
|
( cvv cmpt2 cv wcel wa coprab df-mpt2 vex pm3.2i biantrur oprabbii eqtr4i
|
|
wceq ) ABEEDFAGEHZBGEHZIZCGDQZIZABCJUAABCJABCEEDKUAUBABCTUARSALBLMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The restricted identity expressed with the "maps to" notation.
|
|
(Contributed by FL, 25-Apr-2012.) $)
|
|
mptresid $p |- ( x e. A |-> x ) = ( _I |` A ) $=
|
|
( vy cv cmpt wcel weq wa copab cid cres df-mpt opabresid eqtri ) ABADZEOB
|
|
FCAGHACIJBKACBOLACBMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $. $d y C $.
|
|
fvmptex.1 $e |- F = ( x e. A |-> B ) $.
|
|
fvmptex.2 $e |- G = ( x e. A |-> ( _I ` B ) ) $.
|
|
$( Express a function ` F ` whose value ` B ` may not always be a set in
|
|
terms of another function ` G ` for which sethood is guaranteed. (Note
|
|
that ` ( _I `` B ) ` is just shorthand for
|
|
` if ( B e. _V , B , (/) ) ` , and it is always a set by ~ fvex .) Note
|
|
also that these functions are not the same; wherever ` B ( C ) ` is not
|
|
a set, ` C ` is not in the domain of ` F ` (so it evaluates to the empty
|
|
set), but ` C ` is in the domain of ` G ` , and ` G ( C ) ` is defined
|
|
to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.)
|
|
(Revised by Mario Carneiro, 23-Apr-2014.) $)
|
|
fvmptex $p |- ( F ` C ) = ( G ` C ) $=
|
|
( vy wcel cfv wceq csb cid cv cmpt nfcv cbvmpt eqtri c0 nfcsb1v nffv fvex
|
|
csbeq1 csbeq1a fvmpti fveq2d fvmpt eqtr4d wn cdm sseli con3i ndmfv dmmpti
|
|
dmmptss syl eleq2i sylnbir pm2.61i ) DBJZDEKZDFKZLVAVBADCMZNKZVCIDAIOZCMZ
|
|
VDBEAVFDCUDZEABCPIBVGPGAIBCVGICQAVFCUAZAVFCUEZRSUFIDVGNKZVEBFVFDLVGVDNVHU
|
|
GFABCNKZPIBVKPHAIBVLVKIVLQAVGNANQVIUBAOVFLCVGNVJUGRSVDNUCUHUIVAUJZVBTVCVM
|
|
DEUKZJZUJVBTLVOVAVNBDABCEGUPULUMDEUNUQVADFUKZJVCTLVPBDABVLFCNUCHUOURDFUNU
|
|
SUIUT $.
|
|
$}
|
|
|
|
${
|
|
$d x D $.
|
|
fvmptf.1 $e |- F/_ x A $.
|
|
fvmptf.2 $e |- F/_ x C $.
|
|
fvmptf.3 $e |- ( x = A -> B = C ) $.
|
|
fvmptf.4 $e |- F = ( x e. D |-> B ) $.
|
|
$( Value of a function given by an ordered-pair class abstraction. This
|
|
version of ~ fvmptg uses bound-variable hypotheses instead of distinct
|
|
variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by
|
|
Mario Carneiro, 15-Oct-2016.) $)
|
|
fvmptf $p |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C ) $=
|
|
( wcel cfv wceq cvv elex cv wi nfel1 cmpt nfmpt1 nfcxfr nffv eleq1d fveq2
|
|
nfeq nfim eqeq12d imbi12d fvmpt2 ex vtoclgaf syl5 imp ) BELZDGLZBFMZDNZUP
|
|
DOLZUOURDGPCOLZAQZFMZCNZRUSURRABEHUSURAADOISAUQDABFAFAECTKAECUAUBHUCIUFUG
|
|
VABNZUTUSVCURVDCDOJUDVDVBUQCDVABFUEJUHUIVAELUTVCAECOFKUJUKULUMUN $.
|
|
|
|
$( The value of a function given by an ordered-pair class abstraction is
|
|
the empty set when the class it would otherwise map to is a proper
|
|
class. This version of ~ fvmptn uses bound-variable hypotheses instead
|
|
of distinct variable conditions. (Contributed by NM, 21-Oct-2003.)
|
|
(Revised by Mario Carneiro, 11-Sep-2015.) $)
|
|
fvmptnf $p |- ( -. C e. _V -> ( F ` A ) = (/) ) $=
|
|
( cvv wcel wn cdm cfv c0 wceq dmmptss sseli cid cmpt eqid fvmptex fvex cv
|
|
nfcv nffv fveq2d fvmptf mpan2 syl5eq fvprc sylan9eq expcom ndmfv pm2.61d1
|
|
syl5 ) DKLMZBFNZLZBFOZPQZUTBELZURVBUSEBAECFJRSVCURVBVCURVADTOZPVCVABAECTO
|
|
ZUAZOZVDAECBFVFJVFUBZUCVCVDKLVGVDQDTUDABVEVDEVFKGADTATUFHUGAUEBQCDTIUHVHU
|
|
IUJUKDTULUMUNUQBFUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x C $. $d x D $.
|
|
fvmptn.1 $e |- ( x = D -> B = C ) $.
|
|
fvmptn.2 $e |- F = ( x e. A |-> B ) $.
|
|
$( This somewhat non-intuitive theorem tells us the value of its function
|
|
is the empty set when the class ` C ` it would otherwise map to is a
|
|
proper class. This is a technical lemma that can help eliminate
|
|
redundant sethood antecedents otherwise required by ~ fvmptg .
|
|
(Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro,
|
|
9-Sep-2013.) $)
|
|
fvmptn $p |- ( -. C e. _V -> ( F ` D ) = (/) ) $=
|
|
( nfcv fvmptnf ) AECDBFAEIADIGHJ $.
|
|
|
|
$( A mapping always evaluates to a subset of the substituted expression in
|
|
the mapping, even if this is a proper class, or we are out of the
|
|
domain. (Contributed by Mario Carneiro, 13-Feb-2015.) $)
|
|
fvmptss2 $p |- ( F ` D ) C_ C $=
|
|
( cdm wcel cfv wss cvv wa cv wceq eleq1d dmmpt elrab2 c0 fvmptg syl sylbi
|
|
eqimss wn ndmfv 0ss a1i eqsstrd pm2.61i ) EFIZJZEFKZDLZULEBJDMJZNZUNCMJUO
|
|
AEBUKAOEPCDMGQABCFHRSUPUMDPUNAECDBMFGHUAUMDUDUBUCULUEZUMTDEFUFTDLUQDUGUHU
|
|
IUJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d y C $. $d x D $. $d x y ph $.
|
|
f1od.1 $e |- F = ( x e. A |-> C ) $.
|
|
${
|
|
f1od.2 $e |- ( ( ph /\ x e. A ) -> C e. W ) $.
|
|
f1od.3 $e |- ( ( ph /\ y e. B ) -> D e. X ) $.
|
|
f1od.4 $e |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) ) $.
|
|
$( Describe an implicit one-to-one onto function. (Contributed by Mario
|
|
Carneiro, 12-May-2014.) $)
|
|
f1od $p |- ( ph -> F : A -1-1-onto-> B ) $=
|
|
( wfn ccnv wcel wral ralrimiva copab wf1o fnmpt cmpt eqid wceq opabbidv
|
|
cv wa df-mpt eqtri cnveqi cnvopab 3eqtr4g fneq1d mpbird dff1o4 sylanbrc
|
|
syl ) AHDOZHPZEOZDEHUAAFIQZBDRUSAVBBDLSBDFHIKUBURAVACEGUCZEOZAGJQZCERVD
|
|
AVECEMSCEGVCJVCUDUBURAEUTVCABUGZDQCUGZFUEUHZCBTZVGEQVFGUEUHZCBTUTVCAVHV
|
|
JCBNUFUTVHBCTZPVIHVKHBDFUCVKKBCDFUIUJUKVHBCULUJCBEGUIUMUNUODEHUPUQ $.
|
|
$}
|
|
|
|
f1o2d.2 $e |- ( ( ph /\ x e. A ) -> C e. B ) $.
|
|
f1o2d.3 $e |- ( ( ph /\ y e. B ) -> D e. A ) $.
|
|
f1o2d.4 $e |- ( ( ph /\ ( x e. A /\ y e. B ) ) ->
|
|
( x = D <-> y = C ) ) $.
|
|
$( Describe an implicit one-to-one onto function. (Contributed by Mario
|
|
Carneiro, 12-May-2014.) $)
|
|
f1o2d $p |- ( ph -> F : A -1-1-onto-> B ) $=
|
|
( cv wcel wceq wa wi eleq1a syl impr biimpar exp42 com34 imp32 jcai com23
|
|
biimpa impbida f1od ) ABCDEFGHEDIJKABMZDNZCMZFOZPZULENZUJGOZPZAUNPUOUPAUK
|
|
UMUOAUKPFENUMUOQJFEULRSTAUKUMUOUPQAUKUOUMUPAUKUOUMUPAUKUOPPZUPUMLUAUBUCUD
|
|
UEAUQPUKUMAUOUPUKAUOPGDNUPUKQKGDUJRSTAUOUPUKUMQAUOUKUPUMAUKUOUPUMQAUKUOUP
|
|
UMURUPUMLUGUBUFUCUDUEUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( Alternate definition of ` Swap ` as an operator abstraction.
|
|
(Contributed by SF, 23-Feb-2015.) $)
|
|
dfswap3 $p |- Swap = { <. <. x , y >. , z >. | z = <. y , x >. } $=
|
|
( vw cswap cv cop wceq wa wex copab coprab df-swap dfoprab2 eqtr4i ) EDFA
|
|
FZBFZGHCFQPGHZIBJAJDCKRABCLDCABMRABCDNO $.
|
|
|
|
$( Alternate definition of ` Swap ` as an operator mapping. (Contributed
|
|
by SF, 23-Feb-2015.) $)
|
|
dfswap4 $p |- Swap = ( x e. _V , y e. _V |-> <. y , x >. ) $=
|
|
( vz cswap cv cop wceq coprab cvv cmpt2 dfswap3 mpt2v eqtr4i ) DCEBEAEFZG
|
|
ABCHABIINJABCKABCNLM $.
|
|
$}
|
|
|
|
${
|
|
$d v w x y z A $. $d v w y z B $. $d v w z C $. $d v w x y z D $.
|
|
fmpt2x.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( Functionality, domain and codomain of a class given by the "maps to"
|
|
notation, where ` B ( x ) ` is not constant but depends on ` x ` .
|
|
(Contributed by NM, 29-Dec-2014.) $)
|
|
fmpt2x $p |- ( A. x e. A A. y e. B C e. D <->
|
|
F : U_ x e. A ( { x } X. B ) --> D ) $=
|
|
( vz vw vv cv csb wcel wral wceq eleq1d wa nfv nfcsb1v csn ciun c1st c2nd
|
|
cxp wf cfv cop op1std csbeq1d op2ndd csbeq2dv eqtrd raliunxp cmpt2 coprab
|
|
cmpt nfcri nfan nfeq2 nfcv nfcsb wb eleq1 adantr csbeq1a eleq2d sylan9bbr
|
|
anbi12d sylan9eqr eqeq2d cbvoprab12 df-mpt2 3eqtr4i mpt2mptx bitr3i nfel1
|
|
fmpt nfral cbvral raleqbidv syl5bb nfxp sneq xpeq12d cbviun feq2i 3bitr4i
|
|
vex ) AILZBJLZEMZMZFNZJAWJDMZOZICOZICWJUAZWOUEZUBZFGUFZEFNZBDOZACOACALZUA
|
|
ZDUEZUBZFGUFWQAKLZUCUGZBXHUDUGZEMZMZFNZKWTOXAXMWNKIJCWOXHWJWKUHPZXLWMFXNX
|
|
LAWJXKMWMXNAXIWJXKWJWKXHIWIZJWIZUIUJXNAWJXKWLXNBXJWKEWJWKXHXOXPUKUJULUMZQ
|
|
UNKWTFXLGABCDEUOZIJCWOWMUOZGKWTXLUQXDCNZBLZDNZRZXHEPZRZABKUPWJCNZWKWONZRZ
|
|
XHWMPZRZIJKUPXRXSYEYJABKIJYEISYEJSYHYIAYFYGAYFASAJWOAWJDTZURUSAXHWMAWJWLT
|
|
ZUTUSYHYIBYHBSBXHWMBAWJWLBWJVABWKETZVBUTUSXDWJPZYAWKPZRZYCYHYDYIYPXTYFYBY
|
|
GYNXTYFVCYOXDWJCVDVEYOYBWKDNYNYGYAWKDVDYNDWOWKAWJDVFZVGVHVIYPEWMXHYOYNEWL
|
|
WMBWKEVFZAWJWLVFZVJVKVIVLABKCDEVMIJKCWOWMVMVNHIJKCWOXLWMXQVOVNVRVPXCWPAIC
|
|
XCISWNAJWOYKAWMFYLVQVSXCWLFNZJDOYNWPXBYTBJDXBJSBWLFYMVQYOEWLFYRQVTYNYTWNJ
|
|
DWOYQYNWLWMFYSQWAWBVTXGWTFGAICXFWSIXFVAAWRWOAWRVAYKWCYNXEWRDWOXDWJWDYQWEW
|
|
FWGWH $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $. $d D x y $.
|
|
fmpt2.1 $e |- F = ( x e. A , y e. B |-> C ) $.
|
|
$( Functionality, domain and range of a class given by the "maps to"
|
|
notation. (Contributed by FL, 17-May-2010.) $)
|
|
fmpt2 $p |- ( A. x e. A A. y e. B C e. D <-> F : ( A X. B ) --> D ) $=
|
|
( wcel wral cv csn cxp ciun wf fmpt2x iunxpconst feq2i bitri ) EFIBDJACJA
|
|
CAKLDMNZFGOCDMZFGOABCDEFGHPTUAFGACDQRS $.
|
|
|
|
$( Functionality and domain of a class given by the "maps to" notation.
|
|
(Contributed by FL, 17-May-2010.) $)
|
|
fnmpt2 $p |- ( A. x e. A A. y e. B C e. V -> F Fn ( A X. B ) ) $=
|
|
( wcel wral cvv cxp wfn elex ralimi wf fmpt2 dffn2 bitr4i sylib ) EGIZBDJ
|
|
ZACJEKIZBDJZACJZFCDLZMZUBUDACUAUCBDEGNOOUEUFKFPUGABCDEKFHQUFFRST $.
|
|
|
|
fnmpt2i.2 $e |- C e. _V $.
|
|
$( Functionality and domain of a class given by the "maps to" notation.
|
|
(Contributed by FL, 17-May-2010.) $)
|
|
fnmpt2i $p |- F Fn ( A X. B ) $=
|
|
( cvv wcel wral cxp wfn rgen2w fnmpt2 ax-mp ) EIJZBDKACKFCDLMQABCDHNABCDE
|
|
FIGOP $.
|
|
|
|
$( Domain of a class given by the "maps to" notation. (Contributed by FL,
|
|
17-May-2010.) $)
|
|
dmmpt2 $p |- dom F = ( A X. B ) $=
|
|
( cxp wfn cdm wceq fnmpt2i fndm ax-mp ) FCDIZJFKPLABCDEFGHMPFNO $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Set construction lemmas
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c (x) $. $( Tail cross product $)
|
|
$c Fix $. $( Fixed points $)
|
|
$c Image $. $( Image function $)
|
|
$c Cup $. $( Cup function $)
|
|
$c Disj $. $( Disjointedness relationship. $)
|
|
$c AddC $. $( Cardinal sum function. $)
|
|
$c Ins2 $. $( Second insertion operation $)
|
|
$c Ins3 $. $( Third insertion operation $)
|
|
$c Ins4 $. $( Fourth insertion operation $)
|
|
$c SI_3 $. $( Triple singleton image. $)
|
|
$c Funs $. $( Class of functions. $)
|
|
$c Fns $. $( Function with domain relationship. $)
|
|
$c PProd $. $( Parallel product. $)
|
|
$c Cross $. $( Cross product function. $)
|
|
$c Compose $. $( Composition function. $)
|
|
$c Pw1Fn $. $( The unit power class function. $)
|
|
$c FullFun $. $( The full function operation. $)
|
|
$c Dom $. $( The domain function. $)
|
|
$c Ran $. $( The range function. $)
|
|
|
|
$( Extend the definition of a class to include the tail cross product. $)
|
|
ctxp $a class ( A (x) B ) $.
|
|
|
|
$( Define the tail cross product of two classes. Definition from [Holmes]
|
|
p. 40. See ~ brtxp for membership. (Contributed by SF, 9-Feb-2015.) $)
|
|
df-txp $a |- ( A (x) B ) = ( ( `' 1st o. A ) i^i ( `' 2nd o. B ) ) $.
|
|
|
|
$( Extend the definition of a class to include the parallel product
|
|
operation. $)
|
|
cpprod $a class PProd ( A , B ) $.
|
|
|
|
$( Define the parallel product operation. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
df-pprod $a |- PProd ( A , B ) = ( ( A o. 1st ) (x) ( B o. 2nd ) ) $.
|
|
|
|
$( Extend the definition of a class to include the fixed points of a
|
|
relationship. $)
|
|
cfix $a class Fix A $.
|
|
|
|
$( Define the fixed points of a relationship. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
df-fix $a |- Fix A = ran ( A i^i _I ) $.
|
|
|
|
$( Extend the definition of a class to include the cup function. $)
|
|
ccup $a class Cup $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the cup function. (Contributed by SF, 9-Feb-2015.) $)
|
|
df-cup $a |- Cup = ( x e. _V , y e. _V |-> ( x u. y ) ) $.
|
|
$}
|
|
|
|
$( Extend the definition of a class to include the disjoint relationship. $)
|
|
cdisj $a class Disj $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the relationship of all disjoint sets. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
df-disj $a |- Disj = { <. x , y >. | ( x i^i y ) = (/) } $.
|
|
$}
|
|
|
|
$( Extend the definition of a class to include the cardinal sum function. $)
|
|
caddcfn $a class AddC $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the function representing cardinal sum. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
df-addcfn $a |- AddC = ( x e. _V , y e. _V |-> ( x +c y ) ) $.
|
|
$}
|
|
|
|
$( Extend the definition of a class to include the compostion function. $)
|
|
ccompose $a class Compose $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the composition function. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
df-compose $a |- Compose = ( x e. _V , y e. _V |-> ( x o. y ) ) $.
|
|
$}
|
|
|
|
$( Extend the definition of a class to include the second insertion
|
|
operation. $)
|
|
cins2 $a class Ins2 A $.
|
|
|
|
$( Define the second insertion operation. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
df-ins2 $a |- Ins2 A = ( _V (x) A ) $.
|
|
|
|
$( Extend the definition of a class to include the third insertion
|
|
operation. $)
|
|
cins3 $a class Ins3 A $.
|
|
|
|
$( Define the third insertion operation. (Contributed by SF, 9-Feb-2015.) $)
|
|
df-ins3 $a |- Ins3 A = ( A (x) _V ) $.
|
|
|
|
$( Extend the definition of a class to include the image function. $)
|
|
cimage $a class Image A $.
|
|
|
|
$( Define the image function of a class. (Contributed by SF, 9-Feb-2015.)
|
|
(Revised by Scott Fenton, 19-Apr-2021.) $)
|
|
df-image $a |- Image A =
|
|
~ ( ( Ins2 _S (+) Ins3 ( _S o. `' SI A ) ) " 1c ) $.
|
|
|
|
$( Extend the definition of a class to include the fourth insertion
|
|
operation. $)
|
|
cins4 $a class Ins4 A $.
|
|
|
|
$( Define the fourth insertion operation. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
df-ins4 $a |- Ins4 A =
|
|
( `' ( 1st (x) ( ( 1st o. 2nd ) (x) ( ( 1st o. 2nd ) o. 2nd ) ) ) " A ) $.
|
|
|
|
$( Extend the definition of a class to include the triple singleton image. $)
|
|
csi3 $a class SI_3 A $.
|
|
|
|
$( Define the triple singleton image. (Contributed by SF, 9-Feb-2015.) $)
|
|
df-si3 $a |- SI_3 A =
|
|
( ( SI 1st (x) ( SI ( 1st o. 2nd ) (x) SI ( 2nd o. 2nd ) ) ) " ~P1 A ) $.
|
|
|
|
$( Extend the definition of a class to include the set of all functions. $)
|
|
cfuns $a class Funs $.
|
|
|
|
$( Define the class of all functions. (Contributed by SF, 9-Feb-2015.) $)
|
|
df-funs $a |- Funs = { f | Fun f } $.
|
|
|
|
$( Extend the definition of a class to include the function with domain
|
|
relationship. $)
|
|
cfns $a class Fns $.
|
|
|
|
${
|
|
$d f a $.
|
|
$( Define the function with domain relationship. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
df-fns $a |- Fns = { <. f , a >. | f Fn a } $.
|
|
$}
|
|
|
|
$( Extend the definition of a class to include the cross product function. $)
|
|
ccross $a class Cross $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the cross product function. (Contributed by SF, 9-Feb-2015.) $)
|
|
df-cross $a |- Cross = ( x e. _V , y e. _V |-> ( x X. y ) ) $.
|
|
$}
|
|
|
|
$( Extend the definition of a class to include the unit power class
|
|
function. $)
|
|
cpw1fn $a class Pw1Fn $.
|
|
|
|
$( Define the function that takes a singleton to the unit power class of its
|
|
member. This function is defined in such a way as to ensure
|
|
stratification. (Contributed by SF, 9-Feb-2015.) $)
|
|
df-pw1fn $a |- Pw1Fn = ( x e. 1c |-> ~P1 U. x ) $.
|
|
|
|
$( Extend the definition of a class to include the full function
|
|
operation. $)
|
|
cfullfun $a class FullFun F $.
|
|
|
|
$( Define the full function operator. This is a function over ` _V ` that
|
|
agrees with the function value of ` F ` at every point. (Contributed by
|
|
SF, 9-Feb-2015.) $)
|
|
df-fullfun $a |- FullFun F =
|
|
( ( ( _I o. F ) \ ( ~ _I o. F ) ) u.
|
|
( ~ dom ( ( _I o. F ) \ ( ~ _I o. F ) ) X. { (/) } ) ) $.
|
|
|
|
$( Extend the definition of a class to include the domain function. $)
|
|
cdomfn $a class Dom $.
|
|
|
|
$( Define the domain function. This is a function wrapper for the domain
|
|
operator. (Contributed by Scott Fenton, 9-Aug-2019.) $)
|
|
df-domfn $a |- Dom = ( x e. _V |-> dom x ) $.
|
|
|
|
$( Extend the definition of a class to include the range function. $)
|
|
cranfn $a class Ran $.
|
|
|
|
$( Define the range function. This is a function wrapper for the range
|
|
operator. (Contributed by Scott Fenton, 9-Aug-2019.) $)
|
|
df-ranfn $a |- Ran = ( x e. _V |-> ran x ) $.
|
|
|
|
${
|
|
$d A x y z w $. $d B x y z w $. $d R x y z w $.
|
|
brsnsi.1 $e |- A e. _V $.
|
|
brsnsi.2 $e |- B e. _V $.
|
|
$( Binary relationship of singletons in a singleton image. (Contributed by
|
|
SF, 9-Feb-2015.) $)
|
|
brsnsi $p |- ( { A } SI R { B } <-> A R B ) $=
|
|
( vx vy vz vw csn wbr cv wceq w3a wex snex eqeq1 eqcom vex bitri csi brab
|
|
sneqb syl6bb 3anbi1d 2exbidv 3anbi2d df-si breq1 breq2 ceqsex2v ) AJZBJZC
|
|
UAZKFLZAMZGLZBMZUOUQCKZNZGOFOZABCKZHLZUOJZMZILZUQJZMZUSNZGOFOUPVHUSNZGOFO
|
|
VAHIULUMUNAPBPVCULMZVIVJFGVKVEUPVHUSVKVEULVDMZUPVCULVDQVLVDULMUPULVDRUOAF
|
|
SUCTUDUEUFVFUMMZVJUTFGVMVHURUPUSVMVHUMVGMZURVFUMVGQVNVGUMMURUMVGRUQBGSUCT
|
|
UDUGUFHIFGCUHUBUSAUQCKVBFGABDEUOAUQCUIUQBACUJUKT $.
|
|
|
|
$( Ordered pair membership of singletons in a singleton image.
|
|
(Contributed by SF, 9-Feb-2015.) $)
|
|
opsnelsi $p |- ( <. { A } , { B } >. e. SI R <-> <. A , B >. e. R ) $=
|
|
( csn csi wbr cop wcel brsnsi df-br 3bitr3i ) AFZBFZCGZHABCHNOIPJABICJABC
|
|
DEKNOPLABCLM $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d R x $. $d R y $. $d x y $.
|
|
brsnsi1.1 $e |- A e. _V $.
|
|
$( Binary relationship of a singleton to an arbitrary set in a singleton
|
|
image. (Contributed by SF, 9-Mar-2015.) $)
|
|
brsnsi1 $p |- ( { A } SI R B <-> E. x ( B = { x } /\ A R x ) ) $=
|
|
( vy csn csi wbr cv wceq w3a wex wa brsi excom eqcom vex bitri exbii
|
|
sneqb 3anbi1i 3anass breq1 anbi2d ceqsexv ) BGZCDHIUGFJZGZKZCAJZGKZUHUKDI
|
|
ZLZAMFMZULBUKDIZNZAMZFAUGCDOUOUNFMZAMURUNFAPUSUQAUSUHBKZULUMNZNZFMUQUNVBF
|
|
UNUTULUMLVBUJUTULUMUJUIUGKUTUGUIQUHBFRUASUBUTULUMUCSTVAUQFBEUTUMUPULUHBUK
|
|
DUDUEUFSTSS $.
|
|
|
|
$( Binary relationship of an arbitrary set to a singleton in a singleton
|
|
image. (Contributed by SF, 9-Mar-2015.) $)
|
|
brsnsi2 $p |- ( B SI R { A } <-> E. x ( B = { x } /\ x R A ) ) $=
|
|
( vy csn csi wbr cv wceq w3a wex wa brsi 3anass exbii 19.42v sneqb bitri
|
|
eqcom anbi1i breq2 ceqsexv anbi2i ) CBGZDHICAJZGKZUFFJZGKZUGUIDIZLZFMZAMU
|
|
HUGBDIZNZAMAFCUFDOUMUOAUMUHUJUKNZNZFMZUOULUQFUHUJUKPQURUHUPFMZNUOUHUPFRUS
|
|
UNUHUSUIBKZUKNZFMUNUPVAFUJUTUKUJBUIKUTBUIESBUIUATUBQUKUNFBEUIBUGDUCUDTUET
|
|
TQT $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $. $d R x $.
|
|
brco1st.1 $e |- A e. _V $.
|
|
brco1st.2 $e |- B e. _V $.
|
|
$( Binary relationship of composition with ` 1st ` . (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
brco1st $p |- ( <. A , B >. ( R o. 1st ) C <-> A R C ) $=
|
|
( vx cop c1st ccom wbr cv wa wex wceq brco opbr1st eqcom bitri anbi1i
|
|
exbii breq1 ceqsexv 3bitri ) ABHZCDIJKUEGLZIKZUFCDKZMZGNUFAOZUHMZGNACDKZG
|
|
UECDIPUIUKGUGUJUHUGAUFOUJABUFEFQAUFRSTUAUHULGAEUFACDUBUCUD $.
|
|
|
|
$( Binary relationship of composition with ` 2nd ` . (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
brco2nd $p |- ( <. A , B >. ( R o. 2nd ) C <-> B R C ) $=
|
|
( vx cop c2nd ccom wbr cv wa wex wceq brco opbr2nd eqcom bitri anbi1i
|
|
exbii breq1 ceqsexv 3bitri ) ABHZCDIJKUEGLZIKZUFCDKZMZGNUFBOZUHMZGNBCDKZG
|
|
UECDIPUIUKGUGUJUHUGBUFOUJABUFEFQBUFRSTUAUHULGBFUFBCDUBUCUD $.
|
|
$}
|
|
|
|
$( Equality theorem for tail cross product. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
txpeq1 $p |- ( A = B -> ( A (x) C ) = ( B (x) C ) ) $=
|
|
( wceq c1st ccnv ccom c2nd cin ctxp coeq2 ineq1d df-txp 3eqtr4g ) ABDZEFZAG
|
|
ZHFCGZIPBGZRIACJBCJOQSRABPKLACMBCMN $.
|
|
|
|
$( Equality theorem for tail cross product. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
txpeq2 $p |- ( A = B -> ( C (x) A ) = ( C (x) B ) ) $=
|
|
( wceq c1st ccnv ccom c2nd cin ctxp coeq2 ineq2d df-txp 3eqtr4g ) ABDZEFCGZ
|
|
HFZAGZIPQBGZICAJCBJORSPABQKLCAMCBMN $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d B y $. $d B z $. $d C x $. $d C z $. $d R t $.
|
|
$d R x $. $d R y $. $d R z $. $d S t $. $d S x $. $d S y $. $d S z $.
|
|
$d t x $. $d t y $. $d t z $. $d x y $. $d x z $. $d y z $.
|
|
|
|
$( Trinary relationship over a tail cross product. (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
trtxp $p |- ( A ( R (x) S ) <. B , C >. <-> ( A R B /\ A S C ) ) $=
|
|
( vy vz vt cop wbr cvv wcel wa brex wb cv breq2 wex weq 3bitri ctxp opexb
|
|
vx anbi2i sylib anim12i anandi sylibr wi wceq breq1 anbi12d bibi12d opeq1
|
|
imbi2d breq2d anbi1d opeq2 anbi2d c1st ccnv ccom c2nd cin breqi brin brco
|
|
df-txp ancom brcnv vex opbr1st equcom bitri exbii ceqsexv opbr2nd anbi12i
|
|
anbi1i vtocl2g vtoclg imp pm5.21nii ) ABCIZDEUAZJZAKLZBKLZCKLZMZMZABDJZAC
|
|
EJZMZWFWGWDKLZMWKAWDWENWOWJWGBCUBUDUEWNWGWHMZWGWIMZMWKWLWPWMWQABDNACENUFW
|
|
GWHWIUGUHWGWJWFWNOZWJUCPZWDWEJZWSBDJZWSCEJZMZOZUIWJWRUIUCAKWSAUJZXDWRWJXE
|
|
WTWFXCWNWSAWDWEUKXEXAWLXBWMWSABDUKWSACEUKULUMUOWSFPZGPZIZWEJZWSXFDJZWSXGE
|
|
JZMZOWSBXGIZWEJZXAXKMZOXDFGBCKKXFBUJZXIXNXLXOXPXHXMWSWEXFBXGUNUPXPXJXAXKX
|
|
FBWSDQUQUMXGCUJZXNWTXOXCXQXMWDWSWEXGCBURUPXQXKXBXAXGCWSEQUSUMXIWSXHUTVAZD
|
|
VBZVCVAZEVBZVDZJWSXHXSJZWSXHYAJZMXLWSXHWEYBDEVHVEWSXHXSYAVFYCXJYDXKYCWSHP
|
|
ZDJZYEXHXRJZMZHRHFSZYFMZHRXJHWSXHXRDVGYHYJHYHYGYFMYJYFYGVIYGYIYFYGXHYEUTJ
|
|
FHSYIYEXHUTVJXFXGYEFVKZGVKZVLFHVMTVSVNVOYFXJHXFYKYEXFWSDQVPTYDWSYEEJZYEXH
|
|
XTJZMZHRHGSZYMMZHRXKHWSXHXTEVGYOYQHYOYNYMMYQYMYNVIYNYPYMYNXHYEVCJGHSYPYEX
|
|
HVCVJXFXGYEYKYLVQGHVMTVSVNVOYMXKHXGYLYEXGWSEQVPTVRTVTWAWBWC $.
|
|
$}
|
|
|
|
$( Ordered triple membership in a tail cross product. (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
oteltxp $p |- ( <. A , <. B , C >. >. e. ( R (x) S ) <->
|
|
( <. A , B >. e. R /\ <. A , C >. e. S ) ) $=
|
|
( cop ctxp wbr wa wcel trtxp df-br anbi12i 3bitr3i ) ABCFZDEGZHABDHZACEHZIA
|
|
OFPJABFDJZACFEJZIABCDEKAOPLQSRTABDLACELMN $.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d R x $. $d R y $. $d S x $.
|
|
$d S y $. $d B w $. $d B z $. $d w x $. $d w y $. $d w z $. $d x y $.
|
|
$d x z $. $d y z $.
|
|
$( Binary relationship over a tail cross product. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
brtxp $p |- ( A ( R (x) S ) B <->
|
|
E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) ) $=
|
|
( vz vw wbr cv c1st ccnv wa c2nd wex cop wceq ccom bitri weq ctxp w3a cin
|
|
brin anbi12i df-txp breqi eeanv 3bitr4i an4 ancom brcnv br1st br2nd eqtr2
|
|
brco vex opth simplbi eqcomd opeq1d wb eqeq1 adantl mpbird exlimivv opeq2
|
|
eqeq2d opeq1 bi2anan9 spc2ev anidms impbii 3bitr2i anbi2i 3anass 2exbii
|
|
syl ) CDEFUAZIZCAJZEIZWADKLZIZMZCBJZFIZWFDNLZIZMZMZBOAOZDWAWFPZQZWBWGUBZB
|
|
OAOCDWCERZWHFRZUCZIZWEAOZWJBOZMZVTWLWSCDWPIZCDWQIZMXBCDWPWQUDXCWTXDXAACDW
|
|
CEUPBCDWHFUPUESCDVSWREFUFUGWEWJABUHUIWKWOABWKWBWGMZWDWIMZMZWOWBWDWGWIUJXE
|
|
WNMWNXEMXGWOXEWNUKXFWNXEXFDWAGJZPZQZGOZDHJZWFPZQZHOZMXJXNMZHOGOZWNWDXKWIX
|
|
OWDDWAKIXKWADKULGDWAAUQZUMSWIDWFNIXOWFDNULHDWFBUQZUNSUEXJXNGHUHXQWNXPWNGH
|
|
XPWNXMWMQZXPXIXMQZXTDXIXMUOYAXLWAWFYAWAXLYAAHTGBTZWAXHXLWFURUSUTVAVRXNWNX
|
|
TVBXJDXMWMVCVDVEVFWNXQXPWNWNMGHWFWAXSXRYBXJWNHATZXNWNYBXIWMDXHWFWAVGVHYCX
|
|
MWMDXLWAWFVIVHVJVKVLVMVNVOWNWBWGVPUISVQS $.
|
|
$}
|
|
|
|
$( The tail cross product of two sets is a set. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
txpexg $p |- ( ( A e. V /\ B e. W ) -> ( A (x) B ) e. _V ) $=
|
|
( wcel ctxp c1st ccnv ccom c2nd cin cvv df-txp 1stex cnvex coexg mpan 2ndex
|
|
wa inexg syl2an syl5eqel ) ACEZBDEZSABFGHZAIZJHZBIZKZLABMUCUFLEZUHLEZUILEUD
|
|
UELEUCUJGNOUEALCPQUGLEUDUKJROUGBLDPQUFUHLLTUAUB $.
|
|
|
|
${
|
|
txpex.1 $e |- A e. _V $.
|
|
txpex.2 $e |- B e. _V $.
|
|
$( The tail cross product of two sets is a set. (Contributed by SF,
|
|
9-Feb-2015.) $)
|
|
txpex $p |- ( A (x) B ) e. _V $=
|
|
( cvv wcel ctxp txpexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d B a $. $d B b $. $d b x $. $d B x $. $d b y $. $d B y $. $d C a $.
|
|
$d C b $. $d C x $. $d C y $. $d x y $.
|
|
$( Restriction distributes over tail cross product. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
restxp $p |- ( ( A (x) B ) |` C ) = ( ( A |` C ) (x) ( B |` C ) ) $=
|
|
( vx vy va vb ctxp cres cv wbr w3a wex 3anass anbi1i bitri 3bitr4i 2exbii
|
|
wa brres cop wceq wcel anandir anass brtxp 19.41vv biid 3anbi123i eqbrriv
|
|
anbi2i ) DEABHZCIZACIZBCIZHZEJZFJZGJZUAUBZDJZURAKZVAUSBKZLZVACUCZSZGMFMZU
|
|
TVBVESZVCVESZLZGMFMZVAUQUMKZVAUQUPKZVFVJFGUTVBVCSZVESZSZUTVHVISZSVFVJVOVQ
|
|
UTVBVCVEUDUKVFUTVNSZVESVPVDVRVEUTVBVCNOUTVNVEUEPUTVHVINQRVAUQULKZVESVDGMF
|
|
MZVESVLVGVSVTVEFGVAUQABUFOVAUQULCTVDVEFGUGQVMUTVAURUNKZVAUSUOKZLZGMFMVKFG
|
|
VAUQUNUOUFWCVJFGUTUTWAVHWBVIUTUHVAURACTVAUSBCTUIRPQUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d R x y $.
|
|
$( Membership in the fixed points of a relationship. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
elfix $p |- ( A e. Fix R <-> A R A ) $=
|
|
( vx vy cfix wcel cvv wbr elex brex simpld cv eleq1 wceq wb breq12 cid wa
|
|
wex 3bitri anidms cin crn df-fix eleq2i elrn brin ancom ideq anbi1i exbii
|
|
weq vex bitri breq1 ceqsexv vtoclbg pm5.21nii ) ABEZFZAGFZAABHZAUSIVBVAVA
|
|
AABJKCLZUSFZVCVCBHZUTVBCAGVCAUSMVCANVEVBOVCAVCABPUAVDVCBQUBZUCZFZDCULZDLZ
|
|
VCBHZRZDSZVEUSVGVCBUDUEVHVJVCVFHZDSVMDVCVFUFVNVLDVNVKVJVCQHZRVOVKRVLVJVCB
|
|
QUGVKVOUHVOVIVKVJVCCUMZUIUJTUKUNVKVEDVCVPVJVCVCBUOUPTUQUR $.
|
|
$}
|
|
|
|
$( The fixed points of a set form a set. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
fixexg $p |- ( R e. V -> Fix R e. _V ) $=
|
|
( wcel cfix cid cin crn cvv df-fix idex inexg mpan2 rnexg syl syl5eqel ) AB
|
|
CZADAEFZGZHAIPQHCZRHCPEHCSJAEBHKLQHMNO $.
|
|
|
|
${
|
|
fixex.1 $e |- R e. _V $.
|
|
$( The fixed points of a set form a set. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
fixex $p |- Fix R e. _V $=
|
|
( cvv wcel cfix fixexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $.
|
|
op1st2nd.1 $e |- A e. _V $.
|
|
op1st2nd.2 $e |- B e. _V $.
|
|
$( Express equality to an ordered pair via ` 1st ` and ` 2nd ` .
|
|
(Contributed by SF, 12-Feb-2015.) $)
|
|
op1st2nd $p |- ( ( C 1st A /\ C 2nd B ) <-> C = <. A , B >. ) $=
|
|
( vx c1st wbr c2nd wa cop wceq cv wex wi opbr2nd breq1 mpbiri eqid mpbir
|
|
br1st vex biimpi opeq2d eqeq1 imbi12d exlimiv sylbi opbr1st pm3.2i impbii
|
|
imp anbi12d ) CAGHZCBIHZJZCABKZLZUNUOURUNCAFMZKZLZFNUOUROZFCADUAVAVBFVAVB
|
|
UTBIHZUTUQLZOVCUSBAVCUSBLAUSBDFUBPUCUDVAUOVCURVDCUTBIQCUTUQUEUFRUGUHULURU
|
|
PUQAGHZUQBIHZJVEVFVEAALASABADEUITVFBBLBSABBDEPTUJURUNVEUOVFCUQAGQCUQBIQUM
|
|
RUK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $. $d R x $.
|
|
otelins2.1 $e |- B e. _V $.
|
|
$( Ordered triple membership in ` Ins2 ` . (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
otelins2 $p |- ( <. A , <. B , C >. >. e. Ins2 R <-> <. A , C >. e. R ) $=
|
|
( vx cop cins2 wcel cvv elex opexb simplbi syl wceq opeq1 eleq1d vex opex
|
|
cv ctxp wa df-ins2 eleq2i oteltxp bitri mpbiran vtoclbg pm5.21nii ) ABCGZ
|
|
GZDHZIZAJIZACGZDIZUMUKJIZUNUKULKUQUNUJJIAUJLMNUPUOJIZUNUODKURUNCJIACLMNFT
|
|
ZUJGZULIZUSCGZDIZUMUPFAJUSAOZUTUKULUSAUJPQVDVBUODUSACPQVAUSBGJIZVCUSBFRES
|
|
VAUTJDUAZIVEVCUBULVFUTDUCUDUSBCJDUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d C x $. $d R x $.
|
|
otelins3.1 $e |- C e. _V $.
|
|
$( Ordered triple membership in ` Ins3 ` . (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
otelins3 $p |- ( <. A , <. B , C >. >. e. Ins3 R <-> <. A , B >. e. R ) $=
|
|
( vx cop cins3 wcel cvv elex opexb simplbi syl wceq opeq1 eleq1d vex opex
|
|
cv ctxp wa df-ins3 eleq2i oteltxp bitri mpbiran2 vtoclbg pm5.21nii ) ABCG
|
|
ZGZDHZIZAJIZABGZDIZUMUKJIZUNUKULKUQUNUJJIAUJLMNUPUOJIZUNUODKURUNBJIABLMNF
|
|
TZUJGZULIZUSBGZDIZUMUPFAJUSAOZUTUKULUSAUJPQVDVBUODUSABPQVAVCUSCGJIZUSCFRE
|
|
SVAUTDJUAZIVCVEUBULVFUTDUCUDUSBCDJUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d A x y t $. $d B x y t $. $d R x y t $.
|
|
brimage.1 $e |- A e. _V $.
|
|
brimage.2 $e |- B e. _V $.
|
|
$( Binary relationship over the image function. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
brimage $p |- ( A Image R B <-> B = ( R " A ) ) $=
|
|
( vx vt vy cv wcel cima cop csset wn wbr wex bitri wa exbii 3bitri wb wal
|
|
cins2 csi ccnv ccom csymdif c1c wceq cimage csn elima1c elsymdif otelins2
|
|
cins3 vex opelssetsn otelins3 brcnv anbi1i 19.41v bitr4i excom anass snex
|
|
brsnsi2 breq1 anbi2d ancom brssetsn syl6bb ceqsexv opelco 3bitr4i bibi12i
|
|
elima2 xchbinx exnal con2bii dfcleq df-image breqi df-br elcompl 3bitr4ri
|
|
ccompl opex ) FIZBJZWHCAKZJZUAZFUBZABLZMUCZMCUDZUEZUFZUOZUGZUHKZJZNZBWJUI
|
|
ABCUJZOZXBWMXBWHUKZWNLZWTJZFPWLNZFPWMNFWNWTULXHXIFXHXGWOJZXGWSJZUAWLXGWOW
|
|
SUMXJWIXKWKXJXFBLMJWIXFABMDUNWHBFUPZEUQQXKXFALWRJZWKXFABWREURXFGIZWQOZXNA
|
|
MOZRZGPZHIZAJZXSWHCOZRZHPZXMWKXRXNXSUKZUIZYARZXPRZHPZGPYGGPZHPYCXQYHGXQYF
|
|
HPZXPRYHXOYJXPXOXNXFWPOYJXFXNWPUSHWHXNCXLVFQUTYFXPHVAVBSYGGHVCYIYBHYIYEYA
|
|
XPRZRZGPYBYGYLGYEYAXPVDSYKYBGYDXSVEYEYKYAYDAMOZRZYBYEXPYMYAXNYDAMVGVHYNYM
|
|
YARYBYAYMVIYMXTYAXSAHUPDVJUTQVKVLQSTGXFAMWQVMHWHCAVPVNQVOVQSWLFVRTVSFBWJV
|
|
TXEABXAWFZOWNYOJXCABXDYOCWAWBABYOWCWNXAABDEWGWDTWE $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a p $. $d a x $. $d A x $. $d a y $. $d a z $. $d b p $.
|
|
$d b x $. $d B x $. $d b y $. $d B y $. $d b z $. $d B z $. $d C x $.
|
|
$d C z $. $d D a $. $d D b $. $d D p $. $d D x $. $d D y $. $d D z $.
|
|
$d p x $. $d p y $. $d p z $. $d R p $. $d R x $. $d R y $. $d R z $.
|
|
$d x y $. $d x z $. $d y z $.
|
|
oqelins4.4 $e |- D e. _V $.
|
|
$( Ordered quadruple membership in ` Ins4 ` . (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
oqelins4 $p |- ( <. A , <. B , <. C , D >. >. >. e. Ins4 R
|
|
<-> <. A , <. B , C >. >. e. R ) $=
|
|
( vx vp vb va cop wcel cvv wa bitri wceq opeq1 c1st wbr wex vy cins4 elex
|
|
vz opexb anbi2i simplbi anim2i sylbi sylib wb cv wi eleq1d bibi12d imbi2d
|
|
syl opeq2d opeq2 c2nd ccom ctxp ccnv cima df-ins4 eleq2i wrex brcnv brtxp
|
|
w3a weq 3ancoma 3anass vex opex opbr1st equcom anbi1i 3bitri exbii 19.42v
|
|
eqeq2d anbi1d exbidv ceqsexv ancom 3anrot brco2nd biid 3anbi123i ceqsex2v
|
|
2exbii rexbii elima risset 3bitr4i vtocl2g vtoclg imp pm5.21nii ) ABCDKZK
|
|
ZKZEUBZLZAMLZBMLZCMLZNZNZABCKZKZELZXEXCMLZXJXCXDUCXNXFXGXAMLZNZNZXJXNXFXB
|
|
MLZNXQAXBUEXRXPXFBXAUEUFOXPXIXFXOXHXGXOXHDMLCDUEUGUHUHUIUQXMXLMLZXJXLEUCX
|
|
SXFXKMLZNXJAXKUEXTXIXFBCUEUFOUJXFXIXEXMUKZXIGULZXBKZXDLZYBXKKZELZUKZUMXIY
|
|
AUMGAMYBAPZYGYAXIYHYDXEYFXMYHYCXCXDYBAXBQUNYHYEXLEYBAXKQUNUOUPYBUAULZUDUL
|
|
ZDKZKZKZXDLZYBYIYJKZKZELZUKYBBYKKZKZXDLZYBBYJKZKZELZUKYGUAUDBCMMYIBPZYNYT
|
|
YQUUCUUDYMYSXDUUDYLYRYBYIBYKQURUNUUDYPUUBEUUDYOUUAYBYIBYJQURUNUOYJCPZYTYD
|
|
UUCYFUUEYSYCXDUUEYRXBYBUUEYKXABYJCDQURURUNUUEUUBYEEUUEUUAXKYBYJCBUSURUNUO
|
|
YNYMRRUTVAZUUFUTVAZVBZVBZVCZEVDZLZYQXDUUKYMEVEVFHULZYMUUJSZHEVGUUMYPPZHEV
|
|
GUULYQUUNUUOHEUUNYMUUMUUISZUUMYBIULZKZPZYMUUQUUHSZNZITZUUOUUMYMUUIVHUUPUU
|
|
MJULZUUQKZPZYMUVCRSZUUTVJZITZJTJGVKZUVEUUTNZITZNZJTUVBJIYMUUMRUUHVIUVHUVL
|
|
JUVHUVIUVJNZITUVLUVGUVMIUVGUVFUVEUUTVJUVFUVJNUVMUVEUVFUUTVLUVFUVEUUTVMUVF
|
|
UVIUVJUVFGJVKUVIYBYLUVCGVNZYIYKUAVNZYJDUDVNZFVOZVOZVPGJVQOVRVSVTUVIUVJIWA
|
|
OVTUVKUVBJYBUVNUVIUVJUVAIUVIUVEUUSUUTUVIUVDUURUUMUVCYBUUQQWBWCWDWEVSUVBUU
|
|
QYOPZUUSNZITUUOUVAUVTIUVAUUTUUSNUVTUUSUUTWFUUTUVSUUSUUTUUQUUMUVCKZPZYMUUM
|
|
UUFSZYMUVCUUGSZVJZJTHTHUAVKZJUDVKZUWBVJZJTHTUVSHJYMUUQUUFUUGVIUWEUWHHJUWE
|
|
UWCUWDUWBVJUWHUWBUWCUWDWGUWCUWFUWDUWGUWBUWBUWCYLUUMRSUAHVKUWFYBYLUUMRUVNU
|
|
VRWHYIYKUUMUVOUVQVPUAHVQVSUWDYLUVCUUFSZUDJVKZUWGYBYLUVCUUFUVNUVRWHUWIYKUV
|
|
CRSUWJYIYKUVCRUVOUVQWHYJDUVCUVPFVPOUDJVQVSUWBWIWJOWLUWBUUQYIUVCKZPUVSHJYI
|
|
YJUVOUVPUWFUWAUWKUUQUUMYIUVCQWBUWGUWKYOUUQUVCYJYIUSWBWKVSVROVTUUSUUOIYOYI
|
|
YJUVOUVPVOUVSUURYPUUMUUQYOYBUSWBWEOVSWMHYMUUJEWNHYPEWOWPOWQWRWSWT $.
|
|
$}
|
|
|
|
$( ` Ins2 ` preserves sethood. (Contributed by SF, 9-Mar-2015.) $)
|
|
ins2exg $p |- ( A e. V -> Ins2 A e. _V ) $=
|
|
( wcel cins2 cvv ctxp df-ins2 vvex txpexg mpan syl5eqel ) ABCZADEAFZEAGEECL
|
|
MECHEAEBIJK $.
|
|
|
|
$( ` Ins3 ` preserves sethood. (Contributed by SF, 22-Feb-2015.) $)
|
|
ins3exg $p |- ( A e. V -> Ins3 A e. _V ) $=
|
|
( wcel cins3 cvv ctxp df-ins3 vvex txpexg mpan2 syl5eqel ) ABCZADAEFZEAGLEE
|
|
CMECHAEBEIJK $.
|
|
|
|
${
|
|
insex.1 $e |- A e. _V $.
|
|
$( ` Ins2 ` preserves sethood. (Contributed by SF, 12-Feb-2015.) $)
|
|
ins2ex $p |- Ins2 A e. _V $=
|
|
( cvv wcel cins2 ins2exg ax-mp ) ACDAECDBACFG $.
|
|
|
|
$( ` Ins3 ` preserves sethood. (Contributed by SF, 12-Feb-2015.) $)
|
|
ins3ex $p |- Ins3 A e. _V $=
|
|
( cvv wcel cins3 ins3exg ax-mp ) ACDAECDBACFG $.
|
|
|
|
$( ` Ins4 ` preserves sethood. (Contributed by SF, 12-Feb-2015.) $)
|
|
ins4ex $p |- Ins4 A e. _V $=
|
|
( cins4 c1st c2nd ccom ctxp ccnv cima cvv df-ins4 1stex 2ndex txpex cnvex
|
|
coex imaex eqeltri ) ACDDEFZSEFZGZGZHZAIJAKUCAUBDUALSTDELMPZSEUDMPNNOBQR
|
|
$.
|
|
$}
|
|
|
|
$( The image function of a set is a set. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
imageexg $p |- ( A e. V -> Image A e. _V ) $=
|
|
( wcel cimage csset cins2 csi ccnv ccom cins3 csymdif c1c cima cvv df-image
|
|
ccompl siexg ssetex mpan 3syl cnvexg coexg ins3exg ins2ex 1cex imaexg mpan2
|
|
symdifexg complexg syl5eqel ) ABCZADEFZEAGZHZIZJZKZLMZPZNAOUKUQNCZURNCZUSNC
|
|
UKUONCZUPNCZUTUKUMNCUNNCZVBABQUMNUAENCVDVBREUNNNUBSTUONUCULNCVCUTERUDULUPNN
|
|
UHSTUTLNCVAUEUQLNNUFUGURNUITUJ $.
|
|
|
|
${
|
|
imageex.1 $e |- A e. _V $.
|
|
$( The image function of a set is a set. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
imageex $p |- Image A e. _V $=
|
|
( cvv wcel cimage imageexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d p x $. $d p y $. $d p z $. $d R p $. $d R x $. $d R y $. $d R z $.
|
|
$d S p $. $d S x $. $d S y $. $d S z $. $d x y $. $d x z $. $d y z $.
|
|
$( The domain of a tail cross product is the intersection of the domains of
|
|
its arguments. (Contributed by SF, 18-Feb-2015.) $)
|
|
dmtxp $p |- dom ( R (x) S ) = ( dom R i^i dom S ) $=
|
|
( vx vp vy vz ctxp cdm cin cv wbr wex wa wcel cop exbii bitri bicomi eldm
|
|
vex wceq w3a brtxp exrot3 3anass 19.41v opex isseti biantrur elin anbi12i
|
|
2exbii eeanv 3bitr4i eqriv ) CABGZHZAHZBHZIZCJZDJZUPKZDLZVAEJZAKZVAFJZBKZ
|
|
MZFLELZVAUQNVAUTNZVDVBVEVGOZUAZVFVHUBZDLZFLELZVJVDVNFLELZDLVPVCVQDEFVAVBA
|
|
BUCPVNDEFUDQVOVIEFVOVMDLZVIMZVIVOVMVIMZDLVSVNVTDVMVFVHUEPVMVIDUFQVIVSVRVI
|
|
DVLVEVGETFTUGUHUIRQULQDVAUPSVKVAURNZVAUSNZMZVJVAURUSUJWCVFELZVHFLZMZVJWAW
|
|
DWBWEEVAASFVABSUKVJWFVFVHEFUMRQQUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d F t $. $d F x $. $d F y $. $d F z $. $d R t $. $d R x $. $d R y $.
|
|
$d R z $. $d S t $. $d S x $. $d S y $. $d S z $. $d t x $. $d t y $.
|
|
$d t z $. $d x y $. $d x z $. $d y z $.
|
|
txpcofun.1 $e |- Fun F $.
|
|
$( Composition distributes over tail cross product in the case of a
|
|
function. (Contributed by SF, 18-Feb-2015.) $)
|
|
txpcofun $p |- ( ( R (x) S ) o. F ) = ( ( R o. F ) (x) ( S o. F ) ) $=
|
|
( vx vt vy vz ccom cv cop wceq wex wcel wb ax-mp wbr wa breq1 opelco ctxp
|
|
cvv vex opeqex cdm cfv dmcoss opeldm sseldi pm4.71ri anbi1i anass ceqsexv
|
|
fvex anbi12i eqcom funbrfvb syl5bb anbi1d exbidv syl6bbr anbi12d syl5rbbr
|
|
wfun mpan pm5.32i 3bitrri 19.41v wi funbrfv trtxp syl exbii 3bitr4i bitri
|
|
eldm oteltxp opeq2 eleq1d bibi12d mpbiri exlimivv eqrelriv ) EFABUAZCIZAC
|
|
IZBCIZUAZFJZGJZHJZKZLZHMGMZEJZWIKZWENZWPWHNZOZWIUBNWNFUCGHWIUBUDPWMWSGHWM
|
|
WSWOWLKZWENZWTWHNZOWOCUEZNZWOCUFZWJAQZXEWKBQZRZRZWOWJKWFNZWOWKKWGNZRZXAXB
|
|
XLXDXJRZXKRXDXLRXIXJXMXKXJXDXJWFUEXCWOACUGWOWJWFUHUIUJUKXDXJXKULXDXLXHXHW
|
|
IXELZWIWJAQZRZFMZXNWIWKBQZRZFMZRXDXLXQXFXTXGXOXFFXEWOCUNZWIXEWJASUMXRXGFX
|
|
EYAWIXEWKBSUMUOXDXQXJXTXKXDXQWOWICQZXORZFMXJXDXPYCFXDXNYBXOXNXEWILZXDYBWI
|
|
XEUPCVDZXDYDYBODWOWICUQVEURZUSUTFWOWJACTVAXDXTYBXRRZFMXKXDXSYGFXDXNYBXRYF
|
|
USUTFWOWKBCTVAVBVCVFVGXAYBWIWLWDQZRZFMZXIFWOWLWDCTYBXHRZFMYBFMZXHRYJXIYBX
|
|
HFVHYIYKFYBYHXHYBYDYHXHOYEYBYDVIDWOWICVJPXHXEWLWDQYDYHXEWJWKABVKXEWIWLWDS
|
|
VCVLVFVMXDYLXHFWOCVPUKVNVOWOWJWKWFWGVQVNWMWQXAWRXBWMWPWTWEWIWLWOVRZVSWMWP
|
|
WTWHYMVSVTWAWBPWC $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a c $. $d a d $. $d a x $. $d a y $. $d a z $. $d b c $.
|
|
$d b d $. $d b x $. $d b y $. $d b z $. $d c d $. $d c x $. $d c y $.
|
|
$d c z $. $d d x $. $d d y $. $d d z $. $d F a $. $d F b $. $d F c $.
|
|
$d F d $. $d F x $. $d F y $. $d F z $. $d G a $. $d G b $. $d G c $.
|
|
$d G d $. $d G x $. $d G y $. $d G z $. $d x y $. $d x z $. $d y z $.
|
|
$( If ` F ` and ` G ` are functions, then their tail cross product is a
|
|
function over the intersection of their domains. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
fntxp $p |- ( ( F Fn A /\ G Fn B ) -> ( F (x) G ) Fn ( A i^i B ) ) $=
|
|
( vx vy vz va vb vc vd wfun cdm wceq wa wfn cv wbr weq wex cin wi wal cop
|
|
w3a brtxp anbi12i ee4anv bitr4i an6 fununiq 3expib im2anan9 eqeq12 syl6bb
|
|
ctxp opth imbi2d syl5ibrcom exp4a syl5bi exlimdvv alrimiv alrimivv dffun2
|
|
3impd sylibr dmtxp ineq12 syl5eq anim12i an4s df-fn 3imtr4i ) CLZCMZANZOZ
|
|
DLZDMZBNZOZOCDUPZLZWCMZABUAZNZOZCAPZDBPZOWCWFPVOVSVQWAWHVOVSOZWDVQWAOZWGW
|
|
KEQZFQZWCRZWMGQZWCRZOZFGSZUBZGUCZFUCEUCWDWKXAEFWKWTGWRWNHQZIQZUDZNZWMXBCR
|
|
ZWMXCDRZUEZWPJQZKQZUDZNZWMXICRZWMXJDRZUEZOZKTJTZITHTZWKWSWRXHITHTZXOKTJTZ
|
|
OXRWOXSWQXTHIWMWNCDUFJKWMWPCDUFUGXHXOHIJKUHUIWKXQWSHIWKXPWSJKXPXEXLOZXFXM
|
|
OZXGXNOZUEWKWSXEXFXGXLXMXNUJWKYAYBYCWSWKYAYBYCWSWKYBYCOZWSUBYAYDHJSZIKSZO
|
|
ZUBVOYBYEVSYCYFVOXFXMYEWMXBXICUKULVSXGXNYFWMXCXJDUKULUMYAWSYGYDYAWSXDXKNY
|
|
GWNXDWPXKUNXBXCXIXJUQUOURUSUTVFVAVBVBVAVCVDEFGWCVEVGWLWEVPVTUAWFCDVHVPAVT
|
|
BVIVJVKVLWIVRWJWBCAVMDBVMUGWCWFVMVN $.
|
|
$}
|
|
|
|
${
|
|
$d A p x $. $d B x p $. $d C x p $. $d R x p $.
|
|
otsnelsi3.1 $e |- A e. _V $.
|
|
otsnelsi3.2 $e |- B e. _V $.
|
|
otsnelsi3.3 $e |- C e. _V $.
|
|
$( Ordered triple membership in triple singleton image. (Contributed by
|
|
SF, 12-Feb-2015.) $)
|
|
otsnelsi3 $p |- ( <. { A } , <. { B } , { C } >. >. e. SI_3 R <->
|
|
<. A , <. B , C >. >. e. R ) $=
|
|
( vp vx csn cop wcel c1st csi c2nd wceq wa wbr wex 3bitri df-si3 elimapw1
|
|
csi3 ccom ctxp cpw1 cima cv wrex eleq2i oteltxp vex opsnelsi df-br bitr4i
|
|
cproj2 opelco cproj1 opeq breq1i proj1ex proj2ex eqcom anbi1i exbii breq1
|
|
opbr2nd ceqsexv bitri anbi12i op1st2nd 3bitr4ri opex rexbii risset ) AJZB
|
|
JZCJZKZKZDUCZLVTMNZMOUDZNZOOUDZNZUEZUEZDUFUGZLHUHZJZVTKWHLZHDUIZABCKZKZDL
|
|
ZWAWIVTDUAUJHVTWHDUBWMWJWOPZHDUIWPWLWQHDWLWKVPKWBLZWKVSKWGLZQWJAMRZWJWNOR
|
|
ZQWQWKVPVSWBWGUKWRWTWSXAWRWJAKMLWTWJAMHULZEUMWJAMUNUOWSWKVQKWDLZWKVRKWFLZ
|
|
QWJUPZBMRZXECORZQZXAWKVQVRWDWFUKXCXFXDXGXCWJBKWCLWJIUHZORZXIBMRZQZISZXFWJ
|
|
BWCXBFUMIWJBMOUQXMXIXEPZXKQZISXFXLXOIXJXNXKXJWJURZXEKZXIORXEXIPXNWJXQXIOW
|
|
JUSZUTXPXEXIWJXBVAZWJXBVBZVGXEXIVCTZVDVEXKXFIXEXTXIXEBMVFVHVITXDWJCKWELXJ
|
|
XICORZQZISZXGWJCWEXBGUMIWJCOOUQYDXNYBQZISXGYCYEIXJXNYBYAVDVEYBXGIXEXTXIXE
|
|
COVFVHVITVJXQWNORXEWNPXAXHXPXEWNXSXTVGWJXQWNOXRUTBCXEFGVKVLTVJAWNWJEBCFGV
|
|
MVKTVNHWODVOUOT $.
|
|
$}
|
|
|
|
${
|
|
si3ex.1 $e |- A e. _V $.
|
|
$( ` SI_3 ` preserves sethood. (Contributed by SF, 12-Feb-2015.) $)
|
|
si3ex $p |- SI_3 A e. _V $=
|
|
( csi3 c1st csi c2nd ccom ctxp cpw1 cima cvv df-si3 1stex siex 2ndex coex
|
|
txpex pw1ex imaex eqeltri ) ACDEZDFGZEZFFGZEZHZHZAIZJKALUGUHUAUFDMNUCUEUB
|
|
DFMOPNUDFFOOPNQQABRST $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d R y $. $d T y $. $d x y $.
|
|
releqel.1 $e |- T e. _V $.
|
|
releqel.2 $e |- ( <. { y } , T >. e. R <-> y e. A ) $.
|
|
$( Lemma to turn a membership condition into an equality condition.
|
|
(Contributed by SF, 9-Mar-2015.) $)
|
|
releqel $p |- ( <. x , T >. e. ~ ( ( Ins3 _S (+) Ins2 R ) " 1c ) <->
|
|
x = A ) $=
|
|
( cv cop csset cins3 cins2 csymdif c1c wcel wn wb wex vex bitri cima wceq
|
|
wel wal csn elima1c elsymdif otelins3 opelssetsn otelins2 bibi12i xchbinx
|
|
ccompl exbii exnal 3bitrri con1bii opex elcompl dfcleq 3bitr4i ) AHZEIZJK
|
|
ZDLZMZNUAZOZPBAUCZBHZCOZQZBUDZVCVGUMOVBCUBVMVHVHVJUEZVCIZVFOZBRVLPZBRVMPB
|
|
VCVFUFVPVQBVPVOVDOZVOVEOZQVLVOVDVEUGVRVIVSVKVRVNVBIJOVIVNVBEJFUHVJVBBSASZ
|
|
UITVSVNEIDOVKVNVBEDVTUJGTUKULUNVLBUOUPUQVCVGVBEVTFURUSBVBCUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A z $. $d R x $. $d R y $. $d R z $. $d V y $. $d V z $.
|
|
$d x y $. $d x z $. $d y z $.
|
|
releqmpt.1 $e |- ( <. { y } , x >. e. R <-> y e. V ) $.
|
|
$( Equality condition for a mapping. (Contributed by SF, 9-Mar-2015.) $)
|
|
releqmpt $p |- ( ( A X. _V ) i^i
|
|
`' ~ ( ( Ins3 _S (+) Ins2 R ) " 1c ) ) = ( x e. A |-> V ) $=
|
|
( vz cvv cxp csset cins3 cins2 csymdif c1c cv wcel wa cop vex bitri copab
|
|
cima ccompl ccnv wceq cmpt elin mpbiran2 opelcnv releqel anbi12i opabbi2i
|
|
cin opelxp df-mpt eqtr4i ) CHIZJKDLMNUBUCZUDZUMZAOZCPZGOZEUEZQZAGUAACEUFV
|
|
EAGUTVAVCRZUTPVFUQPZVFUSPZQVEVFUQUSUGVGVBVHVDVGVBVCHPGSVAVCCHUNUHVHVCVARU
|
|
RPVDVAVCURUIGBEDVAASFUJTUKTULAGCEUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d A w $. $d A x $. $d A y $. $d B w $. $d B x $. $d B y $. $d R w $.
|
|
$d R x $. $d R y $. $d R z $. $d V w $. $d V z $. $d w x $. $d w y $.
|
|
$d w z $. $d x y $. $d x z $. $d y z $.
|
|
releqmpt2.1 $e |- ( <. { z } , <. x , y >. >. e. R <-> z e. V ) $.
|
|
$( Equality condition for a mapping operation. (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
releqmpt2 $p |- ( ( ( A X. B ) X. _V ) \
|
|
( ( Ins2 _S (+) Ins3 R ) " 1c ) ) = ( x e. A , y e. B |-> V ) $=
|
|
( vw cxp cvv csset cv wcel wa cop wn vex opelxp bitri cins2 cins3 csymdif
|
|
c1c cima cdif wceq coprab cmpt2 eldif mpbiran2 wel wal dfcleq csn elima1c
|
|
wb elsymdif opex otelins2 opelssetsn otelins3 bibi12i xchbinx exbii exnal
|
|
wex 3bitri con2bii bitr2i anbi12i oprabbi2i df-mpt2 eqtr4i ) DEJZKJZLUAZF
|
|
UBZUCZUDUEZUFZAMZDNBMZENOZIMZGUGZOZABIUHABDEGUIWGABIWAWBWCPZWEPZWANWIVPNZ
|
|
WIVTNZQZOWGWIVPVTUJWJWDWLWFWJWHVONZWDWJWMWEKNIRZWHWEVOKSUKWBWCDESTWFCIULZ
|
|
CMZGNZUQZCUMZWLCWEGUNWKWSWKWPUOZWIPZVSNZCVGWRQZCVGWSQCWIVSUPXBXCCXBXAVQNZ
|
|
XAVRNZUQWRXAVQVRURXDWOXEWQXDWTWEPLNWOWTWHWELWBWCARBRUSUTWPWECRWNVATXEWTWH
|
|
PFNWQWTWHWEFWNVBHTVCVDVEWRCVFVHVIVJVKTVLABIDEGVMVN $.
|
|
$}
|
|
|
|
${
|
|
mptexlem.1 $e |- A e. _V $.
|
|
mptexlem.2 $e |- R e. _V $.
|
|
$( Lemma for the existence of a mapping. (Contributed by SF,
|
|
9-Mar-2015.) $)
|
|
mptexlem $p |- ( ( A X. _V ) i^i
|
|
`' ~ ( ( Ins3 _S (+) Ins2 R ) " 1c ) ) e. _V $=
|
|
( cvv cxp csset cins3 cins2 csymdif c1c cima ccompl ccnv vvex xpex ssetex
|
|
ins3ex ins2ex symdifex 1cex imaex complex cnvex inex ) AEFGHZBIZJZKLZMZNA
|
|
ECOPUJUIUHKUFUGGQRBDSTUAUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
mpt2exlem.1 $e |- A e. _V $.
|
|
mpt2exlem.2 $e |- B e. _V $.
|
|
mpt2exlem.3 $e |- R e. _V $.
|
|
$( Lemma for the existence of a double mapping. (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
mpt2exlem $p |- ( ( ( A X. B ) X. _V ) \
|
|
( ( Ins2 _S (+) Ins3 R ) " 1c ) ) e. _V $=
|
|
( cxp cvv csset cins2 cins3 csymdif cima xpex vvex ssetex ins2ex symdifex
|
|
c1c ins3ex 1cex imaex difex ) ABGZHGIJZCKZLZSMUDHABDENONUGSUEUFIPQCFTRUAU
|
|
BUC $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The value of the little cup function. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
cupvalg $p |- ( ( A e. V /\ B e. W ) -> ( A Cup B ) = ( A u. B ) ) $=
|
|
( vx vy wcel cvv ccup co cun wceq elex unexg uneq1 df-cup ovmpt2g mpd3an3
|
|
cv uneq2 syl2an ) ACGAHGZBHGZABIJABKZLZBDGACMBDMUBUCUDHGUEABHHNEFABHHESZF
|
|
SZKUDIAUGKHUFAUGOUGBATEFPQRUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The cup function is a function over the universe.
|
|
(Contributed by SF, 11-Feb-2015.) (Revised by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
fncup $p |- Cup Fn _V $=
|
|
( vx vy ccup cvv cxp wfn cv cun df-cup vex unex fnmpt2i xpvv fneq2i mpbi
|
|
) CDDEZFCDFABDDAGZBGZHCABIQRAJBJKLPDCMNO $.
|
|
$}
|
|
|
|
$( Binary relationship form of the cup function. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
brcupg $p |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. Cup C <-> C = ( A u. B
|
|
) ) ) $=
|
|
( wcel wa cop ccup cfv wceq wbr cun cvv wfn wb fncup opexg fnbrfvb sylancr
|
|
co cupvalg eqeq1d df-ov eqeq1i eqcom 3bitr3g bitr3d ) ADFBEFGZABHZIJZCKZUJC
|
|
ILZCABMZKZUIINOUJNFULUMPQABDERNUJCISTUIABIUAZCKUNCKULUOUIUPUNCABDEUBUCUPUKC
|
|
ABIUDUEUNCUFUGUH $.
|
|
|
|
${
|
|
brcup.1 $e |- A e. _V $.
|
|
brcup.2 $e |- B e. _V $.
|
|
$( Binary relationship form of the cup function. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
brcup $p |- ( <. A , B >. Cup C <-> C = ( A u. B ) ) $=
|
|
( cvv wcel cop ccup wbr cun wceq wb brcupg mp2an ) AFGBFGABHCIJCABKLMDEAB
|
|
CFFNO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The little cup function is a set. (Contributed by SF, 11-Feb-2015.) $)
|
|
cupex $p |- Cup e. _V $=
|
|
( vx vy vz ccup cvv cxp csset cins3 cun cv cop wcel wo wel vex opelssetsn
|
|
bitri elun vvex ssetex cins2 c1c cima cdif cmpt2 df-cup otelins3 otelins2
|
|
csymdif csn orbi12i 3bitr4i releqmpt2 eqtr4i ins3ex ins2ex unex mpt2exlem
|
|
eqeltri ) DEEFEFGUAZGHZUTIZHUIUBUCUDZEDABEEAJZBJZIZUEVCABUFABCEEVBVFCJZUJ
|
|
ZVDVEKKZVALZVIUTLZMCANZCBNZMVIVBLVGVFLVJVLVKVMVJVHVDKGLVLVHVDVEGBOZUGVGVD
|
|
COZAOZPQVKVHVEKGLVMVHVDVEGVPUHVGVEVOVNPQUKVIVAUTRVGVDVERULUMUNEEVBSSVAUTG
|
|
TUOGTUPUQURUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( The value of the composition function. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
composevalg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( A Compose B ) = ( A o. B ) ) $=
|
|
( vx vy wcel wa cvv ccom ccompose co wceq elex adantr adantl coexg coeq1
|
|
cv coeq2 df-compose ovmpt2g syl3anc ) ACGZBDGZHAIGZBIGZABJZIGABKLUHMUDUF
|
|
UEACNOUEUGUDBDNPABCDQEFABIIESZFSZJUHKAUJJIUIAUJRUJBATEFUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The compose function is a function over the universe. (Contributed
|
|
by Scott Fenton, 19-Apr-2021.) $)
|
|
composefn $p |- Compose Fn _V $=
|
|
( vx vy vz ccompose cvv wfn cv wcel ccom wceq coprab copab weu vex eueq1
|
|
wa coex a1i fnoprab wb cmpt2 df-compose df-mpt2 eqtri df-xp eqtr3i fneq1
|
|
cxp xpvv fneq2 sylan9bb mp2an mpbir ) DEFZAGZEHBGZEHPZCGUOUPIZJZPABCKZUQ
|
|
ABLZFZUQUSABCUSCMUQCURUOUPANBNQORSDUTJZEVAJZUNVBTDABEEURUAUTABUBABCEEURU
|
|
CUDEEUHEVAUIABEEUEUFVCUNUTEFVDVBEDUTUGEVAUTUJUKULUM $.
|
|
$}
|
|
|
|
$( Binary relationship form of the compose function. (Contributed by Scott
|
|
Fenton, 19-Apr-2021.) $)
|
|
brcomposeg $p |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. Compose C <->
|
|
( A o. B ) = C ) ) $=
|
|
( wcel wa cop ccompose cfv wceq wbr ccom cvv wfn wb composefn opexg fnbrfvb
|
|
sylancr co df-ov composevalg syl5eqr eqeq1d bitr3d ) ADFBEFGZABHZIJZCKZUHCI
|
|
LZABMZCKUGINOUHNFUJUKPQABDERNUHCISTUGUIULCUGUIABIUAULABIUBABDEUCUDUEUF $.
|
|
|
|
${
|
|
$d x y z w t u v $.
|
|
$( The compose function is a set. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
composeex $p |- Compose e. _V $=
|
|
( vx vy vw vu vt vv cvv csset cins2 c1c cv wbr wex wcel cop vex otelins2
|
|
wa bitri ins2ex vz ccompose cxp c1st ccnv c2nd cin csi3 cins4 cswap cima
|
|
cid cins3 csymdif cdif ccom cmpt2 df-compose copab wceq csn elopab df-co
|
|
eleq2i elima1c elin opex oqelins4 otsnelsi3 opelxp mpbiran df-br 3bitr2i
|
|
brcnv anbi12i op1st2nd 3bitri snex wel df-clel opelssetsn exbii 3bitr4ri
|
|
brswap2 ideq otelins3 releqmpt2 eqtr4i vvex 1stex cnvex xpex 2ndex si3ex
|
|
inex ins4ex swapex ssetex 1cex imaex idex ins3ex mpt2exlem eqeltri ) UBG
|
|
GUCGUCHIZGUDUEZUCZUFUEZIZUGZUHZUIZUJUHZUIZXEIZIZIZUGZJUKZIZULUHZUIZHUMZI
|
|
ZIZIZIZUGZJUKZUGZJUKZUGZJUKZJUKZUMUNJUKUOZGUBABGGAKZBKZUPZUQYOABURABUAGG
|
|
YNYRUAKZCKZDKZYQLZUUAEKZYPLZRZDMZCEUSZNYSYTUUCOUTZUUFRZEMZCMZYSYRNYSVAZY
|
|
PYQOZOZYNNZUUFCEYSVBYRUUGYSCEDYPYQVCVDUUOYTVAZUUNOZYMNZCMUUKCUUNYMVEUURU
|
|
UJCUURUUCVAZUUQOZYLNZEMUUJEUUQYLVEUVAUUIEUVAUUTXLNZUUTYKNZRUUIUUTXLYKVFU
|
|
VBUUHUVCUUFUVBUUSUUPUULOOXKNUUCYTYSOZOZXJNZUUHUUSUUPUULUUMXKYPYQAPZBPZVG
|
|
ZVHUUCYTYSXJEPZCPZUAPVIUVFUVEXGNZUVEXINZRYSYTUDLZYSUUCUFLZRUUHUVEXGXIVFU
|
|
VLUVNUVMUVOUVLUVDXFNZYTYSXFLUVNUVLUUCGNUVPUVJUUCUVDGXFVJVKYTYSXFVLYTYSUD
|
|
VNVMUVMUUCYSOXHNUUCYSXHLUVOUUCYTYSXHUVKQUUCYSXHVLUUCYSUFVNVMVOYTUUCYSUVK
|
|
UVJVPVQVQUVCUUAVAZUUTOZYJNZDMUUFDUUTYJVEUVSUUEDUVSUVRXTNZUVRYINZRUUEUVRX
|
|
TYIVFUVTUUBUWAUUDUVTUVQUUQOZXSNZUUBUVQUUSUUQXSUUCVRZQYTUUAOZYQNUUCUWEUTZ
|
|
EBVSZRZEMZUUBUWCEUWEYQVTYTUUAYQVLUWCUUSUWBOZXRNZEMUWIEUWBXRVEUWKUWHEUWKU
|
|
WJXNNZUWJXQNZRUWHUWJXNXQVFUWLUWFUWMUWGUWLUUSUVQUUPOOXMNZUWFUUSUVQUUPUUNX
|
|
MUULUUMYSVRZUVIVGZVHUWNUUCUUAYTOZOUJNUUCUWQUJLUWFUUCUUAYTUJUVJDPZUVKVIUU
|
|
CUWQUJVLUUCUUAYTUWRUVKWDVMSUWMUUTXPNUUSUUNOXONZUWGUUSUVQUUQXPUUAVRZQUUSU
|
|
UPUUNXOYTVRZQUWSUUSUUMOXENUUSYQOHNUWGUUSUULUUMXEUWOQUUSYPYQHUVGQUUCYQUVJ
|
|
UVHWAVQVQVOSWBSWCSUUAUUCOZYPNFKZUXBUTZFAVSZRZFMZUUDUWAFUXBYPVTUUAUUCYPVL
|
|
UWAUXCVAZUVROZYHNZFMUXGFUVRYHVEUXJUXFFUXJUXIYBNZUXIYGNZRUXFUXIYBYGVFUXKU
|
|
XDUXLUXEUXKUXHUVQUUSOOYANZUXDUXHUVQUUSUUQYAUUPUUNUXAUWPVGVHUXMUXCUXBOULN
|
|
UXCUXBULLUXDUXCUUAUUCULFPZUWRUVJVIUXCUXBULVLUXCUXBUUAUUCUWRUVJVGWEVMSUXL
|
|
UXHUUTOYFNUXHUUQOYENZUXEUXHUVQUUTYFUWTQUXHUUSUUQYEUWDQUXOUXHUUNOYDNUXHUU
|
|
MOYCNZUXEUXHUUPUUNYDUXAQUXHUULUUMYCUWOQUXPUXHYPOHNUXEUXHYPYQHUVHWFUXCYPU
|
|
XNUVGWASVQVQVOSWBSWCVOSWBSVOSWBSWBSWCWGWHGGYNWIWIYMJYLJXLYKXKXJXGXIGXFWI
|
|
UDWJWKWLXHUFWMWKTWOWNWPYJJXTYIXSXRJXNXQXMUJWQWNWPXPXOXEHWRTTTTWOWSWTTYHJ
|
|
YBYGYAULXAWNWPYFYEYDYCHWRXBTTTTWOWSWTWOWSWTWOWSWTWSWTXCXD $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The binary relationship form of the ` Disj ` relationship. (Contributed
|
|
by SF, 11-Feb-2015.) $)
|
|
brdisjg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( A Disj B <-> ( A i^i B ) = (/) ) ) $=
|
|
( vx vy cv cin c0 wceq cdisj ineq1 eqeq1d ineq2 df-disj brabg ) EGZFGZHZI
|
|
JARHZIJABHZIJEFABCDKQAJSTIQARLMRBJTUAIRBANMEFOP $.
|
|
$}
|
|
|
|
${
|
|
brdisj.1 $e |- A e. _V $.
|
|
brdisj.2 $e |- B e. _V $.
|
|
$( The binary relationship form of the ` Disj ` relationship. (Contributed
|
|
by SF, 11-Feb-2015.) $)
|
|
brdisj $p |- ( A Disj B <-> ( A i^i B ) = (/) ) $=
|
|
( cvv wcel cdisj wbr cin c0 wceq wb brdisjg mp2an ) AEFBEFABGHABIJKLCDABE
|
|
EMN $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The disjointedness relationship is a set. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
disjex $p |- Disj e. _V $=
|
|
( vx vy vz cdisj csset ctxp c1c cima ccompl cv wel wn cop wcel wex wa vex
|
|
opelssetsn bitri ssetex cvv cin c0 wceq df-disj wral wrex oteltxp anbi12i
|
|
copab csn exbii elima1c df-rex 3bitr4i con2bii disj opex elcompl 3bitr4ri
|
|
dfrex2 opabbi2i eqtr4i txpex 1cex imaex complex eqeltri ) DEEFZGHZIZUADAJ
|
|
ZBJZUBUCUDZABUJVKABUEVNABVKCBKZLCVLUFZVLVMMZVJNZLVNVQVKNVRVPVRVOCVLUGZVPL
|
|
CJZUKZVQMVINZCOCAKZVOPZCOVRVSWBWDCWBWAVLMENZWAVMMENZPWDWAVLVMEEUHWEWCWFVO
|
|
VTVLCQZAQZRVTVMWGBQZRUISULCVQVIUMVOCVLUNUOVOCVLVASUPCVLVMUQVQVJVLVMWHWIUR
|
|
USUTVBVCVJVIGEETTVDVEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d x y z a b p $.
|
|
$( The cardinal addition function exists. (Contributed by SF,
|
|
12-Feb-2015.) $)
|
|
addcfnex $p |- AddC e. _V $=
|
|
( vx vy vz va vb vp cvv csset cdisj c1st c2nd ccup cv cop wcel wa wex wbr
|
|
wrex bitri caddcfn cxp cins2 cins3 ccom ctxp cima cin csi3 cins4 c1c cdif
|
|
csymdif cplc cmpt2 df-addcfn csn c0 wceq cun wel elin otelins2 opelssetsn
|
|
snex 3bitri oqelins4 otsnelsi3 otelins3 df-br brdisj 3bitr2i trtxp anbi2i
|
|
vex anass 3bitr4i brco br1st 19.42v bitr4i exbii excom exancom opex breq2
|
|
weq ceqsexv anbi1i 19.41v wb breq1 brco1st opbr2nd syl6bb exlimiv pm5.32i
|
|
sylbi opeq2 breq2d rexbii elima risset brcup anbi12i df-rex eladdc rexcom
|
|
op1st2nd elima1c releqmpt2 eqtr4i ssetex ins2ex disjex ins3ex 1stex 2ndex
|
|
vvex coex txpex cupex imaex inex si3ex ins4ex 1cex mpt2exlem eqeltri ) UA
|
|
GGUBGUBHUCZYJUCZYKIUDZJJUEZKJUEZKUFZUFZLUGZUHZUIZUJZUHZUKUGZUJZUHZUKUGZUD
|
|
UMUKUGULZGUAABGGAMZBMZUNZUOUUFABUPABCGGUUEUUICMZUQZUUGUUHNZNZUUEOZDMZEMZU
|
|
HURUSZUUJUUOUUPUTUSZPZDUUGSZEUUHSZUUJUUIOZUUPUQZUUMNZUUDOZEQEBVAZUUTPZEQU
|
|
UNUVAUVEUVGEUVEUVDYKOZUVDUUCOZPUVGUVDYKUUCVBUVHUVFUVIUUTUVHUVCUULNYJOUVCU
|
|
UHNHOUVFUVCUUKUULYJUUJVEZVCUVCUUGUUHHAVOZVCUUPUUHEVOZBVOZVDVFUVIUVCUUKUUG
|
|
NZNZUUBOZUUTUVCUUKUUGUUHUUBUVMVGUUOUQZUVONZUUAOZDQDAVAZUUSPZDQUVPUUTUVSUW
|
|
ADUVSUVRYKOZUVRYTOZPUWAUVRYKYTVBUWBUVTUWCUUSUWBUVQUVNNYJOUVQUUGNHOUVTUVQU
|
|
VCUVNYJUUPVEVCUVQUUKUUGHUVJVCUUOUUGDVOZUVKVDVFUWCUVQUVCUUKNNYSOUUOUUPUUJN
|
|
ZNZYROZUUSUVQUVCUUKUUGYSUVKVGUUOUUPUUJYRUWDUVLCVOZVHUWGUWFYLOZUWFYQOZPUUS
|
|
UWFYLYQVBUWIUUQUWJUURUWIUUOUUPNZIOUUOUUPIRUUQUUOUUPUUJIUWHVIUUOUUPIVJUUOU
|
|
UPUWDUVLVKVLUWJUWKUUJNZLOZUWKUUJLRUURFMZUWFYPRZFLSUWNUWLUSZFLSUWJUWMUWOUW
|
|
PFLUWOUWNUUOYMRZUWNUUPYNRZPZUWNUUJKRZPZUWNUWKJRZUWTPUWPUWQUWNUWEYORZPUWQU
|
|
WRUWTPZPUWOUXAUXCUXDUWQUWNUUPUUJYNKVMVNUWNUUOUWEYMYOVMUWQUWRUWTVPVQUWSUXB
|
|
UWTUWSBEWGZUWNUUOUUHNZJRZPBQZUXBUWSUXGBQZUWRPUXGUWRPZBQZUXHUWQUXIUWRUWQUW
|
|
NUUGJRZUUGUUOJRZPZAQUXLUUGUXFUSZPZBQZAQZUXIAUWNUUOJJVRUXNUXQAUXNUXLUXOBQZ
|
|
PUXQUXMUXSUXLBUUGUUOUWDVSVNUXLUXOBVTWAWBUXRUXPAQZBQUXIUXPABWCUXTUXGBUXTUX
|
|
OUXLPAQUXGUXLUXOAWDUXLUXGAUXFUUOUUHUWDUVMWEZUUGUXFUWNJWFWHTWBTVFWIUXGUWRB
|
|
WJUXKUXGUXEPZBQUXHUXJUYBBUXGUWRUXEUXGUWNUXFUUJNZUSZCQUWRUXEWKZCUWNUXFUYAV
|
|
SUYDUYECUYDUWRUYCUUPYNRZUXEUWNUYCUUPYNWLUYFUXFUUPKRUXEUXFUUJUUPKUYAUWHWMU
|
|
UOUUHUUPUWDUVMWNTWOWPWRWQWBUXGUXEBWDTVLUXGUXBBUUPUVLUXEUXFUWKUWNJUUHUUPUU
|
|
OWSWTWHTWIUWKUUJUWNUUOUUPUWDUVLWEUWHXIVFXAFUWFYPLXBFUWLLXCVQUWKUUJLVJUUOU
|
|
UPUUJUWDUVLXDVLXETVFXETWBDUVOUUAXJUUSDUUGXFVQTXETWBEUUMUUDXJUUTEUUHXFVQUV
|
|
BUUSEUUHSDUUGSUVAUUJUUGUUHDEXGUUSDEUUGUUHXHTWAXKXLGGUUEXSXSUUDUKYKUUCYJHX
|
|
MXNXNZUUBUUAUKYKYTUYGYSYRYLYQIXOXPYPLYMYOJJXQXQXTYNKKJXRXQXTXRYAYAYBYCYDY
|
|
EYFYDYGYCYFYDYGYCYHYI $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( ` AddC ` is a function over the universe. (Contributed by SF,
|
|
2-Mar-2015.) (Revised by Scott Fenton, 19-Apr-2021.) $)
|
|
addcfn $p |- AddC Fn _V $=
|
|
( vx vy caddcfn cvv cxp wfn cplc df-addcfn vex addcex fnmpt2i xpvv fneq2i
|
|
cv mpbi ) CDDEZFCDFABDDANZBNZGCABHQRAIBIJKPDCLMO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
braddcfn.1 $e |- A e. _V $.
|
|
braddcfn.2 $e |- B e. _V $.
|
|
$( Binary relationship form of the ` AddC ` function. (Contributed by SF,
|
|
2-Mar-2015.) $)
|
|
braddcfn $p |- ( <. A , B >. AddC C <-> ( A +c B ) = C ) $=
|
|
( vx vy cop caddcfn wbr cfv wceq cplc cvv wfn wcel wb addcfn mp2an cv co
|
|
opex fnbrfvb addceq1 addceq2 df-addcfn addcex ovmpt2 eqtr3i eqeq1i bitr3i
|
|
df-ov ) ABHZCIJZUMIKZCLZABMZCLINOUMNPUPUNQRABDEUBNUMCIUCSUOUQCABIUAZUOUQA
|
|
BIULANPBNPURUQLDEFGABNNFTZGTZMUQIAUTMUSAUTUDUTBAUEFGUFABDEUGUHSUIUJUK $.
|
|
$}
|
|
|
|
$( The membership relationship is a proper class. This theorem together with
|
|
~ vvex demonstrates the basic idea behind New Foundations: since
|
|
` x e. y ` is not a stratified relationship, then it does not have a
|
|
realization as a set of ordered pairs, but since ` x = x ` is stratified,
|
|
then it does have a realization as a set. (Contributed by SF,
|
|
20-Feb-2015.) $)
|
|
epprc $p |- -. _E e. _V $=
|
|
( vx cep cvv wcel cv wnel cab wn df-nel mpbi cfix ccompl wel wbr elfix epel
|
|
ru bitri notbii vex elcompl 3bitr4i abbi2i fixexg complexg syl syl5eqelr
|
|
mto ) BCDZAEZUJFZAGZCDZULCFUMHAQULCIJUIULBKZLZCUKAUOUJUNDZHAAMZHUJUODUKUPUQ
|
|
UPUJUJBNUQUJBOAAPRSUJUNATUAUJUJIUBUCUIUNCDUOCDBCUDUNCUEUFUGUH $.
|
|
|
|
${
|
|
$d f x y z p q $.
|
|
$( The class of all functions forms a set. (Contributed by SF,
|
|
18-Feb-2015.) $)
|
|
funsex $p |- Funs e. _V $=
|
|
( vf vx vy vz vp vq c1st c2nd c1c csset cid cv wbr wex wn cop df-br bitri
|
|
wcel wa cfuns ccnv ctxp csi3 cima cins4 cins2 cin cins3 cdif cvv wfun cab
|
|
ccompl df-funs weq wal csn elima1c snex vex opex elcompl eldif opelssetsn
|
|
otelins2 wrex wceq elin oqelins4 oteltxp opelcnv bitr4i otsnelsi3 anbi12i
|
|
wi ancom bitr2i op1st2nd 3bitri anbi12ci exbii sneq breq2d ceqsexv rexbii
|
|
3bitr4i elima risset otelins3 sneqb 3bitr3i notbii exanali con2bii dffun3
|
|
ideq alex abbi2i eqtr4i 1stex cnvex 2ndex txpex si3ex imaex ins4ex ins2ex
|
|
1cex inex ssetex idex ins3ex difex complex eqeltri ) UAGUBZHGUCZUDZUCZIUE
|
|
ZUFZHUGZUGZUHZJUEZUGZKUIZUJZIUEZUNZIUEZUNZIUEZUNZUKUAALZULZAUMYOAUOYQAYOY
|
|
PYOSZBLZCLZYPMZCDUPZVPCUQZDNZBUQZYQYPYNSZOUUDOZBNZOYRUUEUUFUUHUUFYSURZYPP
|
|
ZYMSZBNUUHBYPYMUSUUKUUGBUUKUUJYLSZOUUGUUJYLUUIYPYSUTZAVAZVBZVCUULUUDUULDL
|
|
ZURZUUJPZYKSZDNUUDDUUJYKUSUUSUUCDUURYJSZOUUAUUBOZTZCNZOUUSUUCUUTUVCUUTYTU
|
|
RZUURPZYISZCNUVCCUURYIUSUVFUVBCUVFUVEYGSZUVEYHSZOZTUVBUVEYGYHVDUVGUUAUVIU
|
|
VAUVGUVDUUJPZYFSZUUAUVDUUQUUJYFUUPUTZVFYSYTPZURZYPPZJSZUVMYPSUVKUUAUVMYPY
|
|
SYTBVAZCVAZVBZUUNVEELZUVJYEMZEJVGUVTUVOVHZEJVGUVKUVPUWAUWBEJUWAUVTUVJPZYE
|
|
SZUVTUVNGMZUVTYPHMZTZUWBUVTUVJYEQUWDUWCYBSZUWCYDSZTUWGUWCYBYDVIUWHUWEUWIU
|
|
WFUWHUVTUVDUUIPZPZYASZUWEUVTUVDUUIYPYAUUNVJUWLFLZURZUWKPXTSZFNUWMUVMVHZUV
|
|
TUWNGMZTZFNUWEFUWKXTUSUWOUWRFUWOUWNUVTPXQSZUWNUWJPXSSZTUWRUWNUVTUWJXQXSVK
|
|
UWSUWQUWTUWPUWSUVTUWNPGSUWQUWNUVTGVLUVTUWNGQVMUWTUWMYTYSPPXRSZUWPUWMYTYSX
|
|
RFVAUVRUVQVNUXAUWMYTPHSZUWMYSPGSZTZUWMYSGMZUWMYTHMZTZUWPUWMYTYSHGVKUXGUXC
|
|
UXBTUXDUXEUXCUXFUXBUWMYSGQUWMYTHQVOUXCUXBVQVRYSYTUWMUVQUVRVSVTRWARWBUWQUW
|
|
EFUVMUVSUWPUWNUVNUVTGUWMUVMWCWDWEVTRUVTUUJPYCSUVTYPPHSUWIUWFUVTUUIYPHUUMV
|
|
FUVTUVDUUJYCYTUTVFUVTYPHQWGVORUVNYPUVTUVMUTUUNVSVTWFEUVJYEJWHEUVOJWIWGYSY
|
|
TYPQWGRUVHUUBUVHUVDUUQPKSZUUBUVDUUQUUJKUUOWJUVDUUQKMUVDUUQVHUXHUUBUVDUUQU
|
|
VLWQUVDUUQKQYTUUPUVRWKWLRWMVORWBRWMUURYJUUQUUJUVLUUOVBVCUVCUUCUUAUUBCWNWO
|
|
WGWBRWMRWBRWMYPYNUUNVCUUDBWRWGBCDYPWPVMWSWTYNYMIYLYKIYJYIIYGYHYFYEJYBYDYA
|
|
XTIXQXSGXAXBXRHGXCXAXDXEXDXIXFXGYCHXCXHXHXJXKXFXHKXLXMXNXIXFXOXIXFXOXIXFX
|
|
OXP $.
|
|
$}
|
|
|
|
${
|
|
$d F f $.
|
|
elfuns.1 $e |- F e. _V $.
|
|
$( Membership in the set of all functions. (Contributed by SF,
|
|
23-Feb-2015.) $)
|
|
elfuns $p |- ( F e. Funs <-> Fun F ) $=
|
|
( vf cv wfun cfuns funeq df-funs elab2 ) CDZEAECAFBJAGCHI $.
|
|
$}
|
|
|
|
${
|
|
$d F f $.
|
|
$( Membership in the set of all functions. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
elfunsg $p |- ( F e. V -> ( F e. Funs <-> Fun F ) ) $=
|
|
( vf cv cfuns wcel wfun eleq1 funeq vex elfuns vtoclbg ) CDZEFMGAEFAGCABM
|
|
AEHMAIMCJKL $.
|
|
$}
|
|
|
|
$( Membership in the set of all functions implies functionhood. (Contributed
|
|
by Scott Fenton, 31-Jul-2019.) $)
|
|
elfunsi $p |- ( F e. Funs -> Fun F ) $=
|
|
( cfuns wcel wfun elfunsg ibi ) ABCADABEF $.
|
|
|
|
${
|
|
$d f a $.
|
|
$( The function with domain relationship exists. (Contributed by SF,
|
|
23-Feb-2015.) $)
|
|
fnsex $p |- Fns e. _V $=
|
|
( vf va cfns cfuns cvv cxp c1st cimage cin wfn copab df-fns cop wcel wfun
|
|
cv wa cdm wceq vex opelxp mpbiran2 elfuns bitri cima eqcom eqeq1i brimage
|
|
dfdm4 wbr df-br bitr3i 3bitr4ri anbi12i elin df-fn opabbi2i eqtr4i funsex
|
|
3bitr4i vvex xpex 1stex imageex inex eqeltri ) CDEFZGHZIZECAPZBPZJZABKVIA
|
|
BLVLABVIVJVKMZVGNZVMVHNZQVJOZVJRZVKSZQVMVINVLVNVPVOVRVNVJDNZVPVNVSVKENBTZ
|
|
VJVKDEUAUBVJATZUCUDGVJUEZVKSVKWBSZVRVOWBVKUFVQWBVKVJUIUGVOVJVKVHUJWCVJVKV
|
|
HUKVJVKGWAVTUHULUMUNVMVGVHUOVJVKUPUTUQURVGVHDEUSVAVBGVCVDVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d A a b f $. $d F a b f $.
|
|
brfns.1 $e |- F e. _V $.
|
|
$( Binary relationship form of ` Fns ` relationship. (Contributed by SF,
|
|
23-Feb-2015.) $)
|
|
brfns $p |- ( F Fns A <-> F Fn A ) $=
|
|
( va vf vb cfns wbr cvv wcel wfn brex simprd cdm eqcomd dmexg ax-mp fneq2
|
|
fndm cv syl6eqel breq2 vex fneq1 df-fns brab vtoclbg pm5.21nii ) BAGHZAIJ
|
|
ZBAKZUIBIJZUJBAGLMUKABNZIUKUMAABSOULUMIJCBIPQUABDTZGHBUNKZUIUKDAIUNABGUBU
|
|
NABRETZFTZKBUQKUOEFBUNGCDUCUQUPBUDUQUNBREFUEUFUGUH $.
|
|
$}
|
|
|
|
$( Equality theorem for parallel product. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
pprodeq1 $p |- ( A = B -> PProd ( A , C ) = PProd ( B , C ) ) $=
|
|
( wceq c1st ccom c2nd ctxp cpprod coeq1 txpeq1 syl df-pprod 3eqtr4g ) ABDZA
|
|
EFZCGFZHZBEFZQHZACIBCIOPSDRTDABEJPSQKLACMBCMN $.
|
|
|
|
$( Equality theorem for parallel product. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
pprodeq2 $p |- ( A = B -> PProd ( C , A ) = PProd ( C , B ) ) $=
|
|
( wceq c1st ccom c2nd ctxp cpprod coeq1 txpeq2 syl df-pprod 3eqtr4g ) ABDZC
|
|
EFZAGFZHZPBGFZHZCAICBIOQSDRTDABGJQSPKLCAMCBMN $.
|
|
|
|
${
|
|
$d a w $. $d a x $. $d A x $. $d a y $. $d A y $. $d a z $. $d B y $.
|
|
$d C w $. $d C x $. $d C y $. $d C z $. $d D w $. $d D x $. $d D y $.
|
|
$d R a $. $d R w $. $d R x $. $d R y $. $d R z $. $d S a $. $d S w $.
|
|
$d S x $. $d S y $. $d S z $. $d w x $. $d w y $. $d w z $. $d x y $.
|
|
$d x z $. $d y z $.
|
|
$( A quadratic relationship over a parallel product. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
qrpprod $p |- ( <. A , B >. PProd ( R , S ) <. C , D >. <->
|
|
( A R C /\ B S D ) ) $=
|
|
( vx vy va cop wbr cvv wcel wa wb cv wceq breq1 bibi12d bitri cpprod brex
|
|
vz vw opexb anbi12i sylib anim12i an4 sylibr wi opeq1 breq1d anbi1d opeq2
|
|
imbi2d anbi2d breq2 c1st ccom c2nd ctxp df-pprod breqi trtxp weq wex brco
|
|
breq2d vex opbr1st eqcom anbi1i exbii ceqsexv opbr2nd vtocl2g pm5.21nii
|
|
imp ) ABJZCDJZEFUAZKZALMZBLMZNZCLMZDLMZNZNZACEKZBDFKZNZWCVTLMZWALMZNWJVTW
|
|
AWBUBWNWFWOWIABUECDUEUFUGWMWDWGNZWEWHNZNWJWKWPWLWQACEUBBDFUBUHWDWEWGWHUIU
|
|
JWFWIWCWMOZWIGPZHPZJZWAWBKZWSCEKZWTDFKZNZOZUKWIAWTJZWAWBKZWKXDNZOZUKWIWRU
|
|
KGHABLLWSAQZXFXJWIXKXBXHXEXIXKXAXGWAWBWSAWTULUMXKXCWKXDWSACERUNSUPWTBQZXJ
|
|
WRWIXLXHWCXIWMXLXGVTWAWBWTBAUOUMXLXDWLWKWTBDFRUQSUPXAUCPZUDPZJZWBKZWSXMEK
|
|
ZWTXNFKZNZOXACXNJZWBKZXCXRNZOXFUCUDCDLLXMCQZXPYAXSYBYCXOXTXAWBXMCXNULVIYC
|
|
XQXCXRXMCWSEURUNSXNDQZYAXBYBXEYDXTWAXAWBXNDCUOVIYDXRXDXCXNDWTFURUQSXPXAXM
|
|
EUSUTZKZXAXNFVAUTZKZNZXSXPXAXOYEYGVBZKYIXAXOWBYJEFVCVDXAXMXNYEYGVETYFXQYH
|
|
XRYFIGVFZIPZXMEKZNZIVGZXQYFXAYLUSKZYMNZIVGYOIXAXMEUSVHYQYNIYPYKYMYPGIVFYK
|
|
WSWTYLGVJZHVJZVKWSYLVLTVMVNTYMXQIWSYRYLWSXMERVOTYHIHVFZYLXNFKZNZIVGZXRYHX
|
|
AYLVAKZUUANZIVGUUCIXAXNFVAVHUUEUUBIUUDYTUUAUUDHIVFYTWSWTYLYRYSVPWTYLVLTVM
|
|
VNTUUAXRIWTYSYLWTXNFRVOTUFTVQVQVSVR $.
|
|
$}
|
|
|
|
$( The parallel product of two sets is a set. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
pprodexg $p |- ( ( A e. V /\ B e. W ) -> PProd ( A , B ) e. _V ) $=
|
|
( wcel wa cpprod c1st ccom c2nd cvv df-pprod 1stex coexg mpan2 2ndex txpexg
|
|
ctxp syl2an syl5eqel ) ACEZBDEZFABGAHIZBJIZRZKABLUAUCKEZUDKEZUEKEUBUAHKEUFM
|
|
AHCKNOUBJKEUGPBJDKNOUCUDKKQST $.
|
|
|
|
${
|
|
pprodex.1 $e |- A e. _V $.
|
|
pprodex.2 $e |- B e. _V $.
|
|
$( The parallel product of two sets is a set. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
pprodex $p |- PProd ( A , B ) e. _V $=
|
|
( cvv wcel cpprod pprodexg mp2an ) AEFBEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d A w $. $d A x $. $d A y $. $d A z $. $d B w $. $d B x $. $d B y $.
|
|
$d B z $. $d R w $. $d R x $. $d R y $. $d R z $. $d S w $. $d S x $.
|
|
$d S y $. $d S z $. $d w x $. $d w y $. $d w z $. $d x y $. $d x z $.
|
|
$d y z $.
|
|
$( Binary relationship over a parallel product. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
brpprod $p |- ( A PProd ( R , S ) B <->
|
|
E. x E. y E. z E. w ( A = <. x , y >. /\ B = <. z , w >. /\
|
|
( x R z /\ y S w ) ) ) $=
|
|
( wbr c1st ccom cv cop w3a wex wa anbi1i 19.41v exbii bitri c2nd df-pprod
|
|
cpprod ctxp wceq breqi brtxp brco an32 br1st breq1 brco2nd syl6bb pm5.32i
|
|
3bitr2i bitr3i bitr2i 3bitri anbi2i 3anass 3ancoma 2exbii 19.42vv 3bitr4i
|
|
vex anass exrot4 ) EFGHUCZIEFGJKZHUAKZUDZIFCLZDLZMUEZEVLVIIZEVMVJIZNZDOCO
|
|
ZEALZBLZMZUEZVNVSVLGIZVTVMHIZPZNZDOCOBOAOZEFVHVKGHUBUFCDEFVIVJUGVRWFBOAOZ
|
|
DOCOWGVQWHCDVNVOVPPZPVNWBWEPZBOZAOZPZVQWHWIWLVNWIEVSJIZWCPZAOZVPPWOVPPZAO
|
|
WLVOWPVPAEVLGJUHQWOVPARWQWKAWQWNVPPZWCPWBWDPZBOZWCPZWKWNWCVPUIWRWTWCWRWBB
|
|
OZVPPWBVPPZBOWTWNXBVPBEVSAVEZUJQWBVPBRXCWSBWBVPWDWBVPWAVMVJIWDEWAVMVJUKVS
|
|
VTVMHXDBVEULUMUNSUOQWKWSWCPZBOXAWJXEBWJWBWCPWDPXEWBWCWDVFWBWCWDUIUPSWSWCB
|
|
RUQURSUOUSVNVOVPUTWHVNWJPZBOAOWMWFXFABWFVNWBWENXFWBVNWEVAVNWBWEUTTVBVNWJA
|
|
BVCTVDVBWFCDABVGTUR $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. $d A d $.
|
|
$d a x $. $d A x $. $d B a $. $d B b $. $d b c $. $d B c $. $d b d $.
|
|
$d B d $. $d b x $. $d B x $. $d c d $. $d c x $. $d d x $. $d a t $.
|
|
$d A t $. $d a u $. $d A u $. $d b t $. $d B t $. $d b u $. $d B u $.
|
|
$d c t $. $d c u $. $d d t $. $d d u $. $d t u $. $d t x $. $d u x $.
|
|
$( The domain of a parallel product. (Contributed by SF, 24-Feb-2015.) $)
|
|
dmpprod $p |- dom PProd ( A , B ) = ( dom A X. dom B ) $=
|
|
( va vb vx vc vd vt vu cdm cv cop wceq wbr wa wex vex 2exbii wcel eldm
|
|
cpprod cxp opex isseti 19.41v mpbiran df-br w3a weq brpprod 19.42vv eqcom
|
|
3anass opth bitri anbi1i df-3an 3bitr4i breq1 anbi1d anbi2d 2exbidv exbii
|
|
ceqsex2v 3bitri exrot3 anbi12i brxp eeanv eqbrriv ) CDABUAZJZAJZBJZUBZEKZ
|
|
FKZGKZLZMZCKZVQANZDKZVRBNZOZOZEPZGPFPZWEGPFPZWAWCVLNZWAWCVONZWGWEFGWGVTEP
|
|
WEEVSVQVRFQGQUCUDVTWEEUEUFRWJWAWCLZVLSWLVPVKNZEPZWHWAWCVLUGEWLVKTWNWFGPFP
|
|
ZEPWHWMWOEWMWLHKZIKZLZMZVTWPVQANZWQVRBNZOZUHZGPFPZIPHPHCUIZIDUIZVTXBOZGPF
|
|
PZUHZIPHPWOHIFGWLVPABUJXDXIHIXEXFOZXGOZGPFPXJXHOXDXIXJXGFGUKXCXKFGXCWSXGO
|
|
XKWSVTXBUMWSXJXGWSWRWLMXJWLWRULWPWQWAWCUNUOUPUORXEXFXHUQURRXHVTWBXAOZOZGP
|
|
FPWOHIWAWCCQDQXEXGXMFGXEXBXLVTXEWTWBXAWPWAVQAUSUTVAVBXFXMWFFGXFXLWEVTXFXA
|
|
WDWBWQWCVRBUSVAVAVBVDVEVCWFEFGVFUOVEWAVMSZWCVNSZOWBFPZWDGPZOWKWIXNXPXOXQF
|
|
WAATGWCBTVGWAWCVMVNVHWBWDFGVIURURVJ $.
|
|
$}
|
|
|
|
$( The converse of a parallel product. (Contributed by SF, 24-Feb-2015.) $)
|
|
cnvpprod $p |- `' PProd ( A , B ) = PProd ( `' A , `' B ) $=
|
|
( c1st ccnv ccom cin cpprod cnvin cnvco cnvcnv coeq12i coass 3eqtri ineq12i
|
|
c2nd eqtri ctxp df-pprod df-txp cnveqi 3eqtr4i ) CDZACEZEZODZBOEZEZFZDZUBAD
|
|
ZCEZEZUEBDZOEZEZFZABGZDUJUMGZUIUDDZUGDZFUPUDUGHUSULUTUOUSUCDZUBDZEUBUJEZCEU
|
|
LUBUCIVAVCVBCACICJKUBUJCLMUTUFDZUEDZEUEUMEZOEUOUEUFIVDVFVEOBOIOJKUEUMOLMNPU
|
|
QUHUQUCUFQUHABRUCUFSPTURUKUNQUPUJUMRUKUNSPUA $.
|
|
|
|
$( The range of a parallel product. (Contributed by SF, 24-Feb-2015.) $)
|
|
rnpprod $p |- ran PProd ( A , B ) = ( ran A X. ran B ) $=
|
|
( cpprod ccnv cdm cxp cnvpprod dmeqi dmpprod eqtri dfrn4 xpeq12i 3eqtr4i
|
|
crn ) ABCZDZEZADZEZBDZEZFZONANZBNZFQRTCZEUBPUEABGHRTIJOKUCSUDUAAKBKLM $.
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. $d A d $.
|
|
$d a e $. $d A e $. $d a f $. $d A f $. $d a g $. $d A g $. $d a h $.
|
|
$d A h $. $d a x $. $d A x $. $d a y $. $d A y $. $d a z $. $d A z $.
|
|
$d B a $. $d B b $. $d b c $. $d B c $. $d b d $. $d B d $. $d b e $.
|
|
$d B e $. $d b f $. $d B f $. $d b g $. $d B g $. $d b h $. $d B h $.
|
|
$d b x $. $d B x $. $d b y $. $d B y $. $d b z $. $d B z $. $d c d $.
|
|
$d c e $. $d c f $. $d c g $. $d c h $. $d c x $. $d c y $. $d c z $.
|
|
$d d e $. $d d f $. $d d g $. $d d h $. $d d x $. $d d y $. $d d z $.
|
|
$d e f $. $d e g $. $d e h $. $d e x $. $d e y $. $d e z $. $d F a $.
|
|
$d F b $. $d F c $. $d F d $. $d F e $. $d F f $. $d f g $. $d F g $.
|
|
$d f h $. $d F h $. $d f x $. $d F x $. $d f y $. $d F y $. $d f z $.
|
|
$d F z $. $d G a $. $d G b $. $d G c $. $d G d $. $d G e $. $d G f $.
|
|
$d G g $. $d g h $. $d G h $. $d g x $. $d G x $. $d g y $. $d G y $.
|
|
$d g z $. $d G z $. $d h x $. $d h y $. $d h z $. $d x y $. $d x z $.
|
|
$d y z $.
|
|
$( Functionhood law for parallel product. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
fnpprod $p |- ( ( F Fn A /\ G Fn B ) -> PProd ( F , G ) Fn ( A X. B ) ) $=
|
|
( vy vz va vb vc vd ve vf vg vh wceq wa cv wbr wi wex wfun cdm cpprod cxp
|
|
vx wfn weq wal cop w3a ee4anv 2exbii brpprod anbi12i 3bitr4ri an42 3expib
|
|
fununiq eqcomd im2anan9 syl5bi exp3acom23 wb breq1 bi2anan9 adantr syl6bb
|
|
eqeq2 opth imbi2d adantl imbi12d syl5ibrcom exp3a 3impd com23 eqeq1 eqcom
|
|
bitri 3anbi1d imp3a exlimdvv alrimiv alrimivv dffun2 sylibr xpeq12 syl5eq
|
|
dmpprod anim12i an4s df-fn 3imtr4i ) CUAZCUBZAOZPZDUAZDUBZBOZPZPCDUCZUAZX
|
|
BUBZABUDZOZPZCAUFZDBUFZPXBXEUFWNWRWPWTXGWNWRPZXCWPWTPZXFXJUEQZEQZXBRZXLFQ
|
|
ZXBRZPZEFUGZSZFUHZEUHUEUHXCXJXTUEEXJXSFXQXLGQZHQZUIZOZXMIQZJQZUIZOZYAYECR
|
|
ZYBYFDRZPZUJZXLKQZLQZUIZOZXOMQZNQZUIZOZYMYQCRZYNYRDRZPZUJZPZNTMTZJTITZLTK
|
|
TZHTGTZXJXRYLJTITZUUDNTMTZPZLTKTZHTGTUUJHTGTZUUKLTKTZPUUIXQUUJUUKGHKLUKUU
|
|
HUUMGHUUGUULKLYLUUDIJMNUKULULXNUUNXPUUOGHIJXLXMCDUMKLMNXLXOCDUMUNUOXJUUHX
|
|
RGHXJUUGXRKLXJUUFXRIJXJUUEXRMNXJYLUUDXRXJYDYHYKUUDXRSZXJYDYHYKUUPSZXJUUQY
|
|
DYHPZYKKGUGZLHUGZPZYTUUCUJZYGXOOZSZSXJUVBYKUVCXJUVAYTUUCYKUVCSZXJUVAYTUUC
|
|
UVESZXJUVFUVAYTPZYAYQCRZYBYRDRZPZYKIMUGZJNUGZPZSZSXJYKUVJUVMYKUVJPYIUVHPZ
|
|
UVIYJPZPXJUVMYIYJUVHUVIUPWNUVOUVKWRUVPUVLWNYIUVHUVKYAYEYQCURUQWRUVIYJUVLW
|
|
RUVIYJUJYRYFYBYRYFDURUSUQUTVAVBUVGUUCUVJUVEUVNUVAUUCUVJVCYTUUSUUAUVHUUTUU
|
|
BUVIYMYAYQCVDYNYBYRDVDVEVFYTUVEUVNVCUVAYTUVCUVMYKYTUVCYGYSOUVMXOYSYGVHYEY
|
|
FYQYRVIVGVJVKVLVMVNVOVPUURUUPUVDYKUURUUDUVBXRUVCYDUUDUVBVCYHYDYPUVAYTUUCY
|
|
DYPYCYOOZUVAXLYCYOVQUVQYOYCOUVAYCYOVRYMYNYAYBVIVSVGVTVFYHXRUVCVCYDXMYGXOV
|
|
QVKVLVJVMVNVOWAWBWBWBWBVAWCWDUEEFXBWEWFXKXDWOWSUDXECDWIWOAWSBWGWHWJWKXHWQ
|
|
XIXACAWLDBWLUNXBXEWLWM $.
|
|
$}
|
|
|
|
$( The parallel product of two bijections is a bijection. (Contributed by
|
|
SF, 24-Feb-2015.) $)
|
|
f1opprod $p |- ( ( F : A -1-1-onto-> C /\ G : B -1-1-onto-> D ) ->
|
|
PProd ( F , G ) : ( A X. B ) -1-1-onto-> ( C X. D ) ) $=
|
|
( wfn ccnv wa cpprod cxp wf1o fnpprod cnvpprod fneq1i sylibr anim12i dff1o4
|
|
an4s anbi12i 3imtr4i ) EAGZEHZCGZIZFBGZFHZDGZIZIEFJZABKZGZUJHZCDKZGZIZACELZ
|
|
BDFLZIUKUNUJLUBUFUDUHUPUBUFIULUDUHIZUOABEFMUSUCUGJZUNGUOCDUCUGMUNUMUTEFNOPQ
|
|
SUQUEURUIACERBDFRTUKUNUJRUA $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The value of the cross product function. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
ovcross $p |- ( ( A e. V /\ B e. W ) -> ( A Cross B ) = ( A X. B ) ) $=
|
|
( vx vy wcel cvv ccross co cxp wceq xpexg cv xpeq1 xpeq2 df-cross ovmpt2g
|
|
elex mpd3an3 syl2an ) ACGAHGZBHGZABIJABKZLZBDGACSBDSUBUCUDHGUEABHHMEFABHH
|
|
ENZFNZKUDIAUGKHUFAUGOUGBAPEFQRTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The cross product function is a function over ` ( _V X. _V ) `
|
|
(Contributed by SF, 24-Feb-2015.) $)
|
|
fncross $p |- Cross Fn _V $=
|
|
( vx vy ccross cvv cxp wfn cv df-cross vex xpex fnmpt2i xpvv fneq2i mpbi
|
|
) CDDEZFCDFABDDAGZBGZECABHPQAIBIJKODCLMN $.
|
|
$}
|
|
|
|
$( The domain of the cross product function. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
dmcross $p |- dom Cross = _V $=
|
|
( ccross cvv wfn cdm wceq fncross fndm ax-mp ) ABCADBEFBAGH $.
|
|
|
|
$( Binary relationship over the cross product function. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
brcrossg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( <. A , B >. Cross C <-> C = ( A X. B ) ) ) $=
|
|
( wcel wa ccross co wceq cop wbr cxp cfv eqcom df-ov eqeq1i bitri cvv wfn
|
|
wb fncross opexg fnbrfvb sylancr syl5bb ovcross eqeq2d bitr3d ) ADFBEFGZCAB
|
|
HIZJZABKZCHLZCABMZJULUMHNZCJZUJUNULUKCJUQCUKOUKUPCABHPQRUJHSTUMSFUQUNUAUBAB
|
|
DEUCSUMCHUDUEUFUJUKUOCABDEUGUHUI $.
|
|
|
|
${
|
|
brcross.1 $e |- A e. _V $.
|
|
brcross.2 $e |- B e. _V $.
|
|
$( Binary relationship over the cross product function. (Contributed by
|
|
SF, 24-Feb-2015.) $)
|
|
brcross $p |- ( <. A , B >. Cross C <-> C = ( A X. B ) ) $=
|
|
( cvv wcel cop ccross wbr cxp wceq wb brcrossg mp2an ) AFGBFGABHCIJCABKLM
|
|
DEABCFFNO $.
|
|
$}
|
|
|
|
${
|
|
$d a b x y z $.
|
|
$( The function mapping ` x ` and ` y ` to their cross product is a set.
|
|
(Contributed by SF, 11-Feb-2015.) $)
|
|
crossex $p |- Cross e. _V $=
|
|
( vx vy va vb cvv cxp csset c1st c2nd c1c cv cop wrex wex wa otelins2 vex
|
|
wcel 3bitri wbr ccross cins2 ccnv cin csi3 cins4 cima cins3 csymdif cmpt2
|
|
vz cdif df-cross wceq csn rexcom elxp2 wel elin snex opelssetsn otsnelsi3
|
|
oqelins4 df-br brcnv 3bitr2i mpbiran anbi12i op1st2nd bitri exbii elima1c
|
|
opelxp df-rex 3bitr4i 3bitr4ri releqmpt2 eqtr4i ssetex ins2ex 1stex cnvex
|
|
vvex 2ndex xpex inex si3ex ins4ex 1cex imaex mpt2exlem eqeltri ) UAEEFEFG
|
|
UBZWMUBZWNHUCZUBZEIUCZFZUDZUEZUFZUDZJUGZUFZUDZJUGZUHUIJUGULZEUAABEEAKZBKZ
|
|
FZUJXGABUMABUKEEXFXJUKKZCKZDKZLUNZDXIMCXHMXNCXHMZDXIMZXKXJRXKUOZXHXILZLZX
|
|
FRZXNCDXHXIUPCDXKXHXIUQXMUOZXSLZXERZDNDBURZXOOZDNXTXPYCYEDYCYBWNRZYBXDRZO
|
|
YEYBWNXDUSYFYDYGXOYFYAXRLWMRYAXILGRYDYAXQXRWMXKUTZPYAXHXIGAQZPXMXIDQZBQZV
|
|
ASYGYAXQXHLZLZXCRZXOYAXQXHXIXCYKVCXLUOZYMLZXBRZCNCAURZXNOZCNYNXOYQYSCYQYP
|
|
WNRZYPXARZOYSYPWNXAUSYTYRUUAXNYTYOYLLWMRYOXHLGRYRYOYAYLWMXMUTPYOXQXHGYHPX
|
|
LXHCQZYIVASUUAYOYAXQLLWTRXLXMXKLZLZWSRZXNYOYAXQXHWTYIVCXLXMXKWSUUBYJUKQVB
|
|
UUEUUDWPRZUUDWRRZOXKXLHTZXKXMITZOXNUUDWPWRUSUUFUUHUUGUUIUUFXLXKLWORXLXKWO
|
|
TUUHXLXMXKWOYJPXLXKWOVDXLXKHVEVFUUGUUCWQRZXMXKWQTUUIUUGXLERUUJUUBXLUUCEWQ
|
|
VMVGXMXKWQVDXMXKIVEVFVHXLXMXKUUBYJVISSVHVJVKCYMXBVLXNCXHVNVOVJVHVJVKDXSXE
|
|
VLXODXIVNVOVPVQVREEXFWCWCXEJWNXDWMGVSVTVTZXCXBJWNXAUUKWTWSWPWRWOHWAWBVTEW
|
|
QWCIWDWBWEWFWGWHWFWIWJWHWFWIWJWKWL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
pw1fnval.1 $e |- A e. _V $.
|
|
$( The value of the unit power class function. (Contributed by SF,
|
|
25-Feb-2015.) $)
|
|
pw1fnval $p |- ( Pw1Fn ` { A } ) = ~P1 A $=
|
|
( vx csn c1c wcel cpw1fn cfv cpw1 wceq snel1c cv unieq unisn syl6eq pw1eq
|
|
cuni syl df-pw1fn pw1ex fvmpt ax-mp ) ADZEFUCGHAIZJABKCUCCLZQZIZUDEGUEUCJ
|
|
ZUFAJUGUDJUHUFUCQAUEUCMABNOUFAPRCSABTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y t $.
|
|
$( The unit power class function is a set. (Contributed by SF,
|
|
25-Feb-2015.) $)
|
|
pw1fnex $p |- Pw1Fn e. _V $=
|
|
( vx vy vt cpw1fn c1c cvv csset cid cima cv csn cop wcel wex wceq wbr vex
|
|
wa bitri 1cex cxp cins3 ccnv csi ctxp cpw1 cins2 csymdif ccompl cuni cmpt
|
|
cin df-pw1fn oteltxp snex ideq df-br eqcom sneqb 3bitr3i wel brsnsi brcnv
|
|
brssetsn opelssetsn anbi12i exbii elima1c eluni 3bitr4i elimapw11c df-rex
|
|
ancom wrex elpw1 releqmpt eqtr4i idex ssetex cnvex siex txpex imaex pw1ex
|
|
mptexlem eqeltri ) DEFUAGUBHGUCZUDZGUEZEIZUEZEUFZIZUGUHEIUIUCULZFDAEAJZUJ
|
|
ZUFZUKWNAUMABEWMWQCJZKZKZBJZKZWOLZLWKMZCNWRWPMZXAWSOZRZCNZXCWMMXAWQMZXDXG
|
|
CXDXFXERZXGXDWTXBLHMZWTWOLZWJMZRXJWTXBWOHWJUNXKXFXMXEWTXBHPWTXBOZXKXFWTXB
|
|
XAUOUPWTXBHUQXNXBWTOXFWTXBURXAWSBQZUSSUTXBXLLWIMZBNCBVAZBAVAZRZBNXMXEXPXS
|
|
BXPXBWTLWHMZXCGMZRXSXBWTWOWHGUNXTXQYAXRXBWTWHPXAWSWGPZXTXQXAWSWGXOWRUOVBX
|
|
BWTWHUQYBWSXAGPXQXAWSGVCWRXACQXOVDSUTXAWOXOAQVEVFSVGBXLWIVHBWRWOVIVJVFSXF
|
|
XEVMSVGCXCWKVKXIXFCWPVNXHCXAWPVOXFCWPVLSVJVPVQEWMTWKWLHWJVRWIEWHGWGGVSVTW
|
|
AVSWBTWCWBETWDWCWEWF $.
|
|
$}
|
|
|
|
$( Functionhood statement for ` Pw1Fn ` (Contributed by SF, 25-Feb-2015.) $)
|
|
fnpw1fn $p |- Pw1Fn Fn 1c $=
|
|
( vx cv cuni cpw1 cvv wcel cpw1fn c1c wfn df-pw1fn fnmpt vex uniex a1i mprg
|
|
pw1ex ) ABZCZDZEFZGHIAHAHSGEAJKTQHFRQALMPNO $.
|
|
|
|
${
|
|
brpw1fn.1 $e |- A e. _V $.
|
|
$( Binary relationship form of ` Pw1Fn ` (Contributed by SF,
|
|
25-Feb-2015.) $)
|
|
brpw1fn $p |- ( { A } Pw1Fn B <-> B = ~P1 A ) $=
|
|
( csn cpw1fn cfv wceq cpw1 wbr pw1fnval eqeq1i c1c wcel wb fnpw1fn snel1c
|
|
wfn fnbrfvb mp2an eqcom 3bitr3i ) ADZEFZBGZAHZBGUBBEIZBUEGUCUEBACJKELQUBL
|
|
MUDUFNOACPLUBBERSUEBTUA $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a x $. $d a y $. $d b x $. $d b y $. $d x y $. $d x z $.
|
|
$d y z $.
|
|
$( ` Pw1Fn ` is a one-to-one function with domain ` 1c ` and range
|
|
` ~P 1c ` . (Contributed by SF, 26-Feb-2015.) $)
|
|
pw1fnf1o $p |- Pw1Fn : 1c -1-1-onto-> ~P 1c $=
|
|
( vx vy vz va vb c1c cpw1fn wceq cv cfv cpw1 wss wex cvv wa vex exbii csn
|
|
wcel el1c cpw wf1o wfn crn wi wral fnpw1fn cuni wrex df-pw1fn rnmpt sspw1
|
|
weq cab df1c2 sseq2i ssv biantrur 3bitr4i elpw df-rex anbi1i 19.41v excom
|
|
bitr4i unieq unisn syl6eq pw1eq eqeq2d ceqsexv bitri 3bitri abbi2i eqtr4i
|
|
snex syl anbi12i eeanv pw111 biimpi fveq2 pw1fnval eqeqan12d eqeq12 sneqb
|
|
a1i syl6bb 3imtr4d exlimivv sylbi rgen2a dff1o6 mpbir3an ) FFUAZGUBGFUCGU
|
|
DZWOHAIZGJZBIZGJZHZABUMZUEZBFUFAFUFUGWPWSWQUHZKZHZAFUIZBUNWOABFXEGAUJUKXG
|
|
BWOWSFLZWSCIZKZHZCMZWSWOSXGWSNKZLXINLZXKOZCMXHXLCWSNBPZULFXMWSUOUPXKXOCXN
|
|
XKXIUQURQUSWSFXPUTXGWQFSZXFOZAMWQXIRZHZXFOZCMZAMZXLXFAFVAXRYBAXRXTCMZXFOY
|
|
BXQYDXFCWQTVBXTXFCVCVEQYCYAAMZCMXLYAACVDYEXKCXFXKAXSXIVPXTXEXJWSXTXDXIHXE
|
|
XJHXTXDXSUHXIWQXSVFXICPVGVHXDXIVIVQVJVKQVLVMUSVNVOXCABFXQWSFSZOZWQDIZRZHZ
|
|
WSEIZRZHZOZEMDMZXCYGYJDMZYMEMZOYOXQYPYFYQDWQTEWSTVRYJYMDEVSVEYNXCDEYNYHKZ
|
|
YKKZHZDEUMZXAXBYTUUAUEYNYTUUAYHYKVTWAWGYJYMWRYRWTYSYJWRYIGJYRWQYIGWBYHDPZ
|
|
WCVHYMWTYLGJYSWSYLGWBYKEPWCVHWDYNXBYIYLHUUAWQYIWSYLWEYHYKUUBWFWHWIWJWKWLA
|
|
BFWOGWMWN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d F x $. $d F y $. $d x y $.
|
|
$( Lemma for ~ fnfullfun . Binary relationship over part one of the full
|
|
function definition. (Contributed by SF, 9-Mar-2015.) $)
|
|
fnfullfunlem1 $p |- ( A ( ( _I o. F ) \ ( ~ _I o. F ) ) B <->
|
|
( A F B /\ A. x ( A F x -> x = B ) ) ) $=
|
|
( vy cid ccom wbr cvv wcel cv wceq wi wal wa brex simprd wb breq2 wn cdif
|
|
ccompl adantr weq eqeq2 imbi2d albidv anbi12d bibi12d brdif coi2 wex brco
|
|
breqi cop df-br vex opex elcompl ideq bitr3i xchbinx bitri anbi2i exanali
|
|
exbii 3bitrri con1bii anbi12i vtoclg pm5.21nii ) BCFDGZFUBZDGZUAZHZCIJZBC
|
|
DHZBAKZDHZVSCLZMZANZOZVPBIJZVQBCVOPQVRVQWCVRWEVQBCDPQUCBEKZVOHZBWFDHZVTAE
|
|
UDZMZANZOZRVPWDRECIWFCLZWGVPWLWDWFCBVOSWMWHVRWKWCWFCBDSWMWJWBAWMWIWAVTWFC
|
|
VSUEUFUGUHUIWGBWFVLHZBWFVNHZTZOWLBWFVLVNUJWNWHWPWKBWFVLDDUKUNWKWOWOVTVSWF
|
|
VMHZOZAULVTWITZOZAULWKTABWFVMDUMWRWTAWQWSVTWQVSWFUOZVMJZWSVSWFVMUPXBXAFJZ
|
|
WIXAFVSWFAUQEUQZURUSXCVSWFFHWIVSWFFUPVSWFXDUTVAVBVCVDVFVTWIAVEVGVHVIVCVJV
|
|
K $.
|
|
$}
|
|
|
|
${
|
|
$d F x y z $.
|
|
$( Lemma for ~ fnfullfun . Part one of the full function operator yields a
|
|
function. (Contributed by SF, 9-Mar-2015.) $)
|
|
fnfullfunlem2 $p |- Fun ( ( _I o. F ) \ ( ~ _I o. F ) ) $=
|
|
( vx vy vz cid ccom ccompl cdif cv wbr wa weq wi wal dffun2 fnfullfunlem1
|
|
wfun sp impcom ad2ant2rl syl2anb gen2 mpgbir ) EAFEGAFHZQBIZCIZUDJZUEDIZU
|
|
DJZKCDLZMZDNCNBBCDUDOUKCDUGUEUFAJZUEUHAJZDCLMDNZKUMULUJMZCNZKUJUIDUEUFAPC
|
|
UEUHAPULUPUJUNUMUPULUJUOCRSTUAUBUC $.
|
|
$}
|
|
|
|
$( The full function operator yields a function over ` _V ` . (Contributed
|
|
by SF, 9-Mar-2015.) $)
|
|
fnfullfun $p |- FullFun F Fn _V $=
|
|
( cfullfun cvv wfn cid ccom ccompl cdif cdm c0 csn cxp cun wa cin wceq wfun
|
|
fnfullfunlem2 funfn mp2an mpbi 0ex fnconstg ax-mp pm3.2i incompl df-fullfun
|
|
wcel fnun wb uncompl eqcomi fneq1 fneq2 sylan9bb mpbir ) ABZCDZEAFEGAFHZUSI
|
|
ZGZJKLZMZUTVAMZDZUSUTDZVBVADZNUTVAOJPVEVFVGUSQVFARUSSUAJCUHVGUBVAJCUCUDUEUT
|
|
UFUTVAUSVBUITUQVCPZCVDPZURVEUJAUGVDCUTUKULVHURVCCDVIVECUQVCUMCVDVCUNUOTUP
|
|
$.
|
|
|
|
$( The full function of a set is a set. (Contributed by SF, 9-Mar-2015.) $)
|
|
fullfunexg $p |- ( F e. V -> FullFun F e. _V ) $=
|
|
( wcel cfullfun cid ccom ccompl cdif cdm c0 csn cxp cun cvv df-fullfun idex
|
|
coexg mpan complex syl2anc difexg dmexg complexg 3syl snex sylancl syl5eqel
|
|
xpexg unexg ) ABCZADEAFZEGZAFZHZUNIZGZJKZLZMZNAOUJUNNCZURNCZUSNCUJUKNCZUMNC
|
|
ZUTENCUJVBPEANBQRULNCUJVCEPSULANBQRUKUMNNUATZUJUPNCZUQNCVAUJUTUONCVEVDUNNUB
|
|
UONUCUDJUEUPUQNNUHUFUNURNNUITUG $.
|
|
|
|
${
|
|
fullfunex.1 $e |- F e. _V $.
|
|
$( The full function of a set is a set. (Contributed by SF,
|
|
9-Mar-2015.) $)
|
|
fullfunex $p |- FullFun F e. _V $=
|
|
( cvv wcel cfullfun fullfunexg ax-mp ) ACDAECDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d F y $. $d F z $. $d x y $. $d x z $. $d y z $.
|
|
$( Lemma for ~ fvfullfun . Calculate the domain of part one of the full
|
|
function definition. (Contributed by SF, 9-Mar-2015.) $)
|
|
fvfullfunlem1 $p |- dom ( ( _I o. F ) \ ( ~ _I o. F ) ) =
|
|
{ x | E! y x F y } $=
|
|
( vz cv wbr weu cid ccom ccompl cdif cdm wcel wex weq wi wal wa exbii nfv
|
|
eldm fnfullfunlem1 wsb eu1 breq2 sbie equcom imbi12i anbi2i bitr2i 3bitri
|
|
albii abbi2i ) AEZBEZCFZBGZAHCIHJCIKZLZUNUSMUNUOURFZBNUPUNDEZCFZDBOZPZDQZ
|
|
RZBNZUQBUNURUAUTVFBDUNUOCUBSUQUPUPBDUCZBDOZPZDQZRZBNVGUPBDUPDTUDVLVFBVKVE
|
|
UPVJVDDVHVBVIVCUPVBBDVBBTUOVAUNCUEUFBDUGUHULUISUJUKUM $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d F y $. $d F z $. $d x y $. $d x z $. $d y z $.
|
|
$( Lemma for ~ fvfullfun . Part one of the full function definition is a
|
|
subset of the function. (Contributed by SF, 9-Mar-2015.) $)
|
|
fvfullfunlem2 $p |- ( ( _I o. F ) \ ( ~ _I o. F ) ) C_ F $=
|
|
( vx vy vz cid ccom ccompl cdif wss cv cop wcel wal wbr weq fnfullfunlem1
|
|
wi wa simpl df-br bitr3i 3imtr3i gen2 ssrel mpbir ) EAFEGAFHZAIBJZCJZKZUF
|
|
LZUIALZQZCMBMULBCUGUHANZUGDJANDCOQDMZRZUMUJUKUMUNSUOUGUHUFNUJDUGUHAPUGUHU
|
|
FTUAUGUHATUBUCBCUFAUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d F y $. $d F z $. $d x y $. $d x z $. $d y z $.
|
|
$( Lemma for ~ fvfullfun . Part one of the full function definition agrees
|
|
with the set itself over its domain. (Contributed by SF,
|
|
9-Mar-2015.) $)
|
|
fvfullfunlem3 $p |- ( A e. dom ( ( _I o. F ) \ ( ~ _I o. F ) ) ->
|
|
( ( ( _I o. F ) \ ( ~ _I o. F ) ) ` A ) = ( F ` A ) ) $=
|
|
( vx vy vz cid ccom wcel cfv wss wceq cv wbr wal weu fvfullfunlem1 abeq2i
|
|
wa brres cvv ccompl cdif cdm cres wfun weq wi dffun2 anbi2i bitri adantrl
|
|
tz6.12-1 adantl eqtr3d adantlr syl2anb gen2 cxp cin fvfullfunlem2 crn ssv
|
|
mpgbir ssdmrn xpss2 ax-mp sstri ssini df-res sseqtr4i funssfv mp3an12
|
|
fvres ) AFBGFUABGUBZUCZHZABVOUDZIZAVNIZABIVQUEZVNVQJVPVRVSKVTCLZDLZVQMZWA
|
|
ELZVQMZRDEUFZUGZENDNCCDEVQUHWGDEWCWAWBBMZWAWDBMZEOZRZWIWHDOZRZWFWEWCWHWAV
|
|
OHZRWKWAWBBVOSWNWJWHWJCVOCEBPQUIUJWEWIWNRWMWAWDBVOSWNWLWIWLCVOCDBPQUIUJWH
|
|
WMWFWJWHWMRWABIZWBWDWHWLWOWBKWIDWAWBBULUKWMWOWDKWHDWAWDBULUMUNUOUPUQVCVNB
|
|
VOTURZUSVQVNBWPBUTVNVOVNVAZURZWPVNVDWQTJWRWPJWQVBWQTVOVEVFVGVHBVOVIVJAVQV
|
|
NVKVLAVOBVMUN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d F x $. $d F y $. $d x y $.
|
|
$( The value of the full function definition agrees with the function value
|
|
everywhere. (Contributed by SF, 9-Mar-2015.) $)
|
|
fvfullfun $p |- ( FullFun F ` A ) = ( F ` A ) $=
|
|
( vx vy cvv wcel cfv wceq cv fveq2 cid ccom ccompl c0 wfn wa mp3an12 mpan
|
|
0ex eqtr4d cfullfun eqeq12d cdm csn cxp cun df-fullfun fveq1i cin incompl
|
|
cdif wfun fnfullfunlem2 funfn fnconstg ax-mp fvun1 fvfullfunlem3 eqtrd wn
|
|
mpbi vex elcompl sylbir wbr fvfullfunlem1 abeq2i tz6.12-2 sylnbi fvconst2
|
|
fvun2 weu pm2.61i eqtri vtoclg fvprc ) AEFZABUAZGZABGZHZCIZVRGZWBBGZHWACA
|
|
EWBAHWCVSWDVTWBAVRJWBABJUBWCWBKBLKMBLUKZWEUCZMZNUDUEZUFZGZWDWBVRWIBUGUHWB
|
|
WFFZWJWDHWKWJWBWEGZWDWFWGUINHZWKWJWLHZWFUJZWEWFOZWHWGOZWMWKPWNWEULWPBUMWE
|
|
UNVAZNEFWQSWGNEUOUPZWFWGWEWHWBUQQRWBBURUSWKUTZWJWBWHGZWDWTWBWGFZWJXAHZWBW
|
|
FCVBVCZWMXBXCWOWPWQWMXBPXCWRWSWFWGWEWHWBVKQRVDWTWDNXAWKWBDIBVEDVLZWDNHXEC
|
|
WFCDBVFVGDWBBVHVIWTXBXANHXDWGNWBSVJVDTTVMVNVOVQUTVSNVTAVRVPABVPTVM $.
|
|
$}
|
|
|
|
$( Binary relationship of the full function operation. (Contributed by SF,
|
|
9-Mar-2015.) $)
|
|
brfullfung $p |- ( A e. V -> ( A FullFun F B <-> ( F ` A ) = B ) ) $=
|
|
( wcel cvv cfullfun wbr cfv wceq wb elex fvfullfun eqeq1i fnfullfun fnbrfvb
|
|
wfn mpan syl5rbbr syl ) ADEAFEZABCGZHZACIZBJZKADLUEAUBIZBJZUAUCUFUDBACMNUBF
|
|
QUAUGUCKCOFABUBPRST $.
|
|
|
|
${
|
|
brfullfun.1 $e |- A e. _V $.
|
|
$( Binary relationship of the full function operation. (Contributed by SF,
|
|
9-Mar-2015.) $)
|
|
brfullfun $p |- ( A FullFun F B <-> ( F ` A ) = B ) $=
|
|
( cvv wcel cfullfun wbr cfv wceq wb brfullfung ax-mp ) AEFABCGHACIBJKDABC
|
|
ELM $.
|
|
$}
|
|
|
|
${
|
|
brfullfunop.1 $e |- A e. _V $.
|
|
brfullfunop.2 $e |- B e. _V $.
|
|
$( Binary relationship of the full function operation over an ordered
|
|
pair. (Contributed by SF, 9-Mar-2015.) $)
|
|
brfullfunop $p |- ( <. A , B >. FullFun F C <-> ( A F B ) = C ) $=
|
|
( cop cfullfun wbr cfv wceq co opex brfullfun df-ov eqeq1i bitr4i ) ABGZC
|
|
DHIRDJZCKABDLZCKRCDABEFMNTSCABDOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Calculate the value of the domain function. (Contributed by Scott
|
|
Fenton, 9-Aug-2019.) $)
|
|
fvdomfn $p |- ( A e. V -> ( Dom ` A ) = dom A ) $=
|
|
( vx wcel cvv cdomfn cfv cdm wceq elex dmexg cv df-domfn fvmptg mpdan syl
|
|
dmeq ) ABDAEDZAFGAHZIZABJRSEDTAEKCACLZHSEEFUAAQCMNOP $.
|
|
|
|
$( Calculate the value of the range function. (Contributed by Scott
|
|
Fenton, 9-Aug-2019.) $)
|
|
fvranfn $p |- ( A e. V -> ( Ran ` A ) = ran A ) $=
|
|
( vx wcel cvv cranfn cfv crn wceq elex rnexg cv df-ranfn fvmptg mpdan syl
|
|
rneq ) ABDAEDZAFGAHZIZABJRSEDTAEKCACLZHSEEFUAAQCMNOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( The domain function is stratified. (Contributed by Scott Fenton,
|
|
9-Aug-2019.) $)
|
|
domfnex $p |- Dom e. _V $=
|
|
( vx vy vz vw cdomfn cvv csset cswap cins2 cin c1c cima cv csn cop wex wa
|
|
wcel vex bitri cxp cins3 csi3 cins4 csymdif ccompl ccnv cdm cmpt df-domfn
|
|
wceq wel oqelins4 wbr otsnelsi3 df-br brswap2 3bitr2i otelins2 opelssetsn
|
|
elin snex anbi12i exbii elima1c df-clel eldm2 releqmpt eqtr4i vvex swapex
|
|
3bitr4i si3ex ins4ex ssetex ins2ex inex 1cex imaex mptexlem eqeltri ) EFF
|
|
UAGUBHUCZUDZGIZIZJZKLZKLZIUEKLUFUGJZFEAFAMZUHZUIWIAUJABFWHWKCMZNZBMZNZWJO
|
|
ZOZWGRZCPWNWLOZWJRZCPWPWHRWNWKRWRWTCDMZNZWQOZWFRZDPXAWSUKZDAULZQZDPWRWTXD
|
|
XGDXDXCWCRZXCWERZQXGXCWCWEVAXHXEXIXFXHXBWMWOOOWBRZXEXBWMWOWJWBASZUMXJXAWL
|
|
WNOZOHRXAXLHUNXEXAWLWNHDSZCSZBSZUOXAXLHUPXAWLWNXNXOUQURTXIXBWPOWDRZXFXBWM
|
|
WPWDWLVBUSXPXBWJOGRXFXBWOWJGWNVBUSXAWJXMXKUTTTVCTVDDWQWFVEDWSWJVFVLVDCWPW
|
|
GVECWNWJVGVLVHVIFWHVJWGKWFKWCWEWBHVKVMVNWDGVOVPVPVQVRVSVRVSVTWA $.
|
|
|
|
$( The range function is stratified. (Contributed by Scott Fenton,
|
|
9-Aug-2019.) $)
|
|
ranfnex $p |- Ran e. _V $=
|
|
( vx vy vz vw cranfn cvv csset cid cins2 cin c1c cima cv csn cop wcel wex
|
|
wa vex bitri cxp cins3 csi3 cins4 csymdif ccnv crn cmpt df-ranfn wceq wel
|
|
ccompl elin oqelins4 wbr otsnelsi3 df-br opex 3bitr2i otelins2 opelssetsn
|
|
ideq snex anbi12i exbii elima1c df-clel 3bitr4i releqmpt eqtr4i vvex idex
|
|
elrn2 si3ex ins4ex ssetex ins2ex inex 1cex imaex mptexlem eqeltri ) EFFUA
|
|
GUBHUCZUDZGIZIZJZKLZKLZIUEKLULUFJZFEAFAMZUGZUHWJAUIABFWIWLCMZNZBMZNZWKOZO
|
|
ZWHPZCQWMWOOZWKPZCQWQWIPWOWLPWSXACDMZNZWROZWGPZDQXBWTUJZDAUKZRZDQWSXAXEXH
|
|
DXEXDWDPZXDWFPZRXHXDWDWFUMXIXFXJXGXIXCWNWPOOWCPZXFXCWNWPWKWCASZUNXKXBWTOH
|
|
PXBWTHUOXFXBWMWOHDSZCSZBSZUPXBWTHUQXBWTWMWOXNXOURVBUSTXJXCWQOWEPZXGXCWNWQ
|
|
WEWMVCUTXPXCWKOGPXGXCWPWKGWOVCUTXBWKXMXLVATTVDTVEDWRWGVFDWTWKVGVHVECWQWHV
|
|
FCWOWKVMVHVIVJFWIVKWHKWGKWDWFWCHVLVNVOWEGVPVQVQVRVSVTVSVTWAWB $.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Closure operation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c Clos1 $.
|
|
|
|
$( Extend the definition of a class to include the closure operation. $)
|
|
cclos1 $a class Clos1 ( A , R ) $.
|
|
|
|
${
|
|
$d S a $. $d R a $.
|
|
$( Define the closure operation. A modified version of the definition from
|
|
[Rosser] p. 245. (Contributed by SF, 11-Feb-2015.) $)
|
|
df-clos1 $a |- Clos1 ( S , R ) =
|
|
|^| { a | ( S C_ a /\ ( R " a ) C_ a ) } $.
|
|
$}
|
|
|
|
${
|
|
$d R a $. $d S a $. $d T a $.
|
|
$( Equality law for closure. (Contributed by SF, 11-Feb-2015.) $)
|
|
clos1eq1 $p |- ( S = T -> Clos1 ( S , R ) = Clos1 ( T , R ) ) $=
|
|
( va wceq cv wss cima wa cab cint cclos1 sseq1 anbi1d abbidv syl df-clos1
|
|
inteq 3eqtr4g ) BCEZBDFZGZAUAHUAGZIZDJZKZCUAGZUCIZDJZKZBALCALTUEUIEUFUJET
|
|
UDUHDTUBUGUCBCUAMNOUEUIRPABDQACDQS $.
|
|
|
|
$( Equality law for closure. (Contributed by SF, 11-Feb-2015.) $)
|
|
clos1eq2 $p |- ( R = T -> Clos1 ( S , R ) = Clos1 ( S , T ) ) $=
|
|
( va wceq cv wss cima wa cab cclos1 imaeq1 sseq1d anbi2d abbidv inteq syl
|
|
cint df-clos1 3eqtr4g ) ACEZBDFZGZAUBHZUBGZIZDJZRZUCCUBHZUBGZIZDJZRZBAKBC
|
|
KUAUGULEUHUMEUAUFUKDUAUEUJUCUAUDUIUBACUBLMNOUGULPQABDSCBDST $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d R a $. $d R b $. $d S a $.
|
|
clos1ex.1 $e |- S e. _V $.
|
|
clos1ex.2 $e |- R e. _V $.
|
|
$( The closure of a set under a set is a set. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
clos1ex $p |- Clos1 ( S , R ) e. _V $=
|
|
( va vb cv wss cima cvv csset wcel wbr vex brsset wex imaex bitri ssetex
|
|
wa cclos1 cab cint df-clos1 csn cimage ccom cfix cin elin elimasn 3bitr2i
|
|
df-br elfix brco brimage anbi1i exbii breq1 syl6bb ceqsexv anbi12i abbi2i
|
|
cop wceq snex imageex coex fixex inex eqeltrri intex eqeltri ) BAUABEGZHZ
|
|
AVNIZVNHZTZEUBZUCJABEUDVSKBUEZIZKAUFZUGZUHZUIZVSJVREWEVNWELVNWALZVNWDLZTV
|
|
RVNWAWDUJWFVOWGVQWFBVNVDKLBVNKMVOKBVNUKBVNKUMBVNCENZOULWGVNVNWCMZVQVNWCUN
|
|
WIVNFGZWBMZWJVNKMZTZFPZVQFVNVNKWBUOWNWJVPVEZWLTZFPVQWMWPFWKWOWLVNWJAWHFNU
|
|
PUQURWLVQFVPAVNDWHQZWOWLVPVNKMVQWJVPVNKUSVPVNWQWHOUTVARRRVBRVCWAWDKVTSBVF
|
|
QWCKWBSADVGVHVIVJVKVLVM $.
|
|
$}
|
|
|
|
${
|
|
$d S s r $. $d R s r $.
|
|
$( The closure of a set under a set is a set. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
clos1exg $p |- ( ( S e. V /\ R e. W ) -> Clos1 ( S , R ) e. _V ) $=
|
|
( vs vr cclos1 cvv wcel wceq clos1eq1 eleq1d clos1eq2 vex clos1ex vtocl2g
|
|
cv ) EQZFQZGZHIBSGZHIBAGZHIEFBACDRBJTUAHSRBKLSAJUAUBHSBAMLSRENFNOP $.
|
|
$}
|
|
|
|
${
|
|
$d R a $. $d S a $.
|
|
clos1base.1 $e |- C = Clos1 ( S , R ) $.
|
|
$( The initial set of a closure is a subset of the closure. Theorem
|
|
IX.5.13 of [Rosser] p. 246. (Contributed by SF, 13-Feb-2015.) $)
|
|
clos1base $p |- S C_ C $=
|
|
( va cv wss cima wa cab cint ssmin cclos1 df-clos1 eqtr2i sseqtri ) CCEFZ
|
|
GBQHQGZIEJKZARECLACBMSDBCENOP $.
|
|
|
|
${
|
|
$d A x $. $d B x $. $d B y $. $d C x $. $d C y $. $d R x $.
|
|
$d R y $. $d x y $. $d a x $. $d a y $. $d A y $. $d a z $.
|
|
$d R z $. $d x z $. $d y z $.
|
|
|
|
$( If a class is connected to an element of a closure via ` R ` , then it
|
|
is a member of the closure. Theorem IX.5.14 of [Rosser] p. 246.
|
|
(Contributed by SF, 13-Feb-2015.) $)
|
|
clos1conn $p |- ( ( A e. C /\ A R B ) -> B e. C ) $=
|
|
( vx vy va vz cvv wcel wa wbr cv wi wceq eleq1 breq1 wss adantl anbi12d
|
|
brex imbi1d breq2 anbi2d imbi12d cima wel wal wrex rspcev sylibr ancoms
|
|
elima ssel syl5 exp3a com12 adantld a2d alimdv cab cint cclos1 df-clos1
|
|
eqtri eleq2i vex elintab bitri 3imtr4g impcom vtocl2g mpcom ) AKLBKLMZA
|
|
CLZABDNZMZBCLZVRVPVQABDUCUAGOZCLZWAHOZDNZMZWCCLZPVQAWCDNZMZWFPVSVTPGHAB
|
|
KKWAAQZWEWHWFWIWBVQWDWGWAACRWAAWCDSUBUDWCBQZWHVSWFVTWJWGVRVQWCBADUEUFWC
|
|
BCRUGWDWBWFWDEIOZTZDWKUHZWKTZMZGIUIZPZIUJZWOHIUIZPZIUJZWBWFWDWQWTIWDWOW
|
|
PWSWDWNWPWSPZWLWNWDXBWNWDWPWSWDWPMWCWMLZWNWSWPWDXCWPWDMJOZWCDNZJWKUKXCX
|
|
EWDJWAWKXDWAWCDSULJWCDWKUOUMUNWMWKWCUPUQURUSUTVAVBWBWAWOIVCVDZLWRCXFWAC
|
|
EDVEXFFDEIVFVGZVHWOIWAGVIVJVKWFWCXFLXACXFWCXGVHWOIWCHVIVJVKVLVMVNVO $.
|
|
$}
|
|
|
|
$}
|
|
|
|
${
|
|
$d S a $. $d R a $.
|
|
clos1induct.1 $e |- S e. _V $.
|
|
clos1induct.2 $e |- R e. _V $.
|
|
clos1induct.3 $e |- C = Clos1 ( S , R ) $.
|
|
${
|
|
$d C a x z $. $d R x z $. $d X a x z $.
|
|
$( Inductive law for closure. If the base set is a subset of ` X ` , and
|
|
` X ` is closed under ` R ` , then the closure is a subset of ` X ` .
|
|
Theorem IX.5.15 of [Rosser] p. 247. (Contributed by SF,
|
|
11-Feb-2015.) $)
|
|
clos1induct $p |- ( ( X e. V /\ S C_ X /\
|
|
A. x e. C A. z ( ( x e. X /\ x R z ) -> z e. X ) ) ->
|
|
C C_ X ) $=
|
|
( va wcel cvv wss cv wa wi wal albii bitri cin wral cima cclos1 clos1ex
|
|
wbr eqeltri inexg mpan2 clos1base ssin biimpi wex elima2 imbi12i df-ral
|
|
impexp wb clos1conn biantrud adantrl pm5.74i bitr3i ancom anbi1i imbi1i
|
|
elin anass 19.23v 3bitr2i bitr4i dfss2 ralcom4 3bitr4i biimpri w3a cint
|
|
cab df-clos1 eqtri sseq2 imaeq2 id sseq12d anbi12d elabg biimprd 3impib
|
|
wceq intss1 syl syl5eqss inss1 syl6ss syl3an ) GFLZGCUAZMLZEGNZEWQNZAOZ
|
|
GLZXABOZDUFZPZXCGLZQZBRACUBZDWQUCZWQNZCGNWPCMLWRCEDUDZMJDEHIUEUGGCFMUHU
|
|
IWSECNZWTCDEJUJWSXLPWTEGCUKULUIXJXHXCXILZXCWQLZQZBRXGACUBZBRXJXHXOXPBXO
|
|
XAWQLZXDPZAUMZXFXCCLZPZQZXPXMXSXNYAAXCDWQUNXCGCVGUOXPXACLZXEPZYAQZARZXR
|
|
YAQZARYBXPYCXGQZARYFXGACUPYHYEAYHYDXFQYEYCXEXFUQYDXFYAYCXDXFYAURXBYCXDP
|
|
XTXFXAXCCDEJUSUTVAVBVCSTYGYEAXRYDYAXRYCXBPZXDPYDXQYIXDXQXBYCPYIXAGCVGXB
|
|
YCVDTVEYCXBXDVHTVFSXRYAAVIVJVKSBXIWQVLXGABCVMVNVOWRWTXJVPZCWQGYJCEKOZNZ
|
|
DYKUCZYKNZPZKVRZVQZWQCXKYQJDEKVSVTYJWQYPLZYQWQNWRWTXJYRWRYRWTXJPZYOYSKW
|
|
QMYKWQWIZYLWTYNXJYKWQEWAYTYMXIYKWQYKWQDWBYTWCWDWEWFWGWHWQYPWJWKWLGCWMWN
|
|
WO $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d C y $. $d C z $. $d ch x $. $d ph y $. $d ph z $.
|
|
$d ps x $. $d R y $. $d R z $. $d S x $. $d th x $. $d x y $.
|
|
$d x z $. $d y z $.
|
|
clos1is.1 $e |- S e. _V $.
|
|
clos1is.2 $e |- R e. _V $.
|
|
clos1is.3 $e |- C = Clos1 ( S , R ) $.
|
|
clos1is.4 $e |- { x | ph } e. _V $.
|
|
clos1is.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
|
|
clos1is.6 $e |- ( x = z -> ( ph <-> ch ) ) $.
|
|
clos1is.7 $e |- ( x = A -> ( ph <-> th ) ) $.
|
|
clos1is.8 $e |- ( x e. S -> ph ) $.
|
|
clos1is.9 $e |- ( ( y e. C /\ y R z /\ ps ) -> ch ) $.
|
|
$( Induction scheme for closures. Hypotheses one through three set up
|
|
existence properties, hypothesis four sets up stratification, hypotheses
|
|
five through seven set up implicit substitution, and hypotheses eight
|
|
and nine set up the base and induction steps. (Contributed by SF,
|
|
13-Feb-2015.) $)
|
|
clos1is $p |- ( A e. C -> th ) $=
|
|
( wcel cab cvv wss cv wbr wa wal wral ssab mpgbir 3expib vex anbi1i ancom
|
|
wi elab bitri 3imtr4g alrimiv rgen clos1induct mp3an sseli elabg mpbid )
|
|
HIUAHAEUBZUADIVGHVGUCUAKVGUDZFUEZVGUAZVIGUEZJUFZUGZVKVGUAZUPZGUHZFIUIIVGU
|
|
DOVHEUEKUAAUPEAEKUJSUKVPFIVIIUAZVOGVQVLBUGZCVMVNVQVLBCTULVMBVLUGVRVJBVLAB
|
|
EVIFUMPUQUNBVLUOURACEVKGUMQUQUSUTVAFGIJKUCVGLMNVBVCVDADEHIRVEVF $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d C w $. $d C x $. $d C y $. $d C z $. $d R w $.
|
|
$d R x $. $d R y $. $d R z $. $d S w $. $d S y $. $d S z $. $d w x $.
|
|
$d w y $. $d w z $. $d x y $. $d x z $. $d y z $.
|
|
clos1basesuc.1 $e |- S e. _V $.
|
|
clos1basesuc.2 $e |- R e. _V $.
|
|
clos1basesuc.3 $e |- C = Clos1 ( S , R ) $.
|
|
$( A member of a closure is either in the base set or connected to another
|
|
member by ` R ` . Theorem IX.5.16 of [Rosser] p. 248. (Contributed by
|
|
SF, 13-Feb-2015.) $)
|
|
clos1basesuc $p |- ( A e. C <-> ( A e. S \/ E. x e. C x R A ) ) $=
|
|
( vy vz vw wcel cv wbr wrex wo cab eleq1 breq2 rexbidv cima cun cvv abid2
|
|
eqcomi df-ima uneq12i unab eqtri cclos1 clos1ex eqeltri unex eqeltrri weq
|
|
imaex orbi12d wceq wa wi clos1base sseli breq1 rspcev clos1conn rexlimiva
|
|
orc ex syl jaoi impcom cbvrexv sylibr olcd 3adant1 clos1is impbii ) BCLZB
|
|
ELZAMZBDNZACOZPZIMZELZVTWDDNZACOZPZJMZELZVTWIDNZACOZPZKMZELZVTWNDNZACOZPZ
|
|
WCIJKBCDEFGHEDCUAZUBZWHIQZUCWTWEIQZWGIQZUBXAEXBWSXCXBEIEUDUEIADCUFUGWEWGI
|
|
UHUIEWSFDCGCEDUJUCHDEFGUKULUPUMUNIJUOZWEWJWGWLWDWIERXDWFWKACWDWIVTDSTUQIK
|
|
UOZWEWOWGWQWDWNERXEWFWPACWDWNVTDSTUQWDBURZWEVSWGWBWDBERXFWFWAACWDBVTDSTUQ
|
|
WEWGVGWIWNDNZWMWRWICLZXGWMUSZWQWOXIWDWNDNZICOZWQWMXGXKWJXGXKUTZWLWJXHXLEC
|
|
WICDEHVAZVBXHXGXKXJXGIWICWDWIWNDVCVDVHZVIWKXLACVTCLWKUSXHXLVTWICDEHVEXNVI
|
|
VFVJVKWPXJAICVTWDWNDVCVLVMVNVOVPVSVRWBECBXMVBWAVRACVTBCDEHVEVFVJVQ $.
|
|
|
|
$d S x $.
|
|
$( A closure is equal to the base set together with the image of the
|
|
closure under ` R ` . Theorem X.4.37 of [Rosser] p. 303. (Contributed
|
|
by SF, 10-Mar-2015.) $)
|
|
clos1baseima $p |- C = ( S u. ( R " C ) ) $=
|
|
( vx vy cima cun cv wcel wbr wrex elima orbi2i elun clos1basesuc 3bitr4ri
|
|
wo eqriv ) GACBAIZJZGKZCLZUDUBLZTUEHKUDBMHANZTUDUCLUDALUFUGUEHUDBAOPUDCUB
|
|
QHUDABCDEFRSUA $.
|
|
$}
|
|
|
|
${
|
|
$d A x s r $. $d S x s r $. $d R x s r $. $d C x s r $.
|
|
clos1basesucg.1 $e |- C = Clos1 ( S , R ) $.
|
|
$( A member of a closure is either in the base set or connected to another
|
|
member by ` R ` . Theorem IX.5.16 of [Rosser] p. 248. (Contributed by
|
|
Scott Fenton, 31-Jul-2019.) $)
|
|
clos1basesucg $p |- ( ( S e. V /\ R e. W ) ->
|
|
( A e. C <-> ( A e. S \/ E. x e. C x R A ) ) ) $=
|
|
( vs vr wcel cclos1 cv wbr wrex wo wb wceq eleq2d bibi12d rexeqdv orbi12d
|
|
wa clos1eq1 eleq2 clos1eq2 breq rexeqbidv orbi2d vex clos1basesuc vtocl2g
|
|
eqid eleq2i rexeqi orbi2i 3bitr4g ) EFKDGKUCBEDLZKZBEKZAMZBDNZAUROZPZBCKU
|
|
TVBACOZPBIMZJMZLZKZBVFKZVABVGNZAVHOZPZQBEVGLZKZUTVKAVNOZPZQUSVDQIJEDFGVFE
|
|
RZVIVOVMVQVRVHVNBVGVFEUDZSVRVJUTVLVPVFEBUEVRVKAVHVNVSUAUBTVGDRZVOUSVQVDVT
|
|
VNURBVGEDUFZSVTVPVCUTVTVKVBAVNURWAVABVGDUGUHUITABVHVGVFIUJJUJVHUMUKULCURB
|
|
HUNVEVCUTVBACURHUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d a x y z $.
|
|
$( The finite cardinals as expressed via the closure operation. Theorem
|
|
X.1.3 of [Rosser] p. 276. (Contributed by SF, 12-Feb-2015.) $)
|
|
dfnnc3 $p |- Nn = Clos1 ( { 0c } , ( x e. _V |-> ( x +c 1c ) ) ) $=
|
|
( va vy vz c0c cv wcel c1c cplc wral wa cab cint wss cvv wi wal wceq 1cex
|
|
vex csn cmpt cima cnnc cclos1 0cex snss wel cfv dfss2 ralcom4 wrex wfn wb
|
|
eqid fnmpt addcex a1i mprg ssv fvelimab mp2an imbi1i r19.23v bitr4i albii
|
|
addceq1 fvmpt ax-mp eqeq1i eqcom bitri eleq1 ceqsalv ralbii anbi12i abbii
|
|
3bitr2ri inteqi df-nnc df-clos1 3eqtr4i ) EBFZGZCFZHIZWCGZCWCJZKZBLZMEUAZ
|
|
WCNZAOAFZHIZUBZWCUCZWCNZKZBLZMUDWKWOUEWJWSWIWRBWDWLWHWQEWCUFUGWQDFZWPGZDB
|
|
UHZPZDQZWEWOUIZWTRZXBPZDQZCWCJZWHDWPWCUJXIXGCWCJZDQXDXGCDWCUKXCXJDXCXFCWC
|
|
ULZXBPXJXAXKXBWOOUMZWCONXAXKUNWNOGZXLAOAOWNWOOWOUOZUPXMWMOGWMHATSUQURUSWC
|
|
UTCOWCWTWOVAVBVCXFXBCWCVDVEVFVEXHWGCWCXHWTWFRZXBPZDQWGXGXPDXFXOXBXFWFWTRX
|
|
OXEWFWTWEOGXEWFRCTZAWEWNWFOWOWMWEHVGXNWEHXQSUQZVHVIVJWFWTVKVLVCVFXBWGDWFX
|
|
RWTWFWCVMVNVLVOVRVPVQVSCBVTWOWKBWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d C x $. $d C y $. $d R x $. $d R y $. $d S x $. $d S y $. $d x y $.
|
|
clos1nrel.1 $e |- S e. _V $.
|
|
clos1nrel.2 $e |- R e. _V $.
|
|
clos1nrel.3 $e |- C = Clos1 ( S , R ) $.
|
|
$( The value of a closure when the base set is not related to anything in
|
|
` R ` . (Contributed by SF, 13-Mar-2015.) $)
|
|
clos1nrel $p |- ( ( R " S ) = (/) -> C = S ) $=
|
|
( vx vy cima c0 wceq cv wcel wbr wa wi wal wral wss cvv wn eq0 wrex elima
|
|
rspe sylibr con3i pm2.21d alimi sylbi ralrimivw clos1induct syl clos1base
|
|
ssid mp3an12 a1i eqssd ) BCIZJKZACUTGLZCMVAHLZBNZOZVBCMZPZHQZGARZACSZUTVG
|
|
GAUTVBUSMZUAZHQVGHUSUBVKVFHVKVDVEVDVJVDVCGCUCVJVCGCUEGVBBCUDUFUGUHUIUJUKC
|
|
TMCCSVHVIDCUOGHABCTCDEFULUPUMCASUTABCFUNUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d R x y $. $d C x y $.
|
|
clos10.1 $e |- R e. _V $.
|
|
clos10.2 $e |- C = Clos1 ( (/) , R ) $.
|
|
$( The value of a closure over an empty base set. (Contributed by Scott
|
|
Fenton, 31-Jul-2019.) $)
|
|
clos10 $p |- C = (/) $=
|
|
( vx vy c0 cvv wcel wss cv wbr wa wi wal wral 0ex 0ss noel pm2.21i adantr
|
|
ax-gen rgenw clos1induct mp3an eqssi ) AGGHIGGJEKZGIZUGFKZBLZMUIGIZNZFOZE
|
|
APAGJQGRUMEAULFUHUKUJUHUKUGSTUAUBUCEFABGHGQCDUDUEARUF $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Orderings
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Basic ordering relationships
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c Trans $. $( Transitive relationships $)
|
|
$c Ref $. $( Reflextive relationships $)
|
|
$c Antisym $. $( Antisymmetric relationships $)
|
|
$c Po $. $( Po orderings $)
|
|
$c Connex $. $( Connected relationships $)
|
|
$c Or $. $( Or orderings $)
|
|
$c Fr $. $( Founded relationships $)
|
|
$c We $. $( We-orderings $)
|
|
$c Ext $. $( Extensional relationships $)
|
|
$c Sym $. $( Symmetric relationships. $)
|
|
$c Er $. $( Equivalence relationships. $)
|
|
|
|
$( Extend the definition of a class to include the set of all transitive
|
|
relationships. $)
|
|
ctrans $a class Trans $.
|
|
|
|
$( Extend the definition of a class to include the set of all reflexive
|
|
relationships. $)
|
|
cref $a class Ref $.
|
|
|
|
$( Extend the definition of a class to include the set of all antisymmetric
|
|
relationships. $)
|
|
cantisym $a class Antisym $.
|
|
|
|
$( Extend the definition of a class to include the set of all partial
|
|
orderings. $)
|
|
cpartial $a class Po $.
|
|
|
|
$( Extend the definition of a class to include the set of all connected
|
|
relationships. $)
|
|
cconnex $a class Connex $.
|
|
|
|
$( Extend the definition of a class to include the set of all strict linear
|
|
orderings. $)
|
|
cstrict $a class Or $.
|
|
|
|
$( Extend the definition of a class to include the set of all founded
|
|
relationships. $)
|
|
cfound $a class Fr $.
|
|
|
|
$( Extend the definition of a class to include the set of all well-ordered
|
|
relationships. $)
|
|
cwe $a class We $.
|
|
|
|
$( Extend the definition of a class to include the set of all extensional
|
|
relationships. $)
|
|
cext $a class Ext $.
|
|
|
|
$( Extend the definition of a class to include the symmetric
|
|
relationships. $)
|
|
csym $a class Sym $.
|
|
|
|
$( Extend the definition of a class to include the equivalence
|
|
relationships. $)
|
|
cer $a class Er $.
|
|
|
|
${
|
|
$d r a x y z $.
|
|
$( Define the set of all transitive relationships over a base set.
|
|
(Contributed by SF, 19-Feb-2015.) $)
|
|
df-trans $a |- Trans = { <. r , a >. |
|
|
A. x e. a A. y e. a A. z e. a ( ( x r y /\ y r z ) -> x r z ) } $.
|
|
|
|
$( Define the set of all reflexive relationships over a base set.
|
|
(Contributed by SF, 19-Feb-2015.) $)
|
|
df-ref $a |- Ref = { <. r , a >. | A. x e. a x r x } $.
|
|
|
|
$( Define the set of all antisymmetric relationships over a base set.
|
|
(Contributed by SF, 19-Feb-2015.) $)
|
|
df-antisym $a |- Antisym = { <. r , a >. |
|
|
A. x e. a A. y e. a ( ( x r y /\ y r x ) -> x = y ) } $.
|
|
|
|
$( Define the set of all partial orderings over a base set. (Contributed
|
|
by SF, 19-Feb-2015.) $)
|
|
df-partial $a |- Po = ( ( Ref i^i Trans ) i^i Antisym ) $.
|
|
|
|
$( Define the set of all connected relationships over a base set.
|
|
(Contributed by SF, 19-Feb-2015.) $)
|
|
df-connex $a |- Connex = { <. r , a >. |
|
|
A. x e. a A. y e. a ( x r y \/ y r x ) } $.
|
|
|
|
$( Define the set of all strict orderings over a base set. (Contributed by
|
|
SF, 19-Feb-2015.) $)
|
|
df-strict $a |- Or = ( Po i^i Connex ) $.
|
|
|
|
$( Define the set of all founded relationships over a base set.
|
|
(Contributed by SF, 19-Feb-2015.) $)
|
|
df-found $a |- Fr = { <. r , a >. |
|
|
A. x ( ( x C_ a /\ x =/= (/) ) ->
|
|
E. z e. x A. y e. x ( y r z -> y = z ) ) } $.
|
|
|
|
$( Define the set of all well orderings over a base set. (Contributed by
|
|
SF, 19-Feb-2015.) $)
|
|
df-we $a |- We = ( Or i^i Fr ) $.
|
|
|
|
$( Define the set of all extensional relationships over a base set.
|
|
(Contributed by SF, 19-Feb-2015.) $)
|
|
df-ext $a |- Ext = { <. r , a >. |
|
|
A. x e. a A. y e. a ( A. z e. a ( z r x <-> z r y ) -> x = y ) } $.
|
|
|
|
$( Define the set of all symmetric relationships over a base set.
|
|
(Contributed by SF, 19-Feb-2015.) $)
|
|
df-sym $a |- Sym = { <. r , a >. |
|
|
A. x e. a A. y e. a ( x r y -> y r x ) } $.
|
|
|
|
$( Define the set of all equivalence relationships over a base set.
|
|
(Contributed by SF, 19-Feb-2015.) $)
|
|
df-er $a |- Er = ( Sym i^i Trans ) $.
|
|
$}
|
|
|
|
${
|
|
$d a p q r t x y z $.
|
|
|
|
$( The class of all transitive relationships is a set. (Contributed by SF,
|
|
19-Feb-2015.) $)
|
|
transex $p |- Trans e. _V $=
|
|
( vx vy vr vz va vq csset c2nd c1st c1c wbr wa wcel otelins2 bitri 3bitri
|
|
cop wex df-br anbi12i ctrans cins2 cvv ctxp csi3 cins4 cin cima cxp cins3
|
|
cdif ccompl cv wi wral copab df-trans wn vex opex elcompl wrex opelssetsn
|
|
csn wel elin elima1c snex eldif opelxp mpbiran oqelins4 otsnelsi3 oteltxp
|
|
wceq ancom bitr4i op1st2nd exbii df-clel 3bitr4i otelins3 notbii rexanali
|
|
df-rex bitr3i rexnal opabbi2i eqtr4i ssetex ins2ex vvex 2ndex 1stex txpex
|
|
con2bii si3ex ins4ex inex 1cex imaex xpex ins3ex difex complex eqeltri )
|
|
UAGUBZXGUBZXHUBZUCHIUDZUEZUFZXHUGZJUHZUFZUIZXLGUJZUBZUBZUBZUGZJUHZUGZXOUB
|
|
ZUKZUGZJUHZUGZJUHZUGZJUHZULZUCUAAUMZBUMZCUMZKZYNDUMZYOKZLZYMYQYOKZUNDEUMZ
|
|
UOZBUUAUOZAUUAUOZCEUPYLABDCEUQUUDCEYLYOUUAQZYLMUUEYKMZURUUDUUEYKYOUUACUSZ
|
|
EUSZUTZVAUUFUUDUUFUUCURZAUUAVBZUUDURYMVDZUUEQZYJMZARAEVEZUUJLZARUUFUUKUUN
|
|
UUPAUUNUUMXGMZUUMYIMZLUUPUUMXGYIVFUUQUUOUURUUJUUQUULUUAQGMUUOUULYOUUAGUUG
|
|
NYMUUAAUSZUUHVCOUURYNVDZUUMQZYHMZBRBEVEZUUBURZLZBRZUUJBUUMYHVGUVBUVEBUVBU
|
|
VAXHMZUVAYGMZLUVEUVAXHYGVFUVGUVCUVHUVDUVGUUTUUEQXGMUUTUUAQGMUVCUUTUULUUEX
|
|
GYMVHZNUUTYOUUAGUUGNYNUUABUSZUUHVCPUVHYQVDZUVAQZYFMZDRDEVEZYSYTURZLZLZDRZ
|
|
UVDDUVAYFVGUVMUVQDUVMUVLXIMZUVLYEMZLUVQUVLXIYEVFUVSUVNUVTUVPUVSUVKUUMQZXH
|
|
MUVKUUEQXGMZUVNUVKUUTUUMXHYNVHZNUVKUULUUEXGUVINUWBUVKUUAQGMUVNUVKYOUUAGUU
|
|
GNYQUUADUSZUUHVCOPUVTUVLYCMZUVLYDMZURZLUVPUVLYCYDVIUWEYSUWGUVOUWEUVLXPMZU
|
|
VLYBMZLYSUVLXPYBVFUWHYPUWIYRUWHUVAXOMZUUTUULYOQZQZXNMZYPUWHUVKUCMUWJYQVHZ
|
|
UVKUVAUCXOVJVKUUTUULYOUUAXNUUHVLUVKUWLQZXMMZDRYQYMYNQZVOZDCVEZLZDRZUWMYPU
|
|
WPUWTDUWPUWOXLMZUWOXHMZLUWTUWOXLXHVFUXBUWRUXCUWSUXBUVKUUTUULQQXKMYQYNYMQZ
|
|
QXJMZUWRUVKUUTUULYOXKUUGVLYQYNYMXJUWDUVJUUSVMUXEYQYNQZHMZYQYMQZIMZLZYQYMI
|
|
KZYQYNHKZLZUWRYQYNYMHIVNUXJUXIUXGLUXMUXGUXIVPUXKUXIUXLUXGYQYMISYQYNHSTVQY
|
|
MYNYQUUSUVJVRPPUXCUVKUWKQZXGMUVKYOQGMUWSUVKUUTUWKXGUWCNUVKUULYOGUVINYQYOU
|
|
WDUUGVCPTOVSDUWLXMVGYPUWQYOMUXAYMYNYOSDUWQYOVTOWAPFUMZVDZUVLQZYAMZFRUXOYN
|
|
YQQZVOZFCVEZLZFRZUWIYRUXRUYBFUXRUXQXLMZUXQXTMZLUYBUXQXLXTVFUYDUXTUYEUYAUY
|
|
DUXPUVKUUTQQXKMUXOUXFQXJMZUXTUXPUVKUUTUUMXKUULUUEUVIUUIUTVLUXOYQYNXJFUSZU
|
|
WDUVJVMUYFUXOYQQHMZUXOYNQIMZLZUXOYNIKZUXOYQHKZLZUXTUXOYQYNHIVNUYJUYIUYHLU
|
|
YMUYHUYIVPUYKUYIUYLUYHUXOYNISUXOYQHSTVQYNYQUXOUVJUWDVRPPUYEUXPUVAQXSMUXPU
|
|
UMQXRMZUYAUXPUVKUVAXSUWNNUXPUUTUUMXRUWCNUYNUXPUUEQXQMUXPYOQGMUYAUXPUULUUE
|
|
XQUVINUXPYOUUAGUUHWBUXOYOUYGUUGVCPPTOVSFUVLYAVGYRUXSYOMUYCYNYQYOSFUXSYOVT
|
|
OWATOUWFYTUWFUWAXOMUXNXNMZYTUVKUUTUUMXOUWCNUVKUULYOUUAXNUUHVLUUTUXNQZXMMZ
|
|
BRYNYMYQQZVOZBCVEZLZBRZUYOYTUYQVUABUYQUYPXLMZUYPXHMZLVUAUYPXLXHVFVUCUYSVU
|
|
DUYTVUCUUTUVKUULQQXKMZUYSUUTUVKUULYOXKUUGVLVUEYNUXHQXJMZYNYMIKZYNYQHKZLZU
|
|
YSYNYQYMXJUVJUWDUUSVMUXSHMZUXDIMZLVUKVUJLVUFVUIVUJVUKVPYNYQYMHIVNVUGVUKVU
|
|
HVUJYNYMISYNYQHSTWAYMYQYNUUSUWDVRPOVUDUWLXGMUUTYOQGMUYTUUTUVKUWKXGUWNNUUT
|
|
UULYOGUVINYNYOUVJUUGVCPTOVSBUXNXMVGYTUYRYOMVUBYMYQYOSBUYRYOVTOWAPWCTOTOVS
|
|
UVRUVPDUUAVBUVDUVPDUUAWEYSYTDUUAWDWFPTOVSUVFUVDBUUAVBUUJUVDBUUAWEUUBBUUAW
|
|
GWFPTOVSAUUEYJVGUUJAUUAWEWAUUCAUUAWGOWPVQWHWIYKYJJXGYIGWJWKZYHJXHYGXGVULW
|
|
KZYFJXIYEXHVUMWKYCYDXPYBUCXOWLXNXMJXLXHXKXJHIWMWNWOWQWRZVUMWSWTXAWRZXBYAJ
|
|
XLXTVUNXSXRXQGWJXCWKWKWKWSWTXAWSXOVUOWKXDWSWTXAWSWTXAWSWTXAXEXF $.
|
|
|
|
$( The class of all reflexive relationships is a set. (Contributed by SF,
|
|
11-Mar-2015.) $)
|
|
refex $p |- Ref e. _V $=
|
|
( vx vr va vp cref c1st c2nd csset ctxp c1c cv wbr cop wcel wn vex wa wex
|
|
df-br bitri cin csi cima ccompl cvv wral df-ref opex elcompl wel wrex csn
|
|
copab elima1c oteltxp snex wceq opsnelsi elin anbi12i op1st2nd opelssetsn
|
|
3bitr2i exbii df-clel bitr4i xchbinx anbi12ci df-rex rexnal con2bii 1stex
|
|
opabbi2i eqtr4i 2ndex inex siex ssetex txpex 1cex imaex complex eqeltri )
|
|
EFGUAZUBZHIZJUCZUDZHIZJUCZUDZUEEAKZWLBKZLZACKZUFZBCUMWKABCUGWPBCWKWMWOMZW
|
|
KNWQWJNZOWPWQWJWMWOBPZCPZUHUIWRWPWRACUJZWNOZQZARZXBAWOUKWPOWRWLULZWQMWINZ
|
|
ARXDAWQWIUNXFXCAXFXEWMMZWHNZXEWOMHNZQXCXEWMWOWHHUOXHXBXIXAXHXGWGNZWNXGWGX
|
|
EWMWLUPWSUHUIXJDKZWLWLMZUQZDBUJZQZDRZWNXJXKULZXGMWFNZDRXPDXGWFUNXRXODXRXQ
|
|
XEMWENZXQWMMHNZQXOXQXEWMWEHUOXSXMXTXNXSXKWLMZWDNZXMXKWLWDDPZAPZURYBYAFNZY
|
|
AGNZQXKWLFLZXKWLGLZQXMYAFGUSYGYEYHYFXKWLFSXKWLGSUTWLWLXKYDYDVAVCTXKWMYCWS
|
|
VBUTTVDTWNXLWMNXPWLWLWMSDXLWMVETVFVGWLWOYDWTVBVHTVDTXBAWOVIWNAWOVJVCVKVFV
|
|
MVNWJWIJWHHWGWFJWEHWDFGVLVOVPVQVRVSVTWAWBVRVSVTWAWBWC $.
|
|
|
|
$( The class of all antisymmetric relationships is a set. (Contributed by
|
|
SF, 11-Mar-2015.) $)
|
|
antisymex $p |- Antisym e. _V $=
|
|
( vx vy vr va vp csset c2nd c1st cin c1c cid wbr wa cop wcel bitri 3bitri
|
|
wex df-br anbi12i cantisym cins2 ctxp csi3 cins4 cima cins3 ccompl cvv cv
|
|
cdif weq wi wral copab df-antisym vex opex elcompl wrex csn elin otelins2
|
|
wn wel opelssetsn snex oqelins4 eldif wceq otsnelsi3 oteltxp ancom bitr4i
|
|
op1st2nd exbii elima1c df-clel 3bitr4i ideq 3bitr2i otelins3 eqcom notbii
|
|
sneqb df-rex rexanali rexnal eqtr4i ssetex ins2ex 2ndex 1stex txpex si3ex
|
|
con2bii opabbi2i ins4ex inex 1cex imaex idex ins3ex difex complex eqeltri
|
|
) UAFUBZXGUBZGHUCZUDZUEZXHIZJUFZKUDZUEZXHIZJUFZIZKUGZUKZUEZIZJUFZIZJUFZUH
|
|
ZUIUAAUJZBUJZCUJZLZYHYGYILZMZABULZUMBDUJZUNZAYNUNZCDUOYFABCDUPYPCDYFYIYNN
|
|
ZYFOYQYEOZVDYPYQYEYIYNCUQZDUQZURUSYRYPYRYOVDZAYNUTZYPVDYGVAZYQNZYDOZARADV
|
|
EZUUAMZARYRUUBUUEUUGAUUEUUDXGOZUUDYCOZMUUGUUDXGYCVBUUHUUFUUIUUAUUHUUCYNNF
|
|
OUUFUUCYIYNFYSVCYGYNAUQZYTVFPUUIYLYMVDZMZBYNUTZUUAYHVAZUUDNZYBOZBRBDVEZUU
|
|
LMZBRUUIUUMUUPUURBUUPUUOXHOZUUOYAOZMUURUUOXHYAVBUUSUUQUUTUULUUSUUNYQNXGOU
|
|
UNYNNFOUUQUUNUUCYQXGYGVGZVCUUNYIYNFYSVCYHYNBUQZYTVFQUUTUUNUUCYINZNZXTOUVD
|
|
XROZUVDXSOZVDZMUULUUNUUCYIYNXTYTVHUVDXRXSVIUVEYLUVGUUKUVEUVDXMOZUVDXQOZMY
|
|
LUVDXMXQVBUVHYJUVIYKEUJZVAZUVDNZXLOZERUVJYGYHNZVJZECVEZMZERZUVHYJUVMUVQEU
|
|
VMUVLXKOZUVLXHOZMUVQUVLXKXHVBUVSUVOUVTUVPUVSUVKUUNUUCNZNZXJOUVJYHYGNZNZXI
|
|
OZUVOUVKUUNUUCYIXJYSVHUVJYHYGXIEUQZUVBUUJVKUWEUVJYHNGOZUVJYGNHOZMZUVJYGHL
|
|
ZUVJYHGLZMZUVOUVJYHYGGHVLUWIUWHUWGMUWLUWGUWHVMUWJUWHUWKUWGUVJYGHSUVJYHGST
|
|
VNYGYHUVJUUJUVBVOQQUVTUVKUVCNXGOUVKYINFOUVPUVKUUNUVCXGYHVGVCUVKUUCYIFUVAV
|
|
CUVJYIUWFYSVFQZTPVPEUVDXLVQYJUVNYIOUVRYGYHYISEUVNYIVRPVSUVLXPOZERUVJUWCVJ
|
|
ZUVPMZERZUVIYKUWNUWPEUWNUVLXOOZUVTMUWPUVLXOXHVBUWRUWOUVTUVPUWRUWBXNOZUWOU
|
|
VKUUNUUCYIXNYSVHUWSUWDKOUVJUWCKLUWOUVJYHYGKUWFUVBUUJVKUVJUWCKSUVJUWCYHYGU
|
|
VBUUJURVTWAPUWMTPVPEUVDXPVQYKUWCYIOUWQYHYGYISEUWCYIVRPVSTPUVFYMUVFUWAKOUU
|
|
NUUCKLZYMUUNUUCYIKYSWBUUNUUCKSUWTUUNUUCVJUUCUUNVJYMUUNUUCUVAVTUUNUUCWCYGY
|
|
HUUJWEQWAWDTQTPVPBUUDYBVQUULBYNWFVSYLYMBYNWGPTPVPAYQYDVQUUAAYNWFVSYOAYNWH
|
|
PWPVNWQWIYEYDJXGYCFWJWKZYBJXHYAXGUXAWKZXTXRXSXMXQXLJXKXHXJXIGHWLWMWNWOWRU
|
|
XBWSWTXAXPJXOXHXNKXBWOWRUXBWSWTXAWSKXBXCXDWRWSWTXAWSWTXAXEXF $.
|
|
|
|
$( The class of all connected relationships is a set. (Contributed by SF,
|
|
11-Mar-2015.) $)
|
|
connexex $p |- Connex e. _V $=
|
|
( vx vy vr va vp csset cswap c1c cima cid cv cop wcel wn vex wex otelins2
|
|
wa bitri 3bitri cconnex cins2 csi3 cins4 cin cun cdif ccompl cvv wbr wral
|
|
wo copab df-connex opex elcompl wel wrex csn elima1c elin opelssetsn snex
|
|
eldif oqelins4 elun wceq otsnelsi3 brswap2 bitr3i anbi12i df-clel 3bitr4i
|
|
df-br orbi12i notbii df-rex rexnal 3bitr2i con2bii bitr4i opabbi2i eqtr4i
|
|
exbii ideq ssetex ins2ex swapex si3ex ins4ex inex 1cex imaex idex complex
|
|
unex difex eqeltri ) UAFUBZWSUBZGUCZUDZWTUEZHIZJUCZUDZWTUEZHIZUFZUDZUGZHI
|
|
ZUEZHIZUHZUIUAAKZBKZCKZUJZXQXPXRUJZULZBDKZUKZAYBUKZCDUMXOABCDUNYDCDXOXRYB
|
|
LZXOMYEXNMZNYDYEXNXRYBCOZDOZUOUPYFYDYFADUQZYCNZRZAPZYJAYBURYDNYFXPUSZYELZ
|
|
XMMZAPYLAYEXMUTYOYKAYOYNWSMZYNXLMZRYKYNWSXLVAYPYIYQYJYPYMYBLFMYIYMXRYBFYG
|
|
QXPYBAOZYHVBSYQBDUQZYANZRZBPZYTBYBURYJYQXQUSZYNLZXKMZBPUUBBYNXKUTUUEUUABU
|
|
UEUUDWTMZUUDXJMZNZRUUAUUDWTXJVDUUFYSUUHYTUUFUUCYELWSMUUCYBLFMYSUUCYMYEWSX
|
|
PVCZQUUCXRYBFYGQXQYBBOZYHVBTUUGYAUUGUUCYMXRLZLZXIMUULXDMZUULXHMZULYAUUCYM
|
|
XRYBXIYHVEUULXDXHVFUUMXSUUNXTEKZUSZUULLZXCMZEPUUOXPXQLZVGZECUQZRZEPZUUMXS
|
|
UURUVBEUURUUQXBMZUUQWTMZRUVBUUQXBWTVAUVDUUTUVEUVAUVDUUPUUCYMLLZXAMUUOXQXP
|
|
LZLZGMZUUTUUPUUCYMXRXAYGVEUUOXQXPGEOZUUJYRVHUVIUUOUVGGUJUUTUUOUVGGVNUUOXQ
|
|
XPUUJYRVIVJTUVEUUPUUKLWSMUUPXRLFMUVAUUPUUCUUKWSXQVCQUUPYMXRFUUIQUUOXRUVJY
|
|
GVBTZVKSWDEUULXCUTXSUUSXRMUVCXPXQXRVNEUUSXRVLSVMUUQXGMZEPUUOUVGVGZUVARZEP
|
|
ZUUNXTUVLUVNEUVLUUQXFMZUVERUVNUUQXFWTVAUVPUVMUVEUVAUVPUVFXEMUVHJMZUVMUUPU
|
|
UCYMXRXEYGVEUUOXQXPJUVJUUJYRVHUVQUUOUVGJUJUVMUUOUVGJVNUUOUVGXQXPUUJYRUOWE
|
|
VJTUVKVKSWDEUULXGUTXTUVGXRMUVOXQXPXRVNEUVGXRVLSVMVOTVPVKSWDSYTBYBVQYABYBV
|
|
RVSVKSWDSYJAYBVQYCAYBVRVSVTWAWBWCXNXMHWSXLFWFWGZXKHWTXJWSUVRWGZXIXDXHXCHX
|
|
BWTXAGWHWIWJUVSWKWLWMXGHXFWTXEJWNWIWJUVSWKWLWMWPWJWQWLWMWKWLWMWOWR $.
|
|
|
|
$( The class of all founded relationships is a set. (Contributed by SF,
|
|
19-Feb-2015.) $)
|
|
foundex $p |- Fr e. _V $=
|
|
( vx va vy vz vr vt csset cid c1c c0 cvv cv wa cop wcel wn vex wex 3bitri
|
|
bitri cfound cins3 cins2 csi3 cins4 cin cima cdif ccompl csn cxp ctxp crn
|
|
wss wne wbr weq wi wral wrex wal copab df-found elcompl elrn2 oteltxp wel
|
|
opex elin otelins3 opelssetsn snex otelins2 eldif wceq oqelins4 otsnelsi3
|
|
df-br bitr3i anbi12i exbii elima1c df-clel 3bitr4i notbii df-rex rexanali
|
|
ideq sneqb con2bii bitr4i brsset opelxp mpbiran2 elsn necon3bbii anbi12ci
|
|
exanali opabbi2i eqtr4i ssetex ins3ex ins2ex idex si3ex ins4ex inex imaex
|
|
1cex difex complex vvex xpex txpex rnex eqeltri ) UAGUBZXQUCZHUDZUEZGUCZU
|
|
CZUCZUFZIUGZHUBZUHZUFZIUGZUIZUFZIUGZUIZGJUJZUIZKUKZUFZULZUMZUIZKUAALZBLZU
|
|
NZUUAJUOZMZCLZDLZELZUPZCDUQZURCUUAUSZDUUAUTZURAVAZEBVBYTACDEBVCUUMEBYTUUH
|
|
UUBNZYTOUUNYSOZPUUMUUNYSUUHUUBEQZBQZVHVDUUOUUMUUOUUAUUNNYROZARUUEUULPZMZA
|
|
RUUMPAUUNYRVEUURUUTAUURUUAUUHNZYMOZUUAUUBNZYQOZMUUTUUAUUHUUBYMYQVFUVBUUSU
|
|
VDUUEUVBUVAYLOZPUUSUVAYLUUAUUHAQZUUPVHZVDUVEUULUUGUJZUVANZYKOZDRDAVGZUUKM
|
|
ZDRUVEUULUVJUVLDUVJUVIXQOZUVIYJOZMUVLUVIXQYJVIUVMUVKUVNUUKUVMUVHUUANGOUVK
|
|
UVHUUAUUHGUUPVJUUGUUADQZUVFVKTUVNUVIYIOZPUUKUVIYIUVHUVAUUGVLZUVGVHVDUVPUU
|
|
KUVPUUIUUJPZMZCUUAUTZUUKPUUFUJZUVINZYHOZCRCAVGZUVSMZCRUVPUVTUWCUWECUWCUWB
|
|
XROZUWBYGOZMUWEUWBXRYGVIUWFUWDUWGUVSUWFUWAUVANXQOUWAUUANGOUWDUWAUVHUVAXQU
|
|
VQVMUWAUUAUUHGUUPVJUUFUUACQZUVFVKSUWGUWBYEOZUWBYFOZPZMUVSUWBYEYFVNUWIUUIU
|
|
WKUVRFLZUJZUWBNZYDOZFRUWLUUFUUGNZVOZFEVGZMZFRZUWIUUIUWOUWSFUWOUWNXTOZUWNY
|
|
COZMUWSUWNXTYCVIUXAUWQUXBUWRUXAUWMUWAUVHNZNXSOUWLUWPNHOZUWQUWMUWAUVHUVAXS
|
|
UVGVPUWLUUFUUGHFQZUWHUVOVQUXDUWLUWPHUPUWQUWLUWPHVRUWLUWPUUFUUGUWHUVOVHWHV
|
|
SSUXBUWMUVINYBOUWMUVANYAOZUWRUWMUWAUVIYBUUFVLVMUWMUVHUVAYAUVQVMUXFUWMUUHN
|
|
GOUWRUWMUUAUUHGUVFVMUWLUUHUXEUUPVKTSVTTWAFUWBYDWBUUIUWPUUHOUWTUUFUUGUUHVR
|
|
FUWPUUHWCTWDUWJUUJUWJUXCHOZUUJUWAUVHUVAHUVGVJUXGUWAUVHHUPZUUJUWAUVHHVRUXH
|
|
UWAUVHVOUUJUWAUVHUVQWHUUFUUGUWHWITVSTWEVTTVTTWACUVIYHWBUVSCUUAWFWDUUIUUJC
|
|
UUAWGTWJWKVTTWADUVAYKWBUUKDUUAWFWDWETUVDUVCGOZUVCYPOZMUUEUVCGYPVIUXIUUCUX
|
|
JUUDUXIUUAUUBGUPUUCUUAUUBGVRUUAUUBUVFUUQWLVSUXJUUAYOOZUUAYNOZPUUDUXJUXKUU
|
|
BKOUUQUUAUUBYOKWMWNUUAYNUVFVDUXLUUAJAJWOWPSVTTWQTWAUUEUULAWRSWJWKWSWTYSYR
|
|
YMYQYLYKIXQYJGXAXBZYIYHIXRYGXQUXMXCYEYFYDIXTYCXSHXDXEXFYBYAGXAXCXCXCXGXIX
|
|
HHXDXBXJXGXIXHXKXGXIXHXKGYPXAYOKYNJVLXKXLXMXGXNXOXKXP $.
|
|
|
|
$( The class of all extensional relationships is a set. (Contributed by
|
|
SF, 19-Feb-2015.) $)
|
|
extex $p |- Ext e. _V $=
|
|
( vz vx vr vy va vp csset cins2 cid c1c cv cop wcel wn vex otelins2 bitri
|
|
wex wa ins2ex cext csi3 cins4 cin cima cins3 csymdif ccompl cdif cvv wral
|
|
wbr wb weq wi copab df-ext opex elcompl wrex csn wel elin opelssetsn snex
|
|
eldif elsymdif wceq elima1c oqelins4 otsnelsi3 df-br ideq 3bitr2i anbi12i
|
|
3bitri exbii df-clel 3bitr4i otelins3 bibi12i xchbinx df-rex dfral2 sneqb
|
|
bitr4i equcom notbii rexanali rexnal con2bii opabbi2i eqtr4i ssetex si3ex
|
|
idex ins4ex inex 1cex imaex ins3ex symdifex complex difex eqeltri ) UAGHZ
|
|
XFHZXGHZIUBZUCZXGUDZJUEZUCZHZXJGUFZHZHZHZUDZJUEZUGZUDZJUEZUHZIUFZUIZUDZJU
|
|
EZUDZJUEZUHZUJUAAKZBKZCKZULZYLDKZYNULZUMZAEKZUKZBDUNZUODYSUKZBYSUKZCEUPYK
|
|
BDACEUQUUCCEYKYNYSLZYKMUUDYJMZNUUCUUDYJYNYSCOZEOZURZUSUUEUUCUUEUUBNZBYSUT
|
|
ZUUCNYMVAZUUDLZYIMZBRBEVBZUUISZBRUUEUUJUUMUUOBUUMUULXFMZUULYHMZSUUOUULXFY
|
|
HVCUUPUUNUUQUUIUUPUUKYSLGMUUNUUKYNYSGUUFPYMYSBOZUUGVDQUUQYTUUANZSZDYSUTZU
|
|
UIYPVAZUULLZYGMZDRDEVBZUUTSZDRUUQUVAUVDUVFDUVDUVCXGMZUVCYFMZSUVFUVCXGYFVC
|
|
UVGUVEUVHUUTUVGUVBUUDLXFMUVBYSLGMUVEUVBUUKUUDXFYMVEZPUVBYNYSGUUFPYPYSDOZU
|
|
UGVDVPUVHUVCYDMZUVCYEMZNZSUUTUVCYDYEVFUVKYTUVMUUSUVKYRNZAYSUTZNYTUVKUVCYC
|
|
MZUVOUVCYCUVBUULYPVEZUUKUUDUVIUUHURZURUSYLVAZUVCLZYBMZARAEVBZUVNSZARUVPUV
|
|
OUWAUWCAUWAUVTXHMZUVTYAMZSUWCUVTXHYAVCUWDUWBUWEUVNUWDUVSUULLZXGMUVSUUDLXF
|
|
MZUWBUVSUVBUULXGUVQPUVSUUKUUDXFUVIPUWGUVSYSLGMUWBUVSYNYSGUUFPYLYSAOZUUGVD
|
|
QVPUWEUVTXNMZUVTXTMZUMYRUVTXNXTVGUWIYOUWJYQUWIUWFXMMZYOUVSUVBUULXMUVQPUVS
|
|
UUKYNLZLZXLMZFKZYLYMLZVHZFCVBZSZFRZUWKYOUWNUWOVAZUWMLZXKMZFRUWTFUWMXKVIUX
|
|
CUWSFUXCUXBXJMZUXBXGMZSUWSUXBXJXGVCUXDUWQUXEUWRUXDUXAUVSUUKLLXIMZUWQUXAUV
|
|
SUUKYNXIUUFVJUXFUWOUWPLIMUWOUWPIULUWQUWOYLYMIFOZUWHUURVKUWOUWPIVLUWOUWPYL
|
|
YMUWHUURURVMVNQUXEUXAUWLLXFMUXAYNLGMZUWRUXAUVSUWLXFYLVEZPUXAUUKYNGUVIPUWO
|
|
YNUXGUUFVDZVPVOQVQQUVSUUKYNYSXLUUGVJYOUWPYNMUWTYLYMYNVLFUWPYNVRQVSQUXAUVT
|
|
LZXSMZFRUWOYLYPLZVHZUWRSZFRZUWJYQUXLUXOFUXLUXKXJMZUXKXRMZSUXOUXKXJXRVCUXQ
|
|
UXNUXRUWRUXQUXAUVSUVBLLXIMZUXNUXAUVSUVBUULXIUVRVJUXSUWOUXMLIMUWOUXMIULUXN
|
|
UWOYLYPIUXGUWHUVJVKUWOUXMIVLUWOUXMYLYPUWHUVJURVMVNQUXRUXAUVCLXQMUXAUULLXP
|
|
MZUWRUXAUVSUVCXQUXIPUXAUVBUULXPUVQPUXTUXAUUDLXOMUXHUWRUXAUUKUUDXOUVIPUXAY
|
|
NYSGUUGVTUXJVPVPVOQVQFUVTXSVIYQUXMYNMUXPYLYPYNVLFUXMYNVRQVSWAWBVOQVQAUVCY
|
|
BVIUVNAYSWCVSWBYRAYSWDWFUVLUUAUVLUVBUUKLIMUVBUUKIULZUUAUVBUUKUUDIUUHVTUVB
|
|
UUKIVLUYAUVBUUKVHDBUNUUAUVBUUKUVIVMYPYMUVJWEDBWGVPVNWHVOQVOQVQDUULYGVIUUT
|
|
DYSWCVSYTUUADYSWIQVOQVQBUUDYIVIUUIBYSWCVSUUBBYSWJQWKWFWLWMYJYIJXFYHGWNTZY
|
|
GJXGYFXFUYBTZYDYEYCYBJXHYAXGUYCTXNXTXMXLXKJXJXGXIIWPWOWQZUYCWRWSWTWQTXSJX
|
|
JXRUYDXQXPXOGWNXATTTWRWSWTXBWRWSWTXCIWPXAXDWRWSWTWRWSWTXCXE $.
|
|
|
|
$( The class of all symmetric relationships is a set. (Contributed by SF,
|
|
20-Feb-2015.) $)
|
|
symex $p |- Sym e. _V $=
|
|
( vx vy vr va vp csset cswap cin c1c cid cv cop wn vex wex otelins2 bitri
|
|
wcel wa anbi12i csym cins2 csi3 cins4 cima cdif ccompl cvv wbr wral copab
|
|
wi df-sym opex elcompl wrex csn wel elin opelssetsn 3bitri oqelins4 eldif
|
|
snex otsnelsi3 df-br brswap2 3bitr2i exbii elima1c df-clel 3bitr4i notbii
|
|
df-rex rexanali rexnal con2bii bitr4i opabbi2i eqtr4i ssetex ins2ex si3ex
|
|
wceq ideq swapex ins4ex inex 1cex imaex idex difex complex eqeltri ) UAFU
|
|
BZWOUBZGUCZUDZWPHZIUEZJUCZUDZWPHZIUEZUFZUDZHZIUEZHZIUEZUGZUHUAAKZBKZCKZUI
|
|
ZXMXLXNUIZULBDKZUJZAXQUJZCDUKXKABCDUMXSCDXKXNXQLZXKRXTXJRZMXSXTXJXNXQCNZD
|
|
NZUNUOYAXSYAXRMZAXQUPZXSMXLUQZXTLZXIRZAOADURZYDSZAOYAYEYHYJAYHYGWORZYGXHR
|
|
ZSYJYGWOXHUSYKYIYLYDYKYFXQLFRYIYFXNXQFYBPXLXQANZYCUTQYLXOXPMZSZBXQUPZYDXM
|
|
UQZYGLZXGRZBOBDURZYOSZBOYLYPYSUUABYSYRWPRZYRXFRZSUUAYRWPXFUSUUBYTUUCYOUUB
|
|
YQXTLWORYQXQLFRYTYQYFXTWOXLVDZPYQXNXQFYBPXMXQBNZYCUTVAUUCYQYFXNLZLZXERUUG
|
|
WTRZUUGXDRZMZSYOYQYFXNXQXEYCVBUUGWTXDVCUUHXOUUJYNEKZUQZUUGLZWSRZEOUUKXLXM
|
|
LZWDZECURZSZEOZUUHXOUUNUUREUUNUUMWRRZUUMWPRZSUURUUMWRWPUSUUTUUPUVAUUQUUTU
|
|
ULYQYFLLZWQRZUUPUULYQYFXNWQYBVBUVCUUKXMXLLZLZGRUUKUVDGUIUUPUUKXMXLGENZUUE
|
|
YMVEUUKUVDGVFUUKXMXLUUEYMVGVHQUVAUULUUFLWORUULXNLFRUUQUULYQUUFWOXMVDPUULY
|
|
FXNFUUDPUUKXNUVFYBUTVAZTQVIEUUGWSVJXOUUOXNRUUSXLXMXNVFEUUOXNVKQVLUUIXPUUM
|
|
XCRZEOUUKUVDWDZUUQSZEOZUUIXPUVHUVJEUVHUUMXBRZUVASUVJUUMXBWPUSUVLUVIUVAUUQ
|
|
UVLUVBXARZUVIUULYQYFXNXAYBVBUVMUVEJRUUKUVDJUIUVIUUKXMXLJUVFUUEYMVEUUKUVDJ
|
|
VFUUKUVDXMXLUUEYMUNWEVHQUVGTQVIEUUGXCVJXPUVDXNRUVKXMXLXNVFEUVDXNVKQVLVMTV
|
|
ATQVIBYGXGVJYOBXQVNVLXOXPBXQVOQTQVIAXTXIVJYDAXQVNVLXRAXQVPQVQVRVSVTXJXIIW
|
|
OXHFWAWBZXGIWPXFWOUVNWBZXEWTXDWSIWRWPWQGWFWCWGUVOWHWIWJXCIXBWPXAJWKWCWGUV
|
|
OWHWIWJWLWGWHWIWJWHWIWJWMWN $.
|
|
$}
|
|
|
|
$( The class of all partial orderings is a set. (Contributed by SF,
|
|
11-Mar-2015.) $)
|
|
partialex $p |- Po e. _V $=
|
|
( cpartial cref ctrans cin cantisym df-partial refex transex inex antisymex
|
|
cvv eqeltri ) ABCDZEDKFMEBCGHIJIL $.
|
|
|
|
$( The class of all strict orderings is a set. (Contributed by SF,
|
|
19-Feb-2015.) $)
|
|
strictex $p |- Or e. _V $=
|
|
( cstrict cpartial cconnex cin df-strict partialex connexex inex eqeltri
|
|
cvv ) ABCDJEBCFGHI $.
|
|
|
|
$( The class of all well orderings is a set. (Contributed by SF,
|
|
19-Feb-2015.) $)
|
|
weex $p |- We e. _V $=
|
|
( cwe cstrict cfound cin cvv df-we strictex foundex inex eqeltri ) ABCDEFBC
|
|
GHIJ $.
|
|
|
|
$( The class of all equivalence relationships is a set. (Contributed by SF,
|
|
20-Feb-2015.) $)
|
|
erex $p |- Er e. _V $=
|
|
( cer csym ctrans cin cvv df-er symex transex inex eqeltri ) ABCDEFBCGHIJ
|
|
$.
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d a z $. $d A z $. $d R a $. $d R r $. $d r x $. $d R x $. $d r y $.
|
|
$d R y $. $d r z $. $d R z $. $d X x $. $d x y $. $d X y $. $d x z $.
|
|
$d X z $. $d Y y $. $d y z $. $d Y z $. $d Z z $.
|
|
trd.1 $e |- ( ph -> R Trans A ) $.
|
|
trd.2 $e |- ( ph -> X e. A ) $.
|
|
trd.3 $e |- ( ph -> Y e. A ) $.
|
|
trd.4 $e |- ( ph -> Z e. A ) $.
|
|
trd.5 $e |- ( ph -> X R Y ) $.
|
|
trd.6 $e |- ( ph -> Y R Z ) $.
|
|
$( Transitivity law in natural deduction form. (Contributed by SF,
|
|
20-Feb-2015.) $)
|
|
trd $p |- ( ph -> X R Z ) $=
|
|
( vx vy vz wbr cv wa wi wral vr va ctrans cvv wcel brex wceq breq anbi12d
|
|
wb imbi12d ralbidv 2ralbidv raleq raleqbi1dv df-trans brabg syl ibi breq1
|
|
anbi1d breq2 imbi1d anbi2d rspc3v syl3anc mpd mp2and ) ADECPZEFCPZDFCPZKL
|
|
AMQZNQZCPZVMOQZCPZRZVLVOCPZSZOBTZNBTZMBTZVIVJRZVKSZACBUCPZWBGWEWBWECUDUEB
|
|
UDUERWEWBUJCBUCUFVLVMUAQZPZVMVOWFPZRZVLVOWFPZSZOUBQZTZNWLTMWLTVSOWLTZNWLT
|
|
ZMWLTWBUAUBCBUDUDUCWFCUGZWMWNMNWLWLWPWKVSOWLWPWIVQWJVRWPWGVNWHVPVLVMWFCUH
|
|
VMVOWFCUHUIVLVOWFCUHUKULUMWOWAMWLBWNVTNWLBVSOWLBUNUOUOMNOUAUBUPUQURUSURAD
|
|
BUEEBUEFBUEWBWDSHIJVSWDDVMCPZVPRZDVOCPZSVIEVOCPZRZWSSMNODEFBBBVLDUGZVQWRV
|
|
RWSXBVNWQVPVLDVMCUTVAVLDVOCUTUKVMEUGZWRXAWSXCWQVIVPWTVMEDCVBVMEVOCUTUIVCV
|
|
OFUGZXAWCWSVKXDWTVJVIVOFECVBVDVOFDCVBUKVEVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d a z $.
|
|
$d R a $. $d R r $. $d r x $. $d R x $. $d r y $. $d R y $. $d r z $.
|
|
$d R z $. $d X x $. $d x y $. $d X y $. $d x z $. $d X z $. $d y z $.
|
|
frd.1 $e |- ( ph -> R Fr A ) $.
|
|
frd.2 $e |- ( ph -> X e. V ) $.
|
|
frd.3 $e |- ( ph -> X C_ A ) $.
|
|
frd.4 $e |- ( ph -> X =/= (/) ) $.
|
|
$( Founded relationship in natural deduction form. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
frd $p |- ( ph -> E. y e. X A. z e. X ( z R y -> z = y ) ) $=
|
|
( vx va vr wss c0 cv wi wa cvv wne wbr weq wral wrex wcel wal cfound brex
|
|
wceq breq imbi1d rexralbidv imbi2d albidv sseq2 anbi1d df-found brabg syl
|
|
wb ibi sseq1 neeq1 anbi12d raleq rexeqbi1dv imbi12d spcgv sylc mp2and ) A
|
|
GDOZGPUAZCQZBQZEUBZCBUCZRZCGUDZBGUEZJKAGFUFLQZDOZWAPUAZSZVRCWAUDZBWAUEZRZ
|
|
LUGZVLVMSZVTRZIAEDUHUBZWHHWKWHWKETUFDTUFSWKWHVAEDUHUIWAMQZOZWCSZVNVONQZUB
|
|
ZVQRZCWAUDBWAUEZRZLUGWNWFRZLUGWHNMEDTTUHWOEUJZWSWTLXAWRWFWNXAWQVRBCWAWAXA
|
|
WPVPVQVNVOWOEUKULUMUNUOWLDUJZWTWGLXBWNWDWFXBWMWBWCWLDWAUPUQULUOLCBNMURUSU
|
|
TVBUTWGWJLGFWAGUJZWDWIWFVTXCWBVLWCVMWAGDVCWAGPVDVEWEVSBWAGVRCWAGVFVGVHVIV
|
|
JVK $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d a z $. $d A z $. $d ph z $. $d R a $. $d R r $. $d r x $.
|
|
$d R x $. $d r y $. $d R y $. $d r z $. $d R z $. $d X x $. $d x y $.
|
|
$d X y $. $d x z $. $d X z $. $d Y y $. $d y z $. $d Y z $.
|
|
extd.1 $e |- ( ph -> R Ext A ) $.
|
|
extd.2 $e |- ( ph -> X e. A ) $.
|
|
extd.3 $e |- ( ph -> Y e. A ) $.
|
|
extd.4 $e |- ( ( ph /\ z e. A ) -> ( z R X <-> z R Y ) ) $.
|
|
$( Extensional relationship in natural deduction form. (Contributed by SF,
|
|
20-Feb-2015.) $)
|
|
extd $p |- ( ph -> X = Y ) $=
|
|
( vx vy wcel cv wbr wb wral wi wceq cvv vr weq jca cext brex breq bibi12d
|
|
va wa ralbidv imbi1d 2ralbidv raleq raleqbi1dv df-ext brabg syl ralrimiva
|
|
ibi breq2 bibi1d eqeq1 imbi12d bibi2d eqeq2 rspc2v syl3c ) AECMZFCMZUIBNZ
|
|
KNZDOZVJLNZDOZPZBCQZKLUBZRZLCQZKCQZVJEDOZVJFDOZPZBCQZEFSZAVHVIHIUCADCUDOZ
|
|
VTGWFVTWFDTMCTMUIWFVTPDCUDUEVJVKUANZOZVJVMWGOZPZBUHNZQZVQRZLWKQKWKQVOBWKQ
|
|
ZVQRZLWKQZKWKQVTUAUHDCTTUDWGDSZWMWOKLWKWKWQWLWNVQWQWJVOBWKWQWHVLWIVNVJVKW
|
|
GDUFVJVMWGDUFUGUJUKULWPVSKWKCWOVRLWKCWKCSWNVPVQVOBWKCUMUKUNUNKLBUAUHUOUPU
|
|
QUSUQAWCBCJURVRWDWERWAVNPZBCQZEVMSZRKLEFCCVKESZVPWSVQWTXAVOWRBCXAVLWAVNVK
|
|
EVJDUTVAUJVKEVMVBVCVMFSZWSWDWTWEXBWRWCBCXBVNWBWAVMFVJDUTVDUJVMFEVEVCVFVG
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d R a $. $d R r $. $d r x $. $d R x $. $d r y $. $d R y $. $d X x $.
|
|
$d x y $. $d X y $. $d Y y $.
|
|
symd.1 $e |- ( ph -> R Sym A ) $.
|
|
symd.2 $e |- ( ph -> X e. A ) $.
|
|
symd.3 $e |- ( ph -> Y e. A ) $.
|
|
symd.4 $e |- ( ph -> X R Y ) $.
|
|
$( Symmetric relationship in natural deduction form. (Contributed by SF,
|
|
20-Feb-2015.) $)
|
|
symd $p |- ( ph -> Y R X ) $=
|
|
( vx vy vr va wcel cv wbr wi wral csym cvv wa jca brex wceq breq 2ralbidv
|
|
wb imbi12d raleq raleqbi1dv df-sym brabg syl ibi breq1 breq2 rspc2v syl3c
|
|
) ADBNZEBNZUAJOZKOZCPZVBVACPZQZKBRZJBRZDECPZEDCPZAUSUTGHUBACBSPZVGFVJVGVJ
|
|
CTNBTNUAVJVGUGCBSUCVAVBLOZPZVBVAVKPZQZKMOZRJVORVEKVORZJVORVGLMCBTTSVKCUDZ
|
|
VNVEJKVOVOVQVLVCVMVDVAVBVKCUEVBVAVKCUEUHUFVPVFJVOBVEKVOBUIUJJKLMUKULUMUNU
|
|
MIVEVHVIQDVBCPZVBDCPZQJKDEBBVADUDVCVRVDVSVADVBCUOVADVBCUPUHVBEUDVRVHVSVIV
|
|
BEDCUPVBEDCUOUHUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d a z $. $d A z $. $d ph x $. $d ph y $. $d ph z $. $d R a $.
|
|
$d R r $. $d r x $. $d R x $. $d r y $. $d R y $. $d r z $. $d R z $.
|
|
$d x y $. $d x z $. $d y z $.
|
|
trrd.1 $e |- ( ph -> R e. V ) $.
|
|
trrd.2 $e |- ( ph -> A e. W ) $.
|
|
trrd.3 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) /\
|
|
( x R y /\ y R z ) ) -> x R z ) $.
|
|
$( Deduce transitivity from its properties. (Contributed by SF,
|
|
22-Feb-2015.) $)
|
|
trrd $p |- ( ph -> R Trans A ) $=
|
|
( vr va ctrans wbr cv wa wral wcel breq wi w3a df-3an 3exp exp3a ralrimdv
|
|
syl5bir ralrimivv wb wceq anbi12d imbi12d ralbidv 2ralbidv raleq df-trans
|
|
raleqbi1dv brabg syl2anc mpbird ) AFENOZBPZCPZFOZVCDPZFOZQZVBVEFOZUAZDERZ
|
|
CERZBERZAVJBCEEAVBESZVCESZQZVIDEAVOVEESZVIVOVPQVMVNVPUBZAVIVMVNVPUCAVQVGV
|
|
HKUDUGUEUFUHAFGSEHSVAVLUIIJVBVCLPZOZVCVEVROZQZVBVEVROZUAZDMPZRZCWDRBWDRVI
|
|
DWDRZCWDRZBWDRVLLMFEGHNVRFUJZWEWFBCWDWDWHWCVIDWDWHWAVGWBVHWHVSVDVTVFVBVCV
|
|
RFTVCVEVRFTUKVBVEVRFTULUMUNWGVKBWDEWFVJCWDEVIDWDEUOUQUQBCDLMUPURUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d ph x $.
|
|
$d R a $. $d R r $. $d r x $. $d R x $.
|
|
refrd.1 $e |- ( ph -> R e. V ) $.
|
|
refrd.2 $e |- ( ph -> A e. W ) $.
|
|
refrd.3 $e |- ( ( ph /\ x e. A ) -> x R x ) $.
|
|
$( Deduce reflexitiviy from its properties. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
refrd $p |- ( ph -> R Ref A ) $=
|
|
( vr va cref wbr cv wral ralrimiva wcel wb wceq breq ralbidv raleq df-ref
|
|
brabg syl2anc mpbird ) ADCLMZBNZUHDMZBCOZAUIBCIPADEQCFQUGUJRGHUHUHJNZMZBK
|
|
NZOUIBUMOUJJKDCEFLUKDSULUIBUMUHUHUKDTUAUIBUMCUBBJKUCUDUEUF $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d R a $. $d R r $.
|
|
$d r x $. $d R x $. $d X x $.
|
|
refd.1 $e |- ( ph -> R Ref A ) $.
|
|
refd.2 $e |- ( ph -> X e. A ) $.
|
|
$( Natural deduction form of reflexitivity. (Contributed by SF,
|
|
20-Mar-2015.) $)
|
|
refd $p |- ( ph -> X R X ) $=
|
|
( vx vr va cv wbr wral wcel cref cvv wa wb brex wceq syl breq raleq brabg
|
|
ralbidv df-ref ibi id breq12d rspccv sylc ) AGJZUKCKZGBLZDBMDDCKZACBNKZUM
|
|
EUOUMUOCOMBOMPUOUMQCBNRUKUKHJZKZGIJZLULGURLUMHICBOONUPCSUQULGURUKUKUPCUAU
|
|
DULGURBUBGHIUEUCTUFTFULUNGDBUKDSZUKDUKDCUSUGZUTUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d ph x $. $d ph y $. $d R a $. $d R r $. $d r x $. $d R x $.
|
|
$d r y $. $d R y $. $d x y $.
|
|
antird.1 $e |- ( ph -> R e. V ) $.
|
|
antird.2 $e |- ( ph -> A e. W ) $.
|
|
antird.3 $e |- ( ( ph /\ ( x e. A /\ y e. A ) /\ ( x R y /\ y R x ) ) ->
|
|
x = y ) $.
|
|
$( Deduce antisymmetry from its properties. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
antird $p |- ( ph -> R Antisym A ) $=
|
|
( vr va cantisym wbr cv wa wi wral wcel breq 3expia ralrimivva wb anbi12d
|
|
weq wceq imbi1d 2ralbidv raleq raleqbi1dv df-antisym brabg syl2anc mpbird
|
|
) AEDMNZBOZCOZENZUQUPENZPZBCUEZQZCDRZBDRZAVBBCDDAUPDSUQDSPUTVAJUAUBAEFSDG
|
|
SUOVDUCHIUPUQKOZNZUQUPVENZPZVAQZCLOZRBVJRVBCVJRZBVJRVDKLEDFGMVEEUFZVIVBBC
|
|
VJVJVLVHUTVAVLVFURVGUSUPUQVEETUQUPVEETUDUGUHVKVCBVJDVBCVJDUIUJBCKLUKULUMU
|
|
N $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d R a $. $d R r $. $d r x $. $d R x $. $d r y $. $d R y $. $d X x $.
|
|
$d x y $. $d X y $. $d Y y $.
|
|
antid.1 $e |- ( ph -> R Antisym A ) $.
|
|
antid.2 $e |- ( ph -> X e. A ) $.
|
|
antid.3 $e |- ( ph -> Y e. A ) $.
|
|
antid.4 $e |- ( ph -> X R Y ) $.
|
|
antid.5 $e |- ( ph -> Y R X ) $.
|
|
$( The antisymmetry property. (Contributed by SF, 18-Mar-2015.) $)
|
|
antid $p |- ( ph -> X = Y ) $=
|
|
( vx vy wbr wceq cv wa wi wral cvv wcel vr weq cantisym brex breq anbi12d
|
|
va wb imbi1d 2ralbidv raleq raleqbi1dv df-antisym brabg breq1 breq2 eqeq1
|
|
syl ibi imbi12d eqeq2 rspc2v syl2anc mpd mp2and ) ADECMZEDCMZDENZIJAKOZLO
|
|
ZCMZVJVICMZPZKLUBZQZLBRZKBRZVFVGPZVHQZACBUCMZVQFVTVQVTCSTBSTPVTVQUHCBUCUD
|
|
VIVJUAOZMZVJVIWAMZPZVNQZLUGOZRKWFRVOLWFRZKWFRVQUAUGCBSSUCWACNZWEVOKLWFWFW
|
|
HWDVMVNWHWBVKWCVLVIVJWACUEVJVIWACUEUFUIUJWGVPKWFBVOLWFBUKULKLUAUGUMUNURUS
|
|
URADBTEBTVQVSQGHVOVSDVJCMZVJDCMZPZDVJNZQKLDEBBVIDNZVMWKVNWLWMVKWIVLWJVIDV
|
|
JCUOVIDVJCUPUFVIDVJUQUTVJENZWKVRWLVHWNWIVFWJVGVJEDCUPVJEDCUOUFVJEDVAUTVBV
|
|
CVDVE $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d ph x $. $d ph y $. $d R a $. $d R r $. $d r x $. $d R x $.
|
|
$d r y $. $d R y $. $d x y $.
|
|
connexrd.1 $e |- ( ph -> R e. V ) $.
|
|
connexrd.2 $e |- ( ph -> A e. W ) $.
|
|
connexrd.3 $e |- ( ( ph /\ x e. A /\ y e. A ) -> ( x R y \/ y R x ) ) $.
|
|
$( Deduce connectivity from its properties. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
connexrd $p |- ( ph -> R Connex A ) $=
|
|
( vr va cconnex wbr cv wo wral wcel 3expib breq ralrimivv wb wceq orbi12d
|
|
2ralbidv raleq raleqbi1dv df-connex brabg syl2anc mpbird ) AEDMNZBOZCOZEN
|
|
ZUNUMENZPZCDQZBDQZAUQBCDDAUMDRUNDRUQJSUAAEFRDGRULUSUBHIUMUNKOZNZUNUMUTNZP
|
|
ZCLOZQBVDQUQCVDQZBVDQUSKLEDFGMUTEUCZVCUQBCVDVDVFVAUOVBUPUMUNUTETUNUMUTETU
|
|
DUEVEURBVDDUQCVDDUFUGBCKLUHUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d R a $. $d R r $. $d r x $. $d R x $. $d r y $. $d R y $. $d X x $.
|
|
$d x y $. $d X y $. $d Y y $.
|
|
connexd.1 $e |- ( ph -> R Connex A ) $.
|
|
connexd.2 $e |- ( ph -> X e. A ) $.
|
|
connexd.3 $e |- ( ph -> Y e. A ) $.
|
|
$( The connectivity property. (Contributed by SF, 18-Mar-2015.) $)
|
|
connexd $p |- ( ph -> ( X R Y \/ Y R X ) ) $=
|
|
( vx vy vr va cconnex wbr wo cv wral cvv wcel wceq wa wb orbi12d 2ralbidv
|
|
brex breq raleq raleqbi1dv df-connex brabg syl ibi wi breq1 breq2 syl2anc
|
|
rspc2v syl5 mpd ) ACBMNZDECNZEDCNZOZFUTIPZJPZCNZVEVDCNZOZJBQZIBQZAVCUTVJU
|
|
TCRSBRSUAUTVJUBCBMUEVDVEKPZNZVEVDVKNZOZJLPZQIVOQVHJVOQZIVOQVJKLCBRRMVKCTZ
|
|
VNVHIJVOVOVQVLVFVMVGVDVEVKCUFVEVDVKCUFUCUDVPVIIVOBVHJVOBUGUHIJKLUIUJUKULA
|
|
DBSEBSVJVCUMGHVHVCDVECNZVEDCNZOIJDEBBVDDTVFVRVGVSVDDVECUNVDDVECUOUCVEETVR
|
|
VAVSVBVEEDCUOVEEDCUNUCUQUPURUS $.
|
|
$}
|
|
|
|
$( Equivalence relationship as symmetric, transitive relationship.
|
|
(Contributed by SF, 22-Feb-2015.) $)
|
|
ersymtr $p |- ( R Er A <-> ( R Sym A /\ R Trans A ) ) $=
|
|
( cer wbr csym ctrans cin wa df-er breqi brin bitri ) BACDBAEFGZDBAEDBAFDHB
|
|
ACMIJBAEFKL $.
|
|
|
|
$( Partial ordering as reflexive, transitive, antisymmetric relationship.
|
|
(Contributed by SF, 12-Mar-2015.) $)
|
|
porta $p |- ( R Po A <-> ( R Ref A /\ R Trans A /\ R Antisym A ) ) $=
|
|
( cref ctrans cin cantisym wbr wa cpartial w3a brin anbi1i bitri df-partial
|
|
breqi df-3an 3bitr4i ) BACDEZFEZGZBACGZBADGZHZBAFGZHZBAIGUAUBUDJTBARGZUDHUE
|
|
BARFKUFUCUDBACDKLMBAISNOUAUBUDPQ $.
|
|
|
|
$( Linear ordering as partial, connected relationship. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
sopc $p |- ( R Or A <-> ( R Po A /\ R Connex A ) ) $=
|
|
( cstrict wbr cpartial cconnex cin wa df-strict breqi brin bitri ) BACDBAEF
|
|
GZDBAEDBAFDHBACMIJBAEFKL $.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d A z $. $d ch x $. $d ps y $. $d ps z $.
|
|
$d R y $. $d R z $. $d th x $. $d x y $. $d x z $. $d y z $.
|
|
frds.1 $e |- { x | ps } e. _V $.
|
|
frds.2 $e |- ( x = y -> ( ps <-> ch ) ) $.
|
|
frds.3 $e |- ( x = z -> ( ps <-> th ) ) $.
|
|
frds.4 $e |- ( ph -> R Fr A ) $.
|
|
frds.5 $e |- ( ph -> E. x e. A ps ) $.
|
|
$( Substitution schema verson of ~ frd . (Contributed by SF,
|
|
19-Mar-2015.) $)
|
|
frds $p |- ( ph -> E. y e. A ( ch /\ A. z e. A ( ( th /\ z R y ) -> z = y )
|
|
) ) $=
|
|
( cv wi wcel wa wrex cvv wbr weq cab wral cin dfrab2 df-rab eqtr3i cfound
|
|
crab brex syl simprd inexg sylancr syl5eqelr wss ssab2 a1i wex wne df-rex
|
|
c0 sylib abn0 sylibr frd eleq1 anbi12d rexab anass exbii bitri wal impexp
|
|
imbi2i bitr4i albii ralab df-ral 3bitr4i rexbii ) AGOZFOZIUAZGFUBZPZGEOZH
|
|
QZBRZEUCZUDZFWKSZCDWERWFPZGHUDZRZFHSZAFGHITWKMAWKBEUCZHUEZTBEHUJWSWKBEHUF
|
|
BEHUGUHAWRTQHTQZWSTQJAITQZWTAIHUIUAXAWTRMIHUIUKULUMWRHTTUNUOUPWKHUQABEHUR
|
|
USAWJEUTZWKVCVAABEHSXBNBEHVBVDWJEVEVFVGWOFWKSZWDHQZWPRZFUTZWMWQXCXDCRZWOR
|
|
ZFUTXFWJXGWOFEEFUBWIXDBCWHWDHVHKVIVJXHXEFXDCWOVKVLVMWLWOFWKWCHQZDRZWGPZGV
|
|
NXIWNPZGVNWLWOXKXLGXKXIDWGPZPXLXIDWGVOWNXMXIDWEWFVOVPVQVRWJXJWGGEEGUBWIXI
|
|
BDWHWCHVHLVIVSWNGHVTWAWBWPFHVBWAVD $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d A z $. $d ph x $. $d ph y $. $d ph z $.
|
|
$d R x $. $d R y $. $d R z $. $d x y $. $d x z $. $d y z $.
|
|
pod.1 $e |- ( ph -> R e. V ) $.
|
|
pod.2 $e |- ( ph -> A e. W ) $.
|
|
pod.3 $e |- ( ( ph /\ x e. A ) -> x R x ) $.
|
|
pod.4 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) /\
|
|
( x R y /\ y R z ) ) -> x R z ) $.
|
|
pod.5 $e |- ( ( ph /\ ( x e. A /\ y e. A ) /\ ( x R y /\ y R x ) ) ->
|
|
x = y ) $.
|
|
$( A reflexive, transitive, and anti-symmetric ordering is a partial
|
|
ordering. (Contributed by SF, 22-Feb-2015.) $)
|
|
pod $p |- ( ph -> R Po A ) $=
|
|
( cref wbr ctrans cantisym cpartial refrd trrd antird porta syl3anbrc ) A
|
|
FENOFEPOFEQOFEROABEFGHIJKSABCDEFGHIJLTABCEFGHIJMUAEFUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d ph x $. $d ph y $. $d R x $. $d R y $.
|
|
$d x y $. $d A z $. $d ph z $. $d R z $. $d x z $. $d y z $.
|
|
sod.1 $e |- ( ph -> R e. V ) $.
|
|
sod.2 $e |- ( ph -> A e. W ) $.
|
|
sod.3 $e |- ( ( ph /\ x e. A ) -> x R x ) $.
|
|
sod.4 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) /\
|
|
( x R y /\ y R z ) ) -> x R z ) $.
|
|
sod.5 $e |- ( ( ph /\ ( x e. A /\ y e. A ) /\ ( x R y /\ y R x ) ) ->
|
|
x = y ) $.
|
|
sod.6 $e |- ( ( ph /\ x e. A /\ y e. A ) ->
|
|
( x R y \/ y R x ) ) $.
|
|
$( A reflexive, transitive, antisymmetric, and connected relationship is a
|
|
strict ordering. (Contributed by SF, 12-Mar-2015.) $)
|
|
sod $p |- ( ph -> R Or A ) $=
|
|
( cpartial wbr cconnex cstrict pod connexrd sopc sylanbrc ) AFEOPFEQPFERP
|
|
ABCDEFGHIJKLMSABCEFGHIJNTEFUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d A z $. $d ch x $. $d ph y $. $d ph z $.
|
|
$d ps y $. $d ps z $. $d R y $. $d R z $. $d th x $. $d x y $.
|
|
$d x z $. $d y z $.
|
|
weds.1 $e |- { x | ps } e. _V $.
|
|
weds.2 $e |- ( x = y -> ( ps <-> ch ) ) $.
|
|
weds.3 $e |- ( x = z -> ( ps <-> th ) ) $.
|
|
weds.4 $e |- ( ph -> R We A ) $.
|
|
weds.5 $e |- ( ph -> E. x e. A ps ) $.
|
|
$( Any property that holds for some element of a well-ordered set ` A ` has
|
|
an ` R ` minimal element satisfying that property. (Contributed by SF,
|
|
20-Mar-2015.) $)
|
|
weds $p |- ( ph -> E. y e. A ( ch /\ A. z e. A ( th -> y R z ) ) ) $=
|
|
( wbr wa wi cfound cstrict syl cv weq wral wrex cwe cin df-we breqi bitri
|
|
brin simprbi frds wcel impexp cconnex simplbi cpartial sopc adantr simprl
|
|
wo simprr connexd ax1 a1i pm2.27 cref ctrans cantisym porta simp1bi sylbi
|
|
simpr refd adantrl breq1 syl5ibcom syl9r jaod mpd imim2d anassrs ralimdva
|
|
syl5bi anim2d reximdva ) ACDGUAZFUAZIOZPGFUBZQZGHUCZPZFHUDCDWHWGIOZQZGHUC
|
|
ZPZFHUDABCDEFGHIJKLAIHUEOZIHROZMWRIHSOZWSWRIHSRUFZOWTWSPIHUEXAUGUHIHSRUJU
|
|
IZUKTNULAWMWQFHAWHHUMZPZWLWPCXDWKWOGHAXCWGHUMZWKWOQWKDWIWJQZQAXCXEPZPZWOD
|
|
WIWJUNXHXFWNDXHWNWIVAXFWNQZXHHIWHWGAIHUOOZXGAWTXJAWRWTMWRWTWSXBUPTZWTIHUQ
|
|
OZXJHIURZUKTUSAXCXEUTAXCXEVBVCXHWNXIWIWNXIQXHWNXFVDVEWIXFWJXHWNWIWJVFXHWG
|
|
WGIOZWJWNAXEXNXCAXEPHIWGAIHVGOZXEAWTXOXKWTXLXJPXOXMXLXOXJXLXOIHVHOIHVIOHI
|
|
VJVKUSVLTUSAXEVMVNVOWGWHWGIVPVQVRVSVTWAWDWBWCWEWFVT $.
|
|
$}
|
|
|
|
${
|
|
$d R x y z $. $d ph x y z $.
|
|
ord0.1 $e |- ( ph -> R e. V ) $.
|
|
$( Anything partially orders the empty set. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
po0 $p |- ( ph -> R Po (/) ) $=
|
|
( vx vy vz c0 cvv wcel 0ex a1i cv wbr noel pm2.21i adantl w3a wa 3ad2ant2
|
|
3ad2ant1 weq adantr pod ) AEFGHBCIDHIJAKLEMZHJZUEUEBNZAUFUGUEOZPQUFFMZHJZ
|
|
GMZHJZRAUEUKBNZUEUIBNZUIUKBNSUFUJUMULUFUMUHPUATUFUJSAEFUBZUNUIUEBNSUFUOUJ
|
|
UFUOUHPUCTUD $.
|
|
|
|
$( Anything is connected over the empty set. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
connex0 $p |- ( ph -> R Connex (/) ) $=
|
|
( vx vy c0 cvv wcel 0ex a1i cv wbr wo noel pm2.21i 3ad2ant2 connexrd ) AE
|
|
FGBCHDGHIAJKELZGIZASFLZBMUASBMNZUAGITUBSOPQR $.
|
|
|
|
$( Anything totally orders the empty set. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
so0 $p |- ( ph -> R Or (/) ) $=
|
|
( c0 cpartial wbr cconnex cstrict po0 connex0 sopc sylanbrc ) ABEFGBEHGBE
|
|
IGABCDJABCDKEBLM $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a r $. $d A r $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d a z $. $d A z $. $d ph x $. $d ph y $. $d ph z $. $d R a $.
|
|
$d R r $. $d r x $. $d R x $. $d r y $. $d R y $. $d r z $. $d R z $.
|
|
$d x y $. $d x z $. $d y z $.
|
|
iserd.1 $e |- ( ph -> R e. V ) $.
|
|
iserd.2 $e |- ( ph -> A e. W ) $.
|
|
iserd.3 $e |- ( ( ph /\ ( x e. A /\ y e. A ) /\ x R y ) -> y R x ) $.
|
|
iserd.4 $e |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) /\
|
|
( x R y /\ y R z ) ) -> x R z ) $.
|
|
$( A symmetric, transitive relationship is an equivalence relationship.
|
|
(Contributed by SF, 22-Feb-2015.) $)
|
|
iserd $p |- ( ph -> R Er A ) $=
|
|
( vr va csym wbr cv wi wral wcel ctrans wa 3expia ralrimivva wb wceq breq
|
|
cer imbi12d 2ralbidv raleq raleqbi1dv df-sym brabg syl2anc mpbird ersymtr
|
|
trrd sylanbrc ) AFEOPZFEUAPFEUHPAUTBQZCQZFPZVBVAFPZRZCESZBESZAVEBCEEAVAET
|
|
VBETUBVCVDKUCUDAFGTEHTUTVGUEIJVAVBMQZPZVBVAVHPZRZCNQZSBVLSVECVLSZBVLSVGMN
|
|
FEGHOVHFUFZVKVEBCVLVLVNVIVCVJVDVAVBVHFUGVBVAVHFUGUIUJVMVFBVLEVECVLEUKULBC
|
|
MNUMUNUOUPABCDEFGHIJLUREFUQUS $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The identity relationship is an equivalence relationship over the
|
|
universe. (Contributed by SF, 22-Feb-2015.) $)
|
|
ider $p |- _I Er _V $=
|
|
( vx vy vz cid cvv cer wbr wtru wcel idex a1i vvex cv wa weq equcomi ideq
|
|
vex 3imtr4i 3ad2ant3 w3a eqtr anbi12i iserd trud ) DEFGHABCEDEEDEIHJKEEIH
|
|
LKAMZBMZDGZHUGUFDGZUFEIZUGEIZNABOZBAOUHUIABPUFUGBRQZUGUFARQSTUHUGCMZDGZNZ
|
|
HUFUNDGZUJUKUNEIUAULBCOZNACOUPUQUFUGUNUBUHULUOURUMUGUNCRZQUCUFUNUSQSTUDUE
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( The subset relationship partially orders the universe. (Contributed by
|
|
SF, 12-Mar-2015.) $)
|
|
ssetpov $p |- _S Po _V $=
|
|
( vx vy vz csset cvv cpartial wbr wtru wcel ssetex a1i vvex cv wa wss vex
|
|
ssid brsset anbi12i 3ad2ant3 w3a sstr 3imtr4i weq eqss bitr4i biimpri pod
|
|
mpbir trud ) DEFGHABCEDEEDEIHJKEEIHLKAMZUKDGZHUKEIZNULUKUKOUKQUKUKAPZUNRU
|
|
IKUKBMZDGZUOCMZDGZNZHUKUQDGZUMUOEIZUQEIUAUKUOOZUOUQOZNUKUQOUSUTUKUOUQUBUP
|
|
VBURVCUKUOUNBPZRZUOUQVDCPZRSUKUQUNVFRUCTUPUOUKDGZNZHABUDZUMVANVIVHVIVBUOU
|
|
KOZNVHUKUOUEUPVBVGVJVEUOUKVDUNRSUFUGTUHUJ $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Equivalence relations and classes
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c /. $. $( Long slash for quotient set $)
|
|
|
|
$( Extend the definition of a class to include equivalence class. $)
|
|
cec $a class [ A ] R $.
|
|
|
|
$( Extend the definition of a class to include quotient set. $)
|
|
cqs $a class ( A /. R ) $.
|
|
|
|
$( Define the ` R ` -coset of ` A ` . Exercise 35 of [Enderton] p. 61. This
|
|
is called the equivalence class of ` A ` modulo ` R ` when ` R ` is an
|
|
equivalence relation. In this case, ` A ` is a representative (member) of
|
|
the equivalence class ` [ A ] R ` , which contains all sets that are
|
|
equivalent to ` A ` . Definition of [Enderton] p. 57 uses the notation
|
|
` [ A ] ` (subscript) ` R ` , although we simply follow the brackets by
|
|
` R ` since we don't have subscripted expressions. For an alternate
|
|
definition, see ~ dfec2 . (Contributed by set.mm contributors,
|
|
22-Feb-2015.) $)
|
|
df-ec $a |- [ A ] R = ( R " { A } ) $.
|
|
|
|
${
|
|
$d y A $. $d y R $.
|
|
$( Alternate definition of ` R ` -coset of ` A ` . Definition 34 of
|
|
[Suppes] p. 81. (Contributed by set.mm contributors, 22-Feb-2015.) $)
|
|
dfec2 $p |- [ A ] R = { y | A R y } $=
|
|
( cec csn cima cv wbr cab df-ec imasn eqtri ) BCDCBEFBAGCHAIBCJABCKL $.
|
|
$}
|
|
|
|
$( An equivalence class modulo a set is a set. (Contributed by set.mm
|
|
contributors, 24-Jul-1995.) $)
|
|
ecexg $p |- ( R e. B -> [ A ] R e. _V ) $=
|
|
( wcel cec csn cima cvv df-ec snex imaexg mpan2 syl5eqel ) CBDZACECAFZGZHAC
|
|
INOHDPHDAJCOBHKLM $.
|
|
|
|
$( A nonempty equivalence class implies the representative is a set.
|
|
(Contributed by set.mm contributors, 9-Jul-2014.) $)
|
|
ecexr $p |- ( A e. [ B ] R -> B e. _V ) $=
|
|
( cvv wcel csn cima cec c0 wceq n0i wn snprc imaeq2 sylbi ima0 syl6eq nsyl2
|
|
df-ec eleq2s ) BDEZACBFZGZBCHAUCEUCIJUAUCAKUALZUCCIGZIUDUBIJUCUEJBMUBICNOCP
|
|
QRBCST $.
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
$( Define quotient set. ` R ` is usually an equivalence relation.
|
|
Definition of [Enderton] p. 58. (Contributed by set.mm contributors,
|
|
22-Feb-2015.) $)
|
|
df-qs $a |- ( A /. R ) = { y | E. x e. A y = [ x ] R } $.
|
|
$}
|
|
|
|
${
|
|
ersym.1 $e |- ( ph -> R Er A ) $.
|
|
ersym.2 $e |- ( ph -> X e. A ) $.
|
|
ersym.3 $e |- ( ph -> Y e. A ) $.
|
|
ersym.4 $e |- ( ph -> X R Y ) $.
|
|
$( An equivalence relation is symmetric. (Contributed by set.mm
|
|
contributors, 22-Feb-2015.) $)
|
|
ersym $p |- ( ph -> Y R X ) $=
|
|
( cer wbr csym ctrans ersymtr simplbi syl symd ) ABCDEACBJKZCBLKZFRSCBMKB
|
|
CNOPGHIQ $.
|
|
$}
|
|
|
|
${
|
|
ersymb.1 $e |- ( ph -> R Er A ) $.
|
|
ersymb.2 $e |- ( ph -> X e. A ) $.
|
|
ersymb.3 $e |- ( ph -> Y e. A ) $.
|
|
$( An equivalence relation is symmetric. (Contributed by set.mm
|
|
contributors, 30-Jul-1995.) (Revised by set.mm contributors,
|
|
9-Jul-2014.) $)
|
|
ersymb $p |- ( ph -> ( X R Y <-> Y R X ) ) $=
|
|
( wbr wa cer adantr wcel simpr ersym impbida ) ADECIZEDCIZAQJBCDEACBKIZQF
|
|
LADBMZQGLAEBMZQHLAQNOARJBCEDASRFLAUARHLATRGLARNOP $.
|
|
$}
|
|
|
|
${
|
|
ertr.1 $e |- ( ph -> R Er A ) $.
|
|
ertr.2 $e |- ( ph -> X e. A ) $.
|
|
ertr.3 $e |- ( ph -> Y e. A ) $.
|
|
ertr.4 $e |- ( ph -> Z e. A ) $.
|
|
$( An equivalence relation is transitive. (Contributed by set.mm
|
|
contributors, 4-Jun-1995.) (Revised by set.mm contributors,
|
|
9-Jul-2014.) $)
|
|
ertr $p |- ( ph -> ( ( X R Y /\ Y R Z ) -> X R Z ) ) $=
|
|
( wbr wa ctrans cer csym ersymtr simprbi syl adantr wcel simprl simprr ex
|
|
trd ) ADECKZEFCKZLZDFCKAUGLBCDEFACBMKZUGACBNKZUHGUICBOKUHBCPQRSADBTUGHSAE
|
|
BTUGISAFBTUGJSAUEUFUAAUEUFUBUDUC $.
|
|
|
|
${
|
|
ertrd.5 $e |- ( ph -> X R Y ) $.
|
|
ertrd.6 $e |- ( ph -> Y R Z ) $.
|
|
$( A transitivity relation for equivalences. (Contributed by set.mm
|
|
contributors, 9-Jul-2014.) $)
|
|
ertrd $p |- ( ph -> X R Z ) $=
|
|
( cer wbr ctrans csym ersymtr simprbi syl trd ) ABCDEFACBMNZCBONZGUACBP
|
|
NUBBCQRSHIJKLT $.
|
|
|
|
$( A transitivity relation for equivalences. (Contributed by set.mm
|
|
contributors, 9-Jul-2014.) $)
|
|
ertr2d $p |- ( ph -> Z R X ) $=
|
|
( ertrd ersym ) ABCDFGHJABCDEFGHIJKLMN $.
|
|
$}
|
|
|
|
${
|
|
ertr3d.5 $e |- ( ph -> Y R X ) $.
|
|
ertr3d.6 $e |- ( ph -> Y R Z ) $.
|
|
$( A transitivity relation for equivalences. (Contributed by set.mm
|
|
contributors, 9-Jul-2014.) $)
|
|
ertr3d $p |- ( ph -> X R Z ) $=
|
|
( ersym ertrd ) ABCDEFGHIJABCEDGIHKMLN $.
|
|
$}
|
|
|
|
${
|
|
ertr4d.5 $e |- ( ph -> X R Y ) $.
|
|
ertr4d.6 $e |- ( ph -> Z R Y ) $.
|
|
$( A transitivity relation for equivalences. (Contributed by set.mm
|
|
contributors, 9-Jul-2014.) $)
|
|
ertr4d $p |- ( ph -> X R Z ) $=
|
|
( ersym ertrd ) ABCDEFGHIJKABCFEGJILMN $.
|
|
$}
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d ph y $. $d R y $. $d X y $.
|
|
erref.1 $e |- ( ph -> R Er _V ) $.
|
|
erref.2 $e |- ( ph -> dom R = A ) $.
|
|
erref.3 $e |- ( ph -> X e. A ) $.
|
|
$( An equivalence relation is reflexive on its field. Compare Theorem 3M
|
|
of [Enderton] p. 56. (Contributed by set.mm contributors,
|
|
6-May-2013.) $)
|
|
erref $p |- ( ph -> X R X ) $=
|
|
( vy wcel wbr cdm eleq2d cv wex eldm wa cvv cer adantr elex syl vex simpr
|
|
a1i ertr4d ex exlimdv syl5bi sylbird mpd ) ADBIZDDCJZGAUKDCKZIZULAUMBDFLU
|
|
NDHMZCJZHNAULHDCOAUPULHAUPULAUPPZQCDUODACQRJUPESADQIZUPAUKURGDBTUASZUOQIU
|
|
QHUBUDUSAUPUCZUTUEUFUGUHUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d y A $. $d v x B $. $d v w z R $. $d v w x y z $.
|
|
eqer.1 $e |- ( x = y -> A = B ) $.
|
|
eqer.2 $e |- R = { <. x , y >. | A = B } $.
|
|
$( Lemma for ~ eqer . (Contributed by set.mm contributors,
|
|
17-Mar-2008.) $)
|
|
eqerlem $p |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A ) $=
|
|
( cv wceq wsbc csb bitri cvv wcel wb vex ax-mp eqeq2i brabsb sbceq1g nfcv
|
|
wbr sbccom csbief bitr4i sbcbii sbceq2g csbco 3bitri ) CJZDJZGUDZEFKZAULL
|
|
ZBUMLZAULEMZABJZEMZKZBUMLZURAUMEMZKZUNUOBUMLAULLUQUOABULUMGIUAUOABULUMUEN
|
|
UPVABUMUPURFKZVAULOPUPVEQCRAULEFOUBSUTFURAUSEFBRAFUCHUFTUGUHVBURBUMUTMZKZ
|
|
VDUMOPVBVGQDRBUMURUTOUISVFVCURABUMEUJTNUK $.
|
|
|
|
eqer.3 $e |- R e. _V $.
|
|
$( Equivalence relation involving equality of dependent classes ` A ( x ) `
|
|
and ` B ( y ) ` . (Contributed by set.mm contributors, 17-Mar-2008.) $)
|
|
eqer $p |- R Er _V $=
|
|
( vz vw vv cvv wbr wtru wcel cv wa csb wceq eqerlem cer id eqcomd 3imtr4i
|
|
a1i vvex 3ad2ant3 w3a eqtr anbi12i iserd trud ) ELUAMNIJKLELLELONHUELLONU
|
|
FUEIPZJPZEMZNUNUMEMZUMLOZUNLOZQAUMCRZAUNCRZSZUTUSSUOUPVAUSUTVAUBUCABIJCDE
|
|
FGTZABJICDEFGTUDUGUOUNKPZEMZQZNUMVCEMZUQURVCLOUHVAUTAVCCRZSZQUSVGSVEVFUSU
|
|
TVGUIUOVAVDVHVBABJKCDEFGTUJABIKCDEFGTUDUGUKUL $.
|
|
$}
|
|
|
|
$( Equality theorem for equivalence class. (Contributed by set.mm
|
|
contributors, 23-Jul-1995.) $)
|
|
eceq1 $p |- ( A = B -> [ A ] C = [ B ] C ) $=
|
|
( wceq csn cima cec sneq imaeq2d df-ec 3eqtr4g ) ABDZCAEZFCBEZFACGBCGLMNCAB
|
|
HIACJBCJK $.
|
|
|
|
$( Equality theorem for equivalence class. (Contributed by set.mm
|
|
contributors, 23-Jul-1995.) $)
|
|
eceq2 $p |- ( A = B -> [ C ] A = [ C ] B ) $=
|
|
( wceq csn cima cec imaeq1 df-ec 3eqtr4g ) ABDACEZFBKFCAGCBGABKHCAICBIJ $.
|
|
|
|
$( Membership in an equivalence class. Theorem 72 of [Suppes] p. 82.
|
|
(Contributed by set.mm contributors, 9-Jul-2014.) $)
|
|
elec $p |- ( A e. [ B ] R <-> B R A ) $=
|
|
( csn cima wcel cop cec wbr elimasn df-ec eleq2i df-br 3bitr4i ) ACBDEZFBAG
|
|
CFABCHZFBACICBAJPOABCKLBACMN $.
|
|
|
|
${
|
|
$d x y R $.
|
|
$( The range and domain of an equivalence relation are equal. (Contributed
|
|
by Rodolfo Medina, 11-Oct-2010.) $)
|
|
erdmrn $p |- ( R Er _V -> dom R = ran R ) $=
|
|
( vx vy cvv cer wbr cdm crn cv wex wcel id vex ersymb exbidv eldm 3bitr4g
|
|
a1i elrn eqrdv ) ADEFZBAGZAHZUABIZCIZAFZCJUEUDAFZCJUDUBKUDUCKUAUFUGCUADAU
|
|
DUEUALUDDKUABMRUEDKUACMRNOCUDAPCUDASQT $.
|
|
$}
|
|
|
|
${
|
|
ecss.1 $e |- ( ph -> R Er _V ) $.
|
|
ecss.2 $e |- ( ph -> dom R = X ) $.
|
|
$( An equivalence class is a subset of the domain. (Contributed by set.mm
|
|
contributors, 6-Aug-1995.) (Revised by set.mm contributors,
|
|
9-Jul-2014.) $)
|
|
ecss $p |- ( ph -> [ A ] R C_ X ) $=
|
|
( crn cec csn cima df-ec imassrn eqsstri cdm cvv cer wbr wceq erdmrn syl
|
|
eqtr3d syl5sseq ) ACGZBCHZDUDCBIZJUCBCKCUELMACNZUCDACOPQUFUCRECSTFUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x R $. $d x A $.
|
|
$( A representative of a nonempty equivalence class belongs to the domain
|
|
of the equivalence relation. (Contributed by set.mm contributors,
|
|
15-Feb-1996.) (Revised by set.mm contributors, 9-Jul-2014.) $)
|
|
ecdmn0 $p |- ( A e. dom R <-> [ A ] R =/= (/) ) $=
|
|
( vx cv cec wcel wex wbr c0 wne cdm elec exbii n0 eldm 3bitr4ri ) CDZABEZ
|
|
FZCGAQBHZCGRIJABKFSTCQABLMCRNCABOP $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $. $d x ph $.
|
|
erth.1 $e |- ( ph -> R Er _V ) $.
|
|
erth.2 $e |- ( ph -> dom R = X ) $.
|
|
erth.3 $e |- ( ph -> A e. X ) $.
|
|
erth.4 $e |- ( ph -> B e. V ) $.
|
|
$( Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82.
|
|
(Contributed by set.mm contributors, 23-Jul-1995.) (Revised by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
erth $p |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) ) $=
|
|
( vx wbr cec wa cab cvv adantr wcel elex syl cv cer vex a1i simprl simprr
|
|
wceq ertr3d expr ertr expdimp impbid abbidv dfec2 3eqtr4g simpl 3syl elec
|
|
erref sylibr eleq2 syl5ibcom imp sylib ersym impbida ) ABCDLZBDMZCDMZUGZA
|
|
VGNZBKUAZDLZKOCVLDLZKOVHVIVKVMVNKVKVMVNAVGVMVNAVGVMNZNZPDCBVLADPUBLZVOGQA
|
|
CPRZVOACERZVRJCESZTZQABPRZVOABFRZWBIBFSZTZQVLPRZVPKUCZUDAVGVMUEAVGVMUFUHU
|
|
IAVGVNVMAPDBCVLGWEWAWFAWGUDUJUKULUMKBDUNKCDUNUOAVJNZPDCBAVQVJGQWHAVSVRAVJ
|
|
UPZJVTUQWHAWCWBWIIWDUQWHBVIRZCBDLAVJWJABVHRZVJWJABBDLWKAFDBGHIUSBBDURUTVH
|
|
VIBVAVBVCBCDURVDVEVF $.
|
|
$}
|
|
|
|
${
|
|
erth2.1 $e |- ( ph -> R Er _V ) $.
|
|
erth2.2 $e |- ( ph -> dom R = X ) $.
|
|
erth2.3 $e |- ( ph -> A e. V ) $.
|
|
erth2.4 $e |- ( ph -> B e. X ) $.
|
|
$( Basic property of equivalence relations. Compare Theorem 73 of [Suppes]
|
|
p. 82. Assumes membership of the second argument in the domain.
|
|
(Contributed by set.mm contributors, 30-Jul-1995.) (Revised by set.mm
|
|
contributors, 9-Jul-2014.) $)
|
|
erth2 $p |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) ) $=
|
|
( wbr cec wceq cvv wcel elex syl ersymb erth eqcom syl6bb bitrd ) ABCDKCB
|
|
DKZBDLZCDLZMZANDBCGABEOBNOIBEPQACFOCNOJCFPQRAUCUEUDMUFACBDEFGHJISUEUDTUAU
|
|
B $.
|
|
$}
|
|
|
|
${
|
|
erthi.1 $e |- ( ph -> R Er _V ) $.
|
|
erthi.4 $e |- ( ph -> A R B ) $.
|
|
$( Basic property of equivalence relations. Part of Lemma 3N of [Enderton]
|
|
p. 57. (Contributed by set.mm contributors, 30-Jul-1995.) (Revised by
|
|
set.mm contributors, 9-Jul-2014.) $)
|
|
erthi $p |- ( ph -> [ A ] R = [ B ] R ) $=
|
|
( wbr cec wceq crn cdm eqidd wcel breldm syl brelrn erth mpbid ) ABCDGZBD
|
|
HCDHIFABCDDJZDKZEAUALASBUAMFBCDNOASCTMFBCDPOQR $.
|
|
$}
|
|
|
|
${
|
|
ereldm.1 $e |- ( ph -> R Er _V ) $.
|
|
ereldm.2 $e |- ( ph -> dom R = X ) $.
|
|
ereldm.3 $e |- ( ph -> [ A ] R = [ B ] R ) $.
|
|
ereldm.4 $e |- ( ph -> A e. V ) $.
|
|
ereldm.5 $e |- ( ph -> B e. W ) $.
|
|
$( Equality of equivalence classes implies equivalence of domain
|
|
membership. (Contributed by set.mm contributors, 28-Jan-1996.)
|
|
(Revised by set.mm contributors, 9-Jul-2014.) $)
|
|
ereldm $p |- ( ph -> ( A e. X <-> B e. X ) ) $=
|
|
( cdm wcel cec c0 wne neeq1d ecdmn0 eleq2d 3bitr4g 3bitr3d ) ABDMZNZCUCNZ
|
|
BGNCGNABDOZPQCDOZPQUDUEAUFUGPJRBDSCDSUAAUCGBITAUCGCITUB $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $.
|
|
$( Equivalence classes do not overlap. In other words, two equivalence
|
|
classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83.
|
|
(Contributed by set.mm contributors, 15-Jun-2004.) (Revised by Mario
|
|
Carneiro, 9-Jul-2014.) $)
|
|
erdisj $p |- ( R Er _V ->
|
|
( [ A ] R = [ B ] R \/ ( [ A ] R i^i [ B ] R ) = (/) ) ) $=
|
|
( vx cvv cer wbr cec cin c0 wceq wn cv wcel sseli adantl ecexr elec sylib
|
|
syl wex neq0 wa simpl inss1 vex inss2 ertr4d erthi ex exlimdv syl5bi orrd
|
|
a1i orcomd ) CEFGZACHZBCHZIZJKZUQURKZUPUTVAUTLDMZUSNZDUAUPVADUSUBUPVCVADU
|
|
PVCVAUPVCUCZABCUPVCUDZVDECAVBBVEVDVBUQNZAENVCVFUPUSUQVBUQURUEOZPVBACQTVBE
|
|
NVDDUFUNVDVBURNZBENVCVHUPUSURVBUQURUGOZPVBBCQTVCAVBCGZUPVCVFVJVGVBACRSPVC
|
|
BVBCGZUPVCVHVKVIVBBCRSPUHUIUJUKULUMUO $.
|
|
$}
|
|
|
|
$( An equivalence class modulo the identity relation is a singleton.
|
|
(Contributed by set.mm contributors, 24-Oct-2004.) $)
|
|
ecidsn $p |- [ A ] _I = { A } $=
|
|
( cid cec csn cima df-ec imai eqtri ) ABCBADZEIABFIGH $.
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y C $.
|
|
$( Equality theorem for quotient set. (Contributed by set.mm contributors,
|
|
23-Jul-1995.) $)
|
|
qseq1 $p |- ( A = B -> ( A /. C ) = ( B /. C ) ) $=
|
|
( vy vx wceq cv cec wrex cab cqs rexeq abbidv df-qs 3eqtr4g ) ABFZDGEGCHF
|
|
ZEAIZDJQEBIZDJACKBCKPRSDQEABLMEDACNEDBCNO $.
|
|
|
|
$( Equality theorem for quotient set. (Contributed by set.mm contributors,
|
|
23-Jul-1995.) $)
|
|
qseq2 $p |- ( A = B -> ( C /. A ) = ( C /. B ) ) $=
|
|
( vy vx wceq cec wrex cab cqs eceq2 eqeq2d rexbidv abbidv df-qs 3eqtr4g
|
|
cv ) ABFZDQZEQZAGZFZECHZDISTBGZFZECHZDICAJCBJRUCUFDRUBUEECRUAUDSABTKLMNED
|
|
CAOEDCBOP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $. $d x y R $.
|
|
$( Closed form of ~ elqs . (Contributed by Rodolfo Medina,
|
|
12-Oct-2010.) $)
|
|
elqsg $p |- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) ) $=
|
|
( vy cv cec wceq wrex cqs eqeq1 rexbidv df-qs elab2g ) FGZAGDHZIZABJCQIZA
|
|
BJFCBDKEPCIRSABPCQLMAFBDNO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $.
|
|
elqs.1 $e |- B e. _V $.
|
|
$( Membership in a quotient set. (Contributed by set.mm contributors,
|
|
23-Jul-1995.) (Revised by set.mm contributors, 12-Nov-2008.) $)
|
|
elqs $p |- ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) $=
|
|
( cvv wcel cqs cv cec wceq wrex wb elqsg ax-mp ) CFGCBDHGCAIDJKABLMEABCDF
|
|
NO $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $.
|
|
$( Membership in a quotient set. (Contributed by set.mm contributors,
|
|
23-Jul-1995.) $)
|
|
elqsi $p |- ( B e. ( A /. R ) -> E. x e. A B = [ x ] R ) $=
|
|
( cqs wcel cv cec wceq wrex elqsg ibi ) CBDEZFCAGDHIABJABCDMKL $.
|
|
$}
|
|
|
|
${
|
|
$d R x $. $d B x $. $d A x $.
|
|
$( Membership of an equivalence class in a quotient set. (Contributed by
|
|
Jeff Madsen, 10-Jun-2010.) $)
|
|
ecelqsg $p |- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) ) $=
|
|
( vx wcel cec cv wceq wrex cqs eqid eceq1 eqeq2d rspcev mpan2 ecexg elqsg
|
|
cvv wb syl biimpar sylan2 ) BAFZCDFZBCGZEHZCGZIZEAJZUFACKFZUDUFUFIZUJUFLU
|
|
IULEBAUGBIUHUFUFUGBCMNOPUEUKUJUEUFSFUKUJTBDCQEAUFCSRUAUBUC $.
|
|
$}
|
|
|
|
${
|
|
ecelqsi.1 $e |- R e. _V $.
|
|
$( Membership of an equivalence class in a quotient set. (Contributed by
|
|
set.mm contributors, 25-Jul-1995.) (Revised by set.mm contributors,
|
|
9-Jul-2014.) $)
|
|
ecelqsi $p |- ( B e. A -> [ B ] R e. ( A /. R ) ) $=
|
|
( cvv wcel cec cqs ecelqsg mpan ) CEFBAFBCGACHFDABCEIJ $.
|
|
$}
|
|
|
|
${
|
|
ecopqsi.1 $e |- R e. _V $.
|
|
ecopqsi.2 $e |- S = ( ( A X. A ) /. R ) $.
|
|
$( "Closure" law for equivalence class of ordered pairs. (Contributed by
|
|
set.mm contributors, 25-Mar-1996.) $)
|
|
ecopqsi $p |- ( ( B e. A /\ C e. A ) -> [ <. B , C >. ] R e. S ) $=
|
|
( wcel wa cop cxp cec opelxp cqs ecelqsi syl6eleqr sylbir ) BAHCAHIBCJZAA
|
|
KZHZRDLZEHBCAAMTUASDNESRDFOGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z R $.
|
|
$( A quotient set exists. (Contributed by FL, 19-May-2007.) $)
|
|
qsexg $p |- ( ( R e. V /\ A e. W ) -> ( A /. R ) e. _V ) $=
|
|
( vx vy vz wcel csset c1c cima cvv cv wrex csn cop wb wn vex bitri wa cqs
|
|
cins2 ccnv csi cins3 csymdif cpw1 cec wceq cab df-qs elimapw1 wel wal wex
|
|
ccompl elima1c elsymdif snex otelins2 opelssetsn otelins3 wbr df-br brcnv
|
|
bitr3i opsnelsi elec 3bitr4i bibi12i notbii exbii opex alex dfcleq bitr4i
|
|
elcompl rexbii abbi2i eqtr4i ssetex ins2ex cnvexg siexg ins3exg symdifexg
|
|
3syl sylancr 1cex imaexg sylancl complexg syl pw1exg syl2an syl5eqel ) BC
|
|
HZADHZUAABUBZIUCZBUDZUEZUFZUGZJKZUQZAUHZKZLWTEMZFMZBUIZUJZFANZEUKXIFEABUL
|
|
XNEXIXJXIHXKOZXJPZXGHZFANXNFXJXGAUMXQXMFAXQGEUNZGMZXLHZQZGUOZXMXPXFHZRYAR
|
|
ZGUPZRXQYBYCYEYCXSOZXPPZXEHZGUPYEGXPXEURYHYDGYHYGXAHZYGXDHZQZRYDYGXAXDUSY
|
|
KYAYIXRYJXTYIYFXJPIHXRYFXOXJIXKUTZVAXSXJGSZESZVBTYJYFXOPXCHZXTYFXOXJXCYNV
|
|
CXSXKPXBHZXKXSBVDZYOXTYPXSXKXBVDYQXSXKXBVEXSXKBVFVGXSXKXBYMFSVHXSXKBVIVJT
|
|
VKVLTVMTVLXPXFXOXJYLYNVNVRYAGVOVJGXJXLVPVQVSTVTWAWRXGLHZXHLHXILHWSWRXFLHZ
|
|
YRWRXELHZJLHYSWRXALHXDLHZYTIWBWCWRXBLHXCLHUUABCWDXBLWEXCLWFWHXAXDLLWGWIWJ
|
|
XEJLLWKWLXFLWMWNADWOXGXHLLWKWPWQ $.
|
|
$}
|
|
|
|
${
|
|
qsex.1 $e |- R e. _V $.
|
|
qsex.2 $e |- A e. _V $.
|
|
$( A quotient set exists. (Contributed by set.mm contributors,
|
|
14-Aug-1995.) $)
|
|
qsex $p |- ( A /. R ) e. _V $=
|
|
( cvv wcel cqs qsexg mp2an ) BEFAEFABGEFCDABEEHI $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $. $d x V $.
|
|
$( The union of a quotient set. (Contributed by set.mm contributors,
|
|
9-Dec-2008.) $)
|
|
uniqs $p |- ( R e. V -> U. ( A /. R ) = ( R " A ) ) $=
|
|
( vy vx wcel cv cec wceq wrex cab cuni ciun cqs cima wral ecexg ralrimivw
|
|
cvv dfiun2g syl eqcomd df-qs unieqi csn df-ec a1i iuneq2i imaiun 3eqtr2ri
|
|
iunid imaeq2i 3eqtr4g ) BCFZDGEGZBHZIEAJDKZLZEAUPMZABNZLBAOZUNUSURUNUPSFZ
|
|
EAPUSURIUNVBEAUOCBQREDAUPSTUAUBUTUQEDABUCUDUSEABUOUEZOZMBEAVCMZOVAEAUPVDU
|
|
PVDIUOAFUOBUFUGUHEBAVCUIVEABEAUKULUJUM $.
|
|
$}
|
|
|
|
${
|
|
qsss.1 $e |- ( ph -> R Er _V ) $.
|
|
qsss.2 $e |- ( ph -> dom R = A ) $.
|
|
qsss.3 $e |- ( ph -> R e. V ) $.
|
|
$( The union of a quotient set. (Contributed by set.mm contributors,
|
|
11-Jul-2014.) $)
|
|
uniqs2 $p |- ( ph -> U. ( A /. R ) = A ) $=
|
|
( cdm cima crn cqs cuni imadmrn wcel wceq uniqs syl imaeq2d eqtr4d cvv
|
|
cer wbr erdmrn eqtr3d 3eqtr4a ) ACCHZIZCJZBCKLZBCMAUICBIZUGACDNUIUJOGBCDP
|
|
QAUFBCFRSAUFBUHFACTUAUBUFUHOECUCQUDUE $.
|
|
|
|
$( A quotient set is a set of subsets of the base set. (Contributed by
|
|
Mario Carneiro, 9-Jul-2014.) $)
|
|
qsss $p |- ( ph -> ( A /. R ) C_ ~P A ) $=
|
|
( cqs cuni wss cpw wceq uniqs2 eqimss syl sspwuni sylibr ) ABCHZIZBJZRBKJ
|
|
ASBLTABCDEFGMSBNORBPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y R $.
|
|
snec.1 $e |- A e. _V $.
|
|
$( The singleton of an equivalence class. (Contributed by set.mm
|
|
contributors, 29-Jan-1999.) (Revised by set.mm contributors,
|
|
9-Jul-2014.) $)
|
|
snec $p |- { [ A ] R } = ( { A } /. R ) $=
|
|
( vy vx cv cec wceq csn wrex cab cqs eceq1 eqeq2d rexsn abbii df-qs df-sn
|
|
3eqtr4ri ) DFZEFZBGZHZEAIZJZDKTABGZHZDKUDBLUFIUEUGDUCUGEACUAAHUBUFTUAABMN
|
|
OPEDUDBQDUFRS $.
|
|
$}
|
|
|
|
${
|
|
ecqs.1 $e |- R e. _V $.
|
|
$( Equivalence class in terms of quotient set. (Contributed by set.mm
|
|
contributors, 29-Jan-1999.) (Revised by set.mm contributors,
|
|
15-Jan-2009.) $)
|
|
ecqs $p |- [ A ] R = U. ( { A } /. R ) $=
|
|
( cec csn cima cqs cuni df-ec cvv wcel wceq uniqs ax-mp eqtr4i ) ABDBAEZF
|
|
ZPBGHZABIBJKRQLCPBJMNO $.
|
|
$}
|
|
|
|
${
|
|
ecid.1 $e |- A e. _V $.
|
|
$( A set is equal to its converse epsilon coset. (Note: converse epsilon
|
|
is not an equivalence relation.) (Contributed by set.mm contributors,
|
|
13-Aug-1995.) (Revised by set.mm contributors, 9-Jul-2014.) $)
|
|
ecid $p |- [ A ] `' _E = A $=
|
|
( cep ccnv cec csn cima df-ec epini eqtri ) ACDZEKAFGAAKHABIJ $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $.
|
|
$( A set is equal to its quotient set mod converse epsilon. (Note:
|
|
converse epsilon is not an equivalence relation.) (Contributed by
|
|
set.mm contributors, 13-Aug-1995.) (Revised by set.mm contributors,
|
|
9-Jul-2014.) $)
|
|
qsid $p |- ( A /. `' _E ) = A $=
|
|
( vy vx cep ccnv cqs cv cec wceq wrex wcel ecid eqeq2i eqcom bitri rexbii
|
|
vex elqs risset 3bitr4i eqriv ) BADEZFZABGZCGZUBHZIZCAJUEUDIZCAJUDUCKUDAK
|
|
UGUHCAUGUDUEIUHUFUEUDUECQLMUDUENOPCAUDUBBQRCUDASTUA $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x B $. $d x R $. $d x ps $. $d x ch $.
|
|
ectocl.1 $e |- S = ( B /. R ) $.
|
|
ectocl.2 $e |- ( [ x ] R = A -> ( ph <-> ps ) ) $.
|
|
${
|
|
ectocld.3 $e |- ( ( ch /\ x e. B ) -> ph ) $.
|
|
$( Implicit substitution of class for equivalence class. (Contributed by
|
|
set.mm contributors, 9-Jul-2014.) $)
|
|
ectocld $p |- ( ( ch /\ A e. S ) -> ps ) $=
|
|
( wcel cv cec wceq wrex cqs elqsi eleq2s wa wb syl5ibcom rexlimdva syl5
|
|
eqcoms imp ) CEHLZBUGEDMZGNZOZDFPZCBUKEFGQHDFEGRISCUJBDFCUHFLTAUJBKABUA
|
|
UIEJUEUBUCUDUF $.
|
|
$}
|
|
|
|
ectocl.3 $e |- ( x e. B -> ph ) $.
|
|
$( Implicit substitution of class for equivalence class. (Contributed by
|
|
set.mm contributors, 23-Jul-1995.) (Revised by set.mm contributors,
|
|
9-Jul-2014.) $)
|
|
ectocl $p |- ( A e. S -> ps ) $=
|
|
( wtru wcel tru cv adantl ectocld mpan ) KDGLBMABKCDEFGHICNELAKJOPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x R $. $d x A $. $d x B $.
|
|
$( A quotient set doesn't contain the empty set. (Contributed by set.mm
|
|
contributors, 24-Aug-1995.) (Revised by set.mm contributors,
|
|
21-Mar-2007.) $)
|
|
elqsn0 $p |- ( ( dom R = A /\ B e. ( A /. R ) ) -> B =/= (/) ) $=
|
|
( vx cv cec c0 wne cdm wceq eqid neeq1 wcel wa eleq2 biimpar ecdmn0 sylib
|
|
cqs ectocld ) DEZCFZGHZBGHCIZAJZDBACACSZUFKUBBGLUEUAAMZNUAUDMZUCUEUHUGUDA
|
|
UAOPUACQRT $.
|
|
$}
|
|
|
|
$( Membership of an equivalence class in a quotient set. (Contributed by
|
|
set.mm contributors, 30-Jul-1995.) (Revised by set.mm contributors,
|
|
21-Mar-2007.) $)
|
|
ecelqsdm $p |- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A ) $=
|
|
( cdm wceq cec cqs wcel wa c0 wne elqsn0 ecdmn0 sylibr simpl eleqtrd ) CDZA
|
|
EZBCFZACGHZIZBQAUASJKBQHASCLBCMNRTOP $.
|
|
|
|
${
|
|
$d x y A $. $d x B $. $d x y C $. $d x y ph $. $d x y R $.
|
|
qsdisj.1 $e |- ( ph -> R Er _V ) $.
|
|
qsdisj.2 $e |- ( ph -> B e. ( A /. R ) ) $.
|
|
qsdisj.3 $e |- ( ph -> C e. ( A /. R ) ) $.
|
|
$( Members of a quotient set do not overlap. (Contributed by Rodolfo
|
|
Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) $)
|
|
qsdisj $p |- ( ph -> ( B = C \/ ( B i^i C ) = (/) ) ) $=
|
|
( vx vy wcel wceq cin c0 wo cv cec eqeq1d orbi12d wa cqs eqid eqeq1 ineq1
|
|
adantr eqeq2 ineq2 cvv cer wbr ad2antrr erdisj syl ectocld mpdan ) ACBEUA
|
|
ZKCDLZCDMZNLZOZGIPZEQZDLZVBDMZNLZOZUTAICBEUPUPUBZVBCLZVCUQVEUSVBCDUCVHVDU
|
|
RNVBCDUDRSAVABKZTZDUPKZVFAVKVIHUEVBJPZEQZLZVBVMMZNLZOZVFVJJDBEUPVGVMDLZVN
|
|
VCVPVEVMDVBUFVRVOVDNVMDVBUGRSVJVLBKZTEUHUIUJZVQAVTVIVSFUKVAVLEULUMUNUOUNU
|
|
O $.
|
|
$}
|
|
|
|
${
|
|
$d x y z A $. $d x y z B $. $d x y z C $. $d x y z R $. $d x y z ps $.
|
|
ecoptocl.1 $e |- S = ( ( B X. C ) /. R ) $.
|
|
ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
|
|
ecoptocl.3 $e |- ( ( x e. B /\ y e. C ) -> ph ) $.
|
|
$( Implicit substitution of class for equivalence class of ordered pair.
|
|
(Contributed by set.mm contributors, 23-Jul-1995.) $)
|
|
ecoptocl $p |- ( A e. S -> ps ) $=
|
|
( vz cxp cqs wcel cv cec wceq wi wrex elqsi cop eceq1 eqeq2d imbi1d wa wb
|
|
eqid eqcoms syl5ibcom optocl rexlimiv syl eleq2s ) BEFGNZHOZIEUQPEMQZHRZS
|
|
ZMUPUABMUPEHUBUTBMUPECQZDQZUCZHRZSZBTUTBTCDURFGUPUPUIVCURSZVEUTBVFVDUSEVC
|
|
URHUDUEUFVAFPVBGPUGAVEBLABUHVDEKUJUKULUMUNJUO $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w A $. $d z w B $. $d x y z w C $. $d x y z w D $. $d z w S $.
|
|
$d x y z w R $. $d x y ps $. $d z w ch $.
|
|
2ecoptocl.1 $e |- S = ( ( C X. D ) /. R ) $.
|
|
2ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
|
|
2ecoptocl.3 $e |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) ) $.
|
|
2ecoptocl.4 $e |- ( ( ( x e. C /\ y e. D ) /\
|
|
( z e. C /\ w e. D ) ) -> ph ) $.
|
|
$( Implicit substitution of classes for equivalence classes of ordered
|
|
pairs. (Contributed by set.mm contributors, 23-Jul-1995.) $)
|
|
2ecoptocl $p |- ( ( A e. S /\ B e. S ) -> ch ) $=
|
|
( wcel wi cv cop cec wceq imbi2d wa ex ecoptocl com12 impcom ) IMRHMRZCUJ
|
|
BSUJCSFGIJKLMNFTZGTZUALUBIUCBCUJPUDUJUKJRULKRUEZBUMASUMBSDEHJKLMNDTZETZUA
|
|
LUBHUCABUMOUDUNJRUOKRUEUMAQUFUGUHUGUI $.
|
|
$}
|
|
|
|
${
|
|
$d x y z w v u A $. $d z w v u B $. $d v u C $. $d x y z w v u D $.
|
|
$d z w v u S $. $d x y z w v u R $. $d x y ps $. $d z w ch $.
|
|
$d v u th $.
|
|
3ecoptocl.1 $e |- S = ( ( D X. D ) /. R ) $.
|
|
3ecoptocl.2 $e |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) ) $.
|
|
3ecoptocl.3 $e |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) ) $.
|
|
3ecoptocl.4 $e |- ( [ <. v , u >. ] R = C -> ( ch <-> th ) ) $.
|
|
3ecoptocl.5 $e |- ( ( ( x e. D /\ y e. D ) /\
|
|
( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ph ) $.
|
|
$( Implicit substitution of classes for equivalence classes of ordered
|
|
pairs. (Contributed by set.mm contributors, 9-Aug-1995.) $)
|
|
3ecoptocl $p |- ( ( A e. S /\ B e. S /\ C e. S ) -> th ) $=
|
|
( wcel wa wi cop cec wceq imbi2d 3expib ecoptocl com12 2ecoptocl 3impib
|
|
cv ) KPUBZLPUBZMPUBZDUPUQUCUODUOBUDUOCUDUODUDGHIJLMNNOPQGUNZHUNZUEOUFLUGB
|
|
CUOSUHIUNZJUNZUEOUFMUGCDUOTUHUOURNUBUSNUBUCZUTNUBVANUBUCZUCZBVDAUDVDBUDEF
|
|
KNNOPQEUNZFUNZUEOUFKUGABVDRUHVENUBVFNUBUCVBVCAUAUIUJUKULUKUM $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The mapping operation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c ^m $. $( The mapping operation. $)
|
|
$c ^pm $. $( The mapping operation. $)
|
|
|
|
$( Extend the definition of a class to include the mapping operation. (Read
|
|
for ` A ^m B ` , "the set of all functions that map from ` B ` to
|
|
` A ` .) $)
|
|
cmap $a class ^m $.
|
|
|
|
$( Extend the definition of a class to include the partial mapping
|
|
operation. (Read for ` A ^m B ` , "the set of all partial functions that
|
|
map from ` B ` to ` A ` .) $)
|
|
cpm $a class ^pm $.
|
|
|
|
${
|
|
$d x y f $.
|
|
$( Define the mapping operation or set exponentiation. The set of all
|
|
functions that map from ` B ` to ` A ` is written ` ( A ^m B ) ` (see
|
|
~ mapval ). Many authors write ` A ` followed by ` B ` as a superscript
|
|
for this operation and rely on context to avoid confusion other
|
|
exponentiation operations (e.g. Definition 10.42 of [TakeutiZaring]
|
|
p. 95). Other authors show ` B ` as a prefixed superscript, which is
|
|
read " ` A ` pre ` B ` " (e.g. definition of [Enderton] p. 52).
|
|
Definition 8.21 of [Eisenberg] p. 125 uses the notation Map( ` B ` ,
|
|
` A ` ) for our ` ( A ^m B ) ` . The up-arrow is used by Donald Knuth
|
|
for iterated exponentiation (_Science_ 194, 1235-1242, 1976). We adopt
|
|
the first case of his notation (simple exponentiation) and subscript it
|
|
with _m_ to distinguish it from other kinds of exponentiation.
|
|
(Contributed by NM, 15-Nov-2007.) $)
|
|
df-map $a |- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } ) $.
|
|
|
|
$( Define the partial mapping operation. A partial function from ` B ` to
|
|
` A ` is a function from a subset of ` B ` to ` A ` . The set of all
|
|
partial functions from ` B ` to ` A ` is written ` ( A ^pm B ) ` (see
|
|
~ pmvalg ). A notation for this operation apparently does not appear in
|
|
the literature. We use ` ^pm ` to distinguish it from the less general
|
|
set exponentiation operation ` ^m ` ( ~ df-map ) . See ~ mapsspm for
|
|
its relationship to set exponentiation. (Contributed by NM,
|
|
15-Nov-2007.) $)
|
|
df-pm $a |- ^pm = ( x e. _V , y e. _V |->
|
|
{ f e. ~P ( y X. x ) | Fun f } ) $.
|
|
$}
|
|
|
|
$( Note: an alternate way to express partial functions is as follows,
|
|
which would be added after df-fun above. The symbol would be _pfun.gif.
|
|
This definition would be needed only if we need proper class partial
|
|
functions, which seems unlikely. While nice-looking, for ordinary usage
|
|
it would inconveniently require a new set of equality theorems, etc. $)
|
|
$( Define a partial function, which is a function from a subset of the
|
|
domain ` A ` to the codomain ` B ` . The notation of a stroke through
|
|
the arrow is used by the Z language: see, for example,
|
|
~ http://staff.washington.edu/jon/z/dcs.html . $)
|
|
$(
|
|
df-fp $a |- ( F : A -|-> B <-> ( Fun F /\ F C_ ( X X. Y ) ) $.
|
|
$)
|
|
|
|
${
|
|
$d A f $. $d B f $. $d B x $. $d f x $.
|
|
mapexi.1 $e |- A e. _V $.
|
|
mapexi.2 $e |- B e. _V $.
|
|
$( The class of all functions mapping one set to another is a set. Remark
|
|
after Definition 10.24 of [Kunen] p. 31. (Contributed by set.mm
|
|
contributors, 25-Feb-2015.) $)
|
|
mapexi $p |- { f | f : A --> B } e. _V $=
|
|
( vx cfuns c1st cimage ccnv cima cin c2nd cv wcel wceq wbr bitri 3bitr4i
|
|
wa csn cpw wf cab cvv wfun cdm crn wss vex elfuns cop elimasn df-br brcnv
|
|
elin brimage dfdm4 eqeq2i eqcom 3bitr2i anbi12i dfrn5 rexbii elima risset
|
|
wrex rnex elpw wfn df-f df-fn anbi1i abbi2i funsex 1stex cnvex snex imaex
|
|
imageex inex 2ndex pwex eqeltrri ) GHIZJZAUAZKZLZMIZJZBUBZKZLZABCNZUCZCUD
|
|
UEWPCWNWOWIOZWOWMOZTWOUFZWOUGZAPZTZWOUHZBUIZTZWOWNOWPWQXBWRXDWQWOGOZWOWHO
|
|
ZTXBWOGWHUPXFWSXGXAWOCUJZUKXGAWOULWFOAWOWFQZXAWFAWOUMAWOWFUNXIWOAWEQZXAAW
|
|
OWEUOXJAHWOKZPAWTPXAWOAHXHDUQWTXKAWOURUSAWTUTVARVAVBRWRXCWLOZXDFNZWOWKQZF
|
|
WLVGXMXCPZFWLVGWRXLXNXOFWLWOXMWJQXMMWOKZPXNXOWOXMMXHFUJUQXMWOWJUOXCXPXMWO
|
|
VCUSSVDFWOWKWLVEFXCWLVFSXCBWOXHVHVIRVBWOWIWMUPWPWOAVJZXDTXEABWOVKXQXBXDWO
|
|
AVLVMRSVNWIWMGWHVOWFWGWEHVPVTVQAVRVSWAWKWLWJMWBVTVQBEWCVSWAWD $.
|
|
$}
|
|
|
|
${
|
|
$d f A $. $d f B $.
|
|
$( When ` A ` is a proper class, the class of all functions mapping ` A `
|
|
to ` B ` is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed
|
|
by set.mm contributors, 8-Dec-2003.) $)
|
|
mapprc $p |- ( -. A e. _V -> { f | f : A --> B } = (/) ) $=
|
|
( cvv wcel cv wf cab c0 wne wex abn0 cdm fdm dmex syl6eqelr exlimiv sylbi
|
|
vex necon1bi ) ADEZABCFZGZCHZIUDIJUCCKUAUCCLUCUACUCAUBMDABUBNUBCSOPQRT $.
|
|
|
|
$( The class of all partial functions from one set to another is a set.
|
|
(Contributed by set.mm contributors, 15-Nov-2007.) $)
|
|
pmex $p |- ( ( A e. C /\ B e. D ) ->
|
|
{ f | ( Fun f /\ f C_ ( A X. B ) ) } e. _V ) $=
|
|
( wcel wa wfun cxp wss cab cfuns cpw cin cvv df-funs df-pw ineq12i inab
|
|
cv eqtr2i xpexg pwexg funsex inexg mpan 3syl syl5eqel ) ACFBDFGZETZHZUJAB
|
|
IZJZGEKZLULMZNZOUPUKEKZUMEKZNUNLUQUOUREPEULQRUKUMESUAUIULOFUOOFZUPOFZABCD
|
|
UBULOUCLOFUSUTUDLUOOOUEUFUGUH $.
|
|
|
|
$d A a $. $d a b $. $d A b $. $d a f $. $d B b $. $d b f $.
|
|
$( The class of all functions mapping one set to another is a set. Remark
|
|
after Definition 10.24 of [Kunen] p. 31. (Contributed by set.mm
|
|
contributors, 25-Feb-2015.) $)
|
|
mapex $p |- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V ) $=
|
|
( va vb cv cab cvv wcel wceq feq2 abbidv eleq1d feq3 vex mapexi vtocl2g
|
|
wf ) FHZGHZEHZTZEIZJKAUBUCTZEIZJKABUCTZEIZJKFGABCDUAALZUEUGJUJUDUFEUAAUBU
|
|
CMNOUBBLZUGUIJUKUFUHEUBBAUCPNOUAUBEFQGQRS $.
|
|
$}
|
|
|
|
${
|
|
$d f x y $.
|
|
$( Set exponentiation has a universal domain. (Contributed by set.mm
|
|
contributors, 8-Dec-2003.) (Revised by set.mm contributors,
|
|
8-Sep-2013.) (Revised by Scott Fenton, 19-Apr-2019.) $)
|
|
fnmap $p |- ^m Fn _V $=
|
|
( vx vy vf cmap cvv cxp wfn cv wf cab df-map vex mapexi fnmpt2i xpvv mpbi
|
|
fneq2i ) DEEFZGDEGABEEBHZAHZCHICJDABCKSTCBLALMNREDOQP $.
|
|
|
|
$( Partial function exponentiation has a universal domain. (Contributed by
|
|
set.mm contributors, 14-Nov-2013.) (Revised by Scott Fenton,
|
|
19-Apr-2019.) $)
|
|
fnpm $p |- ^pm Fn _V $=
|
|
( vx vy vf cpm cvv cxp wfn cv wfun cpw crab df-pm cfuns cin wcel cab elin
|
|
wa vex mpbi abbi2i df-rab wral wceq elfuns rgenw 3eqtr2i xpex pwex funsex
|
|
wb rabbi inex eqeltrri fnmpt2i xpvv fneq2i ) DEEFZGDEGABEECHZIZCBHZAHZFZJ
|
|
ZKZDABCLVDMNZVEEVFUSVDOUSMOZRZCPVGCVDKZVEVHCVFUSVDMQUAVGCVDUBVGUTUKZCVDUC
|
|
VIVEUDVJCVDUSCSUEUFVGUTCVDULTUGVDMVCVAVBBSASUHUIUJUMUNUOUREDUPUQT $.
|
|
$}
|
|
|
|
${
|
|
$d x y f A $. $d x y f B $.
|
|
$( The value of set exponentiation. ` ( A ^m B ) ` is the set of all
|
|
functions that map from ` B ` to ` A ` . Definition 10.24 of [Kunen]
|
|
p. 24. (Contributed by set.mm contributors, 8-Dec-2003.) (Revised by
|
|
set.mm contributors, 8-Sep-2013.) $)
|
|
mapvalg $p |- ( ( A e. C /\ B e. D ) ->
|
|
( A ^m B ) = { f | f : B --> A } ) $=
|
|
( vx vy wcel wa cv wf cab cvv cmap co wceq mapex ancoms elex abbidv feq3
|
|
wi feq2 df-map ovmpt2g 3expia syl2an mpd ) ACHZBDHZIBAEJZKZELZMHZABNOUMPZ
|
|
UJUIUNBADCEQRUIAMHZBMHZUNUOUBUJACSBDSUPUQUNUOFGABMMGJZFJZUKKZELUMNURAUKKZ
|
|
ELMUSAPUTVAEUSAURUKUATURBPVAULEURBAUKUCTFGEUDUEUFUGUH $.
|
|
|
|
$( The value of the partial mapping operation. ` ( A ^pm B ) ` is the set
|
|
of all partial functions that map from ` B ` to ` A ` . (Contributed by
|
|
set.mm contributors, 15-Nov-2007.) (Revised by set.mm contributors,
|
|
8-Sep-2013.) $)
|
|
pmvalg $p |- ( ( A e. C /\ B e. D ) ->
|
|
( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } ) $=
|
|
( vx vy wcel cvv cpm cv cxp cpw crab wceq elex wa cab pweqd biidd co wfun
|
|
wss df-rab ancom df-pw abeq2i anbi2i bitri abbii eqtri syl5eqel rabeqbidv
|
|
pmex ancoms xpeq2 xpeq1 df-pm ovmpt2g mpd3an3 syl2an ) ACHAIHZBIHZABJUAEK
|
|
ZUBZEBALZMZNZOZBDHACPBDPVBVCVHIHVIVBVCQVHVEVDVFUCZQZERZIVHVDVGHZVEQZERVLV
|
|
EEVGUDVNVKEVNVEVMQVKVMVEUEVMVJVEVJEVGEVFUFUGUHUIUJUKVCVBVLIHBAIIEUNUOULFG
|
|
ABIIVEEGKZFKZLZMZNVHJVEEVOALZMZNIVPAOZVEVEEVRVTWAVQVSVPAVOUPSWAVETUMVOBOZ
|
|
VEVEEVTVGWBVSVFVOBAUQSWBVETUMFGEURUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d f A $. $d f B $.
|
|
mapval.1 $e |- A e. _V $.
|
|
mapval.2 $e |- B e. _V $.
|
|
$( The value of set exponentiation (inference version). ` ( A ^m B ) ` is
|
|
the set of all functions that map from ` B ` to ` A ` . Definition
|
|
10.24 of [Kunen] p. 24. (Contributed by set.mm contributors,
|
|
8-Dec-2003.) $)
|
|
mapval $p |- ( A ^m B ) = { f | f : B --> A } $=
|
|
( cvv wcel cmap co cv wf cab wceq mapvalg mp2an ) AFGBFGABHIBACJKCLMDEABF
|
|
FCNO $.
|
|
$}
|
|
|
|
${
|
|
$d g A $. $d g B $. $d g C $.
|
|
$( Membership relation for set exponentiation. (Contributed by set.mm
|
|
contributors, 17-Oct-2006.) $)
|
|
elmapg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( C e. ( A ^m B ) <-> C : B --> A ) ) $=
|
|
( vg wcel w3a cmap co cv wf cab wb wa mapvalg eleq2d 3adant3 feq1 elabg
|
|
3ad2ant3 bitrd ) ADHZBEHZCFHZICABJKZHZCBAGLZMZGNZHZBACMZUDUEUHULOUFUDUEPU
|
|
GUKCABDEGQRSUFUDULUMOUEUJUMGCFBAUICTUAUBUC $.
|
|
|
|
$( The predicate "is a partial function." (Contributed by set.mm
|
|
contributors, 14-Nov-2013.) $)
|
|
elpmg $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
|
|
( C e. ( A ^pm B ) <-> ( Fun C /\ C C_ ( B X. A ) ) ) ) $=
|
|
( vg wcel w3a cpm co cxp cpw wfun wa wss wb cv crab syl6bb pmvalg 3adant3
|
|
eleq2d funeq elrab elpwg anbi1d ancom 3ad2ant3 bitrd ) ADHZBEHZCFHZICABJK
|
|
ZHZCBALZMZHZCNZOZUSCUPPZOZUKULUOUTQUMUKULOZUOCGRZNZGUQSZHUTVCUNVFCABDEGUA
|
|
UCVEUSGCUQVDCUDUETUBUMUKUTVBQULUMUTVAUSOVBUMURVAUSCUPFUFUGVAUSUHTUIUJ $.
|
|
|
|
$( The predicate "is a partial function." (Contributed by set.mm
|
|
contributors, 31-Dec-2013.) $)
|
|
elpm2g $p |- ( ( A e. V /\ B e. W /\ F e. X ) ->
|
|
( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) ) $=
|
|
( wcel w3a cpm co wfun cxp wss wa cdm wf elpmg funssxp syl6bb ) ADGBEGCFG
|
|
HCABIJGCKCBALMNCOZACPTBMNABCDEFQBACRS $.
|
|
|
|
$( A partial function is a function. (Contributed by Mario Carneiro,
|
|
30-Jan-2014.) $)
|
|
pmfun $p |- ( F e. ( A ^pm B ) -> Fun F ) $=
|
|
( cpm co wcel wfun cxp wss wa cvv wb elovex1 elovex2 id elpmg syl3anc ibi
|
|
simpld ) CABDEZFZCGZCBAHIZUAUBUCJZUAAKFBKFUAUAUDLCABDMCABDNUAOABCKKTPQRS
|
|
$.
|
|
$}
|
|
|
|
$( A mapping is a function, forward direction only with antecedents removed.
|
|
(Contributed by set.mm contributors, 25-Feb-2015.) $)
|
|
elmapi $p |- ( A e. ( B ^m C ) -> A : C --> B ) $=
|
|
( cmap co wcel wf cvv wb elovex1 elovex2 id elmapg syl3anc ibi ) ABCDEZFZCB
|
|
AGZQBHFCHFQQRIABCDJABCDKQLBCAHHPMNO $.
|
|
|
|
${
|
|
$d g A $. $d f g B $. $d g F $.
|
|
elmap.1 $e |- A e. _V $.
|
|
elmap.2 $e |- B e. _V $.
|
|
elmap.3 $e |- F e. _V $.
|
|
$( Membership relation for set exponentiation. (Contributed by set.mm
|
|
contributors, 8-Dec-2003.) $)
|
|
elmap $p |- ( F e. ( A ^m B ) <-> F : B --> A ) $=
|
|
( cvv wcel cmap co wf wb elmapg mp3an ) AGHBGHCGHCABIJHBACKLDEFABCGGGMN
|
|
$.
|
|
|
|
$( Alternate expression for the value of set exponentiation. (Contributed
|
|
by set.mm contributors, 3-Nov-2007.) $)
|
|
mapval2 $p |- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } ) $=
|
|
( vg cmap co cxp cpw cv wfn cab cin wf wa wcel bitri wss dff2 ancom elmap
|
|
vex elin elpw fneq1 elab anbi12i 3bitr4i eqriv ) HABIJZBAKZLZCMZBNZCOZPZB
|
|
AHMZQZUTUNUAZUTBNZRZUTUMSUTUSSZVAVCVBRVDBAUTUBVCVBUCTABUTEFHUEZUDVEUTUOSZ
|
|
UTURSZRVDUTUOURUFVGVBVHVCUTUNVFUGUQVCCUTVFBUPUTUHUIUJTUKUL $.
|
|
|
|
$( The predicate "is a partial function." (Contributed by set.mm
|
|
contributors, 15-Nov-2007.) (Revised by set.mm contributors,
|
|
14-Nov-2013.) $)
|
|
elpm $p |- ( F e. ( A ^pm B ) <-> ( Fun F /\ F C_ ( B X. A ) ) ) $=
|
|
( cvv wcel cpm co wfun cxp wss wa wb elpmg mp3an ) AGHBGHCGHCABIJHCKCBALM
|
|
NODEFABCGGGPQ $.
|
|
|
|
$( The predicate "is a partial function." (Contributed by set.mm
|
|
contributors, 15-Nov-2007.) (Revised by set.mm contributors,
|
|
31-Dec-2013.) $)
|
|
elpm2 $p |- ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) $=
|
|
( cvv wcel cpm co cdm wf wss wa wb elpm2g mp3an ) AGHBGHCGHCABIJHCKZACLRB
|
|
MNODEFABCGGGPQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x f $. $d B x f $.
|
|
$( Set exponentiation is a subset of partial maps. (Contributed by set.mm
|
|
contributors, 15-Nov-2007.) $)
|
|
mapsspm $p |- ( A ^m B ) C_ ( A ^pm B ) $=
|
|
( vx vf cmap co cpm wss c0 wceq 0ss sseq1 mpbiri wne cvv wcel wa wex cab
|
|
cv n0 elovex12 exlimiv sylbi wfun cxp cpw crab fssxp vex elpw sylibr ffun
|
|
wf jca ss2abi df-rab sseqtr4i a1i mapvalg pmvalg 3sstr4d syl pm2.61ine )
|
|
ABEFZABGFZHZVEIVEIJVGIVFHVFKVEIVFLMVEINZAOPBOPQZVGVHCTZVEPZCRVICVEUAVKVIC
|
|
VJABEUBUCUDVIBADTZUNZDSZVLUEZDBAUFZUGZUHZVEVFVNVRHVIVNVLVQPZVOQZDSVRVMVTD
|
|
VMVSVOVMVLVPHVSBAVLUIVLVPDUJUKULBAVLUMUOUPVODVQUQURUSABOODUTABOODVAVBVCVD
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d x f A $. $d x f B $.
|
|
$( Set exponentiation is a subset of the power set of the cross product of
|
|
its arguments. (Contributed by set.mm contributors, 8-Dec-2006.) $)
|
|
mapsspw $p |- ( A ^m B ) C_ ~P ( B X. A ) $=
|
|
( vx vf cmap co cxp cpw wss c0 wceq 0ss sseq1 mpbiri wne cvv wcel wa wex
|
|
cv n0 elovex12 exlimiv sylbi wf cab fssxp vex sylibr abssi mapvalg sseq1d
|
|
elpw syl pm2.61ine ) ABEFZBAGZHZIZUPJUPJKUSJURIURLUPJURMNUPJOZAPQBPQRZUSU
|
|
TCTZUPQZCSVACUPUAVCVACVBABEUBUCUDVAUSBADTZUEZDUFZURIVEDURVEVDUQIVDURQBAVD
|
|
UGVDUQDUHUMUIUJVAUPVFURABPPDUKULNUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d f A $.
|
|
map0e.1 $e |- A e. _V $.
|
|
$( Set exponentiation with an empty exponent is the unit class of the empty
|
|
set. (Contributed by set.mm contributors, 10-Dec-2003.) $)
|
|
map0e $p |- ( A ^m (/) ) = { (/) } $=
|
|
( vf c0 cv wf cab wceq cmap co csn wfn crn wss fn0 anbi1i df-f 0ss rneq
|
|
wa rn0 syl6eq sseq1d mpbiri pm4.71i 3bitr4i abbii mapval df-sn 3eqtr4i
|
|
0ex ) DACEZFZCGULDHZCGADIJDKUMUNCULDLZULMZANZTUNUQTUMUNUOUNUQULOPDAULQUNU
|
|
QUNUQDANARUNUPDAUNUPDMDULDSUAUBUCUDUEUFUGADCBUKUHCDUIUJ $.
|
|
|
|
$( Set exponentiation with an empty base is the empty set, provided the
|
|
exponent is non-empty. Theorem 96 of [Suppes] p. 89. (Contributed by
|
|
set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors,
|
|
19-Mar-2007.) $)
|
|
map0b $p |- ( A =/= (/) -> ( (/) ^m A ) = (/) ) $=
|
|
( vf c0 wne cmap co cv cab 0ex mapval wex wceq abn0 cdm fdm crn wss frn
|
|
wf ss0 syl dm0rn0 sylibr eqtr3d exlimiv sylbi necon1i syl5eq ) ADEDAFGADC
|
|
HZTZCIZDDACJBKULDADULDEUKCLADMZUKCNUKUMCUKUJOZADADUJPUKUJQZDMZUNDMUKUODRU
|
|
PADUJSUOUAUBUJUCUDUEUFUGUHUI $.
|
|
$}
|
|
|
|
${
|
|
$d f x y A $. $d f x y B $.
|
|
map0.1 $e |- A e. _V $.
|
|
map0.2 $e |- B e. _V $.
|
|
$( Set exponentiation is empty iff the base is empty and the exponent is
|
|
not empty. Theorem 97 of [Suppes] p. 89. (Contributed by set.mm
|
|
contributors, 10-Dec-2003.) (Revised by set.mm contributors,
|
|
17-May-2007.) $)
|
|
map0 $p |- ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) $=
|
|
( vf vx cmap co c0 wceq wne wa cv wf cab mapval eqeq1i wcel wex csn snssi
|
|
wss cxp vex fconst mpan snex xpex feq1 spcev 3syl exlimiv n0 abn0 3imtr4i
|
|
fss necon4i sylbi map0e 0ex snid ne0i ax-mp eqnetri neeq1d mpbiri necon2i
|
|
oveq2 jca oveq1 map0b sylan9eq impbii ) ABGHZIJZAIJZBIKZLVOVPVQVOBAEMZNZE
|
|
OZIJVPVNVTIABECDPQAIVTIFMZARZFSVSESZAIKVTIKWBWCFWBWATZAUBZBABWDUCZNZWCWAA
|
|
UABWDWFNWEWGBWAFUDUEBWDAWFUPUFVSWGEWFBWDDWAUGUHBAVRWFUIUJUKULFAUMVSEUNUOU
|
|
QURBIVNIBIJZVNIKAIGHZIKWIITZIACUSIWJRWJIKIUTVAWJIVBVCVDWHVNWIIBIAGVHVEVFV
|
|
GVIVPVQVNIBGHIAIBGVJBDVKVLVM $.
|
|
|
|
$( The value of set exponentiation with a singleton exponent. Theorem 98
|
|
of [Suppes] p. 89. (Contributed by set.mm contributors,
|
|
10-Dec-2003.) $)
|
|
mapsn $p |- ( A ^m { B } ) = { f | E. y e. A f = { <. B , y >. } } $=
|
|
( csn cmap co cv wf cab cop wceq wrex wcel wex cima wss syl5ibcom snex wa
|
|
mapval crn wbr weu wfn ffn sylancl euabsn cdm imadmrn fdm imaeq2d syl5eqr
|
|
snid fneu imasn syl6req eqeq1d exbidv syl5bb mpbid frn sseq1 snss syl6bbr
|
|
vex wfo dffn4 sylib fof syl feq3 fsn syl6ib jcad eximdv mpd df-rex sylibr
|
|
wf1o f1osn f1of ax-mp feq1 mpbiri snssi fss syl2an expcom rexlimiv impbii
|
|
abbii eqtri ) BCGZHIWPBDJZKZDLWQCAJZMGZNZABOZDLBWPDECUAUCWRXBDWRXBWRWSBPZ
|
|
XAUBZAQZXBWRWQUDZWSGZNZAQZXEWRCWSWQUEZAUFZXIWRWQWPUGZCWPPXKWPBWQUHZCFUPAW
|
|
PCWQUQUIXKXJALZXGNZAQWRXIXJAUJWRXOXHAWRXNXFXGWRXFWQWPRZXNWRXFWQWQUKZRXPWQ
|
|
ULWRXQWPWQWPBWQUMUNUOACWQURUSUTVAVBVCWRXHXDAWRXHXCXAWRXFBSZXHXCWPBWQVDXHX
|
|
RXGBSZXCXFXGBVEWSBAVHZVFVGTWRXHWPXGWQKZXAWRWPXFWQKZXHYAWRWPXFWQVIZYBWRXLY
|
|
CXMWPWQVJVKWPXFWQVLVMXFXGWPWQVNTCWSWQFXTVOVPVQVRVSXAABVTWAXAWRABXAXCWRXAY
|
|
AXSWRXCXAYAWPXGWTKZWPXGWTWBYDCWSFXTWCWPXGWTWDWEWPXGWQWTWFWGWSBWHWPXGBWQWI
|
|
WJWKWLWMWNWO $.
|
|
$}
|
|
|
|
${
|
|
$d f A $. $d f B $. $d f C $.
|
|
mapss.1 $e |- A e. _V $.
|
|
mapss.2 $e |- B e. _V $.
|
|
mapss.3 $e |- C e. _V $.
|
|
$( Subset inheritance for set exponentiation. Theorem 99 of [Suppes]
|
|
p. 89. (Contributed by set.mm contributors, 10-Dec-2003.) $)
|
|
mapss $p |- ( A C_ B -> ( A ^m C ) C_ ( B ^m C ) ) $=
|
|
( vf wss cv wf cab cmap co fss expcom ss2abdv mapval 3sstr4g ) ABHZCAGIZJ
|
|
ZGKCBTJZGKACLMBCLMSUAUBGUASUBCABTNOPACGDFQBCGEFQR $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Equinumerosity
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Introduce new constant symbols. $)
|
|
$c ~~ $. $( Equinumerosity relation ("approximately equals" symbol) $)
|
|
|
|
$( Extend class definition to include the equinumerosity relation
|
|
("approximately equals" symbol) $)
|
|
cen $a class ~~ $.
|
|
|
|
${
|
|
$d x y f $.
|
|
$( Define the equinumerosity relation. Definition of [Enderton] p. 129.
|
|
We define ` ~~ ` to be a binary relation rather than a connective, so
|
|
its arguments must be sets to be meaningful. This is acceptable because
|
|
we do not consider equinumerosity for proper classes. We derive the
|
|
usual definition as ~ bren . (Contributed by NM, 28-Mar-1998.) $)
|
|
df-en $a |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y } $.
|
|
$}
|
|
|
|
${
|
|
$d A f x y $. $d B f x y $.
|
|
$( Equinumerosity relationship. (Contributed by SF, 23-Feb-2015.) $)
|
|
bren $p |- ( A ~~ B <-> E. f f : A -1-1-onto-> B ) $=
|
|
( vx vy cen wbr cvv wcel wa wf1o wex brex cdm crn vex eleq1d wceq exbidv
|
|
cv dmex rnex pm3.2i f1odm wfo f1ofo forn syl anbi12d mpbii exlimiv f1oeq2
|
|
f1oeq3 df-en brabg pm5.21nii ) ABFGAHIZBHIZJZABCTZKZCLZABFMVAUSCVAUTNZHIZ
|
|
UTOZHIZJUSVDVFUTCPZUAUTVGUBUCVAVDUQVFURVAVCAHABUTUDQVAVEBHVAABUTUEVEBRABU
|
|
TUFABUTUGUHQUIUJUKDTZETZUTKZCLAVIUTKZCLVBDEABHHFVHARVJVKCVHAVIUTULSVIBRVK
|
|
VACVIBAUTUMSDECUNUOUP $.
|
|
$}
|
|
|
|
${
|
|
$d f g x y $.
|
|
$( The equinumerosity relationship is a set. (Contributed by SF,
|
|
23-Feb-2015.) $)
|
|
enex $p |- ~~ e. _V $=
|
|
( vx vy vf vg cen cfns cswap ccnv ctxp crn cv wex cop wcel elrn2 wa df-br
|
|
wfn wbr bitri cimage cvv wf1o copab df-en vex bitr3i wceq oteltxp opelcnv
|
|
brfns cima dfcnv2 eqeq2i brimage anbi12i exbii cnvex fneq1 ceqsexv dff1o4
|
|
3bitr2ri 3bitr4i opabbi2i eqtr4i fnsex swapex imageex txpex rnex eqeltri
|
|
) EFGUAZHZFIZJZIZJZUBEAKZBKZCKZUCZCLZABUDVQABCUEWBABVQVRVSMZVQNVTWCMVPNZC
|
|
LWBCWCVPOWDWACVTVRMFNZVTVSMZVONZPVTVRRZVTHZVSRZPWDWAWEWHWGWJWEVTVRFSWHVTV
|
|
RFQVRVTCUFZUKUGWGDKZWFMVNNZDLZWJDWFVNOWNWLWIUHZWLVSRZPZDLWJWMWQDWMWLVTMVM
|
|
NZWLVSMFNZPWQWLVTVSVMFUIWRWOWSWPWRVTWLMVLNZWOWLVTVLUJWOWLGVTULZUHVTWLVLSW
|
|
TWIXAWLVTUMUNVTWLGWKDUFZUOVTWLVLQVBTWSWLVSFSWPWLVSFQVSWLXBUKUGUPTUQWPWJDW
|
|
IVTWKURVSWLWIUSUTTTUPVTVRVSFVOUIVRVSVTVAVCUQTVDVEVPFVOVFVNVMFVLGVGVHURVFV
|
|
IVJVIVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d f A $. $d f B $. $d f F $.
|
|
$( The domain and range of a one-to-one, onto function are equinumerous.
|
|
(Contributed by SF, 23-Feb-2015.) $)
|
|
f1oeng $p |- ( ( F e. C /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
|
|
( vf wcel wf1o wa cv wex cen wbr f1oeq1 spcegv imp bren sylibr ) DCFZABDG
|
|
ZHABEIZGZEJZABKLRSUBUASEDCABTDMNOABEPQ $.
|
|
$}
|
|
|
|
${
|
|
f1oen.1 $e |- F e. _V $.
|
|
$( The domain and range of a one-to-one, onto function are equinumerous.
|
|
(Contributed by SF, 19-Jun-1998.) $)
|
|
f1oen $p |- ( F : A -1-1-onto-> B -> A ~~ B ) $=
|
|
( cvv wcel wf1o cen wbr f1oeng mpan ) CEFABCGABHIDABECJK $.
|
|
$}
|
|
|
|
$( Equinumerosity is reflexive. (Contributed by SF, 23-Feb-2015.) $)
|
|
enrflxg $p |- ( A e. V -> A ~~ A ) $=
|
|
( wcel cid cres cvv wf1o cen wbr idex resexg mpan f1oi f1oeng sylancl ) ABC
|
|
ZDAEZFCZAAQGAAHIDFCPRJDAFBKLAMAAFQNO $.
|
|
|
|
${
|
|
enrflx.1 $e |- A e. _V $.
|
|
$( Equinumerosity is reflexive. (Contributed by SF, 23-Feb-2015.) $)
|
|
enrflx $p |- A ~~ A $=
|
|
( cvv wcel cen wbr enrflxg ax-mp ) ACDAAEFBACGH $.
|
|
$}
|
|
|
|
${
|
|
$d A f g $. $d B f g $. $d C f g $.
|
|
|
|
$( Equinumerosity is symmetric. (Contributed by SF, 23-Feb-2015.) $)
|
|
ensymi $p |- ( A ~~ B -> B ~~ A ) $=
|
|
( vf cen wbr wf1o wex bren ccnv f1ocnv vex cnvex f1oen syl exlimiv sylbi
|
|
cv ) ABDEABCQZFZCGBADEZABCHSTCSBARIZFTABRJBAUARCKLMNOP $.
|
|
|
|
$( Equinumerosity is symmetric. (Contributed by SF, 23-Feb-2015.) $)
|
|
ensym $p |- ( A ~~ B <-> B ~~ A ) $=
|
|
( cen wbr ensymi impbii ) ABCDBACDABEBAEF $.
|
|
|
|
$( Equinumerosity is transitive. (Contributed by SF, 23-Feb-2015.) $)
|
|
entr $p |- ( ( A ~~ B /\ B ~~ C ) -> A ~~ C ) $=
|
|
( vf vg cen wbr wa cv wf1o wex bren anbi12i eeanv bitr4i f1oco ancoms vex
|
|
ccom coex f1oen syl exlimivv sylbi ) ABFGZBCFGZHZABDIZJZBCEIZJZHZEKDKZACF
|
|
GZUGUIDKZUKEKZHUMUEUOUFUPABDLBCELMUIUKDENOULUNDEULACUJUHSZJZUNUKUIURABCUJ
|
|
UHPQACUQUJUHERDRTUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Equinumerosity is an equivalence relationship over the universe.
|
|
(Contributed by SF, 23-Feb-2015.) $)
|
|
ener $p |- ~~ Er _V $=
|
|
( vx vy vz cen cvv cer wbr wtru wcel enex a1i vvex cv ensymi 3ad2ant3 w3a
|
|
wa entr iserd trud ) DEFGHABCEDEEDEIHJKEEIHLKAMZBMZDGZHUBUADGUAEIZUBEIZQU
|
|
AUBNOUCUBCMZDGQHUAUFDGUDUEUFEIPUAUBUFROST $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( Equality implies equinumerosity. (Contributed by SF, 30-Apr-1998.) $)
|
|
idssen $p |- _I C_ ~~ $=
|
|
( vx vy cid cen cv wbr cop wcel weq ideq enrflx breq2 mpbii sylbi 3imtr3i
|
|
vex df-br relssi ) ABCDAEZBEZCFZSTDFZSTGZCHUCDHUAABIZUBSTBPJUDSSDFUBSAPKS
|
|
TSDLMNSTCQSTDQOR $.
|
|
$}
|
|
|
|
$( The domain of equinumerosity. (Contributed by SF, 10-May-1998.) $)
|
|
dmen $p |- dom ~~ = _V $=
|
|
( cvv cen cdm wss wceq cid idssen dmi dmss syl5eqssr ax-mp vss mpbi ) ABCZD
|
|
ZNAEFBDZOGPAFCNHFBIJKNLM $.
|
|
|
|
${
|
|
$d f A $.
|
|
$( The empty set is equinumerous only to itself. Exercise 1 of
|
|
[TakeutiZaring] p. 88. (Contributed by SF, 27-May-1998.) $)
|
|
en0 $p |- ( A ~~ (/) <-> A = (/) ) $=
|
|
( vf c0 cen wbr wceq cv wf1o wex bren ccnv f1ocnv f1o00 simprbi syl sylbi
|
|
exlimiv 0ex enrflx breq1 mpbiri impbii ) ACDEZACFZUCACBGZHZBIUDACBJUFUDBU
|
|
FCAUEKZHZUDACUELUHUGCFUDAUGMNOQPUDUCCCDECRSACCDTUAUB $.
|
|
$}
|
|
|
|
${
|
|
$d x y z a b F $.
|
|
fundmen.1 $e |- F e. _V $.
|
|
$( A function is equinumerous to its domain. Exercise 4 of [Suppes]
|
|
p. 98. (Contributed by SF, 23-Feb-2015.) $)
|
|
fundmen $p |- ( Fun F -> dom F ~~ F ) $=
|
|
( vx vy vz va vb wfun cen wbr c1st wceq cvv cv wa wi wal wcel wex bitr4i
|
|
cdm cres wf1o wfn ccnv crn wss ssv wfo 1stfo fofn fnssresb mp2b mpbir a1i
|
|
wb weq cop brcnv brres br1st anbi1i 19.41v 3bitri anbi12i eeanv dffun4 sp
|
|
vex an4 sps sylbi opeq2 syl6 eleq1 bi2anan9 eqeq12 imbi12d biimprcd imp3a
|
|
syl5bi exlimdvv alrimiv alrimivv dffun2 sylibr dfdm4 dfima3 eqtr2i dff1o2
|
|
syl cima syl3anbrc 1stex resex f1oen ensym sylib ) AHZAAUAZIJZWTAIJWSAWTK
|
|
AUBZUCZXAWSXBAUDZXBUEZHZXBUFZWTLZXCXDWSXDAMUGZAUHMMKUIKMUDXDXIUPUJMMKUKMA
|
|
KULUMUNUOWSCNZDNZXEJZXJENZXEJZOZDEUQZPZEQZDQCQXFWSXRCDWSXQEXOXKXJFNZURZLZ
|
|
XKARZOZXMXJGNZURZLZXMARZOZOZGSFSZWSXPXOYCFSZYHGSZOYJXLYKXNYLXLXKXJXBJXKXJ
|
|
KJZYBOZYKXJXKXBUSXKXJKAUTYNYAFSZYBOYKYMYOYBFXKXJCVIZVAVBYAYBFVCTVDXNYFGSZ
|
|
YGOZYLXNXMXJXBJXMXJKJZYGOYRXJXMXBUSXMXJKAUTYSYQYGGXMXJYPVAVBVDYFYGGVCTVEY
|
|
CYHFGVFTWSYIXPFGYIYAYFOZYBYGOZOZWSXPYAYBYFYGVJWSXTARZYEARZOZXTYELZPZUUBXP
|
|
PWSUUEFGUQZUUFWSUUEUUHPZGQZFQZCQUUICFGAVGUUKUUICUUJUUIFUUIGVHVKVKVLXSYDXJ
|
|
VMVNUUGYTUUAXPYTUUAXPPUUGYTUUAUUEXPUUFYAYBUUCYFYGUUDXKXTAVOXMYEAVOVPXKXTX
|
|
MYEVQVRVSVTWKWAWBWAWCWDCDEXEWEWFXHWSWTKAWLXGAWGKAWHWIUOAWTXBWJWMAWTXBKAWN
|
|
BWOWPWKAWTWQWR $.
|
|
$}
|
|
|
|
${
|
|
$d x A $. $d x F $.
|
|
$( A function is equinumerous to its domain. Exercise 4 of [Suppes]
|
|
p. 98. (Contributed by set.mm contributors, 17-Sep-2013.) $)
|
|
fundmeng $p |- ( ( F e. V /\ Fun F ) -> dom F ~~ F ) $=
|
|
( vx wcel wfun cdm cen wbr cv wceq funeq dmeq breq12d imbi12d vex fundmen
|
|
wi id vtoclg imp ) ABDAEZAFZAGHZCIZEZUDFZUDGHZQUAUCQCABUDAJZUEUAUGUCUDAKU
|
|
HUFUBUDAGUDALUHRMNUDCOPST $.
|
|
|
|
$( A relational set is equinumerous to its converse. (Contributed by
|
|
set.mm contributors, 28-Dec-2014.) (Modified by Scott Fenton,
|
|
17-Apr-2021.) $)
|
|
cnven $p |- ( A e. V -> A ~~ `' A ) $=
|
|
( wcel cswap cres cvv ccnv wf1o cen wbr swapex resexg mpan swapres f1oeng
|
|
sylancl ) ABCZDAEZFCZAAGZRHATIJDFCQSKDAFBLMANATFROP $.
|
|
$}
|
|
|
|
$( A function is equinumerate to its domain. (Contributed by Paul Chapman,
|
|
22-Jun-2011.) $)
|
|
fndmeng $p |- ( ( F Fn A /\ F e. C ) -> A ~~ F ) $=
|
|
( wcel wfn cen wbr wfun wceq wa df-fn fundmeng breq1 syl5ibcom impr sylan2b
|
|
cdm ancoms ) CBDZCAEZACFGZTSCHZCQZAIZJUACAKSUBUDUASUBJUCCFGUDUACBLUCACFMNOP
|
|
R $.
|
|
|
|
$( Two singletons are equinumerous. (Contributed by set.mm contributors,
|
|
9-Nov-2003.) $)
|
|
en2sn $p |- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } ) $=
|
|
( wcel wa csn cop wf1o cen wbr f1osng snex f1oen syl ) ACEBDEFAGZBGZABHZGZI
|
|
PQJKABCDLPQSRMNO $.
|
|
|
|
${
|
|
$d f g h A $. $d f g h B $. $d f g h C $. $d f g h D $.
|
|
$( Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.
|
|
(Contributed by set.mm contributors, 11-Jun-1998.) $)
|
|
unen $p |- ( ( ( A ~~ B /\ C ~~ D ) /\
|
|
( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( A u. C ) ~~ ( B u. D ) ) $=
|
|
( vf vg vh cv wf1o wa wex cin c0 wceq cun cen wbr wi vex bren unex f1oeq1
|
|
f1oun spcev syl ex exlimivv imp anbi12i eeanv bitr4i anbi1i 3imtr4i ) ABE
|
|
HZIZCDFHZIZJZFKEKZACLMNBDLMNJZJACOZBDOZGHZIZGKZABPQZCDPQZJZUTJVAVBPQUSUTV
|
|
EURUTVEREFURUTVEURUTJVAVBUNUPOZIZVEABCDUNUPUCVDVJGVIUNUPESFSUAVAVBVCVIUBU
|
|
DUEUFUGUHVHUSUTVHUOEKZUQFKZJUSVFVKVGVLABETCDFTUIUOUQEFUJUKULVAVBGTUM $.
|
|
$}
|
|
|
|
${
|
|
xpsnen.1 $e |- A e. _V $.
|
|
xpsnen.2 $e |- B e. _V $.
|
|
$( A set is equinumerous to its cross-product with a singleton.
|
|
Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by set.mm
|
|
contributors, 23-Feb-2015.) $)
|
|
xpsnen $p |- ( A X. { B } ) ~~ A $=
|
|
( csn cxp cen wbr cdm wcel c0 wne wceq snid ne0i dmxp mp2b wf wfun fconst
|
|
ffun snex xpex fundmen eqbrtrri ensym mpbi ) AABEZFZGHUIAGHUIIZAUIGBUHJUH
|
|
KLUJAMBDNUHBOAUHPQAUHUIRUISUJUIGHABDTAUHUIUAUIAUHCBUBUCUDQUEAUIUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
$( A set is equinumerous to its cross-product with a singleton.
|
|
Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by set.mm
|
|
contributors, 22-Oct-2004.) $)
|
|
xpsneng $p |- ( ( A e. V /\ B e. W ) -> ( A X. { B } ) ~~ A ) $=
|
|
( vx vy cv csn cxp cen wbr wceq xpeq1 id breq12d xpeq2d breq1d vex xpsnen
|
|
sneq vtocl2g ) EGZFGZHZIZUBJKAUDIZAJKABHZIZAJKEFABCDUBALZUEUFUBAJUBAUDMUI
|
|
NOUCBLZUFUHAJUJUDUGAUCBTPQUBUCERFRSUA $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d x y B $.
|
|
endisj.1 $e |- A e. _V $.
|
|
endisj.2 $e |- B e. _V $.
|
|
$( Any two sets are equinumerous to disjoint sets. Exercise 4.39 of
|
|
[Mendelson] p. 255. (Contributed by set.mm contributors,
|
|
16-Apr-2004.) $)
|
|
endisj $p |- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) ) $=
|
|
( c0 csn cxp cen wbr wa cin wceq cv wex 0ex xpsnen snex xpex ccompl breq1
|
|
complex pm3.2i necompl xpnedisj bi2anan9 ineq12 eqeq1d anbi12d spc2ev
|
|
mp2an ) CGUAZHZIZCJKZDGHZIZDJKZLZUOURMZGNZAOZCJKZBOZDJKZLZVCVEMZGNZLZBPAP
|
|
UPUSCUMEGQUCZRDGFQRUDCDUMGVKGUEUFVJUTVBLABUOURCUNEUMSTDUQFGSTVCUONZVEURNZ
|
|
LZVGUTVIVBVLVDUPVMVFUSVCUOCJUBVEURDJUBUGVNVHVAGVCUOVEURUHUIUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
xpcomen.1 $e |- A e. _V $.
|
|
xpcomen.2 $e |- B e. _V $.
|
|
$( Commutative law for equinumerosity of cross product. Proposition
|
|
4.22(d) of [Mendelson] p. 254. (Contributed by set.mm contributors,
|
|
5-Jan-2004.) (Revised by set.mm contributors, 23-Apr-2014.) $)
|
|
xpcomen $p |- ( A X. B ) ~~ ( B X. A ) $=
|
|
( cxp cswap cres wf1o cen wbr ccnv swapres wceq cnvxp f1oeq3 ax-mp swapex
|
|
wb mpbi xpex resex f1oen ) ABEZBAEZFUCGZHZUCUDIJUCUCKZUEHZUFUCLUGUDMUHUFR
|
|
ABNUGUDUCUEOPSUCUDUEFUCQABCDTUAUBP $.
|
|
$}
|
|
|
|
${
|
|
$d x y A $. $d y B $.
|
|
$( Commutative law for equinumerosity of cross product. Proposition
|
|
4.22(d) of [Mendelson] p. 254. (Contributed by set.mm contributors,
|
|
27-Mar-2006.) $)
|
|
xpcomeng $p |- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) ) $=
|
|
( vx vy cv cxp cen wbr wceq xpeq1 xpeq2 breq12d vex xpcomen vtocl2g ) EGZ
|
|
FGZHZSRHZIJASHZSAHZIJABHZBAHZIJEFABCDRAKTUBUAUCIRASLRASMNSBKUBUDUCUEISBAM
|
|
SBALNRSEOFOPQ $.
|
|
$}
|
|
|
|
$( A set is equinumerous to its cross-product with a singleton on the left.
|
|
(Contributed by Stefan O'Rear, 21-Nov-2014.) $)
|
|
xpsnen2g $p |- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ B ) $=
|
|
( wcel wa csn cxp cen wbr cvv snex xpcomeng mpan adantl xpsneng ancoms entr
|
|
syl2anc ) ACEZBDEZFAGZBHZBUBHZIJZUDBIJZUCBIJUAUETUBKEUAUEALUBBKDMNOUATUFBAD
|
|
CPQUCUDBRS $.
|
|
|
|
${
|
|
$d A f $. $d A g $. $d B f $. $d B g $. $d C f $. $d C g $. $d D f $.
|
|
$d D g $. $d f g $.
|
|
$( Equinumerosity law for cross product. Proposition 4.22(b) of
|
|
[Mendelson] p. 254. (Contributed by set.mm contributors, 24-Jul-2004.)
|
|
(Revised by set.mm contributors, 9-Mar-2013.) $)
|
|
xpen $p |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) ~~ ( B X. D ) ) $=
|
|
( vf vg cen wbr wa wf1o wex cxp bren anbi12i eeanv bitr4i cpprod f1opprod
|
|
cv vex pprodex f1oen syl exlimivv sylbi ) ABGHZCDGHZIZABESZJZCDFSZJZIZFKE
|
|
KZACLZBDLZGHZUHUJEKZULFKZIUNUFURUGUSABEMCDFMNUJULEFOPUMUQEFUMUOUPUIUKQZJU
|
|
QACBDUIUKRUOUPUTUIUKETFTUAUBUCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d x t $. $d y t $.
|
|
$( Lemma for ~ xpassen . Compute a projection. (Contributed by Scott
|
|
Fenton, 19-Apr-2021.) $)
|
|
xpassenlem $p |- ( y ( ( 1st o. 1st ) (x) ( ( 2nd o. 1st ) (x) 2nd ) ) x
|
|
<-> ( Proj1 Proj1 y = Proj1 x /\ Proj2 Proj1 y = Proj1 Proj2 x /\
|
|
Proj2 y = Proj2 Proj2 x ) ) $=
|
|
( vt cv cproj1 c1st ccom wbr cproj2 c2nd ctxp wa wceq cop wex opeq breq1i
|
|
brco bitri 3bitri w3a vex proj1ex proj2ex eqcom anbi1i exbii breq1 syl6bb
|
|
opbr1st ceqsexv breq2i trtxp opbr2nd anbi12i 3anass 3bitr4i ) BDZADZEZFFG
|
|
ZHZURUSIZJFGZJKZHZLZUREZEZUTMZVHIZVCEZMZURIZVCIZMZLZLURUSVAVEKZHZVJVMVPUA
|
|
VBVJVFVQVBVHVNNZUTVAHVTCDZFHZWAUTFHZLZCOZVJURVTUTVAURPZQCVTUTFFRWEWAVHMZW
|
|
CLZCOVJWDWHCWBWGWCWBVHWAMWGVHVNWAURBUBZUCZURWIUDZUJVHWAUESZUFUGWCVJCVHWJW
|
|
GWCVHUTFHZVJWAVHUTFUHWMVIVKNZUTFHVJVHWNUTFVHPZQVIVKUTVHWJUCZVHWJUDZUJSUIU
|
|
KSTVFURVLVONZVEHURVLVDHZURVOJHZLVQVCWRURVEVCPULURVLVOVDJUMWSVMWTVPWSVTVLV
|
|
DHWBWAVLJHZLZCOZVMURVTVLVDWFQCVTVLJFRXCWGXALZCOVMXBXDCWBWGXAWLUFUGXAVMCVH
|
|
WJWGXAVHVLJHZVMWAVHVLJUHXEWNVLJHVMVHWNVLJWOQVIVKVLWPWQUNSUIUKSTWTVTVOJHVP
|
|
URVTVOJWFQVHVNVOWJWKUNSUOTUOVSURUTVCNZVRHVGUSXFURVRUSPULURUTVCVAVEUMSVJVM
|
|
VPUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d A a b c x y z t p $. $d B a b c x y z t p $. $d C a b c x y z t p $.
|
|
xpassen.1 $e |- A e. _V $.
|
|
xpassen.2 $e |- B e. _V $.
|
|
xpassen.3 $e |- C e. _V $.
|
|
$( Associative law for equinumerosity of cross product. Proposition
|
|
4.22(e) of [Mendelson] p. 254. (Contributed by SF, 24-Feb-2015.) $)
|
|
xpassen $p |- ( ( A X. B ) X. C ) ~~ ( A X. ( B X. C ) ) $=
|
|
( vy vx vz vp vt c1st c2nd wbr cvv wa wceq wcel wex wrex va cxp ccom ctxp
|
|
vb vc cres wf1o cen wf1 wss wf cv wmo wal wfn cin crn wfo 1stfo fof ax-mp
|
|
cima dffn2 mpbir ssv fnco mp3an 2ndfo fofn fntxp mp2an fneq2i mpbi weq wi
|
|
inidm cproj1 cproj2 w3a xpassenlem cop simp1 eqtr3 syl2an opeq12d 3eqtr4g
|
|
simp2 opeq simp3 syl2anb breq1 mo4 ax-gen dff12 mpbir2an f1ores wb opeqex
|
|
gen2 vex rexcom4 opex ceqsexv brco1st bitri trtxp brco eqcom anbi1i exbii
|
|
opbr1st opbr2nd 3bitri anbi12i 3anass 3bitr4i rexbii bitr3i 3reeanv elxp2
|
|
df-rex opeq1 eqeq2d reeanv 3bitr4ri risset 3anbi123i elima2 opelxp anbi2i
|
|
r19.41v opeq2 eleq1d bibi12d 1stex coex 2ndex txpex xpex mp2b resex f1oen
|
|
mpbiri exlimivv eqrelriv f1oeq3 ) ABUBZCUBZABCUBZUBZLLUCZMLUCZMUDZUDZUUIU
|
|
GZUHZUUIUUKUINUUIUUOUUIVCZUUPUHZUUQOOUUOUJZUUIOUKUUSUUTOOUUOULZGUMZHUMZUU
|
|
ONZGUNZHUOUUOOUPZUVAUUOOOUQZUPZUVFUULOUPZUUNOUPZUVHLOUPZUVKLURZOUKZUVIUVK
|
|
OOLULZOOLUSUVNUTOOLVAVBOLVDVEZUVOUVLVFZOOLLVGVHUUNUVGUPZUVJUUMOUPZMOUPZUV
|
|
QUVSUVKUVMUVROOMUSUVSVIOOMVJVBZUVOUVPOOMLVGVHUVTOOUUMMVKVLUVGOUUNOVQZVMVN
|
|
OOUULUUNVKVLUVGOUUOUWAVMVNOUUOVDVNUVEHUVEUVDIUMZUVCUUONZPGIVOZVPZIUOGUOUW
|
|
EGIUVDUVBVRZVRZUVCVRZQZUWFVSZUVCVSZVRZQZUVBVSZUWKVSZQZVTZUWBVRZVRZUWHQZUW
|
|
RVSZUWLQZUWBVSZUWOQZVTZUWDUWCHGWAHIWAUWQUXEPZUWFUWNWBUWRUXCWBUVBUWBUXFUWF
|
|
UWRUWNUXCUXFUWGUWJWBUWSUXAWBUWFUWRUXFUWGUWSUWJUXAUWQUWIUWTUWGUWSQUXEUWIUW
|
|
MUWPWCUWTUXBUXDWCUWGUWSUWHWDWEUWQUWMUXBUWJUXAQUXEUWIUWMUWPWHUWTUXBUXDWHUW
|
|
JUXAUWLWDWEWFUWFWIUWRWIWGUWQUWPUXDUWNUXCQUXEUWIUWMUWPWJUWTUXBUXDWJUWNUXCU
|
|
WOWDWEWFUVBWIUWBWIWGWKWTUVDUWCGIUVBUWBUVCUUOWLWMVEWNGHOOUUOWOWPUUIVFOOUUI
|
|
UUOWQVLUURUUKQUUSUUQWRUAJUURUUKJUMZORUXGUEUMZUFUMZWBZQZUFSUESUAUMZUXGWBZU
|
|
URRZUXMUUKRZWRZJXAUEUFUXGOWSUXKUXPUEUFUXKUXPUXLUXJWBZUURRZUXQUUKRZWRUXGUU
|
|
IRZUXGUXQUUONZPZJSZUXLARZUXHBRZUXICRZVTZUXRUXSUXGUVCUVBWBZUWBWBZQZUYAPZIC
|
|
TZGBTZHATZJSZHUAVOZHATZGUEVOZGBTZIUFVOZICTZVTZUYCUYGUYOUYPUYRUYTVTZICTZGB
|
|
TZHATZVUBUYOUYMJSZHATVUFUYMHJAXBVUGVUEHAVUGUYLJSZGBTVUEUYLGJBXBVUHVUDGBVU
|
|
HUYKJSZICTVUDUYKIJCXBVUIVUCICVUIUYIUXQUUONZVUCUYAVUJJUYIUYHUWBUVCUVBHXAZG
|
|
XAZXCZIXAZXCUXGUYIUXQUUOWLXDUYIUXLUULNZUYIUXJUUNNZPUYPUYRUYTPZPVUJVUCVUOU
|
|
YPVUPVUQVUOUYHUXLLNUYPUYHUWBUXLLVUMVUNXEUVCUVBUXLVUKVULXLXFVUPUYIUXHUUMNZ
|
|
UYIUXIMNZPVUQUYIUXHUXIUUMMXGVURUYRVUSUYTVURUYIKUMZLNZVUTUXHMNZPZKSVUTUYHQ
|
|
ZVVBPZKSZUYRKUYIUXHMLXHVVCVVEKVVAVVDVVBVVAUYHVUTQVVDUYHUWBVUTVUMVUNXLUYHV
|
|
UTXIXFXJXKVVFUYHUXHMNZUYRVVBVVGKUYHVUMVUTUYHUXHMWLXDUVCUVBUXHVUKVULXMXFXN
|
|
UYHUWBUXIVUMVUNXMXOXFXOUYIUXLUXJUULUUNXGUYPUYRUYTXPXQXFXRXSXRXSXRXSUYPUYR
|
|
UYTHGIABCXTXFUYBUYNJUYJICTZGBTZUYAPZHATVVIHATZUYAPUYNUYBVVIUYAHAYLUYMVVJH
|
|
AUYMVVHUYAPZGBTVVJUYLVVLGBUYJUYAICYLXRVVHUYAGBYLXFXRUXTVVKUYAUXTUXGVUTUWB
|
|
WBZQZICTZKUUHTVUTUUHRZVVOPZKSZVVKKIUXGUUHCYAVVOKUUHYBVVDVVNPZICTZGBTZKSZH
|
|
ATVWAHATZKSVVKVVRVWAHKAXBVVIVWBHAVVIVVTKSZGBTVWBVVHVWDGBVVHVVSKSZICTVWDVW
|
|
EUYJICVVNUYJKUYHVUMVVDVVMUYIUXGVUTUYHUWBYCYDXDXRVVSIKCXBXSXRVVTGKBXBXFXRV
|
|
VQVWCKVVDGBTZVVOPZHATVWFHATZVVOPVWCVVQVWFVVOHAYLVWAVWGHAVVDVVNGIBCYEXRVVP
|
|
VWHVVOHGVUTABYAXJYFXKYFXNXJYFXKUYDUYQUYEUYSUYFVUAHUXLAYGGUXHBYGIUXICYGYHX
|
|
QJUXQUUOUUIYIUYDUXJUUJRZPUYDUYEUYFPZPUXSUYGVWIVWJUYDUXHUXIBCYJYKUXLUXJAUU
|
|
JYJUYDUYEUYFXPXQXQUXKUXNUXRUXOUXSUXKUXMUXQUURUXGUXJUXLYMZYNUXKUXMUXQUUKVW
|
|
KYNYOUUDUUEUUAUUFUURUUKUUIUUPUUGVBVNUUIUUKUUPUUOUUIUULUUNLLYPYPYQUUMMMLYR
|
|
YPYQYRYSYSUUHCABDEYTFYTUUBUUCVB $.
|
|
$}
|
|
|
|
${
|
|
ensn.1 $e |- A e. _V $.
|
|
ensn.2 $e |- B e. _V $.
|
|
$( Two singletons are equinumerous. Theorem XI.1.10 of [Rosser] p. 348.
|
|
(Contributed by SF, 25-Feb-2015.) $)
|
|
ensn $p |- { A } ~~ { B } $=
|
|
( cvv wcel csn cen wbr en2sn mp2an ) AEFBEFAGBGHICDABEEJK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d X x $. $d Y x $.
|
|
$( Lemma for ~ enadj . Calculate equality of differences. (Contributed by
|
|
SF, 25-Feb-2015.) $)
|
|
enadjlem1 $p |- ( ( ( A u. { X } ) = ( B u. { Y } ) /\
|
|
( -. X e. A /\ -. Y e. B ) /\ ( Y e. A /\ X e. B ) ) ->
|
|
( A \ { Y } ) = ( B \ { X } ) ) $=
|
|
( vx csn cun wceq wcel wn wa cdif wne elsni necon3ai ad2antll ssun1 sseli
|
|
wo ad2antrl w3a cv simpl1 eleqtrd elun sylib orel2 ex simp2l eleq1 notbid
|
|
sylc syl5ibrcom necon2ad adantrd jcad eldifsn ssrdv eleqtrrd simp2r eqssd
|
|
3imtr4g ) ACFZGZBDFZGZHZCAIZJZDBIZJZKZDAICBIKZUAZAVELZBVCLZVNEVOVPVNEUBZA
|
|
IZVQDMZKZVQBIZVQCMZKZVQVOIZVQVPIZVNVTWAWBVNVTWAVNVTKZVQVEIZJZWAWGSZWAVSWH
|
|
VNVRWGVQDVQDNOPWFVQVFIZWIWFVQVDVFVRVQVDIZVNVSAVDVQAVCQRTVGVLVMVTUCUDVQBVE
|
|
UEUFWGWAUGULUHVNVRWBVSVNVRVQCVNVRJVQCHZVIVGVIVKVMUIWLVRVHVQCAUJUKUMUNUOUP
|
|
VQADUQZVQBCUQZVBURVNEVPVOVNWCVTWEWDVNWCVRVSVNWCVRVNWCKZVQVCIZJZVRWPSZVRWB
|
|
WQVNWAWPVQCVQCNOPWOWKWRWOVQVFVDWAWJVNWBBVFVQBVEQRTVGVLVMWCUCUSVQAVCUEUFWP
|
|
VRUGULUHVNWAVSWBVNWAVQDVNWAJVQDHZVKVGVIVKVMUTWSWAVJVQDBUJUKUMUNUOUPWNWMVB
|
|
URVA $.
|
|
$}
|
|
|
|
${
|
|
enadj.1 $e |- A e. _V $.
|
|
enadj.2 $e |- B e. _V $.
|
|
enadj.3 $e |- X e. _V $.
|
|
enadj.4 $e |- Y e. _V $.
|
|
$( Equivalence law for adjunction. Theorem XI.1.13 of [Rosser] p. 348.
|
|
(Contributed by SF, 25-Feb-2015.) $)
|
|
enadj $p |- ( ( ( A u. { X } ) = ( B u. { Y } ) /\
|
|
-. X e. A /\ -. Y e. B ) -> A ~~ B ) $=
|
|
( cun wceq wcel wn w3a cen wbr cdif wa cin c0 a1i csn uneq2d eqeq1d eleq1
|
|
sneq notbid 3anbi12d simp1 difeq1d nnsucelrlem2 3ad2ant2 3ad2ant3 3eqtr3d
|
|
wi enrflx syl6eqbr syl6bi wne wo elsni eqcomd necon3ai adantr snid sselii
|
|
ssun2 simpr1 syl5eleqr elun sylib orel2 syl5eleq jca simpl1 simpl2 simpl3
|
|
sylc simprl simprr enadjlem1 syl122anc snex difex breq2 mpbii adantl ensn
|
|
3adant1 incom disjdif eqtri syl22anc simpl3l nnsucelrlem4 simpl3r 3brtr3d
|
|
unen syl mpdan mpd3an3 ex pm2.61ine ) ACUAZIZBDUAZIZJZCAKZLZDBKLZMZABNOZU
|
|
NCDCDJZXKAXEIZXFJZDAKZLZXJMZXLXMXGXOXIXQXJXMXDXNXFXMXCXEACDUEUBUCXMXHXPCD
|
|
AUDUFUGXRABBNXRXNXEPZXFXEPZABXRXNXFXEXOXQXJUHUIXQXOXSAJXJADUJUKXJXOXTBJXQ
|
|
BDUJULUMBFUOUPUQCDURZXKXLYAXKXPCBKZQZXLYAXKQZXPYBYDDXCKZLZXPYEUSZXPYAYFXK
|
|
YECDYEDCDCUTVAVBVCYDDXDKYGYDDXFXDXEXFDXEBVFDHVDVEYAXGXIXJVGZVHDAXCVIVJYEX
|
|
PVKVQYDCXEKZLZYBYIUSZYBYAYJXKYICDCDUTVBVCYDCXFKYKYDCXDXFXCXDCXCAVFCGVDVEY
|
|
HVLCBXEVIVJYIYBVKVQVMYAXKYCMZAXEPZBXCPZJZXLXKYCYOYAXKYCQXGXIXJXPYBYOXGXIX
|
|
JYCVNXGXIXJYCVOXGXIXJYCVPXKXPYBVRXKXPYBVSABCDVTWAWHYLYOQZYMXEIZYNXCIZABNY
|
|
PYMYNNOZXEXCNOZYMXERZSJZYNXCRZSJZYQYRNOYOYSYLYOYMYMNOYSYMAXEEDWBWCUOYMYNY
|
|
MNWDWEWFYTYPDCHGWGTUUBYPUUAXEYMRSYMXEWIXEAWJWKTUUDYPUUCXCYNRSYNXCWIXCBWJW
|
|
KTYMYNXEXCWQWLYPXPYQAJXPYBYAXKYOWMDAWNWRYPYBYRBJXPYBYAXKYOWOCBWNWRWPWSWTX
|
|
AXB $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a p $. $d a x $. $d a y $. $d b x $. $d b y $. $d g p $.
|
|
$d g x $. $d g y $. $d p x $. $d p y $. $d x y $.
|
|
$( Lemma for ~ enpw1 . Set up stratification for the reverse direction.
|
|
(Contributed by SF, 26-Feb-2015.) $)
|
|
enpw1lem1 $p |- { <. x , y >. | { x } g { y } } e. _V $=
|
|
( vp va vb c1st c2nd cv csn wbr wceq wa wex ancom weq bitri exbii ceqsexv
|
|
3bitri ccnv csi ccom cin cima cuni1 copab cvv wcel wrex opex eluni1 elima
|
|
cop vex brin brco brsnsi2 brcnv opbr1st equcom anbi1i sneq eqeq2d opbr2nd
|
|
snex breq2 anbi2i anbi12i op1st2nd rexbii df-br risset bitr2i 1stex cnvex
|
|
opabbi2i siex coex 2ndex inex imaex uni1ex eqeltrri ) GUAZUBZGUCZHUAZUBZH
|
|
UCZUDZCIZUEZUFZAIZJZBIZJZWLKZABUGUHWSABWNWOWQUNZWNUIWTJZWMUIZDIZWPWRUNZLZ
|
|
DWLUJZWSWTWMWOWQAUOZBUOZUKZULXBXCXAWKKZDWLUJXFDXAWKWLUMXJXEDWLXJXCXAWGKZX
|
|
CXAWJKZMXCWPGKZXCWRHKZMXEXCXAWGWJUPXKXMXLXNXKXCEIZGKZXOXAWFKZMZENXOWPLZXP
|
|
MZENXMEXCXAWFGUQXRXTEXRXQXPMXTXPXQOXQXSXPXQXOFIZJZLZYAWTWEKZMZFNFAPZYCMZF
|
|
NXSFWTXOWEXIURYEYGFYEYDYCMYGYCYDOYDYFYCYDWTYAGKAFPYFYAWTGUSWOWQYAXGXHUTAF
|
|
VATVBQRYCXSFWOXGYFYBWPXOYAWOVCVDSTVBQRXPXMEWPWOVFZXOWPXCGVGSTXLXCXOHKZXOX
|
|
AWIKZMZENXOWRLZYIMZENXNEXCXAWIHUQYKYMEYKYIYLMYMYJYLYIYJYCYAWTWHKZMZFNFBPZ
|
|
YCMZFNYLFWTXOWHXIURYOYQFYOYCYPMYQYNYPYCYNWTYAHKBFPYPYAWTHUSWOWQYAXGXHVEBF
|
|
VATVHYCYPOQRYCYLFWQXHYPYBWRXOYAWQVCVDSTVHYIYLOQRYIXNEWRWQVFZXOWRXCHVGSTVI
|
|
WPWRXCYHYRVJTVKQWSXDWLUIXFWPWRWLVLDXDWLVMVNTVQWMWKWLWGWJWFGWEGVOVPVRVOVSW
|
|
IHWHHVTVPVRVTVSWACUOWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d A a b f $. $d B a b f $. $d a g $. $d a w $. $d a x $. $d a y $.
|
|
$d a z $. $d b g $. $d b w $. $d b x $. $d b y $. $d b z $. $d g w $.
|
|
$d g x $. $d g y $. $d g z $. $d w x $. $d w y $. $d w z $. $d x y $.
|
|
$d x z $. $d y z $.
|
|
$( Two classes are equinumerous iff their unit power classes are
|
|
equinumerous. Theorem XI.1.33 of [Rosser] p. 368. (Contributed by SF,
|
|
26-Feb-2015.) $)
|
|
enpw1 $p |- ( A ~~ B <-> ~P1 A ~~ ~P1 B ) $=
|
|
( vx vy vz vw cen wbr cvv wcel cpw1 wceq wex wfun cdm syl sylanbrc sylibr
|
|
cv wal va vb vf vg wa brex pw1exb anbi12i sylib breq1 pw1eq bibi12d breq2
|
|
breq1d breq2d wf1o bren csi wfn ccnv f1ofun funsi f1odm dmsi syl5eq df-fn
|
|
wb wf1 f1of1 wf df-f1 simprbi 3syl cnvsi funeqi wfo f1ofo forn rnsi dfrn4
|
|
crn eqtr3i syl5eqr dff1o4 vex siex f1oen exlimiv sylbi csn wmo weq wi w3a
|
|
copab fununiq sneqb 3expib alrimivv sneq mo4 alrimiv funopab cab wel eldm
|
|
dmopab brelrn eleq2d wrex elpw1 spcev syl6bi rexlimivw com23 mpdi exlimdv
|
|
syl5bi breldm impbid1 bitr3d snelpw1 syl6bb abbi1dv brcnv 3imtr3g cnvopab
|
|
f1ocnv elrn fneq1i enpw1lem1 impbii vtocl2g pm5.21nii ) ABGHZAIJZBIJZUEZA
|
|
KZBKZGHZABGUFUUAYSIJZYTIJZUEYRYSYTGUFUUBYPUUCYQAUGBUGUHUIUASZUBSZGHZUUDKZ
|
|
UUEKZGHZVGAUUEGHZYSUUHGHZVGYOUUAVGUAUBABIIUUDALZUUFUUJUUIUUKUUDAUUEGUJUUL
|
|
UUGYSUUHGUUDAUKUNULUUEBLZUUJYOUUKUUAUUEBAGUMUUMUUHYTYSGUUEBUKUOULUUFUUIUU
|
|
FUUDUUEUCSZUPZUCMUUIUUDUUEUCUQUUOUUIUCUUOUUGUUHUUNURZUPZUUIUUOUUPUUGUSZUU
|
|
PUTZUUHUSZUUQUUOUUPNZUUPOZUUGLZUURUUOUUNNUVAUUDUUEUUNVAUUNVBPUUOUUNOZUUDL
|
|
ZUVCUUDUUEUUNVCUVEUVBUVDKUUGUUNVDUVDUUDUKVEPUUPUUGVFQUUOUUSNZUUSOZUUHLZUU
|
|
TUUOUUNUTZURZNZUVFUUOUUDUUEUUNVHZUVINZUVKUUDUUEUUNVIUVLUUDUUEUUNVJUVMUUDU
|
|
UEUUNVKVLUVIVBVMUUSUVJUUNVNVORUUOUUDUUEUUNVPUUNWAZUUELZUVHUUDUUEUUNVQUUDU
|
|
UEUUNVRUVOUVGUVNKZUUHUUPWAUVPUVGUUNVSUUPVTWBUVNUUEUKWCVMUUSUUHVFQUUGUUHUU
|
|
PWDQUUGUUHUUPUUNUCWEWFWGPWHWIUUIUUGUUHUDSZUPZUDMUUFUUGUUHUDUQUVRUUFUDUVRU
|
|
UDUUECSZWJZDSZWJZUVQHZCDWOZUPZUUFUVRUWDUUDUSZUWDUTZUUEUSZUWEUVRUWDNZUWDOZ
|
|
UUDLUWFUVRUWCDWKZCTUWIUVRUWKCUVRUWCUVTESZWJZUVQHZUEDEWLZWMZETDTUWKUVRUWPD
|
|
EUVRUVQNZUWPUUGUUHUVQVAUWQUWCUWNUWOUWQUWCUWNWNUWBUWMLUWOUVTUWBUWMUVQWPUWA
|
|
UWLDWEWQUIWRPWSUWCUWNDEUWOUWBUWMUVTUVQUWAUWLWTUOXARXBUWCCDXCRUVRUWJUWCDMZ
|
|
CXDUUDUWCCDXGUVRUWRCUUDUVRUWRUVTUUGJZCUAXEUVRUVTUVQOZJZUWRUWSUVRUXAUWRUXA
|
|
UVTUWLUVQHZEMUVRUWREUVTUVQXFUVRUXBUWREUVRUXBUWLUVQWAZJZUWRUVTUWLUVQXHUVRU
|
|
XDUXBUWRUVRUXDUWLUUHJZUXBUWRWMZUVRUXCUUHUWLUVRUUGUUHUVQVPUXCUUHLUUGUUHUVQ
|
|
VQUUGUUHUVQVRPZXIUXEUWLFSZWJZLZFUUEXJUXFFUWLUUEXKUXJUXFFUUEUXJUXBUVTUXIUV
|
|
QHZUWRUWLUXIUVTUVQUMUWCUXKDUXHFWEZDFWLUWBUXIUVTUVQUWAUXHWTUOXLXMXNWIXMXOX
|
|
PXQXRUWCUXADUVTUWBUVQXSWHXTUVRUWTUUGUVTUUGUUHUVQVCZXIYAUVSUUDYBYCYDVEUWDU
|
|
UDVFQUVRUWCDCWOZUUEUSZUWHUVRUXNNZUXNOZUUELUXOUVRUWCCWKZDTUXPUVRUXRDUVRUWC
|
|
UWMUWBUVQHZUEZCEWLZWMZETCTUXRUVRUYBCEUVRUUHUUGUVQUTZUPUYCNZUYBUUGUUHUVQYH
|
|
UUHUUGUYCVAUYDUWBUVTUYCHZUWBUWMUYCHZUEUVTUWMLZUXTUYAUYDUYEUYFUYGUWBUVTUWM
|
|
UYCWPWRUYEUWCUYFUXSUWBUVTUVQYEUWBUWMUVQYEUHUVSUWLCWEWQYFVMWSUWCUXSCEUYAUV
|
|
TUWMUWBUVQUVSUWLWTUNXARXBUWCDCXCRUVRUXQUWCCMZDXDUUEUWCDCXGUVRUYHDUUEUVRUY
|
|
HUWBUUHJZDUBXEUVRUWBUXCJZUYHUYIUVRUYJUYHUYJUWLUWBUVQHZEMUVRUYHEUWBUVQYIUV
|
|
RUYKUYHEUVRUYKUWLUWTJZUYHUWLUWBUVQXSUVRUYLUYKUYHUVRUYLUWLUUGJZUYKUYHWMZUV
|
|
RUWTUUGUWLUXMXIUYMUXJFUUDXJUYNFUWLUUDXKUXJUYNFUUDUXJUYKUXIUWBUVQHZUYHUWLU
|
|
XIUWBUVQUJUWCUYOCUXHUXLCFWLUVTUXIUWBUVQUVSUXHWTUNXLXMXNWIXMXOXPXQXRUWCUYJ
|
|
CUVTUWBUVQXHWHXTUVRUXCUUHUWBUXGXIYAUWAUUEYBYCYDVEUXNUUEVFQUUEUWGUXNUWCCDY
|
|
GYJRUUDUUEUWDWDQUUDUUEUWDCDUDYKWGPWHWIYLYMYN $.
|
|
$}
|
|
|
|
${
|
|
$d A s p x $. $d G s p x $. $d r s p x $.
|
|
enmap2lem1.1 $e |- W = ( s e. ( G ^m A ) |-> ( s o. `' r ) ) $.
|
|
$( Lemma for ~ enmap2 . Set up stratification. (Contributed by SF,
|
|
26-Feb-2015.) $)
|
|
enmap2lem1 $p |- W e. _V $=
|
|
( vx vp c1st c2nd cv cvv ccompose wcel wceq wa cop wbr vex bitri ccnv csn
|
|
cima cxp cin ccom ctxp cmap cres cmpt copab df-mpt opelres wrex trtxp wex
|
|
brco ancom brin brxp mpbiran2 eliniseg anbi2i cnvex op1st2nd 3bitri exbii
|
|
co anbi1i opex breq2 ceqsexv rexbii elima risset 3bitr4i df-br brcomposeg
|
|
wb mp2an eqcom 3bitr2i anbi2ci opabbi2i 3eqtr4i 1stex snex vvex xpex inex
|
|
2ndex imaex coex txpex composeex ovex resex eqeltri ) CIJUAZEKZUAZUBZUCZL
|
|
UDZUEZIUFZJUGZMUCZBAUHVHZUIZLDXIDKZXAUFZUJXKXINZGKZXLOZPZDGUKCXJDGXIXLULF
|
|
XPDGXJXKXNQZXJNXQXHNZXMPXPXKXNXHXIUMXRXOXMXRXKXAQZXNQZMNZXSXNMRZXOHKZXQXG
|
|
RZHMUNYCXTOZHMUNXRYAYDYEHMYDYCXKXFRZYCXNJRZPYCXSIRZYGPYEYCXKXNXFJUOYFYHYG
|
|
YFYCXNIRZXNXKXERZPZGUPXNXSOZYIPZGUPYHGYCXKXEIUQYKYMGYKYJYIPYMYIYJURYJYLYI
|
|
YJXNXKIRZXNXKXDRZPYNXNXAJRZPYLXNXKIXDUSYOYPYNYOXNXCNZYPYOYQXKLNZDSZXNXKXC
|
|
LUTVAJXAXNVBTVCXKXAXNYSWTESVDZVEVFVITVGYIYHGXSXKXAYSYTVJZXNXSYCIVKVLVFVIX
|
|
SXNYCUUAGSVEVFVMHXQXGMVNHXTMVOVPXSXNMVQYBXLXNOZXOYRXALNYBUUBVSYSYTXKXAXNL
|
|
LVRVTXLXNWATWBWCTWDWEXHXIXGMXFJXEIIXDWFXCLWSXBJWKVDXAWGWLWHWIWJWFWMWKWNWO
|
|
WLBAUHWPWQWR $.
|
|
$}
|
|
|
|
${
|
|
$d a s $. $d G s $.
|
|
enmap2lem2.1 $e |- W = ( s e. ( G ^m a ) |-> ( s o. `' r ) ) $.
|
|
$( Lemma for ~ enmap2 . Establish the functionhood and domain of ` W ` .
|
|
(Contributed by SF, 26-Feb-2015.) $)
|
|
enmap2lem2 $p |- W Fn ( G ^m a ) $=
|
|
( cv ccnv ccom cvv wcel cmap co wfn fnmpt vex cnvex coex a1i mprg ) CGZDG
|
|
ZHZIZJKZBAEGLMZNCUFCUFUDBJFOUEUAUFKUAUCCPUBDPQRST $.
|
|
$}
|
|
|
|
${
|
|
$d a s $. $d G s $. $d r s $. $d S s $.
|
|
enmap2lem3.1 $e |- W = ( s e. ( G ^m a ) |-> ( s o. `' r ) ) $.
|
|
$( Lemma for ~ enmap2 . Binary relationship condition over ` W ` .
|
|
(Contributed by SF, 26-Feb-2015.) $)
|
|
enmap2lem3 $p |- ( r : a -1-1-onto-> b ->
|
|
( S W T -> S = ( T o. r ) ) ) $=
|
|
( wbr cv cmap co wcel ccnv ccom wceq wa cvv coeq1 wf1o cdm wfn enmap2lem2
|
|
breldm fndm ax-mp syl6eleq wb fnbrfvb mpan cnvex coexg mpan2 fvmptg mpdan
|
|
cfv vex eqeq1d eqcom syl6bb biimpd sylbird mpcom jca cid f1ococnv1 coeq2d
|
|
cres adantr elmapi fcoi1 syl adantl eqtr2d coass syl6eq eqeq2d syl5ibrcom
|
|
wf expimpd syl5 ) ABDJZACGKZLMZNZBAFKZOZPZQZRWDHKZWGUAZABWGPZQZWCWFWJWCAD
|
|
UBZWEABDUEDWEUCZWOWEQCDEFGIUDZWEDUFUGUHZWFWCWJWRWFWCADUQZBQZWJWPWFWTWCUIW
|
|
QWEABDUJUKWFWTWJWFWTWIBQWJWFWSWIBWFWISNZWSWIQWFWHSNXAWGFURULAWHWESUMUNEAE
|
|
KZWHPWIWESDXBAWHTIUOUPUSWIBUTVAVBVCVDVEWLWFWJWNWLWFRZWNWJAAWHWGPZPZQXCXEA
|
|
VFWDVIZPZAWLXEXGQWFWLXDXFAWDWKWGVGVHVJWFXGAQZWLWFWDCAVTXHACWDVKWDCAVLVMVN
|
|
VOWJWMXEAWJWMWIWGPXEBWIWGTAWHWGVPVQVRVSWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d a s $. $d a x $. $d a y $. $d a z $. $d b x $. $d b y $. $d b z $.
|
|
$d G s $. $d r s $. $d r x $. $d r y $. $d r z $. $d s y $. $d s z $.
|
|
$d W x $. $d W y $. $d W z $. $d x y $. $d x z $. $d y z $.
|
|
enmap2lem4.1 $e |- W = ( s e. ( G ^m a ) |-> ( s o. `' r ) ) $.
|
|
$( Lemma for ~ enmap2 . The converse of ` W ` is a function. (Contributed
|
|
by SF, 26-Feb-2015.) $)
|
|
enmap2lem4 $p |- ( r : a -1-1-onto-> b -> Fun `' W ) $=
|
|
( vy vx vz cv wf1o wbr wa weq wi wal wceq enmap2lem3 brcnv ccnv wfun ccom
|
|
anim12d eqtr3 alrimiv alrimivv dffun2 anbi12i imbi1i albii 2albii sylibr
|
|
syl6 bitri ) EKFKDKZLZHKZIKZBMZJKZUSBMZNZHJOZPZJQZHQIQZBUAZUBZUQVFIHUQVEJ
|
|
UQVCURUSUPUCZRZVAVJRZNVDUQUTVKVBVLURUSABCDEFGSVAUSABCDEFGSUDURVAVJUEUNUFU
|
|
GVIUSURVHMZUSVAVHMZNZVDPZJQZHQIQVGIHJVHUHVQVFIHVPVEJVOVCVDVMUTVNVBUSURBTU
|
|
SVABTUIUJUKULUOUM $.
|
|
$}
|
|
|
|
${
|
|
$d a p $. $d a s $. $d b p $. $d G p $. $d G s $. $d p r $. $d p s $.
|
|
$d r s $. $d W p $.
|
|
enmap2lem5.1 $e |- W = ( s e. ( G ^m a ) |-> ( s o. `' r ) ) $.
|
|
$( Lemma for ~ enmap2 . Calculate the range of ` W ` . (Contributed by
|
|
SF, 26-Feb-2015.) $)
|
|
enmap2lem5 $p |- ( r : a -1-1-onto-> b -> ran W = ( G ^m b ) ) $=
|
|
( vp cv cmap wcel ccom wceq vex coex adantl wf syl wb cvv wf1o crn co wfn
|
|
cfv wral wss enmap2lem2 ccnv coeq1 cnvex fvmpt elmapi f1ocnv f1of syl2anr
|
|
wa elovex1 elmapg mp3an23 mpbird eqeltrd ralrimiva fnfvrnss sylancr coass
|
|
fco wbr syl6eq cid cres f1ococnv2 coeq2d fcoi1 eqtrd fnbrfvb mpbid brelrn
|
|
sylan9eq ex ssrdv eqssd ) EIZFIZDIZUAZBUBZAWDJUCZWFBAWCJUCZUDZHIZBUEZWHKZ
|
|
HWIUFWGWHUGABCDEGUHZWFWMHWIWFWKWIKZUQZWLWKWEUIZLZWHWOWLWRMWFCWKCIZWQLZWRW
|
|
IBWSWKWQUJGWKWQHNZWEDNZUKZOZULPWPWRWHKZWDAWRQZWOWCAWKQWDWCWQQZXFWFWKAWCUM
|
|
WFWDWCWQUAXGWCWDWEUNWDWCWQUORWDWCAWKWQVGUPWOXEXFSZWFWOATKZXHWKAWCJURXIWDT
|
|
KWRTKXHFNXDAWDWRTTTUSUTRPVAVBVCHWIWHBVDVEWFHWHWGWFWKWHKZWKWGKZWFXJUQZWKWE
|
|
LZWKBVHZXKXLXMBUEZWKMZXNXLXOWKWEWQLZLZWKXLXMWIKZXOXRMXLXSWCAXMQZXJWDAWKQZ
|
|
WCWDWEQXTWFWKAWDUMZWCWDWEUOWCWDAWKWEVGUPXJXSXTSZWFXJXIYCWKAWDJURXIWCTKXMT
|
|
KYCENWKWEXAXBOAWCXMTTTUSUTRPVAZCXMWTXRWIBWSXMMWTXMWQLXRWSXMWQUJWKWEWQVFVI
|
|
GWKXQXAWEWQXBXCOOULRWFXJXRWKVJWDVKZLZWKWFXQYEWKWCWDWEVLVMXJYAYFWKMYBWDAWK
|
|
VNRVSVOXLWJXSXPXNSWNYDWIXMWKBVPVEVQXMWKBVRRVTWAWB $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a r $. $d a s $. $d B b $. $d b r $.
|
|
$d C a $. $d C b $. $d C r $. $d C s $. $d r s $.
|
|
$( Set exponentiation preserves equinumerosity in the second argument.
|
|
Theorem XI.1.22 of [Rosser] p. 357. (Contributed by SF,
|
|
26-Feb-2015.) $)
|
|
enmap2 $p |- ( A ~~ B -> ( C ^m A ) ~~ ( C ^m B ) ) $=
|
|
( va vb vr vs cvv wcel wa cen wbr cmap co cv wi wceq oveq2 imbi12d wf1o
|
|
brex breq1 breq1d breq2 breq2d wex bren ccnv ccom cmpt wfn cdm enmap2lem4
|
|
wfun eqid crn dfrn4 enmap2lem5 syl5eqr jca df-fn sylibr enmap2lem2 dff1o4
|
|
mpbiran enmap2lem1 f1oen syl exlimiv sylbi vtocl2g mpcom ) AHIBHIJABKLZCA
|
|
MNZCBMNZKLZABKUADOZEOZKLZCVQMNZCVRMNZKLZPAVRKLZVNWAKLZPVMVPPDEABHHVQAQZVS
|
|
WCWBWDVQAVRKUBWEVTVNWAKVQACMRUCSVRBQZWCVMWDVPVRBAKUDWFWAVOVNKVRBCMRUESVSV
|
|
QVRFOZTZFUFWBVQVRFUGWHWBFWHVTWAGVTGOWGUHUIUJZTZWBWHWIUHZWAUKZWJWHWKUNZWKU
|
|
LZWAQZJWLWHWMWOCWIGFDEWIUOZUMWHWNWIUPWAWIUQCWIGFDEWPURUSUTWKWAVAVBWJWIVTU
|
|
KWLCWIGFDWPVCVTWAWIVDVEVBVTWAWIVQCWIGFWPVFVGVHVIVJVKVL $.
|
|
$}
|
|
|
|
${
|
|
$d A p r s x $. $d G p s x $.
|
|
enmap1lem1.1 $e |- W = ( s e. ( A ^m G ) |-> ( r o. s ) ) $.
|
|
$( Lemma for ~ enmap1 . Set up stratification. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
enmap1lem1 $p |- W e. _V $=
|
|
( vx vp c1st cv cvv c2nd ccompose wcel wceq wa cop wbr bitri 3bitri copab
|
|
ccnv csn cima cxp cin ccom ctxp cmap cres cmpt df-mpt opelres elima trtxp
|
|
wrex wex brco ancom brin vex brxp mpbiran2 eliniseg anbi1i op1st2nd exbii
|
|
co opex breq2 ceqsexv rexbii risset bitr4i df-br brcomposeg mp2an anbi2ci
|
|
wb eqcom opabbi2i 3eqtr4i 1stex cnvex snex imaex vvex xpex inex composeex
|
|
2ndex coex txpex ovex resex eqeltri ) CIUBZEJZUCZUDZKUEZLUFZIUGZLUHZMUDZA
|
|
BUIVHZUJZKDXFWRDJZUGZUKXHXFNZGJZXIOZPZDGUACXGDGXFXIULFXMDGXGXHXKQZXGNXNXE
|
|
NZXJPXMXHXKXEXFUMXOXLXJXOWRXHQZXKMRZXIXKOZXLXOXPXKQZMNZXQXOHJZXNXDRZHMUPZ
|
|
XTHXNXDMUNYCYAXSOZHMUPXTYBYDHMYBYAXHXCRZYAXKLRZPYAXPIRZYFPYDYAXHXKXCLUOYE
|
|
YGYFYEYAXKIRZXKXHXBRZPZGUQXKXPOZYHPZGUQYGGYAXHXBIURYJYLGYJYIYHPYLYHYIUSYI
|
|
YKYHYIXKXHXARZXKXHLRZPXKWRIRZYNPYKXKXHXALUTYMYOYNYMXKWTNZYOYMYPXHKNZDVAZX
|
|
KXHWTKVBVCIWRXKVDSVEWRXHXKEVAZYRVFTVESVGYHYGGXPWRXHYSYRVIZXKXPYAIVJVKTVEX
|
|
PXKYAYTGVAVFTVLHXSMVMVNSXPXKMVOVNWRKNYQXQXRVSYSYRWRXHXKKKVPVQXIXKVTTVRSWA
|
|
WBXEXFXDMXCLXBIXALWTKWQWSIWCWDWRWEWFWGWHWKWIWCWLWKWMWJWFABUIWNWOWP $.
|
|
$}
|
|
|
|
${
|
|
$d A s $. $d G s $.
|
|
enmap1lem2.1 $e |- W = ( s e. ( A ^m G ) |-> ( r o. s ) ) $.
|
|
$( Lemma for ~ enmap1 . Establish functionhood. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
enmap1lem2 $p |- W Fn ( A ^m G ) $=
|
|
( cv ccom cvv wcel cmap co wfn fnmpt vex coex a1i mprg ) EGZDGZHZIJZCABKL
|
|
ZMDUCDUCUACIFNUBTUCJSTEODOPQR $.
|
|
$}
|
|
|
|
${
|
|
$d G s $. $d r s $. $d S s $. $d A s $.
|
|
enmap1lem3.1 $e |- W = ( s e. ( A ^m G ) |-> ( r o. s ) ) $.
|
|
$( Lemma for ~ enmap2 . Binary relationship condition over ` W ` .
|
|
(Contributed by SF, 3-Mar-2015.) $)
|
|
enmap1lem3 $p |- ( r : A -1-1-onto-> B ->
|
|
( S W T -> S = ( `' r o. T ) ) ) $=
|
|
( wbr cmap co wcel cv ccom wceq wa mpan cvv coeq2 wf1o cdm wfn enmap1lem2
|
|
ccnv breldm fndm ax-mp syl6eleq cfv fnbrfvb vex coexg fvmptg mpdan eqeq1d
|
|
wb bitr3d biimpd mpcom jca coass cid f1ococnv1 coeq1d wf elmapi fcoi2 syl
|
|
cres sylan9eq syl5reqr eqeq2d syl5ibcom expimpd syl5 ) CDFJZCAEKLZMZHNZCO
|
|
ZDPZQABVTUAZCVTUEZDOZPZVQVSWBVQCFUBZVRCDFUFFVRUCZWGVRPAEFGHIUDZVRFUGUHUIZ
|
|
VSVQWBWJVSVQWBVSCFUJZDPZVQWBWHVSWLVQUQWIVRCDFUKRVSWKWADVSWASMZWKWAPVTSMVS
|
|
WMHULVTCSVRUMRGCVTGNZOWAVRSFWNCVTTIUNUOUPURUSUTVAWCVSWBWFWCVSQZCWDWAOZPWB
|
|
WFWOWPWDVTOZCOZCWDVTCVBWCVSWRVCAVJZCOZCWCWQWSCABVTVDVEVSEACVFWTCPCAEVGEAC
|
|
VHVIVKVLWBWPWECWADWDTVMVNVOVP $.
|
|
$}
|
|
|
|
${
|
|
$d G s $. $d r s $. $d r x $. $d r y $. $d r z $. $d s y $. $d s z $.
|
|
$d W x $. $d W y $. $d W z $. $d x y $. $d x z $. $d y z $. $d A s $.
|
|
$d A x $. $d A y $. $d A z $. $d B x $. $d B y $. $d B z $.
|
|
enmap1lem4.1 $e |- W = ( s e. ( A ^m G ) |-> ( r o. s ) ) $.
|
|
$( Lemma for ~ enmap2 . The converse of ` W ` is a function. (Contributed
|
|
by SF, 3-Mar-2015.) $)
|
|
enmap1lem4 $p |- ( r : A -1-1-onto-> B -> Fun `' W ) $=
|
|
( vy vx vz cv wf1o wbr wa wi wal ccnv wceq enmap1lem3 brcnv weq wfun ccom
|
|
anim12d eqtr3 alrimiv alrimivv dffun2 anbi12i imbi1i albii 2albii sylibr
|
|
syl6 bitri ) ABFKZLZHKZIKZDMZJKZUSDMZNZHJUAZOZJPZHPIPZDQZUBZUQVFIHUQVEJUQ
|
|
VCURUPQUSUCZRZVAVJRZNVDUQUTVKVBVLABURUSCDEFGSABVAUSCDEFGSUDURVAVJUEUNUFUG
|
|
VIUSURVHMZUSVAVHMZNZVDOZJPZHPIPVGIHJVHUHVQVFIHVPVEJVOVCVDVMUTVNVBUSURDTUS
|
|
VADTUIUJUKULUOUM $.
|
|
$}
|
|
|
|
${
|
|
$d G p $. $d G s $. $d p r $. $d p s $. $d r s $. $d W p $. $d A p $.
|
|
$d A s $. $d B p $.
|
|
enmap1lem5.1 $e |- W = ( s e. ( A ^m G ) |-> ( r o. s ) ) $.
|
|
$( Lemma for ~ enmap2 . Calculate the range of ` W ` . (Contributed by
|
|
SF, 3-Mar-2015.) $)
|
|
enmap1lem5 $p |- ( r : A -1-1-onto-> B -> ran W = ( B ^m G ) ) $=
|
|
( vp cv wf1o cmap wcel ccom wceq coex wf syl2an cvv wb syl crn co wfn cfv
|
|
wral wss enmap1lem2 coeq2 vex fvmpt adantl f1of elmapi fco wfo f1ofo forn
|
|
wa rnex syl6eqelr elovex2 elmapg mp3an3 mpbird eqeltrd ralrimiva fnfvrnss
|
|
sylancr ccnv wbr f1ocnv cdm f1odm dmex cnvex coass f1ococnv2 coeq1d fcoi2
|
|
cid cres sylan9eq syl5eqr eqtrd fnbrfvb mpbid brelrn ex ssrdv eqssd ) ABF
|
|
IZJZDUAZBCKUBZWLDACKUBZUCZHIZDUDZWNLZHWOUEWMWNUFACDEFGUGZWLWSHWOWLWQWOLZU
|
|
RZWRWKWQMZWNXAWRXCNWLEWQWKEIZMZXCWODXDWQWKUHGWKWQFUIZHUIZOZUJUKXBXCWNLZCB
|
|
XCPZWLABWKPCAWQPXJXAABWKULWQACUMCABWKWQUNQWLBRLZCRLZXIXJSZXAWLBWKUAZRWLAB
|
|
WKUOXNBNABWKUPABWKUQTWKXFUSUTWQACKVAXKXLXCRLXMXHBCXCRRRVBVCQVDVEVFHWOWNDV
|
|
GVHWLHWNWMWLWQWNLZWQWMLZWLXOURZWKVIZWQMZWQDVJZXPXQXSDUDZWQNZXTXQYAWKXSMZW
|
|
QXQXSWOLZYAYCNXQYDCAXSPZWLBAXRPZCBWQPZYEXOWLBAXRJYFABWKVKBAXRULTWQBCUMZCB
|
|
AXRWQUNQWLARLZXLYDYESZXOWLAWKVLRABWKVMWKXFVNUTWQBCKVAYIXLXSRLYJXRWQWKXFVO
|
|
XGOZACXSRRRVBVCQVDZEXSXEYCWODXDXSWKUHGWKXSXFYKOUJTXQYCWKXRMZWQMZWQWKXRWQV
|
|
PWLXOYNVTBWAZWQMZWQWLYMYOWQABWKVQVRXOYGYPWQNYHCBWQVSTWBWCWDXQWPYDYBXTSWTY
|
|
LWOXSWQDWEVHWFXSWQDWGTWHWIWJ $.
|
|
$}
|
|
|
|
${
|
|
$d C r $. $d C s $. $d r s $. $d A r $. $d A s $. $d B r $.
|
|
$( Set exponentiation preserves equinumerosity in the first argument.
|
|
Theorem XI.1.23 of [Rosser] p. 357. (Contributed by SF, 3-Mar-2015.) $)
|
|
enmap1 $p |- ( A ~~ B -> ( A ^m C ) ~~ ( B ^m C ) ) $=
|
|
( vr vs cen wbr cv wf1o wex cmap co bren ccom cmpt wfn ccnv wfun crn wceq
|
|
eqid enmap1lem2 a1i enmap1lem4 enmap1lem5 dff1o2 syl3anbrc enmap1lem1 syl
|
|
f1oen exlimiv sylbi ) ABFGABDHZIZDJACKLZBCKLZFGZABDMUNUQDUNUOUPEUOUMEHNOZ
|
|
IZUQUNURUOPZURQRURSUPTUSUTUNACUREDURUAZUBUCABCUREDVAUDABCUREDVAUEUOUPURUF
|
|
UGUOUPURACUREDVAUHUJUIUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
enpw1pw.1 $e |- A e. _V $.
|
|
$( Unit power class and power class commute within equivalence. Theorem
|
|
XI.1.35 of [Rosser] p. 368. (Contributed by SF, 26-Feb-2015.) $)
|
|
enpw1pw $p |- ~P1 ~P A ~~ ~P ~P1 A $=
|
|
( vy vx vz cpw cpw1 cpw1fn wf1o wbr c1c wss ax-mp wceq cv wrex wa wex vex
|
|
wcel cres cen cima wf1 pw1fnf1o f1of1 pw1ss1c f1ores mp2an wb df-ima elpw
|
|
cab sspw1 df-rex df-pw abeq2i anbi1i exbii bitr2i 3bitri csn elpw1 bitr4i
|
|
r19.41v rexcom4 snex breq1 ceqsexv bitri rexbii bitr3i abbi2i eqtr4i mpbi
|
|
brpw1fn f1oeq3 pw1fnex pwex pw1ex resex f1oen ) AFZGZAGZFZHWDUAZIZWDWFUBJ
|
|
WDHWDUCZWGIZWHKKFZHUDZWDKLWJKWKHIWLUEKWKHUFMWCUGKWKWDHUHUIWIWFNWJWHUJWICO
|
|
ZDOZHJZCWDPZDUMWFDCHWDUKWPDWFWNWFTZWNEOZGNZEWCPZWPWQWNWELWRALZWSQZERZWTWN
|
|
WEDSZULEWNAXDUNWTWRWCTZWSQZERXCWSEWCUOXFXBEXEXAWSXAEWCEAUPUQURUSUTVAWPWMW
|
|
DTZWOQZCRWMWRVBZNZWOQZEWCPZCRZWTWOCWDUOXHXLCXHXJEWCPZWOQXLXGXNWOEWMWCVCUR
|
|
XJWOEWCVEVDUSXMXKCRZEWCPWTXKECWCVFXOWSEWCXOXIWNHJZWSWOXPCXIWRVGWMXIWNHVHV
|
|
IWRWNESVPVJVKVLVAVDVMVNWIWFWDWGVQMVOWDWFWGHWDVRWCABVSVTWAWBM $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d B r $. $d r x $. $d r y $. $d x y $. $d r t $. $d t x $.
|
|
$d t y $.
|
|
enprmaplem1.1 $e |- W = ( r e. ( A ^m B ) |-> ( `' r " { x } ) ) $.
|
|
$( Lemma for ~ enprmap . Set up stratification. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
enprmaplem1 $p |- W e. _V $=
|
|
( vy vt csset c1st c2nd ccnv cv cima cop wcel wbr wa vex bitri co cvv cxp
|
|
cmap cins3 csn cres csi ctxp c1c csymdif ccompl cin cmpt wceq wel elima1c
|
|
cins2 oteltxp opsnelsi df-br brres eliniseg anbi2i bitr3i op1st2nd 3bitri
|
|
wex opelssetsn anbi12i exbii opex eleq1 bitr4i releqmpt eqtr4i ovex 1stex
|
|
ceqsexv 2ndex cnvex snex imaex resex siex ssetex txpex mptexlem eqeltri
|
|
1cex ) DBCUDUAZUBUCIUEJKLZAMZUFZNZUGZUHZIUIZUJNZURUKUJNULLUMZUBDEWKEMZLWN
|
|
NZUNWTFEGWKWSXBGMZUFZXAOZWSPZXCWMXAQZXCXBPXFHMZXCWMOZUOZHEUPZRZHVHZXGXFXH
|
|
UFZXEOWRPZHVHXMHXEWRUQXOXLHXOXNXDOWQPZXNXAOIPZRXLXNXDXAWQIUSXPXJXQXKXPXHX
|
|
COWPPZXHXCJQZXHWMKQZRZXJXHXCWPHSZGSZUTXRXHXCWPQZYAXHXCWPVAYDXSXHWOPZRYAXH
|
|
XCJWOVBYEXTXSKWMXHVCVDTVEXCWMXHYCASZVFVGXHXAYBESVIVJTVKTXMXIXAPZXGXKYGHXI
|
|
XCWMYCYFVLXHXIXAVMVSXCWMXAVAVNTXAWMXCVCVNVOVPWKWSBCUDVQWRUJWQIWPJWOVRWLWN
|
|
KVTWAWMWBWCWDWEWFWGWJWCWHWI $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d B r $.
|
|
enprmaplem2.1 $e |- W = ( r e. ( A ^m B ) |-> ( `' r " { x } ) ) $.
|
|
$( Lemma for ~ enprmap . Establish functionhood. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
enprmaplem2 $p |- W Fn ( A ^m B ) $=
|
|
( cv ccnv csn cima cvv wcel cmap co wfn fnmpt vex cnvex snex imaex mprg
|
|
a1i ) EGZHZAGZIZJZKLZDBCMNZOEUIEUIUGDKFPUHUCUILUDUFUCEQRUESTUBUA $.
|
|
$}
|
|
|
|
${
|
|
$d A p $. $d A q $. $d A r $. $d A w $. $d A z $. $d B r $. $d B w $.
|
|
$d B z $. $d p q $. $d p r $. $d p w $. $d p x $. $d p y $. $d p z $.
|
|
$d q r $. $d q w $. $d q x $. $d q y $. $d q z $. $d r x $. $d W p $.
|
|
$d W q $. $d w x $. $d w y $. $d w z $. $d W z $. $d x z $. $d y z $.
|
|
enprmaplem3.1 $e |- W = ( r e. ( A ^m B ) |-> ( `' r " { x } ) ) $.
|
|
$( Lemma for ~ enprmap . The converse of ` W ` is a function.
|
|
(Contributed by SF, 3-Mar-2015.) $)
|
|
enprmaplem3 $p |- ( ( x =/= y /\ A = { x , y } ) -> Fun `' W ) $=
|
|
( vz vp vq vw cv wceq wa wbr weq wi wcel syl sylan wne cpr ccnv wfun cmap
|
|
wal co csn cima brcnv cdm breldm wfn enprmaplem2 ax-mp syl6eleq cfv fnfun
|
|
fndm funbrfv cnveq imaeq1d vex cnvex snex imaex fvmpt jca anim12i syl2anb
|
|
eqtr3d wf elmapi eqtr2 simprll ffn simprlr ffvelrn wo simpllr eleq2d fvex
|
|
elpr syl6bb simprr simplrr eliniseg 3bitr3g biimpd fnbrfvb 3imtr4d eqtr4d
|
|
wb impr expr simplll neneqd adantr ffun fununiq 3expib ancomsd 3syl exp3a
|
|
mtod biimprd nsyld simprl fdm eleqtrrd wex eldm crn brelrn wss sseld syl5
|
|
wn frn breq2 biimpcd orim12d com12 sylbi syl6bi com23 mpdd exlimdv syl5bi
|
|
mpd orel1 sylc jaod sylbid eqfnfvd expcom syl2an an4s alrimiv alrimivv
|
|
dffun2 sylibr ) ALZBLZUAZCUUCUUDUBZMZNZHLZILZEUCZOZUUIJLZUUKOZNZIJPZQZJUF
|
|
ZIUFHUFUUKUDUUHUURHIUUHUUQJUUOUUJCDUEUGZRZUUIUUJUCZUUCUHZUIZMZNZUUMUUSRZU
|
|
UIUUMUCZUVBUIZMZNZNZUUHUUPUULUUJUUIEOZUUMUUIEOZUVKUUNUUIUUJEUJUUIUUMEUJUV
|
|
LUVEUVMUVJUVLUUTUVDUVLUUJEUKZUUSUUJUUIEULEUUSUMZUVNUUSMACDEFGUNZUUSEUSUOZ
|
|
UPZUVLUUJEUQZUUIUVCEUDZUVLUVSUUIMQUVOUVTUVPUUSEURUOZUUJUUIEUTUOUVLUUTUVSU
|
|
VCMUVRFUUJFLZUCZUVBUIZUVCUUSEFIPUWCUVAUVBUWBUUJVAVBGUVAUVBUUJIVCVDUUCVEZV
|
|
FVGSVKVHUVMUVFUVIUVMUUMUVNUUSUUMUUIEULUVQUPZUVMUUMEUQZUUIUVHUVTUVMUWGUUIM
|
|
QUWAUUMUUIEUTUOUVMUVFUWGUVHMUWFFUUMUWDUVHUUSEFJPUWCUVGUVBUWBUUMVAVBGUVGUV
|
|
BUUMJVCVDUWEVFVGSVKVHVIVJUVKUUHUUPUUTUVFUVDUVIUUHUUPQZUUTUVFNDCUUJVLZDCUU
|
|
MVLZNZUVCUVHMZUWHUVDUVINUUTUWIUVFUWJUUJCDVMUUMCDVMVIUUIUVCUVHVNUUHUWKUWLN
|
|
ZUUPUUHUWMNZHDUUJUUMUWNUWIUUJDUMZUUHUWIUWJUWLVOZDCUUJVPSZUWNUWJUUMDUMZUUH
|
|
UWIUWJUWLVQZDCUUMVPSZUWNUUIDRZNZUUIUUJUQZCRZUXCUUIUUMUQZMZUWNUWIUXAUXDUWP
|
|
DCUUIUUJVRTUXBUXDUXCUUCMZUXCUUDMZVSZUXFUXBUXDUXCUUFRUXIUXBCUUFUXCUUEUUGUW
|
|
MUXAVTWAUXCUUCUUDUUIUUJWBWCWDUXBUXGUXFUXHUWNUXAUXGUXFUWNUXAUXGNNUXCUUCUXE
|
|
UWNUXAUXGWEUWNUXAUXGUXEUUCMZUXBUUIUUCUUJOZUUIUUCUUMOZUXGUXJUXBUXKUXLUXBUU
|
|
IUVCRUUIUVHRUXKUXLUXBUVCUVHUUIUUHUWKUWLUXAWFWAUUJUUCUUIWGUUMUUCUUIWGWHZWI
|
|
UWNUWOUXAUXGUXKWMUWQDUUIUUCUUJWJTUWNUWRUXAUXJUXLWMUWTDUUIUUCUUMWJTWKWNWLW
|
|
OUWNUXAUXHUXFUWNUXAUXHNNUXCUUDUXEUWNUXAUXHWEUWNUXAUXHUXEUUDMZUXBUUIUUDUUJ
|
|
OZUUIUUDUUMOZUXHUXNUWNUXAUXOUXPUWNUXAUXONZNZUXLXRZUXLUXPVSZUXPUWNUXAUXOUX
|
|
SUXBUXOUXKUXLUWNUXAUXOUXKXRUXRUXKABPZUXRUUCUUDUUEUUGUWMUXQWPWQUWNUXAUXOUX
|
|
KUYAQUXBUXOUXKUYAUXBUWIUUJUDZUXOUXKNUYAQUWNUWIUXAUWPWRDCUUJWSUYBUXKUXOUYA
|
|
UYBUXKUXOUYAUUIUUCUUDUUJWTXAXBXCXDWNXEWOUXBUXKUXLUXMXFXGWNUXRUUIUUMUKZRZU
|
|
XTUXRUUIDUYCUWNUXAUXOXHUXRUWJUYCDMUWNUWJUXQUWSWRZDCUUMXISXJUYDUUIKLZUUMOZ
|
|
KXKUXRUXTKUUIUUMXLUXRUYGUXTKUXRUYGUYFCRZUXTUYGUYFUUMXMZRUXRUYHUUIUYFUUMXN
|
|
UXRUYICUYFUXRUWJUYICXOUYEDCUUMXSSXPXQUXRUYHUYGUXTUXRUYHUYFUUFRZUYGUXTQZUX
|
|
RCUUFUYFUUEUUGUWMUXQVTWAUYJKAPZKBPZVSZUYKUYFUUCUUDKVCWCUYGUYNUXTUYGUYLUXL
|
|
UYMUXPUYLUYGUXLUYFUUCUUIUUMXTYAUYMUYGUXPUYFUUDUUIUUMXTYAYBYCYDYEYFYGYHYIY
|
|
JUXLUXPYKYLWOUWNUWOUXAUXHUXOWMUWQDUUIUUDUUJWJTUWNUWRUXAUXNUXPWMUWTDUUIUUD
|
|
UUMWJTWKWNWLWOYMYNYJYOYPYQYRYCXQYSYTHIJUUKUUAUUB $.
|
|
$}
|
|
|
|
${
|
|
$d B u $. $d p u $. $d u x $. $d u y $. $d p z $. $d u z $. $d x z $.
|
|
$d y z $.
|
|
enprmaplem4.1 $e |- R = ( u e. B |-> if ( u e. p , x , y ) ) $.
|
|
enprmaplem4.2 $e |- B e. _V $.
|
|
$( Lemma for ~ enprmap . More stratification condition setup.
|
|
(Contributed by SF, 3-Mar-2015.) $)
|
|
enprmaplem4 $p |- R e. _V $=
|
|
( vz cvv cxp cv cpw1 ccompl ccnv wel wcel wa bitri vex csset csymdif cima
|
|
cins3 cun cins2 c1c cin cif cmpt csn cop wn wo elun opelxp snelpw1 anbi2i
|
|
elcompl anbi12i orbi12i opelcnv elif 3bitr4i releqmpt eqtr4i xpex complex
|
|
pw1ex unex cnvex mptexlem eqeltri ) EDJKUAUDFLZALZMZKZVNNZBLZMZKZUEZOZUFU
|
|
BUGUCNOUHZJECDCFPZVOVSUIZUJWDGCIDWCWFCLZILZUKZULZWBQZWEIAPZRZWEUMZIBPZRZU
|
|
NZWIWGULWCQWHWFQWKWJVQQZWJWAQZUNWQWJVQWAUOWRWMWSWPWRWEWIVPQZRWMWGWIVNVPUP
|
|
WTWLWEWHVOUQURSWSWGVRQZWIVTQZRWPWGWIVRVTUPXAWNXBWOWGVNCTUSWHVSUQUTSVASWIW
|
|
GWBVBWEWHVOVSVCVDVEVFDWCHWBVQWAVNVPFTZVOATVIVGVRVTVNXCVHVSBTVIVGVJVKVLVM
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A p $. $d A r $. $d A u $. $d A z $. $d B p $. $d B r $. $d B u $.
|
|
$d B z $. $d p u $. $d p x $. $d p y $. $d p z $. $d R r $. $d r x $.
|
|
$d R z $. $d u x $. $d u y $. $d u z $. $d W p $. $d x z $. $d y z $.
|
|
enprmaplem5.1 $e |- W = ( r e. ( A ^m B ) |-> ( `' r " { x } ) ) $.
|
|
enprmaplem5.2 $e |- R = ( u e. B |-> if ( u e. p , x , y ) ) $.
|
|
enprmaplem5.3 $e |- B e. _V $.
|
|
$( Lemma for ~ enprmap . Establish that ` ~P B ` is a subset of the range
|
|
of ` W ` . (Contributed by SF, 3-Mar-2015.) $)
|
|
enprmaplem5 $p |- ( ( x =/= y /\ A = { x , y } ) -> ~P B C_ ran W ) $=
|
|
( vz cv wceq wa wcel vex cvv syl wne cpr cpw crn wss elpw wbr w3a cfv csn
|
|
ccnv cima cmap co wf wel cif wral wo ifeqor ifex elpr syl5eleqr ralrimivw
|
|
mpbir id fmpt sylib prex eleq1 mpbiri enprmaplem4 elmapg mp3an23 3ad2ant2
|
|
mpbird cnveq imaeq1d cnvex snex imaex fvmpt eliniseg cdm breldm wfn fnmpt
|
|
wb a1i mprg fndm ax-mp syl6eleq fnbrfvb mpan biimprd com12 jcai weq ifbid
|
|
eqeq1d biimpd imp wn simpl1 df-ne wi iffalse eqeq2d eqcoms adantl mt3d ex
|
|
ssel2 3ad2antl3 iftrue eqtrd mpbid impbid syl5bb eqrdv enprmaplem2 3expia
|
|
syl5 brelrn syl6 syl5bi ssrdv ) ANZBNZUAZDYIYJUBZOZPZIEUCZGUDZINZYOQYQEUE
|
|
ZYNYQYPQZYQEIRUFYNYRFYQGUGZYSYKYMYRYTYKYMYRUHZFGUIZYQOZYTUUAUUBFUKZYIUJZU
|
|
LZYQUUAFDEUMUNZQZUUBUUFOYMYKUUHYRYMUUHEDFUOZYMCIUPZYIYJUQZDQZCEURUUIYMUUL
|
|
CEYMUUKYLDUUKYLQUUKYIOUUKYJOUSUUJYIYJUTUUKYIYJUUJYIYJARZBRZVAZVBVEYMVFVCV
|
|
DCEDUUKFKVGVHYMDSQZUUHUUIWHZYMUUPYLSQYIYJVIDYLSVJVKUUPESQFSQUUQLABCEFIKLV
|
|
LZDEFSSSVMVNTVPVOZHFHNZUKZUUEULUUFUUGGUUTFOUVAUUDUUEUUTFVQVRJUUDUUEFUURVS
|
|
YIVTWAWBTUUAMUUFYQMNZUUFQUVBYIFUGZUUAMIUPZFYIUVBWCUUAUVCUVDUVCUVBEQZUVBFU
|
|
IZYIOZPZUUAUVDUVCUVEUVGUVCUVBFWDZEUVBYIFWEFEWFZUVIEOUUKSQZUVJCECEUUKFSKWG
|
|
UVKCNZEQUUOWIWJZEFWKWLWMUVEUVCUVGUVEUVGUVCUVJUVEUVGUVCWHZUVMEUVBYIFWNWOZW
|
|
PWQWRUVHUVDYIYJUQZYIOZUUAUVDUVEUVGUVQUVEUVGUVQUVEUVFUVPYICUVBUUKUVPEFCMWS
|
|
UUJUVDYIYJUVLUVBYQVJWTKUVDYIYJUUMUUNVAWBZXAXBXCUUAUVQUVDUUAUVQPZUVDABWSZU
|
|
VSYKUVTXDYKYMYRUVQXEYIYJXFVHUVQUVDXDZUVTXGZUUAUWBYIUVPUWAYIUVPOZUVTUWAUWC
|
|
UVTUWAUVPYJYIUVDYIYJXHXIXBWQXJXKXLXMYDYDUUAUVDUVCUUAUVDPZUVGUVCUWDUVFUVPY
|
|
IUWDUVEUVFUVPOYRYKUVDUVEYMYQEUVBXNXOZUVRTUVDUVQUUAUVDYIYJXPXKXQUWDUVEUVNU
|
|
WEUVOTXRXMXSXTYAXQUUAUUHUUCYTWHZUUSGUUGWFUUHUWFADEGHJYBUUGFYQGWNWOTXRYCFY
|
|
QGYEYFYGYH $.
|
|
$}
|
|
|
|
${
|
|
$d A p $. $d A r $. $d A s $. $d A u $. $d B p $. $d B r $. $d B s $.
|
|
$d B u $. $d p r $. $d p s $. $d p u $. $d p x $. $d p y $. $d r s $.
|
|
$d r u $. $d r x $. $d r y $. $d s x $. $d s y $. $d u x $. $d u y $.
|
|
$d W p $. $d W s $.
|
|
enprmaplem6.1 $e |- W = ( r e. ( A ^m B ) |-> ( `' r " { x } ) ) $.
|
|
enprmaplem6.2 $e |- B e. _V $.
|
|
$( Lemma for ~ enprmap . The range of ` W ` is ` ~P B ` . (Contributed by
|
|
SF, 3-Mar-2015.) $)
|
|
enprmaplem6 $p |- ( ( x =/= y /\ A = { x , y } ) -> ran W = ~P B ) $=
|
|
( vp vs vu cv wceq wa crn wss wcel cdm wb ccnv wne cpr cpw wbr wex co cfv
|
|
cmap breldm wfn enprmaplem2 fndm syl6eleq fnbrfvb sylancr ibir jca wi w3a
|
|
ax-mp csn cima weq cnveq imaeq1d cnvex snex imaex eqeq1d 3ad2ant3 imassrn
|
|
vex fvmpt df-dm elmapi fdm eqimss syl5eqssr syl5ss sseq1 syl5ibcom sylbid
|
|
wf 3syl 3expia imp3a syl5 exlimdv elrn elpw 3imtr4g ssrdv wel enprmaplem5
|
|
cif cmpt eqid eqssd ) ALZBLZUAZCWSWTUBMZNZEOZDUCZXCIXDXEXCJLZILZEUDZJUEXG
|
|
DPZXGXDQXGXEQXCXHXIJXHXFCDUHUFZQZXFEUGZXGMZNXCXIXHXKXMXHXFERZXJXFXGEUIEXJ
|
|
UJZXNXJMACDEFGUKZXJEULUTUMZXHXMXHXOXKXMXHSXPXQXJXFXGEUNUOUPUQXCXKXMXIXAXB
|
|
XKXMXIURXAXBXKUSZXMXFTZWSVAZVBZXGMZXIXKXAXMYBSXBXKXLYAXGFXFFLZTZXTVBYAXJE
|
|
FJVCYDXSXTYCXFVDVEGXSXTXFJVLVFWSVGVHVMVIVJXRYADPZYBXIXKXAYEXBXKYAXSOZDXSX
|
|
TVKXKYFXFRZDXFVNXKDCXFWCYGDMYGDPXFCDVODCXFVPYGDVQWDVRVSVJYAXGDVTWAWBWEWFW
|
|
GWHJXGEWIXGDIVLWJWKWLABKCDKDKIWMWSWTWOWPZEFIGYHWQHWNWR $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d B r $. $d r x $. $d r y $.
|
|
enprmap.1 $e |- B e. _V $.
|
|
$( A mapping from a two element pair onto a set is equinumerous with the
|
|
power class of the set. Theorem XI.1.28 of [Rosser] p. 360.
|
|
(Contributed by SF, 3-Mar-2015.) $)
|
|
enprmap $p |- ( ( x =/= y /\ A = { x , y } ) -> ( A ^m B ) ~~ ~P B ) $=
|
|
( vr cv wne cpr wceq wa cmap co cpw ccnv csn cima cmpt wf1o cen wfun eqid
|
|
wbr wfn crn enprmaplem2 a1i enprmaplem3 enprmaplem6 syl3anbrc enprmaplem1
|
|
dff1o2 f1oen syl ) AGZBGZHCUOUPIJKZCDLMZDNZFURFGOUOPQRZSZURUSTUCUQUTURUDZ
|
|
UTOUAUTUEUSJVAVBUQACDUTFUTUBZUFUGABCDUTFVCUHABCDUTFVCEUIURUSUTULUJURUSUTA
|
|
CDUTFVCUKUMUN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d B x $. $d B y $. $d C x $. $d C y $. $d P x $. $d P y $.
|
|
$d x y $.
|
|
enprmapc.1 $e |- A e. _V $.
|
|
enprmapc.2 $e |- B e. _V $.
|
|
enprmapc.3 $e |- C e. _V $.
|
|
$( A mapping from a two element pair onto a set is equinumerous with the
|
|
power class of the set. Theorem XI.1.28 of [Rosser] p. 360.
|
|
(Contributed by SF, 3-Mar-2015.) $)
|
|
enprmapc $p |- ( ( A =/= B /\ P = { A , B } ) -> ( P ^m C ) ~~ ~P C ) $=
|
|
( vx vy cv wne cpr wceq wa cmap wi eqeq2d anbi12d imbi1d vtocl co cpw cen
|
|
wbr neeq1 preq1 neeq2 preq2 enprmap ) HJZBKZDUJBLZMZNZDCOUACUBUCUDZPZABKZ
|
|
DABLZMZNZUOPHAEUJAMZUNUTUOVAUKUQUMUSUJABUEVAULURDUJABUFQRSUJIJZKZDUJVBLZM
|
|
ZNZUOPUPIBFVBBMZVFUNUOVGVCUKVEUMVBBUJUGVGVDULDVBBUJUHQRSHIDCGUITT $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d r x $. $d r y $. $d x y $.
|
|
nenpw1pwlem1.1 $e |- S = { x e. A | -. x e. ( r ` { x } ) } $.
|
|
$( Lemma for ~ nenpw1pw . Set up stratification. (Contributed by SF,
|
|
10-Mar-2015.) $)
|
|
nenpw1pwlem1 $p |- ( A e. V -> S e. _V ) $=
|
|
( vy wcel cv csn cfv wn cin cvv csset vex wceq wa wex bitri crab cfullfun
|
|
cab dfrab2 eqtri cdm cuni1 ccompl elcompl wel cop elin wbr snex brfullfun
|
|
eldm2 df-br eqcom 3bitr3i opelssetsn anbi12i exbii eluni1 3bitr4i xchbinx
|
|
fvex clel3 abbi2i fullfunex ssetex inex dmex uni1ex complex eqeltrri mpan
|
|
inexg syl5eqel ) BDHZCAIZVTJZEIZKZHZLZAUCZBMZNCWEABUAWGFWEABUDUEWFNHVSWGN
|
|
HWBUBZOMZUFZUGZUHZWFNWEAWLVTWLHVTWKHZWDVTWKAPZUIWAWJHZGIZWCQZAGUJZRZGSZWM
|
|
WDWOWAWPUKZWIHZGSWTGWAWIUPXBWSGXBXAWHHZXAOHZRWSXAWHOULXCWQXDWRWAWPWHUMWCW
|
|
PQXCWQWAWPWBVTUNUOWAWPWHUQWCWPURUSVTWPWNGPUTVATVBTVTWJWNVCGVTWCWAWBVFVGVD
|
|
VEVHWKWJWIWHOWBEPVIVJVKVLVMVNVOWFBNDVQVPVR $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d A x $. $d r x $. $d A u $. $d r u $. $d S u $. $d u y $.
|
|
$d A y $. $d r y $. $d S y $. $d x y $.
|
|
nenpw1pwlem2.1 $e |- S = { x e. A | -. x e. ( r ` { x } ) } $.
|
|
$( Lemma for ~ nenpw1pw . Establish the main theorem with an extra
|
|
hypothesis. (Contributed by SF, 10-Mar-2015.) $)
|
|
nenpw1pwlem2 $p |- -. ~P1 A ~~ ~P A $=
|
|
( vy vu wbr cv wcel wn wb wrex wex cvv syl csn cfv wceq eleq2d cpw pm5.19
|
|
cpw1 cen a1i nrex nex wf1o bren cdm f1odm vex dmex syl6eqelr pw1exb sylib
|
|
nenpw1pwlem1 wss crab ssrab2 eqsstri elpwg mpbiri crn wfo f1ofo forn elrn
|
|
wa breldm adantl adantr eleqtrd elpw1 wi w3a breq1 3anbi2d id sneq fveq2d
|
|
weq eleq12d notbid elrab2 simp3 biantrurd simp2 wfn f1ofn snelpw1 biimpri
|
|
3ad2ant1 3ad2ant3 fnbrfvb syl2anc mpbird bitr3d syl5bb syl6bi com12 3expa
|
|
reximdva syl5bi mpd ex exlimdv sylbird eximi sylbi mto ) BUCZBUAZUDHZFIZC
|
|
JZXPKZLZFBMZDNZXSDXRFBXRKXOBJZXPUBUEUFUGXNXLXMDIZUHZDNXTXLXMDUIYCXSDYCCXM
|
|
JZXSYCCOJZYDYCBOJZYEYCXLOJYFYCXLYBUJZOXLXMYBUKZYBDULUMUNBUOUPABCODEUQPYEY
|
|
DCBURCAIZYIQZYBRZJZKZABUSBEYMABUTVACBOVBVCPYCYDCYBVDZJZXSYCYNXMCYCXLXMYBV
|
|
EYNXMSXLXMYBVFXLXMYBVGPTYOGIZCYBHZGNYCXSGCYBVHYCYQXSGYCYQXSYCYQVIZYPXLJZX
|
|
SYRYPYGXLYQYPYGJYCYPCYBVJVKYCYGXLSYQYHVLVMYSYPXOQZSZFBMYRXSFYPBVNYRUUAXRF
|
|
BYCYQYAUUAXRVOUUAYCYQYAVPZXRUUAUUBYCYTCYBHZYAVPZXRUUAYQUUCYCYAYPYTCYBVQVR
|
|
XPYAXOYTYBRZJZKZVIZUUDXQYMUUGAXOBCAFWBZYLUUFUUIYIXOYKUUEUUIVSUUIYJYTYBYIX
|
|
OVTWAWCWDEWEUUDUUGUUHXQUUDYAUUGYCUUCYAWFWGUUDUUFXPUUDUUECXOUUDUUECSZUUCYC
|
|
UUCYAWHUUDYBXLWIZYTXLJZUUJUUCLYCUUCUUKYAXLXMYBWJWMYAYCUULUUCUULYAXOBWKWLW
|
|
NXLYTCYBWOWPWQTWDWRWSWTXAXBXCXDXEXFXGXDXHXEXIXJXK $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d A x $. $d r x $.
|
|
$( No unit power class is equinumerous with the corresponding power class.
|
|
Theorem XI.1.6 of [Rosser] p. 347. (Contributed by SF, 10-Mar-2015.) $)
|
|
nenpw1pw $p |- -. ~P1 A ~~ ~P A $=
|
|
( vx vr cv csn cfv wcel wn crab eqid nenpw1pwlem2 ) BABDZLECDFGHBAIZCMJK
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d B b $.
|
|
$( If ` A ` and ` B ` are equinumerous, then so are their power sets.
|
|
Theorem XI.1.36 of [Rosser] p. 369. (Contributed by SF,
|
|
17-Mar-2015.) $)
|
|
enpw $p |- ( A ~~ B -> ~P A ~~ ~P B ) $=
|
|
( va vb cvv wcel cen wbr cpw cv wi wceq pweq imbi12d c0 cmap vn0 vvex 0ex
|
|
co wa brex breq1 breq1d breq2 breq2d cpr enmap2 eqid enprmapc mp2an ensym
|
|
wne vex mpbir entr mpan2 sylancr syl vtocl2g mpcom ) AEFBEFUAABGHZAIZBIZG
|
|
HZABGUBCJZDJZGHZVFIZVGIZGHZKAVGGHZVCVJGHZKVBVEKCDABEEVFALZVHVLVKVMVFAVGGU
|
|
CVNVIVCVJGVFAMUDNVGBLZVLVBVMVEVGBAGUEVOVJVDVCGVGBMUFNVHEOUGZVFPTZVPVGPTZG
|
|
HZVKVFVGVPUHVSVIVQGHZVQVJGHZVKVTVQVIGHZEOUMZVPVPLZWBQVPUIZEOVFVPRSCUNUJUK
|
|
VIVQULUOVSVRVJGHZWAWCWDWFQWEEOVGVPRSDUNUJUKVQVRVJUPUQVIVQVJUPURUSUTVA $.
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Cardinal numbers
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$c NC $. $( The set of cardinal numbers. $)
|
|
$c <_c $. $( Cardinal less than or equal. $)
|
|
$c <c $. $( Cardinal less than. $)
|
|
$c Nc $. $( Cardinality operation. $)
|
|
$c .c $. $( Cardinal multiplication. $)
|
|
$c T_c $. $( Cardinal type raising. $)
|
|
$c 2c $. $( Cardinal two. $)
|
|
$c 3c $. $( Cardinal three. $)
|
|
$c ^c $. $( Cardinal exponentiation. $)
|
|
$c TcFn $. $( Stratified T-raising. $)
|
|
|
|
$( Extend the definition of a class to include the set of cardinal
|
|
numbers. $)
|
|
cncs $a class NC $.
|
|
|
|
$( Extend the definition of a class to include cardinal less than or
|
|
equal. $)
|
|
clec $a class <_c $.
|
|
|
|
$( Extend the definition of a class to include cardinal strict less than. $)
|
|
cltc $a class <c $.
|
|
|
|
$( Extend the definition of a class to include the cardinality operation. $)
|
|
cnc $a class Nc A $.
|
|
|
|
$( Extend the definition of a class to include cardinal multiplication. $)
|
|
cmuc $a class .c $.
|
|
|
|
$( Extend the definition of a class to include cardinal type raising. $)
|
|
ctc $a class T_c A $.
|
|
|
|
$( Extend the definition of a class to include cardinal two. $)
|
|
c2c $a class 2c $.
|
|
|
|
$( Extend the definition of a class to include cardinal three. $)
|
|
c3c $a class 3c $.
|
|
|
|
$( Extend the definition of a class to include cardinal exponentiation. $)
|
|
cce $a class ^c $.
|
|
|
|
$( Extend the definition of a class to include the stratified T-raising
|
|
function. $)
|
|
ctcfn $a class TcFn $.
|
|
|
|
$( Define the set of all cardinal numbers. We define them as equivalence
|
|
classes of sets of the same size. Definition from [Rosser] p. 372.
|
|
(Contributed by Scott Fenton, 24-Feb-2015.) $)
|
|
df-ncs $a |- NC = ( _V /. ~~ ) $.
|
|
|
|
${
|
|
$d x y a b $.
|
|
$( Define cardinal less than or equal. Definition from [Rosser] p. 375.
|
|
(Contributed by Scott Fenton, 24-Feb-2015.) $)
|
|
df-lec $a |- <_c = { <. a , b >. | E. x e. a E. y e. b x C_ y } $.
|
|
$}
|
|
|
|
$( Define cardinal less than. Definition from [Rosser] p. 375. (Contributed
|
|
by Scott Fenton, 24-Feb-2015.) $)
|
|
df-ltc $a |- <c = ( <_c \ _I ) $.
|
|
|
|
$( Define the cardinality operation. This is the unique cardinal number
|
|
containing a given set. Definition from [Rosser] p. 371. (Contributed by
|
|
Scott Fenton, 24-Feb-2015.) $)
|
|
df-nc $a |- Nc A = [ A ] ~~ $.
|
|
|
|
${
|
|
$d m n a b g $.
|
|
$( Define cardinal multiplication. Definition from [Rosser] p. 378.
|
|
(Contributed by Scott Fenton, 24-Feb-2015.) $)
|
|
df-muc $a |- .c = ( m e. NC , n e. NC |->
|
|
{ a | E. b e. m E. g e. n a ~~ ( b X. g ) } ) $.
|
|
$}
|
|
|
|
${
|
|
$d A b x $.
|
|
$( Define the type-raising operation on a cardinal number. This is the
|
|
unique cardinal containing the unit power classes of the elements of the
|
|
given cardinal. Definition adapted from [Rosser] p. 528. (Contributed
|
|
by Scott Fenton, 24-Feb-2015.) $)
|
|
df-tc $a |- T_c A = ( iota b ( b e. NC /\ E. x e. A b = Nc ~P1 x ) ) $.
|
|
$}
|
|
|
|
$( Define cardinal two. This is the set of all sets with two unique
|
|
elements. (Contributed by Scott Fenton, 24-Feb-2015.) $)
|
|
df-2c $a |- 2c = Nc { (/) , _V } $.
|
|
|
|
$( Define cardinal three. This is the set of all sets with three unique
|
|
elements. (Contributed by Scott Fenton, 24-Feb-2015.) $)
|
|
df-3c $a |- 3c = Nc { (/) , _V , ( _V \ { (/) } ) } $.
|
|
|
|
${
|
|
$d n m g a b $.
|
|
$( Define cardinal exponentiation. Definition from [Rosser] p. 381.
|
|
(Contributed by Scott Fenton, 24-Feb-2015.) $)
|
|
df-ce $a |- ^c = ( n e. NC , m e. NC |->
|
|
{ g | E. a E. b ( ~P1 a e. n /\ ~P1 b e. m /\
|
|
g ~~ ( a ^m b ) ) } ) $.
|
|
$}
|
|
|
|
$( Define the stratified T-raising function. (Contributed by Scott Fenton,
|
|
24-Feb-2015.) $)
|
|
df-tcfn $a |- TcFn = ( x e. 1c |-> T_c U. x ) $.
|
|
|
|
$( Cardinality equality law. (Contributed by SF, 24-Feb-2015.) $)
|
|
nceq $p |- ( A = B -> Nc A = Nc B ) $=
|
|
( wceq cen cec cnc eceq1 df-nc 3eqtr4g ) ABCADEBDEAFBFABDGAHBHI $.
|
|
|
|
${
|
|
nceqi.1 $e |- A = B $.
|
|
$( Equality inference for cardinality. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
nceqi $p |- Nc A = Nc B $=
|
|
( wceq cnc nceq ax-mp ) ABDAEBEDCABFG $.
|
|
$}
|
|
|
|
${
|
|
nceqd.1 $e |- ( ph -> A = B ) $.
|
|
$( Equality deduction for cardinality. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
nceqd $p |- ( ph -> Nc A = Nc B ) $=
|
|
( wceq cnc nceq syl ) ABCEBFCFEDBCGH $.
|
|
$}
|
|
|
|
$( The class of all cardinal numbers is a set. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
ncsex $p |- NC e. _V $=
|
|
( cncs cvv cen cqs df-ncs enex vvex qsex eqeltri ) ABCDBEBCFGHI $.
|
|
|
|
${
|
|
$d A a b x $. $d B a b x y $.
|
|
$( Binary relationship form of cardinal less than or equal. (Contributed
|
|
by SF, 24-Feb-2015.) $)
|
|
brlecg $p |- ( ( A e. V /\ B e. W ) ->
|
|
( A <_c B <-> E. x e. A E. y e. B x C_ y ) ) $=
|
|
( vb va cv wss wrex clec rexeq wceq rexbidv df-lec brabg ) AIBIJZBGIZKZAH
|
|
IZKTACKRBDKZACKHGCDEFLTAUACMSDNTUBACRBSDMOABHGPQ $.
|
|
|
|
brlec.1 $e |- A e. _V $.
|
|
brlec.2 $e |- B e. _V $.
|
|
$( Binary relationship form of cardinal less than or equal. (Contributed
|
|
by SF, 24-Feb-2015.) $)
|
|
brlec $p |- ( A <_c B <-> E. x e. A E. y e. B x C_ y ) $=
|
|
( cvv wcel clec wbr cv wss wrex wb brlecg mp2an ) CGHDGHCDIJAKBKLBDMACMNE
|
|
FABCDGGOP $.
|
|
$}
|
|
|
|
$( Binary relationship form of cardinal less than. (Contributed by SF,
|
|
4-Mar-2015.) $)
|
|
brltc $p |- ( A <c B <-> ( A <_c B /\ A =/= B ) ) $=
|
|
( cltc wbr cvv wcel clec wne wa brex simprd adantr cid wn cdif df-ltc breqi
|
|
brdif bitri ideqg necon3bbid anbi2d syl5bb pm5.21nii ) ABCDZBEFZABGDZABHZIZ
|
|
UEAEFZUFABCJKUGUFUHUGUJUFABGJKLUEUGABMDZNZIZUFUIUEABGMOZDUMABCUNPQABGMRSUFU
|
|
LUHUGUFUKABABETUAUBUCUD $.
|
|
|
|
${
|
|
$d a b $. $d a t $. $d a u $. $d a x $. $d a y $. $d b t $. $d b u $.
|
|
$d b x $. $d b y $. $d t u $. $d t x $. $d t y $. $d u x $. $d u y $.
|
|
$d x y $.
|
|
$( Cardinal less than or equal is a set. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
lecex $p |- <_c e. _V $=
|
|
( va vb vx vy vt vu clec csset ccom cv wrex wbr wa csn 2exbii bitri exbii
|
|
wex vex ssetex csi ccnv cvv wss wceq wel r2ex 19.41vv anass df-3an bitr4i
|
|
w3a ancom snex breq1 anbi1d anbi2d ceqsex2v anbi12i 3bitri anbi1i 3bitr3i
|
|
brlec brco brcnv brsi brsset anbi2i anbi2ci 19.42v 19.42vv bitr2i 3bitr2i
|
|
brssetsn exrot4 3bitr4i eqbrriv siex coex cnvex eqeltri ) GHHUAZIZHUBZIZU
|
|
CABGWECJZDJZUDZDBJZKCAJZKZEJZWJHLZFJZWIHLZMZWLWFNZUEZWNWGNZUEZMZWHMZMZFRE
|
|
RZDRCRZWJWIGLWJWIWELZWKCAUFZDBUFZMZWHMZDRCRXEWHCDWJWIUGXDXJCDWPXAMZWHMZFR
|
|
ERXKFRERZWHMXDXJXKWHEFUHXLXCEFWPXAWHUIOXMXIWHXMWRWTWPULZFRERWQWJHLZWSWIHL
|
|
ZMZXIXKXNEFXKXAWPMXNWPXAUMWRWTWPUJUKOWPXOWOMXQEFWQWSWFUNWGUNWRWMXOWOWLWQW
|
|
JHUOUPWTWOXPXOWNWSWIHUOUQURXOXGXPXHWFWJCSZASZVNWGWIDSZBSZVNUSUTVAVBOUKCDW
|
|
JWIXSYAVCXFWJWLWDLZWLWIWCLZMZERXCDRCRZFRZERXEEWJWIWCWDVDYDYFEYDWMWOXBDRCR
|
|
ZMZFRZMWMYHMZFRYFYBWMYCYIWJWLHVEYCWLWNWBLZWOMZFRYIFWLWIHWBVDYLYHFYKYGWOYK
|
|
WRWTWFWGHLZULZDRCRYGCDWLWNHVFYNXBCDYNXAYMMXBWRWTYMUJYMWHXAWFWGXRXTVGVHPOP
|
|
VIQPUSWMYHFVJYJYEFYEWPYGMYJWPXBCDVKWMWOYGUIVLQVMQXCEFCDVOUTVPVQWCWDHWBTHT
|
|
VRVSHTVTVSWA $.
|
|
$}
|
|
|
|
$( Cardinal strict less than is a set. (Contributed by SF, 24-Feb-2015.) $)
|
|
ltcex $p |- <c e. _V $=
|
|
( cltc clec cid cdif cvv df-ltc lecex idex difex eqeltri ) ABCDEFBCGHIJ $.
|
|
|
|
$( The cardinality of a class is a set. (Contributed by SF, 24-Feb-2015.) $)
|
|
ncex $p |- Nc A e. _V $=
|
|
( cnc cen cec cvv df-nc wcel enex ecexg ax-mp eqeltri ) ABACDZEAFCEGLEGHAEC
|
|
IJK $.
|
|
|
|
$( The empty class is not a cardinal number. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
nulnnc $p |- -. (/) e. NC $=
|
|
( c0 wceq cncs wcel eqid wne cvv cen cqs cdm dmen elqsn0 mpan df-ncs eleq2s
|
|
wn necon2bi ax-mp ) AABACDZPAESAAAAFZAGHIZCHJGBAUADTKGAHLMNOQR $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( Membership in the cardinals. (Contributed by SF, 24-Feb-2015.) $)
|
|
elncs $p |- ( A e. NC <-> E. x A = Nc x ) $=
|
|
( cncs wcel cvv cen cqs cnc wceq wex df-ncs eleq2i elex ncex eleq1 mpbiri
|
|
cv exlimiv cec wrex elqsg df-nc eqeq2i exbii rexv syl6bbr pm5.21nii bitri
|
|
bitr4i ) BCDBEFGZDZBAQZHZIZAJZCUJBKLUKBEDZUOBUJMUNUPAUNUPUMEDULNBUMEOPRUP
|
|
UKBULFSZIZAETZUOAEBFEUAUOURAJUSUNURAUMUQBULUBUCUDURAUEUIUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( The cardinality of a set is a cardinal number. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
ncelncs $p |- ( A e. V -> Nc A e. NC ) $=
|
|
( vx wcel cnc cv wceq wex cncs elisset nceq eqcoms eximi syl elncs sylibr
|
|
) ABDZAEZCFZEGZCHZRIDQSAGZCHUACABJUBTCTASASKLMNCROP $.
|
|
$}
|
|
|
|
${
|
|
ncelncsi.1 $e |- A e. _V $.
|
|
$( The cardinality of a set is a cardinal number. (Contributed by SF,
|
|
10-Mar-2015.) $)
|
|
ncelncsi $p |- Nc A e. NC $=
|
|
( cvv wcel cnc cncs ncelncs ax-mp ) ACDAEFDBACGH $.
|
|
$}
|
|
|
|
$( A set is a member of its own cardinal. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
ncidg $p |- ( A e. V -> A e. Nc A ) $=
|
|
( wcel cen cec cnc wbr enrflxg elec sylibr df-nc syl6eleqr ) ABCZAADEZAFMAA
|
|
DGANCABHAADIJAKL $.
|
|
|
|
${
|
|
ncid.1 $e |- A e. _V $.
|
|
$( A set is a member of its own cardinal. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
ncid $p |- A e. Nc A $=
|
|
( cvv wcel cnc ncidg ax-mp ) ACDAAEDBACFG $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( The cardinality of a proper class is the empty set. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
ncprc $p |- ( -. A e. _V -> Nc A = (/) ) $=
|
|
( vx cvv wcel wn cv cnc wal wceq cen cec ecexr df-nc eleq2s con3i alrimiv
|
|
c0 eq0 sylibr ) ACDZEZBFZAGZDZEZBHUCQIUAUEBUDTTUBAJKUCUBAJLAMNOPBUCRS $.
|
|
$}
|
|
|
|
$( Membership in cardinality. (Contributed by SF, 24-Feb-2015.) $)
|
|
elnc $p |- ( A e. Nc B <-> A ~~ B ) $=
|
|
( cnc wcel cvv wa cen wbr elex cec ecexr df-nc eleq2s jca brex eleq2i bitri
|
|
elec cer ener a1i simpr simpl ersymb syl5bb pm5.21nii ) ABCZDZAEDZBEDZFZABG
|
|
HZUHUIUJAUGIUJABGJZUGABGKBLZMNABGOUHBAGHZUKULUHAUMDUOUGUMAUNPABGRQUKEGBAGES
|
|
HUKTUAUIUJUBUIUJUCUDUEUF $.
|
|
|
|
$( Equality of cardinalities. (Contributed by SF, 24-Feb-2015.) $)
|
|
eqncg $p |- ( A e. V -> ( Nc A = Nc B <-> A ~~ B ) ) $=
|
|
( wcel cvv cnc wceq cen wbr wa cec ncidg adantr wb eleq2 adantl mpbid df-nc
|
|
ex a1i syl6eleq ecexr syl brex simprd cer ener dmen elex simpr erth eqeq12i
|
|
wi cdm syl6rbbr pm5.21ndd ) ACDZBEDZAFZBFZGZABHIZUQVAURUQVAJZABHKZDURVCAUTV
|
|
DVCAUSDZAUTDZUQVEVAACLMVAVEVFNUQUSUTAOPQBRZUAABHUBUCSVBURUMUQVBAEDZURABHUDU
|
|
ETUQURVAVBNUQURJZVBAHKZVDGVAVIABHEEHEUFIVIUGTHUNEGVIUHTUQVHURACUIMUQURUJUKU
|
|
SVJUTVDARVGULUOSUP $.
|
|
|
|
${
|
|
eqnc.1 $e |- A e. _V $.
|
|
$( Equality of cardinalities. (Contributed by SF, 24-Feb-2015.) $)
|
|
eqnc $p |- ( Nc A = Nc B <-> A ~~ B ) $=
|
|
( cvv wcel cnc wceq cen wbr wb eqncg ax-mp ) ADEAFBFGABHIJCABDKL $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d X y $.
|
|
$( A cardinal is equal to the cardinality of a set iff it contains the
|
|
set. (Contributed by SF, 24-Feb-2015.) $)
|
|
ncseqnc $p |- ( A e. NC -> ( A = Nc X <-> X e. A ) ) $=
|
|
( vy cncs wcel cv cnc wceq wex elncs cen wbr cvv cec vex ncid eleq2 df-nc
|
|
wb a1i mpbiri syl6eleq ecexr syl brex simpld cer ener cdm dmen id eqeq12i
|
|
erth syl6rbbr pm5.21nii eqcom elnc 3bitr4i eqeq1 3bitr4d exlimiv sylbi )
|
|
ADEACFZGZHZCIABGZHZBAEZSZCAJVEVICVEVDVFHZBVDEZVGVHVJVKSVEVFVDHZBVCKLZVJVK
|
|
VLBMEZVMVLVCBKNZEVNVLVCVFVOVLVCVFEVCVDEVCCOZPVFVDVCQUABRZUBVCBKUCUDVMVNVC
|
|
MEZBVCKUEUFVNVMVOVCKNZHVLVNBVCKMMKMUGLVNUHTKUIMHVNUJTVNUKVRVNVPTUMVFVOVDV
|
|
SVQVCRULUNUOVDVFUPBVCUQURTAVDVFUSAVDBQUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
eqnc2.1 $e |- X e. _V $.
|
|
$( Alternate condition for equality to a cardinality. (Contributed by SF,
|
|
18-Mar-2015.) $)
|
|
eqnc2 $p |- ( A = Nc X <-> ( A e. NC /\ X e. A ) ) $=
|
|
( cnc wceq cncs wcel ncelncsi eleq1 mpbiri ncseqnc biadan2 ) ABDZEZAFGZBA
|
|
GNOMFGBCHAMFIJABKL $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a c $. $d a g $. $d a m $. $d a n $. $d b c $. $d b g $.
|
|
$d b m $. $d b n $. $d c g $. $d g m $. $d g n $. $d M a $. $d M b $.
|
|
$d M m $. $d m n $. $d M n $. $d N a $. $d N b $. $d N g $. $d N n $.
|
|
$d N m $.
|
|
$( The value of cardinal multiplication. (Contributed by SF,
|
|
10-Mar-2015.) $)
|
|
ovmuc $p |- ( ( M e. NC /\ N e. NC ) ->
|
|
( M .c N ) = { a | E. b e. M E. g e. N a ~~ ( b X. g ) } ) $=
|
|
( vc cncs wcel cv cen wbr wrex cvv wceq wa ccross c2nd c1st cop 3bitri vm
|
|
vn cxp cab cmuc ctxp crn cins4 ccnv cins2 cin cima elima df-br elrn2 elin
|
|
co wex oqelins4 elrn trtxp ancom op1st2nd anbi2i exbii opex breq1 ceqsexv
|
|
vex brcross otelins2 brcnv bitr3i anbi12i bitri xpex breq2 rexbii crossex
|
|
abbi2i 2ndex 1stex txpex rnex ins4ex enex cnvex ins2ex inex imaexg ancoms
|
|
mpan sylan syl5eqelr rexeq abbidv rexbidv df-muc ovmpt2g mpd3an3 ) BGHZCG
|
|
HZDIZEIZAIZUCZJKZACLZEBLZDUDZMHBCUEUQXJNXAXBOXJPQRUFZUFZUGZUHZJUIZUJZUJZU
|
|
KZUGZCULZBULZMXIDYAXCYAHXDXCXTKZEBLXIEXCXTBUMYBXHEBYBXDXCSZXTHXEYCXSKZACL
|
|
XHXDXCXTUNAYCXSCUMYDXGACYDXEYCSZXSHZXGXEYCXSUNYFFIZYESZXRHZFURYGXFNZXCYGJ
|
|
KZOZFURXGFYEXRUOYIYLFYIYHXNHZYHXQHZOYLYHXNXQUPYMYJYNYKYMYGXEXDSZSZXMHZXDX
|
|
ESZYGPKZYJYGXEXDXCXMDVIUSYQXCYPXLKZDURXCYRNZXCYGPKZOZDURYSDYPXLUTYTUUCDYT
|
|
UUBXCYOXKKZOUUBUUAOUUCXCYGYOPXKVAUUDUUAUUBUUDXCXEQKZXCXDRKZOUUFUUEOUUAXCX
|
|
EXDQRVAUUEUUFVBXDXEXCEVIZAVIZVCTVDUUBUUAVBTVEUUBYSDYRXDXEUUGUUHVFXCYRYGPV
|
|
GVHTXDXEYGUUGUUHVJTYNYGYCSXPHYGXCSXOHZYKYGXEYCXPUUHVKYGXDXCXOUUGVKUUIYGXC
|
|
XOKYKYGXCXOUNYGXCJVLVMTVNVOVEYKXGFXFXDXEUUGUUHVPYGXFXCJVQVHTVOVRTVRVOVTXB
|
|
XAYAMHZXBXTMHZXAUUJXSMHXBUUKXRXNXQXMXLPXKVSQRWAWBWCWCWDWEXPXOJWFWGWHWHWIW
|
|
DXSCMGWJWLXTBMGWJWMWKWNUAUBBCGGXGAUBIZLZEUAIZLZDUDXJUEUUMEBLZDUDMUUNBNUUO
|
|
UUPDUUMEUUNBWOWPUULCNZUUPXIDUUQUUMXHEBXGAUULCWOWQWPAUAUBDEWRWSWT $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d A z $. $d B x $. $d B y $. $d B z $. $d x y $.
|
|
$d x z $. $d y z $.
|
|
mucnc.1 $e |- A e. _V $.
|
|
mucnc.2 $e |- B e. _V $.
|
|
$( Cardinal multiplication in terms of cardinality. Theorem XI.2.27 of
|
|
[Rosser] p. 378. (Contributed by SF, 10-Mar-2015.) $)
|
|
mucnc $p |- ( Nc A .c Nc B ) = Nc ( A X. B ) $=
|
|
( vx vy vz cnc cv cxp cen wbr wrex cab cncs wcel wceq ncelncsi wa wex cec
|
|
cmuc co ovmuc mp2an df-nc dfec2 elnc anbi12i ensym 2exbii enrflx bi2anan9
|
|
r2ex breq1 xpeq12 breq1d anbi12d spc2ev mpanl12 xpen sylib sylan exlimivv
|
|
entr impbii 3bitr4ri abbii 3eqtrri eqtri ) AHZBHZUBUCZEIZFIZGIZJZKLZGVLMF
|
|
VKMZENZABJZHZVKOPVLOPVMVTQACRBDRGVKVLEFUDUEWBWAKUAWAVNKLZENVTWAUFEWAKUGWC
|
|
VSEVOVKPZVPVLPZSZVRSZGTFTVOAKLZVPBKLZSZVQVNKLZSZGTFTZVSWCWGWLFGWFWJVRWKWD
|
|
WHWEWIVOAUHVPBUHUIVNVQUJUIUKVRFGVKVLUNWCWMAAKLZBBKLZWCWMACULBDULWLWNWOSZW
|
|
CSFGABCDVOAQZVPBQZSZWJWPWKWCWQWHWNWRWIWOVOAAKUOVPBBKUOUMWSVQWAVNKVOAVPBUP
|
|
UQURUSUTWLWCFGWJWAVQKLZWKWCWJVQWAKLWTVOAVPBVAVQWAUJVBWAVQVNVEVCVDVFVGVHVI
|
|
VJ $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( Closure law for cardinal multiplicaton. (Contributed by SF,
|
|
10-Mar-2015.) $)
|
|
muccl $p |- ( ( A e. NC /\ B e. NC ) -> ( A .c B ) e. NC ) $=
|
|
( vx vy cncs wcel wa cv cnc wceq wex co elncs anbi12i eeanv bitr4i oveq12
|
|
cmuc cxp vex mucnc xpex ncelncsi eqeltri syl6eqel exlimivv sylbi ) AEFZBE
|
|
FZGZACHZIZJZBDHZIZJZGZDKCKZABRLZEFZUJUMCKZUPDKZGURUHVAUIVBCAMDBMNUMUPCDOP
|
|
UQUTCDUQUSULUORLZEAULBUORQVCUKUNSZIEUKUNCTZDTZUAVDUKUNVEVFUBUCUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d a b c d m n $.
|
|
$( Cardinal multiplication is a set. (Contributed by SF, 24-Feb-2015.) $)
|
|
mucex $p |- .c e. _V $=
|
|
( vm vn va vb vc vd cncs csset cen c1c wbr wrex cop wcel wex otelins2 vex
|
|
cv wa bitri cmuc cxp cvv cins2 ccnv cins4 cin crn csi3 cima cins3 csymdif
|
|
ccross cdif cab cmpt2 df-muc csn wel elin snex opelssetsn 3bitri oqelins4
|
|
otsnelsi3 wceq elrn2 df-br brcnv brcross 3bitr2i anbi12i exbii xpex breq2
|
|
ceqsexv elima1c df-rex 3bitr4i bitr4i rexcom weq breq1 2rexbidv releqmpt2
|
|
elab eqtr4i ncsex ssetex ins2ex crossex cnvex ins4ex enex inex rnex si3ex
|
|
1cex imaex mpt2exlem eqeltri ) UAGGUBUCUBHUDZXBUDZXCUMUEZUFZIUEZUDZUDZUGZ
|
|
UHZUIZUFZUGZJUJZUFZUGZJUJZUKULJUJUNZUCUAABGGCRZDRZERZUBZIKZEBRZLDARZLZCUO
|
|
ZUPXREABCDUQABFGGXQYGFRZURZYEYDMZMZXQNZYHYBIKZEYDLDYELZYHYGNYAURZYKMZXPNZ
|
|
EOZYMDYELZEYDLZYLYNYREBUSZYSSZEOYTYQUUBEYQYPXCNZYPXONZSUUBYPXCXOUTUUCUUAU
|
|
UDYSUUCYOYJMXBNYOYDMHNUUAYOYIYJXBYHVAZPYOYEYDHAQZPYAYDEQZBQZVBVCUUDYOYIYE
|
|
MZMZXNNZYSYOYIYEYDXNUUHVDXTURZUUJMZXMNZDODAUSZYMSZDOUUKYSUUNUUPDUUNUUMXCN
|
|
ZUUMXLNZSUUPUUMXCXLUTUUQUUOUURYMUUQUULUUIMXBNUULYEMHNUUOUULYOUUIXBYAVAPUU
|
|
LYIYEHUUEPXTYEDQZUUFVBVCUURUULYOYIMMXKNXTYAYHMZMZXJNZYMUULYOYIYEXKUUFVDXT
|
|
YAYHXJUUSUUGFQZVEUVBXSUVAMZXINZCOXSYBVFZYHXSIKZSZCOYMCUVAXIVGUVEUVHCUVEUV
|
|
DXENZUVDXHNZSUVHUVDXEXHUTUVIUVFUVJUVGUVIXSXTYAMZMXDNXSUVKXDKZUVFXSXTYAYHX
|
|
DUVCVDXSUVKXDVHUVLUVKXSUMKUVFXSUVKUMVIXTYAXSUUSUUGVJTVKUVJXSUUTMXGNZUVGXS
|
|
XTUUTXGUUSPUVMXSYHMXFNXSYHXFKUVGXSYAYHXFUUGPXSYHXFVHXSYHIVIVKTVLTVMUVGYMC
|
|
YBXTYAUUSUUGVNXSYBYHIVOVPVCVCVLTVMDUUJXMVQYMDYEVRVSTVLTVMYSEYDVRVTEYKXPVQ
|
|
YMDEYEYDWAVSYFYNCYHUVCCFWBYCYMDEYEYDXSYHYBIWCWDWFVTWEWGGGXQWHWHXPJXCXOXBH
|
|
WIWJWJZXNXMJXCXLUVNXKXJXIXEXHXDUMWKWLWMXGXFIWNWLWJWJWOWPWQWMWOWRWSWMWOWRW
|
|
SWTXA $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d x y $.
|
|
$( Cardinal multiplication commutes. Theorem XI.2.28 of [Rosser] p. 378.
|
|
(Contributed by SF, 10-Mar-2015.) $)
|
|
muccom $p |- ( ( A e. NC /\ B e. NC ) -> ( A .c B ) = ( B .c A ) ) $=
|
|
( vx vy cncs wcel wa cv cnc wceq wex co elncs anbi12i eeanv cxp vex mucnc
|
|
cmuc oveq12 bitr4i cen wbr xpcomen xpex eqnc mpbir 3eqtr4i ancoms 3eqtr4a
|
|
exlimivv sylbi ) AEFZBEFZGZACHZIZJZBDHZIZJZGZDKCKZABSLZBASLZJZUOURCKZVADK
|
|
ZGVCUMVGUNVHCAMDBMNURVACDOUAVBVFCDVBUQUTSLZUTUQSLZVDVEUPUSPZIZUSUPPZIZVIV
|
|
JVLVNJVKVMUBUCUPUSCQZDQZUDVKVMUPUSVOVPUEUFUGUPUSVOVPRUSUPVPVORUHAUQBUTSTV
|
|
AURVEVJJBUTAUQSTUIUJUKUL $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d A z $. $d B x $. $d B y $. $d B z $. $d C x $.
|
|
$d C y $. $d C z $. $d x y $. $d x z $. $d y z $.
|
|
$( Cardinal multiplication associates. Theorem XI.2.29 of [Rosser]
|
|
p. 378. (Contributed by SF, 10-Mar-2015.) $)
|
|
mucass $p |- ( ( A e. NC /\ B e. NC /\ C e. NC ) ->
|
|
( ( A .c B ) .c C ) = ( A .c ( B .c C ) ) ) $=
|
|
( vx vy vz cncs wcel w3a cv cnc wceq wex cmuc co elncs cxp vex xpex mucnc
|
|
3anbi123i eeeanv bitr4i cen wbr xpassen mpbir oveq1i eqtri oveq2i 3eqtr4i
|
|
eqnc wa oveq12 id oveqan12d 3impa 3impb 3eqtr4a exlimiv exlimivv sylbi )
|
|
AGHZBGHZCGHZIZADJZKZLZBEJZKZLZCFJZKZLZIZFMZEMDMZABNOZCNOZABCNOZNOZLZVFVID
|
|
MZVLEMZVOFMZIVRVCWDVDWEVEWFDAPEBPFCPUAVIVLVODEFUBUCVQWCDEVPWCFVPVHVKNOZVN
|
|
NOZVHVKVNNOZNOZVTWBVGVJQZVMQZKZVGVJVMQZQZKZWHWJWMWPLWLWOUDUEVGVJVMDRZERZF
|
|
RZUFWLWOWKVMVGVJWQWRSZWSSULUGWHWKKZVNNOWMWGXAVNNVGVJWQWRTUHWKVMWTWSTUIWJV
|
|
HWNKZNOWPWIXBVHNVJVMWRWSTUJVGWNWQVJVMWRWSSTUIUKVIVLVOVTWHLVIVLUMVOVSWGCVN
|
|
NAVHBVKNUNVOUOUPUQVIVLVOWBWJLVIVLVOUMAVHWAWINVIUOBVKCVNNUNUPURUSUTVAVB $.
|
|
$}
|
|
|
|
${
|
|
$d A p $. $d A q $. $d A r $. $d A x $. $d B p $. $d B q $. $d B r $.
|
|
$d B x $. $d p q $. $d p x $. $d q x $. $d r x $.
|
|
ncdisjun.1 $e |- A e. _V $.
|
|
ncdisjun.2 $e |- B e. _V $.
|
|
$( Cardinality of disjoint union of two sets. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
ncdisjun $p |- ( ( A i^i B ) = (/) -> Nc ( A u. B ) = ( Nc A +c Nc B ) ) $=
|
|
( vr vp vq cin c0 wceq cv wcel cen wbr elnc wf1o ccnv wa cima adantl cplc
|
|
vx cun cnc wex f1ocnv imaundi cdm crn imadmrn a1i f1odm imaeq2d wfo f1ofo
|
|
bren forn syl 3eqtr3d syl5eqr cres wf1 wss f1of1 ssun1 f1ores sylancl vex
|
|
cnvex resex f1oen 3syl sylibr ssun2 wfun df-f1 simprbi imaeq2 ima0 syl6eq
|
|
wf adantr eqtr3d eladdci syl3anc eqeltrrd syl5 exlimdv syl5bi wrex eladdc
|
|
imain simplrl sylib simplrr simpr simpll unen syl22anc syl5ibrcom expimpd
|
|
ex breq1 rexlimdvva impbid syl5bb eqrdv ) ABHZIJZUBABUCZUDZAUDZBUDZUAZUBK
|
|
ZXKLXOXJMNZXIXOXNLZXOXJOXIXPXQXPXOXJEKZPZEUEXIXQXOXJEUPXIXSXQEXSXJXOXRQZP
|
|
ZXIXQXOXJXRUFXIYAXQXIYARZXTASZXTBSZUCZXOXNYAYEXOJXIYAYEXTXJSZXOXTABUGYAXT
|
|
XTUHZSZXTUIZYFXOYHYIJYAXTUJUKYAYGXJXTXJXOXTULUMYAXJXOXTUNYIXOJXJXOXTUOXJX
|
|
OXTUQURUSUTTYBYCXLLZYDXMLZYCYDHZIJYEXNLYAYJXIYAYCAMNZYJYAAYCXTAVAZPZYCAYN
|
|
QZPYMYAXJXOXTVBZAXJVCYOXJXOXTVDZABVEXJXOAXTVFVGAYCYNUFYCAYPYNXTAXREVHVIZC
|
|
VJVIVKVLYCAOVMTYBYDBMNZYKYAYTXIYABYDXTBVAZPZYDBUUAQZPYTYAYQBXJVCUUBYRBAVN
|
|
XJXOBXTVFVGBYDUUAUFYDBUUCUUAXTBYSDVJVIVKVLTYDBOVMYBXTXHSZYLIYAUUDYLJZXIYA
|
|
YQXTQVOZUUEYRYQXJXOXTWAUUFXJXOXTVPVQABXTWLVLTXIUUDIJYAXIUUDXTISIXHIXTVRXT
|
|
VSVTWBWCYCYDXLXMWDWEWFXBWGWHWIXQFKZGKZHIJZXOUUGUUHUCZJZRZGXMWJFXLWJXIXPXO
|
|
XLXMFGWKXIUULXPFGXLXMXIUUGXLLZUUHXMLZRZRZUUIUUKXPUUPUUIRZXPUUKUUJXJMNZUUQ
|
|
UUGAMNZUUHBMNZUUIXIUURUUQUUMUUSXIUUMUUNUUIWMUUGAOWNUUQUUNUUTXIUUMUUNUUIWO
|
|
UUHBOWNUUPUUIWPXIUUOUUIWQUUGAUUHBWRWSXOUUJXJMXCWTXAXDWIXEXFXG $.
|
|
$}
|
|
|
|
$( Cardinal zero is the cardinality of the empty set. Theorem XI.2.7 of
|
|
[Rosser] p. 372. (Contributed by SF, 24-Feb-2015.) $)
|
|
df0c2 $p |- 0c = Nc (/) $=
|
|
( vx c0 cen cec cv wbr cab cnc c0c dfec2 df-nc wceq wcel en0 ensym 3bitr4ri
|
|
el0c abbi2i 3eqtr4ri ) BCDBAEZCFZAGBHIABCJBKUAAITBCFTBLUATIMTNBTOTQPRS $.
|
|
|
|
$( Cardinal zero is a cardinal number. Corollary 1 to theorem XI.2.7 of
|
|
[Rosser] p. 373. (Contributed by SF, 24-Feb-2015.) $)
|
|
0cnc $p |- 0c e. NC $=
|
|
( c0c c0 cnc cncs df0c2 0ex ncelncsi eqeltri ) ABCDEBFGH $.
|
|
|
|
${
|
|
$d f x $. $d f y $. $d f z $. $d x y $. $d x z $. $d y z $.
|
|
$( Cardinal one is a cardinal number. Corollary 2 to theorem XI.2.8 of
|
|
[Rosser] p. 373. (Contributed by SF, 24-Feb-2015.) $)
|
|
1cnc $p |- 1c e. NC $=
|
|
( vx vy vz vf c1c wcel cv cnc wceq wex csn cen wbr cvv vex exlimiv eqeq2d
|
|
crn spcev sylbi cncs cec cab dfec2 df-nc el1c en2sn mp2an breq2 wf1o bren
|
|
mpbiri wf f1of wfo f1ofo forn syl cfv cop wa wi fsn2 rnsnop syl6eq eqeq1d
|
|
rneq fvex sneq eqcoms syl6bi adantl sylc impbii abbi2i 3eqtr4ri snex nceq
|
|
bitri ax-mp elncs mpbir ) EUAFEAGZHZIZAJZEBGZKZHZIZWFWHLUBWHCGZLMZCUCWIEC
|
|
WHLUDWHUEWLCEWKEFWKWCKZIZAJZWLAWKUFWOWLWNWLAWNWLWHWMLMZWGNFWCNFWPBOZAOWGW
|
|
CNNUGUHWKWMWHLUIULPWLWHWKDGZUJZDJWOWHWKDUKWSWODWSWHWKWRUMZWRRZWKIZWOWHWKW
|
|
RUNWSWHWKWRUOXBWHWKWRUPWHWKWRUQURWTWGWRUSZWKFZWRWGXCUTKZIZVAXBWOVBZWGWKWR
|
|
WQVCXFXGXDXFXBXCKZWKIWOXFXAXHWKXFXAXERXHWRXEVGWGXCWQVDVEVFWOWKXHWNWKXHIAX
|
|
CWGWRVHWCXCIWMXHWKWCXCVIQSVJVKVLTVMPTVNVSVOVPWEWJAWHWGVQWCWHIWDWIEWCWHVRQ
|
|
SVTAEWAWB $.
|
|
$}
|
|
|
|
${
|
|
df1c3.1 $e |- A e. _V $.
|
|
$( Cardinal one is the cardinality of a singleton. Theorem XI.2.8 of
|
|
[Rosser] p. 373. (Contributed by SF, 2-Mar-2015.) $)
|
|
df1c3 $p |- 1c = Nc { A } $=
|
|
( c1c csn cnc wceq wcel snel1c cncs wb 1cnc ncseqnc ax-mp mpbir ) CADZEFZ
|
|
OCGZABHCIGPQJKCOLMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Cardinal one is the cardinality of a singleton. Theorem XI.2.8 of
|
|
[Rosser] p. 373. (Contributed by SF, 13-Mar-2015.) $)
|
|
df1c3g $p |- ( A e. V -> 1c = Nc { A } ) $=
|
|
( vx c1c cv csn cnc wceq sneq nceqd eqeq2d vex df1c3 vtoclg ) DCEZFZGZHDA
|
|
FZGZHCABOAHZQSDTPROAIJKOCLMN $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Cardinal multiplication by zero. Theorem XI.2.32 of [Rosser] p. 379.
|
|
(Contributed by SF, 10-Mar-2015.) $)
|
|
muc0 $p |- ( A e. NC -> ( A .c 0c ) = 0c ) $=
|
|
( vx cncs wcel cv cnc wceq wex c0c cmuc co elncs oveq1 c0 cxp nceqi df0c2
|
|
xp0 oveq2i vex 0ex mucnc eqtri 3eqtr4i syl6eq exlimiv sylbi ) ACDABEZFZGZ
|
|
BHAIJKZIGZBALUJULBUJUKUIIJKZIAUIIJMUHNOZFZNFZUMIUNNUHRPUMUIUPJKUOIUPUIJQS
|
|
UHNBTUAUBUCQUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d A x $.
|
|
$( Cardinal multiplication by one. (Contributed by SF, 11-Mar-2015.) $)
|
|
mucid1 $p |- ( A e. NC -> ( A .c 1c ) = A ) $=
|
|
( vx cncs wcel cv cnc wceq wex c1c cmuc co elncs csn cxp vex df1c3 oveq2i
|
|
snex mucnc cen wbr xpsnen xpex eqnc mpbir 3eqtri oveq1 id 3eqtr4a exlimiv
|
|
sylbi ) ACDABEZFZGZBHAIJKZAGZBALUNUPBUNUMIJKZUMUOAUQUMULMZFZJKULURNZFZUMI
|
|
USUMJULBOZPQULURVBULRZSVAUMGUTULTUAULULVBVBUBUTULULURVBVCUCUDUEUFAUMIJUGU
|
|
NUHUIUJUK $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d x y $. $d x z $. $d y z $.
|
|
$( The cardinals are closed under cardinal addition. Theorem XI.2.10 of
|
|
[Rosser] p. 374. (Contributed by SF, 24-Feb-2015.) $)
|
|
ncaddccl $p |- ( ( A e. NC /\ B e. NC ) -> ( A +c B ) e. NC ) $=
|
|
( vx vy vz cncs wcel cv cnc wceq wex elncs wa c0 csn cxp cen mpbir eqcomi
|
|
cplc eeanv ccompl wbr vex 0ex complex xpsnen snex xpex eqnc eqtr addceq12
|
|
mpan2 cun necompl xpnedisj ncdisjun ax-mp unex nceq eqeq2d spcev syl6eqel
|
|
cin syl2an exlimivv sylbir syl2anb ) AFGACHZIZJZCKZBDHZIZJZDKZABTZFGZBFGC
|
|
ALDBLVLVPMVKVOMZDKCKVRVKVOCDUAVSVRCDVKAVINUBZOZPZIZJZBVMNOZPZIZJZVRVOVKVJ
|
|
WCJWDWCVJWCVJJWBVIQUCVIVTCUDZNUEUFZUGWBVIVIWAWIVTUHUIZUJRSAVJWCUKUMVOVNWG
|
|
JWHWGVNWGVNJWFVMQUCVMNDUDZUEUGWFVMVMWEWLNUHUIZUJRSBVNWGUKUMWDWHMVQWCWGTZF
|
|
ABWCWGULWNFGWNEHZIZJZEKZWNWBWFUNZIZJZWRWTWNWBWFVDNJWTWNJVIVMVTNWJNUOUPWBW
|
|
FWKWMUQURSWQXAEWSWBWFWKWMUSWOWSJWPWTWNWOWSUTVAVBUREWNLRVCVEVFVGVH $.
|
|
$}
|
|
|
|
$( The successor of a cardinal is a cardinal. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
peano2nc $p |- ( A e. NC -> ( A +c 1c ) e. NC ) $=
|
|
( cncs wcel c1c cplc 1cnc ncaddccl mpan2 ) ABCDBCADEBCFADGH $.
|
|
|
|
${
|
|
$d A x n $.
|
|
$( A finite cardinal number is a cardinal number. (Contributed by SF,
|
|
24-Feb-2015.) $)
|
|
nnnc $p |- ( A e. Nn -> A e. NC ) $=
|
|
( vx vn cv cncs wcel c0c c1c cplc cab abid2 ncsex eqeltri eleq1 0cnc cnnc
|
|
cvv wi peano2nc a1i finds ) BDZEFZGEFCDZEFZUDHIZEFZAEFBCAUCBJEQBEKLMUBGEN
|
|
UBUDENUBUFENUBAENOUEUGRUDPFUDSTUA $.
|
|
$}
|
|
|
|
$( The finite cardinals are a subset of the cardinals. Theorem XI.2.11 of
|
|
[Rosser] p. 374. (Contributed by SF, 24-Feb-2015.) $)
|
|
nnssnc $p |- Nn C_ NC $=
|
|
( vx cnnc cncs cv nnnc ssriv ) ABCADEF $.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d x y $.
|
|
$( Two cardinals are either disjoint or equal. (Contributed by SF,
|
|
25-Feb-2015.) $)
|
|
ncdisjeq $p |- ( ( A e. NC /\ B e. NC ) ->
|
|
( ( A i^i B ) = (/) \/ A = B ) ) $=
|
|
( vx vy cncs wcel wa cv cnc wceq wex cin c0 wo elncs cen df-nc eqtr mpan2
|
|
cec anbi12i eeanv bitr4i cvv cer wbr erdisj ax-mp wb eqeq12 ineq12 eqeq1d
|
|
ener orbi12d syl2an mpbiri orcomd exlimivv sylbi ) AEFZBEFZGZACHZIZJZBDHZ
|
|
IZJZGZDKCKZABLZMJZABJZNZVBVECKZVHDKZGVJUTVOVAVPCAODBOUAVEVHCDUBUCVIVNCDVI
|
|
VMVLVIVMVLNZVCPTZVFPTZJZVRVSLZMJZNZPUDUEUFWCUMVCVFPUGUHVEAVRJZBVSJZVQWCUI
|
|
VHVEVDVRJWDVCQAVDVRRSVHVGVSJWEVFQBVGVSRSWDWEGZVMVTVLWBAVRBVSUJWFVKWAMAVRB
|
|
VSUKULUNUOUPUQURUS $.
|
|
$}
|
|
|
|
$( If two cardinals have an element in common, then they are equal.
|
|
(Contributed by SF, 25-Feb-2015.) $)
|
|
nceleq $p |- ( ( ( A e. NC /\ B e. NC ) /\ ( X e. A /\ X e. B ) ) ->
|
|
A = B ) $=
|
|
( cncs wcel wa wceq cin c0 wn elin n0i sylbir ncdisjeq ord syl5 imp ) ADEBD
|
|
EFZCAECBEFZABGZSABHZIGZJZRTSCUAEUCCABKUACLMRUBTABNOPQ $.
|
|
|
|
${
|
|
$d A f $. $d A g $. $d A t $. $d B f $. $d B g $. $d B t $. $d f g $.
|
|
$d f t $. $d f x $. $d f y $. $d g t $. $d g x $. $d g y $. $d t x $.
|
|
$d t y $. $d x y $.
|
|
$( Successor is one-to-one over the cardinals. Theorem XI.2.12 of [Rosser]
|
|
p. 375. (Contributed by SF, 25-Feb-2015.) $)
|
|
peano4nc $p |- ( ( A e. NC /\ B e. NC ) ->
|
|
( ( A +c 1c ) = ( B +c 1c ) <-> A = B ) ) $=
|
|
( vg vt vx vf vy cncs wcel wa c1c cplc wceq adantr cnc eqtr2 csn wrex vex
|
|
cv peano2nc wex elncs simpr jca cun ccompl ncid eleq2 mpbiri elsuc biimpi
|
|
syl sylib anim12i reeanv 2rexbii bitri wi ncseqnc bi2anan9 biimpar cen wn
|
|
wbr wel elcompl enadj ancoms ex syl2anb syl5 rexlimivv eqeq12 eqnc syl6bb
|
|
syl5ibr rexlimdvva syl5bir imp sylan2 expr exlimdv syl5bi addceq1 impbid1
|
|
3expb mpd ) AHIZBHIZJZAKLZBKLZMZABMZWKWNWOWKWNJZWLHIZWOWKWQWNWIWQWJAUANNW
|
|
QWLCTZOZMZCUBWPWOCWLUCWPWTWOCWKWNWTWOWNWTJZWKWTWMWSMZJZWOXAWTXBWNWTUDWLWM
|
|
WSPUEWKXCWOXCWRDTZETZQUFZMZEXDUGZRZDARZWRFTZGTZQUFZMZGXKUGZRZFBRZJZWKWOWT
|
|
XJXBXQWTWRWLIZXJWTXSWRWSIZWRCSUHZWLWSWRUIUJXSXJEWRADUKULUMXBWRWMIZXQXBYBX
|
|
TYAWMWSWRUIUJGWRBFUKUNUOXRXGXNJZGXOREXHRZFBRDARZWKWOYEXIXPJZFBRDARXRYDYFD
|
|
FABXGXNEGXHXOUPUQXIXPDFABUPURWKYDWODFABWKXDAIZXKBIZJZJAXDOZMZBXKOZMZJZYDW
|
|
OUSWKYNYIWIYKYGWJYMYHAXDUTBXKUTVAVBYDWOYNXDXKVCVEZYCYOEGXHXOYCXFXMMZXEXHI
|
|
ZXLXOIZJYOWRXFXMPYQEDVFVDZGFVFVDZYPYOUSYRXEXDESZVGXLXKGSZVGYSYTJZYPYOYPUU
|
|
CYOYPYSYTYOXDXKXEXLDSZFSUUAUUBVHWGVIVJVKVLVMYNWOYJYLMYOAYJBYLVNXDXKUUDVOV
|
|
PVQUMVRVSVLVTWAWBWCWDWHVJABKWEWF $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( A cardinal is finite iff it is a subset of ` Fin ` . (Contributed by
|
|
SF, 25-Feb-2015.) $)
|
|
ncssfin $p |- ( A e. NC -> ( A e. Nn <-> A C_ Fin ) ) $=
|
|
( vx vy cncs wcel cnnc cfin wss cuni elssuni cv wrex wal c0c wne wex wceq
|
|
wi c0 wa df-fin syl6sseqr dfss2 elfin imbi2i wb peano1 ne0i r19.37zv mp2b
|
|
wel bitr4i albii bitri nulnnc eleq1 mtbiri necon2ai n0 19.29r pm2.27 nnnc
|
|
sylib adantl nceleq sylanl2 simplr eqeltrd expr rexlimdva expimpd exlimdv
|
|
syld an32s syl5 mpand syl5bi impbid2 ) ADEZAFEZAGHZVTAFIGAFJUAUBWABKZAEZB
|
|
CUKZRZCFLZBMZVSVTWAWCWBGEZRZBMWGBAGUCWIWFBWIWCWDCFLZRZWFWHWJWCCWBUDUENFEF
|
|
SOWFWKUFUGFNUHWCWDCFUIUJULUMUNVSWCBPZWGVTVSASOWLVSASASQVSSDEUOASDUPUQURBA
|
|
USVCWLWGTWCWFTZBPVSVTWCWFBUTVSWMVTBVSWCWFVTVSWCTWEVTCFVSCKZFEZWCWEVTRVSWO
|
|
TZWCTWEWDVTWCWEWDRWPWCWDVAVDWPWCWDVTWPWCWDTZTAWNFWOVSWNDEWQAWNQWNVBAWNWBV
|
|
EVFVSWOWQVGVHVIVMVNVJVKVLVOVPVQVR $.
|
|
$}
|
|
|
|
${
|
|
ncpw1.1 $e |- A e. _V $.
|
|
$( The cardinality of two sets are equal iff their unit power classes have
|
|
the same cardinality. (Contributed by SF, 25-Feb-2015.) $)
|
|
ncpw1 $p |- ( Nc A = Nc B <-> Nc ~P1 A = Nc ~P1 B ) $=
|
|
( cen wbr cpw1 cnc wceq enpw1 eqnc pw1ex 3bitr4i ) ABDEAFZBFZDEAGBGHMGNGH
|
|
ABIABCJMNACKJL $.
|
|
$}
|
|
|
|
${
|
|
ncpwpw1.1 $e |- A e. _V $.
|
|
$( Power class and unit power class commute within cardinality.
|
|
(Contributed by SF, 26-Feb-2015.) $)
|
|
ncpwpw1 $p |- Nc ~P ~P1 A = Nc ~P1 ~P A $=
|
|
( cpw1 cpw cnc wceq cen wbr enpw1pw ensym mpbi pw1ex pwex eqnc mpbir ) AC
|
|
ZDZEADCZEFQRGHZRQGHSABIRQJKQRPABLMNO $.
|
|
$}
|
|
|
|
$( The cardinality of ` 1c ` is equal to that of its power class.
|
|
(Contributed by SF, 26-Feb-2015.) $)
|
|
ncpw1c $p |- Nc ~P 1c = Nc 1c $=
|
|
( cvv cpw1 cpw cnc c1c vvex ncpwpw1 df1c2 pweqi nceqi wceq pwv pw1eq eqtr4i
|
|
ax-mp 3eqtr4i ) ABZCZDACZBZDECZDEDAFGUAREQHIJETEQTHSAKTQKLSAMONJP $.
|
|
|
|
$( One plus one equals two. Theorem *110.64 of [WhiteheadRussell] p. 86.
|
|
This theorem is occasionally useful. (Contributed by SF, 2-Mar-2015.) $)
|
|
1p1e2c $p |- ( 1c +c 1c ) = 2c $=
|
|
( c0 csn cvv cun cnc cplc c2c c1c cin wceq wcel wn 0ex n0i ax-mp vvex elsnc
|
|
mtbir snex df1c3 disjsn mpbir ncdisjun cpr df-2c df-pr nceqi eqtri 3eqtr4ri
|
|
addceq12i ) ABZCBZDZEZUKEZULEZFZGHHFUKULIAJZUNUQJURCUKKZLUSCAJZACKUTLMCANOC
|
|
APQRUKCUAUBUKULASCSUCOGACUDZEUNUEVAUMACUFUGUHHUOHUPAMTCPTUJUI $.
|
|
|
|
$( Two plus one equals three. (Contributed by SF, 2-Mar-2015.) $)
|
|
2p1e3c $p |- ( 2c +c 1c ) = 3c $=
|
|
( c0 cvv cpr csn cdif cun cnc cplc c3c c2c c1c cin wceq wcel vvex ax-mp 0ex
|
|
wn mtbir snex wo wne vn0 eldifsn mpbir2an n0i wa notnoti intnan eldif eleq2
|
|
snid mpbiri pm3.2ni difex elpr disjsn mpbir prex ncdisjun df-3c df-tp nceqi
|
|
mto ctp eqtri df-2c df1c3 addceq12i 3eqtr4ri ) ABCZBADZEZDZFZGZVKGZVNGZHZIJ
|
|
KHVKVNLAMZVPVSMVTVMVKNZRWAVMAMZVMBMZUAWBWCBVMNZWBRWDBBNBAUBOUCBBAUDUEVMBUFP
|
|
WCAVMNZWEABNZAVLNZRZUGWHWFWGAQULUHUIABVLUJSWCWEWFQVMBAUKUMVDUNVMABBVLOATUOZ
|
|
UPSVKVMUQURVKVNABUSVMTUTPIABVMVEZGVPVAWJVOABVMVBVCVFJVQKVRVGVMWIVHVIVJ $.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d x y $.
|
|
$( The cardinal T operation always yields a set. (Contributed by SF,
|
|
2-Mar-2015.) $)
|
|
tcex $p |- T_c A e. _V $=
|
|
( vx vy ctc cv cncs wcel cpw1 cnc wceq wrex cio cvv df-tc iotaex eqeltri
|
|
wa ) ADBEZFGRCEHIJCAKQZBLMCABNSBOP $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d x y $.
|
|
$( Equality theorem for cardinal T operator. (Contributed by SF,
|
|
2-Mar-2015.) $)
|
|
tceq $p |- ( A = B -> T_c A = T_c B ) $=
|
|
( vx vy wceq cv cncs wcel cpw1 cnc wa cio ctc rexeq anbi2d iotabidv df-tc
|
|
wrex 3eqtr4g ) ABEZCFZGHZUADFIJEZDARZKZCLUBUCDBRZKZCLAMBMTUEUGCTUDUFUBUCD
|
|
ABNOPDACQDBCQS $.
|
|
$}
|
|
|
|
${
|
|
$d A w $. $d A x $. $d A y $. $d A z $. $d w x $. $d w y $. $d w z $.
|
|
$d x y $. $d x z $. $d y z $.
|
|
$( Given a cardinal, there is a unique cardinal that contains the unit
|
|
power class of its members. (Contributed by SF, 2-Mar-2015.) $)
|
|
ncspw1eu $p |- ( A e. NC -> E! x e. NC E. y e. A x = Nc ~P1 y ) $=
|
|
( vz vw cncs wcel cv cpw1 cnc wceq wrex wa weq wi wral wex c0 sylib eqeq1
|
|
wreu wne nulnnc eleq1 mtbiri necon2ai n0 pw1ex ncelncsi eqid rspcev mp2an
|
|
vex jctr a1i eximdv mpd rexcom df-rex bitri sylibr reeanv ncseqnc biimpar
|
|
adantrr adantrl eqtr3d ncpw1 3adant2 eqeq2 anbi1d eqtr3 syl6bi rexlimdvva
|
|
w3a syl 3expa syl5bir ralrimivva rexbidv pw1eq eqeq2d cbvrexv syl6bb reu4
|
|
nceqd sylanbrc ) CFGZAHZBHZIZJZKZBCLZAFLZWNDHZEHZIZJZKZECLZMZADNZOZDFPAFP
|
|
WNAFUAWHWJCGZWMAFLZMZBQZWOWHXEBQZXHWHCRUBXIWHCRCRKWHRFGUCCRFUDUEUFBCUGSWH
|
|
XEXGBXEXGOWHXEXFWLFGWLWLKZXFWKWJBUMZUHUIWLUJWMXJAWLFWIWLWLTUKULUNUOUPUQWO
|
|
XFBCLXHWMABFCURXFBCUSUTVAWHXDADFFXBWMWTMZECLBCLWHWIFGWPFGMZMZXCWMWTBECCVB
|
|
XNXLXCBECCWHXMXEWQCGZMZXLXCOZWHXMXPVOWLWSKZXQWHXPXRXMWHXPMZWJJZWQJZKXRXSC
|
|
XTYAWHXECXTKZXOWHYBXECWJVCVDVEWHXOCYAKZXEWHYCXOCWQVCVDVFVGWJWQXKVHSVIXRXL
|
|
WIWSKZWTMXCXRWMYDWTWLWSWIVJVKWIWPWSVLVMVPVQVNVRVSWNXAADFXCWNWPWLKZBCLXAXC
|
|
WMYEBCWIWPWLTVTYEWTBECBENZWLWSWPYFWKWRWJWQWAWFWBWCWDWEWG $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( The cardinal T operation over a cardinal yields a cardinal.
|
|
(Contributed by SF, 2-Mar-2015.) $)
|
|
tccl $p |- ( A e. NC -> T_c A e. NC ) $=
|
|
( vx vy cncs wcel ctc cv cpw1 cnc wceq wrex wa cio wreu ncspw1eu reiotacl
|
|
df-tc syl syl5eqel ) ADEZAFBGZDEUACGHIJCAKZLBMZDCABQTUBBDNUCDEBCAOUBBDPRS
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The defining property of the cardinal T operation. (Contributed by SF,
|
|
2-Mar-2015.) $)
|
|
eqtc $p |- ( A e. NC -> ( T_c A = B <-> E. x e. A B = Nc ~P1 x ) ) $=
|
|
( vy cncs wcel ctc wceq cv cpw1 cnc wrex wa simpr tccl adantr eqeltrrd ex
|
|
wi wb vex pw1ex ncelncsi eleq1 mpbiri rexlimivw a1i wreu ncspw1eu rexbidv
|
|
cio eqeq1 reiota2 sylan2 ancoms df-tc eqeq1i syl6rbbr pm5.21ndd ) BEFZCEF
|
|
ZBGZCHZCAIZJZKZHZABLZUTVCVAUTVCMVBCEUTVCNUTVBEFVCBOPQRVHVASUTVGVAABVGVAVF
|
|
EFVEVDAUAUBUCCVFEUDUEUFUGUTVAVCVHTUTVAMVHDIZEFVIVFHZABLZMDUKZCHZVCVAUTVHV
|
|
MTZUTVAVKDEUHVNDABUIVKVHDECVICHVJVGABVICVFULUJUMUNUOVBVLCABDUPUQURRUS $.
|
|
$}
|
|
|
|
${
|
|
$d A y $. $d B y $.
|
|
$( The unit power class of an element of a cardinal is in the cardinal's T
|
|
raising. (Contributed by SF, 2-Mar-2015.) $)
|
|
pw1eltc $p |- ( ( A e. NC /\ B e. A ) -> ~P1 B e. T_c A ) $=
|
|
( vy cncs wcel wa cpw1 cnc ctc cvv pw1exg ncidg syl adantl wceq wrex eqid
|
|
cv pw1eq nceqd eqeq2d rspcev mpan2 wb eqtc adantr mpbird eleqtrrd ) ADEZB
|
|
AEZFZBGZULHZAIZUJULUMEZUIUJULJEUOBAKULJLMNUKUNUMOZUMCRZGZHZOZCAPZUJVAUIUJ
|
|
UMUMOZVAUMQUTVBCBAUQBOZUSUMUMVCURULUQBSTUAUBUCNUIUPVAUDUJCAUMUEUFUGUH $.
|
|
$}
|
|
|
|
$( The T raising of cardinal zero is still cardinal zero. (Contributed by
|
|
SF, 2-Mar-2015.) $)
|
|
tc0c $p |- T_c 0c = 0c $=
|
|
( c0c ctc cncs wcel c0 wceq 0cnc tccl ax-mp cpw1 pw10 nulel0c pw1eltc mp2an
|
|
eqeltrri nceleq mp4an ) ABZCDZACDZERDEADZRAFTSGAHIGEJZERKTUAUBRDGLAEMNOLRAE
|
|
PQ $.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d x y $.
|
|
$( T raising distributes over addition. (Contributed by SF,
|
|
2-Mar-2015.) $)
|
|
tcdi $p |- ( ( A e. NC /\ B e. NC ) ->
|
|
T_c ( A +c B ) = ( T_c A +c T_c B ) ) $=
|
|
( vx vy cncs wcel wa cv c0 cnc wceq wex cplc ctc cpw1 mp2an ax-mp pw1eltc
|
|
tccl tceq ccompl csn cxp eeanv cen wbr vex complex xpsnen snex xpex mpbir
|
|
0ex eqnc eqeq2i anbi12i 2exbii 3bitr4ri cun ncelncsi ncaddccl cin necompl
|
|
elncs ncid xpnedisj eladdci mp3an pw1un pw1eq pw10 3eqtr3i eqeltri nceleq
|
|
pw1in mp4an addceq12 syl adantr adantl addceq12d 3eqtr4a exlimivv sylbi )
|
|
AEFZBEFZGZACHZIUAZUBZUCZJZKZBDHZIUBZUCZJZKZGZDLCLZABMZNZANZBNZMZKZAWHJZKZ
|
|
BWNJZKZGZDLCLXHCLZXJDLZGWTWGXHXJCDUDWSXKCDWMXHWRXJWLXGAWLXGKWKWHUEUFWHWIC
|
|
UGZIUMUHZUIWKWHWHWJXNWIUJUKZUNULUOWQXIBWQXIKWPWNUEUFWNIDUGZUMUIWPWNWNWOXQ
|
|
IUJUKZUNULUOUPUQWEXLWFXMCAVDDBVDUPURWSXFCDWSWLWQMZNZWLNZWQNZMZXBXEXTEFZYC
|
|
EFZWKWPUSZOZXTFZYGYCFXTYCKXSEFZYDWLEFZWQEFZYIWKXPUTZWPXRUTZWLWQVAPZXSSQYA
|
|
EFZYBEFZYEYJYOYLWLSQYKYPYMWQSQYAYBVAPYIYFXSFZYHYNWKWLFZWPWQFZWKWPVBZIKZYQ
|
|
WKXPVEZWPXRVEZWHWNWIIXOIVCVFZWKWPWLWQVGVHXSYFRPYGWKOZWPOZUSZYCWKWPVIUUEYA
|
|
FZUUFYBFZUUEUUFVBZIKUUGYCFYJYRUUHYLUUBWLWKRPYKYSUUIYMUUCWQWPRPYTOZIOZUUJI
|
|
UUAUUKUULKUUDYTIVJQWKWPVOVKVLUUEUUFYAYBVGVHVMXTYCYGVNVPWSXAXSKXBXTKABWLWQ
|
|
VQXAXSTVRWSXCYAXDYBWMXCYAKWRAWLTVSWRXDYBKWMBWQTVTWAWBWCWD $.
|
|
$}
|
|
|
|
$( T raising does not change cardinal one. (Contributed by SF,
|
|
2-Mar-2015.) $)
|
|
tc1c $p |- T_c 1c = 1c $=
|
|
( c1c ctc cncs wcel c0 csn wceq 1cnc tccl ax-mp cpw1 0ex pw1sn snel1c mp2an
|
|
pw1eltc eqeltrri snex nceleq mp4an ) ABZCDZACDZEFZFZUADUEADUAAGUCUBHAIJHUDK
|
|
ZUEUAELMUCUDADUFUADHELNAUDPOQUDERNUAAUEST $.
|
|
|
|
$( T raising does not change cardinal two. (Contributed by SF,
|
|
2-Mar-2015.) $)
|
|
tc2c $p |- T_c 2c = 2c $=
|
|
( c1c cplc ctc c2c cncs wcel wceq 1cnc tcdi mp2an tc1c addceq12i eqtri tceq
|
|
1p1e2c ax-mp 3eqtr3i ) AABZCZRDCZDSACZUABZRAEFZUCSUBGHHAAIJUAAUAAKKLMRDGSTG
|
|
ORDNPOQ $.
|
|
|
|
$( Two is a finite cardinal. (Contributed by SF, 4-Mar-2015.) $)
|
|
2nnc $p |- 2c e. Nn $=
|
|
( c1c cplc c2c cnnc 1p1e2c wcel 1cnnc peano2 ax-mp eqeltrri ) AABZCDEADFKDF
|
|
GAHIJ $.
|
|
|
|
$( Two is a cardinal number. (Contributed by SF, 3-Mar-2015.) $)
|
|
2nc $p |- 2c e. NC $=
|
|
( c2c cnnc wcel cncs 2nnc nnnc ax-mp ) ABCADCEAFG $.
|
|
|
|
${
|
|
$d A m n $.
|
|
$( The unit power class of a finite set is finite. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
pw1fin $p |- ( A e. Fin -> ~P1 A e. Fin ) $=
|
|
( vn vm cv wcel cnnc wrex cpw1 ncfinraise 3anidm23 rexlimiva simpl reximi
|
|
cfin wa syl elfin 3imtr4i ) ABDZEZBFGZAHZCDEZCFGZANEUBNEUAUCUCOZCFGZUDTUF
|
|
BFSFETUFAACSIJKUEUCCFUCUCLMPBAQCUBQR $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d A m $. $d a n $. $d A n $. $d m n $.
|
|
$( Cardinal T is closed under the natural numbers. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
nntccl $p |- ( A e. Nn -> T_c A e. Nn ) $=
|
|
( vn vm va cnnc wcel cv wex c0 wceq sylib wa wrex cfin elfin syl ad2antrr
|
|
cncs nnnc mpd ctc wne nulnnn eleq1 mtbiri necon2ai cpw1 wel rspcev sylibr
|
|
n0 eleq2 pw1fin wi ad2antlr simprl pw1eltc syl2anc simprr nceleq syl22anc
|
|
tccl simplr eqeltrd expr an32s rexlimdva ex exlimdv ) AEFZBGZAFZBHZAUAZEF
|
|
ZVJAIUBVMVJAIAIJVJIEFUCAIEUDUEUFBAUKKVJVLVOBVJVLVOVJVLLZVKUGZCGZFZCEMZVOV
|
|
PVQNFZVTVPVKNFZWAVPBDUHZDEMWBWCVLDAEDGAVKULUIDVKOUJVKUMPCVQOKVPVSVOCEVJVR
|
|
EFZVLVSVOUNVJWDLZVLVSVOWEVLVSLZLZVNVREWGVNRFZVRRFZVQVNFZVSVNVRJVJWHWDWFVJ
|
|
ARFZWHASZAVBPQWDWIVJWFVRSUOWGWKVLWJVJWKWDWFWLQWEVLVSUPAVKUQURWEVLVSUSVNVR
|
|
VQUTVAVJWDWFVCVDVEVFVGTVHVIT $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a f $. $d a g $. $d a t $. $d b f $. $d b g $. $d b t $.
|
|
$d f t $. $d g t $. $d M a $. $d M b $. $d M g $. $d M t $. $d N a $.
|
|
$d N b $. $d N g $. $d N t $.
|
|
$( Lemma for ~ ovce . Set up stratification for the result. (Contributed
|
|
by SF, 6-Mar-2015.) $)
|
|
ovcelem1 $p |- ( ( N e. V /\ M e. W ) -> { g | E. a E. b ( ~P1
|
|
a e. N /\ ~P1 b e. M /\ g ~~ ( a ^m b ) ) } e. _V ) $=
|
|
( vt vf wcel wa cen wbr wex cpw1fn csset c1c cvv cop bitri cv cpw1 co w3a
|
|
cmap cab ccnv cima cxp cins3 cfns c2nd cimage ccom ctxp csi3 cins2 ccompl
|
|
csymdif cins4 cin crn csn elima1c otelins3 opelcnv opelxp wrex wceq brcnv
|
|
vex brpw1fn rexbii elima risset 3bitr4i anbi12i 3bitri elrn2 elin wel wal
|
|
wf wb wn snex opex elcompl elsymdif opelssetsn otelins2 otsnelsi3 wfn wss
|
|
df-br brfns bitr3i opelco brimage dfrn5 eqeq2i bitr4i brsset rnex ceqsexv
|
|
exbii sseq1 oteltxp bibi12i xchbinx exnal 3bitrri con1bii oqelins4 mapval
|
|
df-f abeq2 ovex breq2 df-3an abbi2i cnvex imaexg mpan xpexg syl2an cnvexg
|
|
pw1fnex ins3exg syl ssetex ins3ex fnsex 2ndex coex ins2ex 1cex mpan2 3syl
|
|
imageex txpex si3ex symdifex imaex complex ins4ex enex inexg syl5eqelr
|
|
inex ) CDJZBEJZKZFUAZUBZCJZGUAZUBZBJZAUAZUUNUUQUEUCZLMZUDZGNZFNZAUFOUGZCU
|
|
HZUVFBUHZUIZUGZUJZPUJZUKPULUMZUNZUOZUPZUQZUSZQUHZURZUTZLUGZUQZUQZVAZVBZVA
|
|
ZQUHZQUHZRUVEAUWIUUTUWIJUUNVCZUUTSZUWHJZFNUVEFUUTUWHVDUWLUVDFUWLUUQVCZUWK
|
|
SZUWGJZGNUVDGUWKUWGVDUWOUVCGUWNUVKJZUWNUWFJZKUUPUUSKZUVBKUWOUVCUWPUWRUWQU
|
|
VBUWPUWMUWJSZUVJJUWJUWMSUVIJZUWRUWMUWJUUTUVJAVKZVEUWMUWJUVIVFUWTUWJUVGJZU
|
|
WMUVHJZKUWRUWJUWMUVGUVHVGUXBUUPUXCUUSHUAZUWJUVFMZHCVHUXDUUOVIZHCVHUXBUUPU
|
|
XEUXFHCUXEUWJUXDOMUXFUXDUWJOVJUUNUXDFVKZVLTVMHUWJUVFCVNHUUOCVOVPUXDUWMUVF
|
|
MZHBVHUXDUURVIZHBVHUXCUUSUXHUXIHBUXHUWMUXDOMUXIUXDUWMOVJUUQUXDGVKZVLTVMHU
|
|
WMUVFBVNHUURBVOVPVQTVRUWQUXDUWNSZUWEJZHNUXDUVAVIZUUTUXDLMZKZHNUVBHUWNUWEV
|
|
SUXLUXOHUXLUXKUWAJZUXKUWDJZKUXOUXKUWAUWDVTUXPUXMUXQUXNUXDUWSSZUVTJZIHWAZU
|
|
UQUUNIUAZWCZWDZIWBZUXPUXMUXSUXRUVSJZWEUYDUXRUVSUXDUWSHVKZUWMUWJUUQWFZUUNW
|
|
FZWGZWGWHUYDUYEUYEUYAVCZUXRSZUVRJZINUYCWEZINUYDWEIUXRUVRVDUYLUYMIUYLUYKUV
|
|
LJZUYKUVQJZWDUYCUYKUVLUVQWIUYNUXTUYOUYBUYNUYJUXDSPJUXTUYJUXDUWSPUYIVEUYAU
|
|
XDIVKZUYFWJTUYOUYJUWSSUVPJUYAUUQUUNSSUVOJZUYBUYJUXDUWSUVPUYFWKUYAUUQUUNUV
|
|
OUYPUXJUXGWLUYAUUQSUKJZUYAUUNSUVNJZKUYAUUQWMZUYAVBZUUNWNZKUYQUYBUYRUYTUYS
|
|
VUBUYRUYAUUQUKMUYTUYAUUQUKWOUUQUYAUYPWPWQUYSUYAUXDUVMMZUXDUUNPMZKZHNUXDVU
|
|
AVIZUXDUUNWNZKZHNVUBHUYAUUNPUVMWRVUEVUHHVUCVUFVUDVUGVUCUXDULUYAUHZVIVUFUY
|
|
AUXDULUYPUYFWSVUAVUIUXDUYAWTXAXBUXDUUNUYFUXGXCVQXFVUGVUBHVUAUYAUYPXDUXDVU
|
|
AUUNXGXEVRVQUYAUUQUUNUKUVNXHUUQUUNUYAXPVPVRXIXJXFUYCIXKXLXMTUXDUWMUWJUUTU
|
|
VTUXAXNUXMUXDUYBIUFZVIUYDUVAVUJUXDUUNUUQIUXGUXJXOXAUYBIUXDXQTVPUXQUXDUWKS
|
|
UWCJUXDUUTSUWBJZUXNUXDUWMUWKUWCUYGWKUXDUWJUUTUWBUYHWKVUKUXDUUTUWBMUXNUXDU
|
|
UTUWBWOUXDUUTLVJWQVRVQTXFUXNUVBHUVAUUNUUQUEXRUXDUVAUUTLXSXEVRVQUWNUVKUWFV
|
|
TUUPUUSUVBXTVPXFTXFTYAUUMUVIRJZUVKRJZUWIRJZUUKUVGRJZUVHRJZVULUULUVFRJZUUK
|
|
VUOOYHYBZUVFCRDYCYDVUQUULVUPVURUVFBREYCYDUVGUVHRRYEYFVULUVJRJVUMUVIRYGUVJ
|
|
RYIYJVUMUWGRJZUWHRJZVUNVUMUWFRJVUSUWEUWAUWDUVTUVSUVRQUVLUVQPYKYLUVPUVOUKU
|
|
VNYMPUVMYKULYNYTYOUUAUUBYPUUCYQUUDUUEUUFUWCUWBLUUGYBYPYPUUJXDUVKUWFRRUUHY
|
|
RVUSQRJZVUTYQUWGQRRYCYRVUTVVAVUNYQUWHQRRYCYRYSYSUUI $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a m $. $d a n $. $d b m $. $d b n $. $d M a $. $d M b $.
|
|
$d M m $. $d m n $. $d N a $. $d N b $. $d N m $. $d N n $. $d a g $.
|
|
$d b g $. $d g m $. $d g n $. $d M g $. $d N g $. $d M n $.
|
|
$( The value of cardinal exponentiation. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
ovce $p |- ( ( N e. NC /\ M e. NC ) ->
|
|
( N ^c M ) = { g | E. a E. b
|
|
( ~P1 a e. N /\ ~P1 b e. M /\ g ~~ ( a ^m b ) ) } ) $=
|
|
( vn vm cncs wcel cv cpw1 co w3a wex cab cvv cce wceq eleq2 2exbidv df-ce
|
|
cmap cen wbr ovcelem1 3anbi1d abbidv 3anbi2d ovmpt2g mpd3an3 ) CHIBHIDJZK
|
|
ZCIZEJZKZBIZAJUKUNUBLUCUDZMZENDNZAOZPICBQLUTRABCHHDEUEFGCBHHULFJZIZUOGJZI
|
|
ZUQMZENDNZAOUTQUMVDUQMZENDNZAOPVACRZVFVHAVIVEVGDEVIVBUMVDUQVACULSUFTUGVCB
|
|
RZVHUSAVJVGURDEVJVDUPUMUQVCBUOSUHTUGAGFDEUAUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d a n $. $d a t $. $d a u $. $d n t $. $d n u $. $d t u $.
|
|
$( Lemma for ~ ceex . Set up part of the stratification. (Contributed by
|
|
SF, 6-Mar-2015.) $)
|
|
ceexlem1 $p |- ( <. { { a } } , n >. e. ( _S o. SI Pw1Fn ) <->
|
|
~P1 a e. n ) $=
|
|
( vu vt cv csn cpw1fn csi wbr csset wa wex cpw1 wceq wel wcel exbii bitri
|
|
snex vex cop ccom brsnsi1 anbi1i 19.41v anass 3bitr2i excom breq1 ceqsexv
|
|
anbi2d brpw1fn brssetsn anbi12i opelco df-clel 3bitr4i ) BEZFZFZCEZGHZIZV
|
|
AAEZJIZKZCLZDEZURMZNZDAOZKZDLZUTVDUAJVBUBPVIVDPVGVAVHFZNZUSVHGIZVEKZKZCLZ
|
|
DLZVMVGVRDLZCLVTVFWACVFVOVPKZDLZVEKWBVEKZDLWAVCWCVEDUSVAGURSUCUDWBVEDUEWD
|
|
VRDVOVPVEUFQUGQVRCDUHRVSVLDVSVPVNVDJIZKZVLVQWFCVNVHSVOVEWEVPVAVNVDJUIUKUJ
|
|
VPVJWEVKURVHBTULVHVDDTATUMUNRQRCUTVDJVBUODVIVDUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a g $. $d a m $. $d a n $. $d a x $. $d b g $. $d b m $.
|
|
$d b n $. $d b x $. $d g m $. $d g n $. $d g x $. $d m n $. $d a f $.
|
|
$d b f $. $d f g $. $d f x $. $d m x $. $d n x $.
|
|
$( Cardinal exponentiation is stratified. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
ceex $p |- ^c e. _V $=
|
|
( vn vm va vb vg vx cncs csset cins2 cen cv wcel wbr wex csn cop otelins2
|
|
wa vex ins2ex vf cce cxp cvv cpw1fn ccom cins3 cfns c2nd cimage ctxp csi3
|
|
csi csymdif c1c cima ccompl cins4 cin crn cpw1 cdif cmap co w3a cab cmpt2
|
|
df-ce snex otelins3 ceexlem1 elimapw11c elin opex oqelins4 otsnelsi3 wceq
|
|
3bitri bitri elrn2 wfn wss brfns brco brimage dfrn5 eqeq2i bitr4i anbi12i
|
|
wf brsset exbii rnex sseq1 ceqsexv df-br trtxp df-f 3bitr4i elmap releqel
|
|
bitr3i ensym ovex breq2 3anass 19.42v weq breq1 3anbi3d 2exbidv releqmpt2
|
|
elab eqtr4i ncsex ssetex pw1fnex siex coex ins3ex fnsex 2ndex txpex si3ex
|
|
imageex symdifex 1cex imaex complex ins4ex enex pw1ex mpt2exlem eqeltri
|
|
inex ) UBGGUCUDUCHIHUEUMZUFZUGZIZYQIZIZIZHUGZUHHUIUJZUFZUKZULZIZUNZUOUPZU
|
|
QZURZJIZIZUSZUTZULZURZUSZUOVAZUPZUSZUUTUPZUGUNUOUPVBZUDUBABGGCKZVAAKZLZDK
|
|
ZVABKZLZEKZUVEUVHVCVDZJMZVEZDNCNZEVFZVGUVDEBACDVHABFGGUVCUVPUVEOZOZFKZOZU
|
|
VFUVIPZPZPZUVBLZCNUVGUVJUVSUVLJMZVEZDNZCNZUWBUVCLUVSUVPLUWDUWGCUWCYSLZUWC
|
|
UVALZRUVGUVJUWERZDNZRZUWDUWGUWIUVGUWJUWLUWIUVRUWAPYRLUVRUVFPYQLUVGUVRUVTU
|
|
WAYRUVSVIZQUVRUVFUVIYQBSZVJACVKVRUWJUVHOZOZUWCPZUUSLZDNUWLDUWCUUSVLUWSUWK
|
|
DUWSUWRUUBLZUWRUURLZRUWKUWRUUBUURVMUWTUVJUXAUWEUWTUWQUWBPUUALUWQUWAPYTLZU
|
|
VJUWQUVRUWBUUAUVQVIQUWQUVTUWAYTUWNQUXBUWQUVIPYQLUVJUWQUVFUVIYQASZQBDVKVSV
|
|
RUXAUWQUVRUVTPPUUQLUWPUVQUVSPZPZUUPLZUWEUWQUVRUVTUWAUUQUVFUVIUXCUWOVNVOUW
|
|
PUVQUVSUUPUVHVIZUVEVIZFSZVPUXFUVKUXEPZUUOLZENUVKUVLVQZUVSUVKJMZRZENUWEEUX
|
|
EUUOVTUXKUXNEUXKUXJUULLZUXJUUNLZRUXNUXJUULUUNVMUXOUXLUXPUXMUXOUVKUWPUVQPZ
|
|
PUUKLUXLUVKUWPUVQUVSUUKUXIVOEUAUVLUUGUXQUWPUVQUXGUXHVNUAKZUVHUVEPZPUUFLZU
|
|
VHUVEUXRWJZUXROUXQPUUGLUXRUVLLUXRUVHUHMZUXRUVEUUEMZRZUXRUVHWAZUXRUTZUVEWB
|
|
ZRUXTUYAUYBUYEUYCUYGUVHUXRUASZWCUYCUXRUVSUUDMZUVSUVEHMZRZFNUVSUYFVQZUVSUV
|
|
EWBZRZFNUYGFUXRUVEHUUDWDUYKUYNFUYIUYLUYJUYMUYIUVSUIUXRUPZVQUYLUXRUVSUIUYH
|
|
UXIWEUYFUYOUVSUXRWFWGWHUVSUVEUXICSZWKWIWLUYMUYGFUYFUXRUYHWMUVSUYFUVEWNWOV
|
|
RWIUXTUXRUXSUUFMUYDUXRUXSUUFWPUXRUVHUVEUHUUEWQXBUVHUVEUXRWRWSUXRUVHUVEUUF
|
|
UYHDSZUYPVPUVEUVHUXRUYPUYQUYHWTWSXAVSUXPUVKUXDPUUMLZUVKUVSJMZUXMUVKUWPUXD
|
|
UUMUXGQUYRUVKUVSPJLUYSUVKUVQUVSJUXHQUVKUVSJWPWHUVKUVSXCVRWIVSWLUXMUWEEUVL
|
|
UVEUVHVCXDUVKUVLUVSJXEWOVRVRWIVSWLVSWIUWCYSUVAVMUWGUVGUWKRZDNUWMUWFUYTDUV
|
|
GUVJUWEXFWLUVGUWKDXGVSWSWLCUWBUVBVLUVOUWHEUVSUXIEFXHZUVNUWFCDVUAUVMUWEUVG
|
|
UVJUVKUVSUVLJXIXJXKXMWSXLXNGGUVCXOXOUVBUUTYSUVAYRYQHYPXPUEXQXRXSZXTTUUSUU
|
|
TUUBUURUUAYTYQVUBTTTUUQUUPUUOUULUUNUUKUUJUUIUOUUCUUHHXPXTUUGUUFUHUUEYAHUU
|
|
DXPUIYBYEXSYCYDTYFYGYHYIYJUUMJYKTTYOWMYDYJYOUOYGYLZYHYOVUCYHYMYN $.
|
|
$}
|
|
|
|
${
|
|
$d A g $. $d A x $. $d A y $. $d g x $. $d g y $. $d M g $. $d M x $.
|
|
$d M y $. $d N g $. $d N x $. $d N y $. $d x y $.
|
|
$( Membership in cardinal exponentiation. Theorem XI.2.38 of [Rosser]
|
|
p. 382. (Contributed by SF, 6-Mar-2015.) $)
|
|
elce $p |- ( ( N e. NC /\ M e. NC ) ->
|
|
( A e. ( N ^c M ) <-> E. x E. y ( ~P1 x e. N /\ ~P1 y e. M /\
|
|
A ~~ ( x ^m y ) ) ) ) $=
|
|
( vg cncs wcel wa cvv cce co cv cpw1 cen wbr w3a wex wi a1i cmap 3ad2ant3
|
|
elex brex simpld exlimivv wb ovce eleq2d wceq breq1 3anbi3d 2exbidv elabg
|
|
cab sylan9bb ex pm5.21ndd ) EGHDGHIZCJHZCEDKLZHZAMZNEHZBMZNDHZCVCVEUALZOP
|
|
ZQZBRARZVBUTSUSCVAUCTVJUTSUSVIUTABVHVDUTVFVHUTVGJHCVGOUDUEUBUFTUSUTVBVJUG
|
|
USVBCVDVFFMZVGOPZQZBRARZFUOZHUTVJUSVAVOCFDEABUHUIVNVJFCJVKCUJZVMVIABVPVLV
|
|
HVDVFVKCVGOUKULUMUNUPUQUR $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a g $. $d a m $. $d a n $. $d a p $. $d b g $. $d b m $.
|
|
$d b n $. $d b p $. $d g m $. $d g n $. $d g p $. $d m n $. $d m p $.
|
|
$d n p $.
|
|
$( Functionhood statement for cardinal exponentiation. (Contributed by SF,
|
|
6-Mar-2015.) $)
|
|
fnce $p |- ^c Fn ( NC X. NC ) $=
|
|
( vn vm vp va vb vg cce cncs cxp wfn cv wcel wa cpw1 cmap cen wbr w3a wex
|
|
co cab wceq coprab copab weu cvv ovcelem1 isset sylib wmo moeq eu5 sylibr
|
|
mpbiran2 fnoprab cmpt2 df-ce df-mpt2 eqtri fneq1i df-xp fneq2i bitri
|
|
mpbir ) GHHIZJZAKZHLBKZHLMZCKDKZNVGLEKZNVHLFKVJVKOTPQRESDSFUAZUBZMABCUCZV
|
|
IABUDZJZVIVMABCVIVMCSZVMCUEZVIVLUFLVQFVHVGHHDEUGCVLUHUIVRVQVMCUJCVLUKVMCU
|
|
LUNUMUOVFVNVEJVPVEGVNGABHHVLUPVNFBADEUQABCHHVLURUSUTVEVOVNABHHVAVBVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a g $. $d b g $. $d M a $. $d M b $. $d M g $.
|
|
$( A condition for cardinal exponentiation being non-empty. Theorem
|
|
XI.2.42 of [Rosser] p. 382. (Contributed by SF, 6-Mar-2015.) $)
|
|
ce0nnul $p |- ( M e. NC -> ( ( M ^c 0c ) =/= (/) <->
|
|
E. a ~P1 a e. M ) ) $=
|
|
( vg vb cncs wcel cv c0c cce co wex cpw1 cmap cen wbr w3a c0 wb wa wceq
|
|
0cnc elce mpan2 exbidv n0 19.42vv 3anass 2exbii nulel0c ovex enrflx pw1eq
|
|
wne 0ex pw10 syl6eq eleq1d adantl id oveq2 breqan12d anbi12d spc2ev mp2an
|
|
biantru 3bitr4ri exbii excom bitri 3bitr4g ) AEFZCGZAHIJZFZCKBGZLAFZDGZLZ
|
|
HFZVLVOVQMJZNOZPZDKZBKZCKZVMQUMVPBKZVKVNWDCVKHEFVNWDRUABDVLHAUBUCUDCVMUEW
|
|
FWCCKZBKWEVPWGBVPVSWASZSZDKCKVPWHDKCKZSWGVPVPWHCDUFWBWICDVPVSWAUGUHWJVPQH
|
|
FZVOQMJZWLNOZWJUIWLVOQMUJZUKWHWKWMSCDWLQWNUNVLWLTZVQQTZSVSWKWAWMWPVSWKRWO
|
|
WPVRQHWPVRQLQVQQULUOUPUQURWOWPVLWLVTWLNWOUSVQQVOMUTVAVBVCVDVEVFVGWCBCVHVI
|
|
VJ $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d M a $.
|
|
$( Inference form of ~ ce0nnul . (Contributed by SF, 9-Mar-2015.) $)
|
|
ce0nnuli $p |- ( ( M e. NC /\ ~P1 A e. M ) -> ( M ^c 0c ) =/= (/) ) $=
|
|
( va cncs wcel cpw1 wa c0c cce co c0 wne cv wex cvv elex sylib wceq pw1eq
|
|
pw1exb eleq1d spcegv mpcom adantl wb ce0nnul adantr mpbird ) BDEZAFZBEZGB
|
|
HIJKLZCMZFZBEZCNZUKUPUIAOEZUKUPUKUJOEUQUJBPATQUOUKCAOUMARUNUJBUMASUAUBUCU
|
|
DUIULUPUEUKBCUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a g $. $d a p $. $d a q $. $d b g $. $d b p $. $d b q $.
|
|
$d g p $. $d g q $. $d M a $. $d M b $. $d M g $. $d N a $. $d N b $.
|
|
$d N g $. $d p q $.
|
|
$( The sum of two cardinals raised to ` 0c ` is non-empty iff each addend
|
|
raised to ` 0c ` is non-empty. Theorem XI.2.43 of [Rosser] p. 383.
|
|
(Contributed by SF, 9-Mar-2015.) $)
|
|
ce0addcnnul $p |- ( ( M e. NC /\ N e. NC ) ->
|
|
( ( ( M +c N ) ^c 0c ) =/= (/) <->
|
|
( ( M ^c 0c ) =/= (/) /\ ( N ^c 0c ) =/= (/) ) ) ) $=
|
|
( vb vg va vp vq cncs wcel wa c0c cce co c0 wne wceq cpw1 wex cnc cvv cin
|
|
cplc cv cun wrex wb ncaddccl ce0nnul eladdc exbii syl6bb ncseqnc bi2anan9
|
|
syl wi biimpar wss ssun1 id syl5sseqr ssun2 jca sspw1 anbi12i eeanv sylib
|
|
vex bitr4i pw1eq eleq1d pw1ex ncid speiv ncelncs mp2b mpbir pm3.2i oveq1d
|
|
weq nceq neeq1d mpbiri ad2ant2l a1d exlimivv impcom oveq1 syl5ibr exlimdv
|
|
rexlimdvva sylbid syl6bbr csn cxp cen wbr vvex xpsnen mpbi snex xpex eqnc
|
|
enpw1 0ex addceq12i oveq1i pw1un vn0 xpnedisj ax-mp pw1in 3eqtr3i eladdci
|
|
pw10 mp3an eqeltri unex spcev ncelncsi eqnetrri addceq12 syl6bir exlimdvv
|
|
mp2an impbid ) AHIZBHIZJZABUBZKLMZNOZAKLMZNOZBKLMZNOZJZYHYKCUCZDUCZUANPZE
|
|
UCZQZYQYRUDZPZJZDBUECAUEZERZYPYHYIHIZYKUUFUFABUGUUGYKUUAYIIZERUUFYIEUHUUH
|
|
UUEEUUAABCDUIUJUKUNYHUUEYPEYHUUDYPCDABYHYQAIZYRBIZJZJAYQSZPZBYRSZPZJZUUDY
|
|
PUOYHUUPUUKYFUUMUUIYGUUOUUJAYQULBYRULUMUPUUDYPUUPUULKLMZNOZUUNKLMZNOZJZUU
|
|
CYSUVAUUCFUCZYTUQZYQUVBQZPZJZGUCZYTUQZYRUVGQZPZJZJZGRFRZYSUVAUOZUUCYQUUAU
|
|
QZYRUUAUQZJZUVMUUCUVOUVPUUCUUBYQUUAYQYRURUUCUSZUTUUCUUBYRUUAYRYQVAUVRUTVB
|
|
UVQUVFFRZUVKGRZJUVMUVOUVSUVPUVTFYQYTCVGZVCGYRYTDVGZVCVDUVFUVKFGVEVHVFUVLU
|
|
VNFGUVLUVAYSUVEUVJUVAUVCUVHUVEUVJJUVAUVDSZKLMZNOZUVISZKLMZNOZJUWEUWHUWEUU
|
|
AUWCIZERZUWIUVDUWCIEFEFVSUUAUVDUWCYTUVBVIVJUVDUVBFVGVKZVLVMUVDTIUWCHIUWEU
|
|
WJUFUWKUVDTVNUWCEUHVOVPUWHUUAUWFIZERZUWLUVIUWFIEGEGVSUUAUVIUWFYTUVGVIVJUV
|
|
IUVGGVGVKZVLVMUVITIUWFHIUWHUWMUFUWNUVITVNUWFEUHVOVPVQUVEUURUWEUVJUUTUWHUV
|
|
EUUQUWDNUVEUULUWCKLYQUVDVTVRWAUVJUUSUWGNUVJUUNUWFKLYRUVIVTVRWAUMWBWCWDWEU
|
|
NWFUUMYMUURUUOYOUUTUUMYLUUQNAUULKLWGWAUUOYNUUSNBUUNKLWGWAUMWHUNWJWIWKYHYP
|
|
YQQZAIZYRQZBIZJZDRCRZYKYHYPUWPCRZUWRDRZJUWTYFYMUXAYGYOUXBACUHBDUHUMUWPUWR
|
|
CDVEWLYHUWSYKCDYHUWSAUWOSZPZBUWQSZPZJZYKYFUXDUWPYGUXFUWRAUWOULBUWQULUMUXG
|
|
YKUXCUXEUBZKLMZNOYQTWMZWNZQZSZYRNWMZWNZQZSZUBZKLMZUXINUXRUXHKLUXMUXCUXQUX
|
|
EUXMUXCPUXLUWOWOWPZUXKYQWOWPUXTYQTUWAWQWRUXKYQXCWSUXLUWOUXKYQUXJUWATWTXAZ
|
|
VKZXBVPUXQUXEPUXPUWQWOWPZUXOYRWOWPUYCYRNUWBXDWRUXOYRXCWSUXPUWQUXOYRUXNUWB
|
|
NWTXAZVKZXBVPXEXFUXSNOZUUAUXRIZERZUXKUXOUDZQZUXRIZUYHUYJUXLUXPUDZUXRUXKUX
|
|
OXGUXLUXMIUXPUXQIUXLUXPUAZNPUYLUXRIUXLUYBVLUXPUYEVLUXKUXOUAZQZNQZUYMNUYNN
|
|
PUYOUYPPYQYRTNWQXHXIUYNNVIXJUXKUXOXKXNXLUXLUXPUXMUXQXMXOXPUYGUYKEUYIUXKUX
|
|
OUYAUYDXQYTUYIPUUAUYJUXRYTUYIVIVJXRXJUXRHIZUYFUYHUFUXMHIUXQHIUYQUXLUYBXSU
|
|
XPUYEXSUXMUXQUGYDUXREUHXJVPXTUXGYJUXINUXGYIUXHKLABUXCUXEYAVRWAWBYBYCWKYE
|
|
$.
|
|
$}
|
|
|
|
${
|
|
$d m t $. $d m n $. $d N m $.
|
|
$( A natural raised to cardinal zero is non-empty. Theorem XI.2.44 of
|
|
[Rosser] p. 383. (Contributed by SF, 9-Mar-2015.) $)
|
|
ce0nn $p |- ( N e. Nn -> ( N ^c 0c ) =/= (/) ) $=
|
|
( vm vn vt cv c0c cce co wne c1c c1st c2nd csn cima wcel wbr oveq1 neeq1d
|
|
c0 wceq cplc ccnv cres cfullfun ccompl cab cvv wn vex elcompl cop wrex wa
|
|
brres eliniseg anbi2i 0cex 3bitri rexbii elima risset 3bitr4i brfullfunop
|
|
necon3bbii bitri abbi2i 1stex 2ndex cnvex snex imaex resex ceex fullfunex
|
|
op1st2nd complex eqeltrri cncs cpw1 0cnc pw10 nulel0c ce0nnuli mp2an cnnc
|
|
weq eqeltri wi nnnc 1cnc 0ex pw1sn snel1c jctr wb ce0addcnnul syl5ibr syl
|
|
mpan2 finds ) BEZFGHZSIZFFGHZSIZCEZFGHZSIZXFJUAZFGHZSIZAFGHZSIBCAKLUBZFMZ
|
|
NZUCZGUDZUBZSMZNZNZUEZXCBUFUGXCBYBXAYBOXAYAOZUHXCXAYABUIZUJYCXBSYCXAFUKZX
|
|
TOZYESXQPXBSTDEZXAXPPZDXTULYGYETZDXTULYCYFYHYIDXTYHYGXAKPZYGXOOZUMYJYGFLP
|
|
ZUMYIYGXAKXOUNYKYLYJLFYGUOUPXAFYGYDUQVOURUSDXAXPXTUTDYEXTVAVBXQSYEUOXAFSG
|
|
YDUQVCURVDVEVFYAXPXTKXOVGXMXNLVHVIFVJVKVLXRXSXQGVMVNVISVJZVKVKVPVQXAFTXBX
|
|
DSXAFFGQRBCWFXBXGSXAXFFGQRXAXITXBXJSXAXIFGQRXAATXBXLSXAAFGQRFVROSVSZFOXEV
|
|
TYNSFWAWBWGSFWCWDXFWEOXFVROZXHXKWHXFWIXHXKYOXHJFGHSIZUMZXHYPJVROZXSVSZJOY
|
|
PWJYSXSMJSWKWLXSYMWMWGXSJWCWDWNYOYRXKYQWOWJXFJWPWSWQWRWT $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a g $. $d A g $. $d A p $. $d A t $.
|
|
$d B a $. $d B b $. $d b g $. $d B g $. $d B p $. $d B t $. $d g p $.
|
|
$d g t $. $d p t $.
|
|
cenc.1 $e |- A e. _V $.
|
|
cenc.2 $e |- B e. _V $.
|
|
$( Cardinal exponentiation in terms of cardinality. Theorem XI.2.39 of
|
|
[Rosser] p. 382. (Contributed by SF, 6-Mar-2015.) $)
|
|
cenc $p |- ( Nc ~P1 A ^c Nc ~P1 B ) = Nc ( A ^m B ) $=
|
|
( vg vp vt va vb cpw1 cnc co cmap cv wcel cen wbr w3a wex wb cce wa enpw1
|
|
elnc bitr4i enmap1 enmap2 entr syl2an syl2anb ancoms sylan 3impa exlimivv
|
|
cncs pw1ex ncelncsi elce mp2an ssriv ncid wceq pw1eq eleq1d adantr adantl
|
|
3imtr4i oveq12 breq2d 3anbi123d spc2ev mp3an12 eqssi ) AJZKZBJZKZUALZABML
|
|
ZKZEVRVTFNZJZVOOZGNZJZVQOZENZWAWDMLZPQZRZGSFSZWGVSPQZWGVROZWGVTOZWJWLFGWC
|
|
WFWIWLWCWFUBWHVSPQZWIWLWCWAAPQZWDBPQZWOWFWCWBVNPQWPWBVNUDWAAUCUEWFWEVPPQW
|
|
QWEVPUDWDBUCUEWPWHAWDMLZPQWRVSPQWOWQWAAWDUFWDBAUGWHWRVSUHUIUJWIWOWLWGWHVS
|
|
UHUKULUMUNVOUOOZVQUOOZWMWKTVNACUPZUQZVPBDUPZUQZFGWGVQVOURUSWGVSUDZVGUTEVT
|
|
VRWLHNZJZVOOZINZJZVQOZWGXFXIMLZPQZRZISHSZWNWMVNVOOZVPVQOZWLXOVNXAVAVPXCVA
|
|
XNXPXQWLRHIABCDXFAVBZXIBVBZUBZXHXPXKXQXMWLXRXHXPTXSXRXGVNVOXFAVCVDVEXSXKX
|
|
QTXRXSXJVPVQXIBVCVDVFXTXLVSWGPXFAXIBMVHVIVJVKVLXEWSWTWMXOTXBXDHIWGVQVOURU
|
|
SVGUTVM $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a g $. $d b g $. $d M a $. $d M b $. $d M g $. $d N a $.
|
|
$d N b $. $d N g $.
|
|
$( Cardinal exponentiation is non-empty iff the two sets raised to zero are
|
|
non-empty. Theorem XI.2.47 of [Rosser] p. 384. (Contributed by SF,
|
|
9-Mar-2015.) $)
|
|
ce0nnulb $p |- ( ( N e. NC /\ M e. NC ) -> ( ( ( N ^c 0c ) =/= (/) /\ ( M
|
|
^c 0c ) =/= (/) ) <-> ( N ^c M ) =/= (/) ) ) $=
|
|
( va vb vg cncs wcel wa c0c cce co wne cpw1 wex ce0nnul bi2anan9 cnc wceq
|
|
c0 cv eeanv syl6bbr ncseqnc cmap cenc ovex ncid ne0i ax-mp eqnetri oveq12
|
|
vex neeq1d mpbiri syl6bir exlimdvv cen wbr w3a elce 3simpa 2eximi exlimdv
|
|
n0 syl6bi syl5bi impbid bitrd ) BFGZAFGZHZBIJKSLZAIJKSLZHZCTZMZBGZDTZMZAG
|
|
ZHZDNCNZBAJKZSLZVKVNVQCNZVTDNZHWBVIVLWEVJVMWFBCOADOPVQVTCDUAUBVKWBWDVKWAW
|
|
DCDVKWABVPQZRZAVSQZRZHZWDVIWHVQVJWJVTBVPUCAVSUCPWKWDWGWIJKZSLWLVOVRUDKZQZ
|
|
SVOVRCULDULUEWMWNGWNSLWMVOVRUDUFUGWNWMUHUIUJWKWCWLSBWGAWIJUKUMUNUOUPWDETZ
|
|
WCGZENVKWBEWCVDVKWPWBEVKWPVQVTWOWMUQURZUSZDNCNWBCDWOABUTWRWACDVQVTWQVAVBV
|
|
EVCVFVGVH $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d M a $. $d M b $. $d N a $. $d N b $.
|
|
$( Biconditional closure law for cardinal exponentiation. Theorem XI.2.48
|
|
of [Rosser] p. 384. (Contributed by SF, 9-Mar-2015.) $)
|
|
ceclb $p |- ( ( M e. NC /\ N e. NC ) ->
|
|
( ( ( M ^c 0c ) =/= (/) /\ ( N ^c 0c ) =/= (/) ) <-> ( M ^c N ) e. NC )
|
|
) $=
|
|
( va vb cncs wcel wa c0c cce co c0 wne cpw1 wex ce0nnul bi2anan9 cnc wceq
|
|
cv ncseqnc syl6bbr oveq12 cmap vex cenc ncelncsi eqeltri syl6eqel syl6bir
|
|
eeanv ovex exlimdvv sylbid nulnnc mtbiri necon2ai ce0nnulb syl5ibr impbid
|
|
eleq1 ) AEFZBEFZGZAHIJKLZBHIJKLZGZABIJZEFZVCVFCSZMZAFZDSZMZBFZGZDNCNZVHVC
|
|
VFVKCNZVNDNZGVPVAVDVQVBVEVRACOBDOPVKVNCDUJUAVCVOVHCDVCVOAVJQZRZBVMQZRZGZV
|
|
HVAVTVKVBWBVNAVJTBVMTPWCVGVSWAIJZEAVSBWAIUBWDVIVLUCJZQEVIVLCUDDUDUEWEVIVL
|
|
UCUKUFUGUHUIULUMVHVFVCVGKLVHVGKVGKRVHKEFUNVGKEUTUOUPBAUQURUS $.
|
|
$}
|
|
|
|
$( Cardinal exponentiation to zero is a cardinal iff it is non-empty.
|
|
Corollary 1 of theorem XI.2.38 of [Rosser] p. 384. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
ce0nulnc $p |- ( M e. NC -> ( ( M ^c 0c ) =/= (/) <-> ( M ^c 0c ) e. NC ) )
|
|
$=
|
|
( vm c0c cce co c0 wne wa cncs wcel cv cpw1 wex nulel0c wceq pw1eq ax-mp wb
|
|
0ex 0cnc pw10 syl6eq eleq1d spcev ce0nnul mpbir biantru ceclb mpan2 syl5bb
|
|
) ACDEZFGZULCCDEFGZHZAIJZUKIJZUMULUMBKZLZCJZBMZFCJZUTNUSVABFSUQFOZURFCVBURF
|
|
LFUQFPUAUBUCUDQCIJZUMUTRTCBUEQUFUGUOVCUNUPRTACUHUIUJ $.
|
|
|
|
${
|
|
$d M a $.
|
|
$( If cardinal exponentiation to zero is a cardinal, then the base is the
|
|
cardinality of some unit power class. Corollary 2 of theorem XI.2.48 of
|
|
[Rosser] p. 384. (Contributed by SF, 9-Mar-2015.) $)
|
|
ce0ncpw1 $p |- ( ( M e. NC /\ ( M ^c 0c ) e. NC ) -> E. a M = Nc ~P1 a ) $=
|
|
( cncs wcel c0c cce co cv cpw1 cnc wceq wex c0 wne nulnnc mtbiri necon2ai
|
|
eleq1 ce0nnul ncseqnc exbidv bitr4d syl5ib imp ) ACDZAEFGZCDZABHIZJKZBLZU
|
|
GUFMNZUEUJUGUFMUFMKUGMCDOUFMCRPQUEUKUHADZBLUJABSUEUIULBAUHTUAUBUCUD $.
|
|
$}
|
|
|
|
$( Closure law for cardinal exponentiation. Corollary 3 of theorem XI.2.48
|
|
of [Rosser] p. 384. (Contributed by SF, 9-Mar-2015.) $)
|
|
cecl $p |- ( ( ( M e. NC /\ N e. NC ) /\
|
|
( ( M ^c 0c ) e. NC /\ ( N ^c 0c ) e. NC ) ) -> ( M ^c N ) e. NC ) $=
|
|
( cncs wcel wa c0c cce co c0 wne nulnnc eleq1 mtbiri necon2ai anim12i ceclb
|
|
wceq syl5ib imp ) ACDBCDEZAFGHZCDZBFGHZCDZEZABGHCDZUEUAIJZUCIJZETUFUBUGUDUH
|
|
UBUAIUAIQUBICDZKUAICLMNUDUCIUCIQUDUIKUCICLMNOABPRS $.
|
|
|
|
$( Reverse closure law for cardinal exponentiation. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
ceclr $p |- ( ( M e. NC /\ N e. NC /\ ( M ^c N ) e. NC ) ->
|
|
( ( M ^c 0c ) e. NC /\ ( N ^c 0c ) e. NC ) ) $=
|
|
( cncs cce co w3a c0c c0 wne wa ceclb biimp3ar wb ce0nulnc bi2anan9 3adant3
|
|
wcel mpbid ) ACQZBCQZABDECQZFAGDEZHIZBGDEZHIZJZUBCQZUDCQZJZSTUFUAABKLSTUFUI
|
|
MUASUCUGTUEUHANBNOPR $.
|
|
|
|
${
|
|
$d m n $. $d m p $. $d m x $. $d m y $. $d n p $. $d n x $. $d n y $.
|
|
$d p x $. $d p y $. $d x y $.
|
|
$( Full functionhood statement for cardinal exponentiation. (Contributed
|
|
by SF, 13-Mar-2015.) $)
|
|
fce $p |- ^c : ( NC X. NC ) --> ( NC u. { (/) } ) $=
|
|
( vp vn vm vx vy cncs c0 cce fnce cv cfv wcel wral co wa wne wex wo bitri
|
|
wceq cxp csn cun wf wfn crn wss wn wi df-ne c0c n0 cpw1 cmap cen wbr elce
|
|
w3a 3simpa ce0nnuli ex im2anan9 syl5 exlimdvv sylbid exlimdv syl5bi ceclb
|
|
sylibd syl5bir elun ovex elsnc orbi2i orcom df-or sylibr rgen2a cop fveq2
|
|
df-ov syl6eqr eleq1d ralxp mpbir fnfvrnss mp2an df-f mpbir2an ) FFUAZFGUB
|
|
ZUCZHUDHWJUEZHUFWLUGZIWMAJZHKZWLLZAWJMZWNIWRBJZCJZHNZWLLZCFMBFMXBBCFWSFLZ
|
|
WTFLZOZXAGTZUHZXAFLZUIZXBXGXAGPZXEXHXAGUJXEXJWSUKHNGPZWTUKHNGPZOZXHXJWOXA
|
|
LZAQXEXMAXAULXEXNXMAXEXNDJZUMWSLZEJZUMWTLZWOXOXQUNNUOUPZURZEQDQXMDEWOWTWS
|
|
UQXEXTXMDEXTXPXROXEXMXPXRXSUSXCXPXKXDXRXLXCXPXKXOWSUTVAXDXRXLXQWTUTVAVBVC
|
|
VDVEVFVGWSWTVHVIVJXBXHXAWKLZRZXIXAFWKVKYBXHXFRZXIYAXFXHXAGWSWTHVLVMVNYCXF
|
|
XHRXIXHXFVOXFXHVPSSSVQVRWQXBABCFFWOWSWTVSZTZWPXAWLYEWPYDHKXAWOYDHVTWSWTHW
|
|
AWBWCWDWEAWJWLHWFWGWJWLHWHWI $.
|
|
$}
|
|
|
|
$( Closure law for cardinal exponentiation when the base is a natural.
|
|
(Contributed by SF, 13-Mar-2015.) $)
|
|
ceclnn1 $p |- ( ( M e. Nn /\ N e. NC /\ ( N ^c 0c ) e. NC ) ->
|
|
( M ^c N ) e. NC ) $=
|
|
( cnnc wcel cncs c0c cce co w3a 3ad2ant1 simp2 c0 wne ce0nn wb ce0nulnc syl
|
|
nnnc mpbid simp3 cecl syl22anc ) ACDZBEDZBFGHEDZIAEDZUDAFGHZEDZUEABGHEDUCUD
|
|
UFUEARZJUCUDUEKUCUDUHUEUCUGLMZUHANUCUFUJUHOUIAPQSJUCUDUETABUAUB $.
|
|
|
|
${
|
|
$d M a $.
|
|
$( The value of non-empty cardinal exponentiation. Theorem XI.2.49 of
|
|
[Rosser] p. 385. (Contributed by SF, 9-Mar-2015.) $)
|
|
ce0 $p |- ( ( M e. NC /\ ( M ^c 0c ) e. NC ) -> ( M ^c 0c ) = 1c ) $=
|
|
( va cncs wcel c0c cce co wa c0 csn cnc c1c wceq cv cpw1 wex ce0ncpw1 vex
|
|
cmap 0ex map0e ovex ncid eqeltrri cenc eleqtrri df0c2 nceqi eqtr4i oveq12
|
|
pw10 mpan2 syl5eleqr exlimiv syl wb ncseqnc adantl mpbird df1c3 syl6eqr )
|
|
ACDZAEFGZCDZHZVCIJZKZLVEVCVGMZVFVCDZVEABNZOKZMZBPVIABQVLVIBVLVFVKIOZKZFGZ
|
|
VCVFVJISGZKZVOVPVFVQVJBRZUAVPVJISUBUCUDVJIVRTUEUFVLEVNMVCVOMEIKVNUGVMIUKU
|
|
HUIAVKEVNFUJULUMUNUOVDVHVIUPVBVCVFUQURUSITUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d A t $. $d A x $. $d A y $. $d t x $. $d t y $. $d x y $.
|
|
$( Membership in cardinal two. (Contributed by SF, 3-Mar-2015.) $)
|
|
el2c $p |- ( A e. 2c <-> E. x E. y ( x =/= y /\ A = { x , y } ) ) $=
|
|
( vt c1c wcel cv csn wceq cun ccompl wa wex c2c df-rex anbi1i exbii bitri
|
|
wrex 3bitri cplc wne elsuc el1c 19.41v bitr4i excom 1p1e2c eleq2i compleq
|
|
cpr snex uneq1 syl6eqr eqeq2d rexeqbidv ceqsexv wn weq elsn equcom notbii
|
|
df-pr vex elcompl df-ne 3bitr4i 3bitr3i ) CEEUAZFZDGZAGZHZIZCVKBGZHZJZIZB
|
|
VKKZSZLZDMZAMZCNFVLVOUBZCVLVOUKZIZLZBMZAMVJVTDESVKEFZVTLZDMZWCBCEDUCVTDEO
|
|
WKWAAMZDMWCWJWLDWJVNAMZVTLWLWIWMVTAVKUDPVNVTAUEUFQWADAUGRTVINCUHUIWBWHAWB
|
|
WFBVMKZSZVOWNFZWFLZBMWHVTWODVMVLULVNVRWFBVSWNVKVMUJVNVQWECVNVQVMVPJWEVKVM
|
|
VPUMVLVOVCUNUOUPUQWFBWNOWQWGBWPWDWFVOVMFZURABUSZURWPWDWRWSWRBAUSWSBVLUTBA
|
|
VARVBVOVMBVDVEVLVOVFVGPQTQVH $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d x y $.
|
|
ce2.1 $e |- A e. _V $.
|
|
$( The value of base two cardinal exponentiation. Theorem XI.2.70 of
|
|
[Rosser] p. 389. (Contributed by SF, 3-Mar-2015.) $)
|
|
ce2 $p |- ( M = Nc ~P1 A -> ( 2c ^c M ) = Nc ~P A ) $=
|
|
( vx vy cpw1 cnc wceq c2c cce co cvv c0 cpr wcel csn cun eqtri wne mpbir
|
|
cpw oveq2 cmap df-pr pw1eq ax-mp pw1un pw1sn 0ex uneq12i eqtr4i cv wa wex
|
|
vvex sneqb necon3bii eqid snex neeq1 neeq2 sylan9bb preq12 eqeq2d anbi12d
|
|
vn0 spc2ev el2c eqeltri cncs wb 2nc ncseqnc oveq1i prex cenc cen enprmapc
|
|
mp2an wbr ovex eqnc syl6eq ) BAFGZHIBJKIWDJKZAUAZGZBWDIJUBWELMNZAUCKZGZWG
|
|
WEWHFZGZWDJKWJIWLWDJIWLHZWKIOZWKLPZMPZNZIWKWOFZWPFZQZWQWKWOWPQZFZWTWHXAHW
|
|
KXBHLMUDWHXAUEUFWOWPUGRWQWOPZWPPZQWTWOWPUDWRXCWSXDLUOUHMUIUHUJUKUKWQIODUL
|
|
ZEULZSZWQXEXFNZHZUMZEUNDUNZWOWPSZWQWQHZXKXLLMSZVFWOWPLMLMUOUPUQTWQURXJXLX
|
|
MUMDEWOWPLUSMUSXEWOHZXFWPHZUMZXGXLXIXMXOXGWOXFSXPXLXEWOXFUTXFWPWOVAVBXQXH
|
|
WQWQXEXFWOWPVCVDVEVGVSDEWQVHTVIIVJOWMWNVKVLIWKVMUFTVNWHALMVOCVPRWJWGHWIWF
|
|
VQVTZXNWHWHHXRVFWHURLMAWHUOUICVRVSWIWFWHAUCWAWBTRWC $.
|
|
$}
|
|
|
|
$( Compute an exponent of the cardinality of one. Theorem 4.3 of [Specker]
|
|
p. 973. (Contributed by SF, 4-Mar-2015.) $)
|
|
ce2nc1 $p |- ( 2c ^c Nc 1c ) = Nc _V $=
|
|
( c2c c1c cnc cce co cvv cpw cpw1 wceq df1c2 nceqi vvex ce2 ax-mp pwv eqtri
|
|
) ABCZDEZFGZCZFCQFHZCIRTIBUAJKFQLMNSFOKP $.
|
|
|
|
$( Compute an exponent of the cardinality of the unit power class of one.
|
|
Theorem 4.4 of [Specker] p. 973. (Contributed by SF, 4-Mar-2015.) $)
|
|
ce2ncpw11c $p |- ( 2c ^c Nc ~P1 1c ) = Nc 1c $=
|
|
( c2c c1c cpw1 cnc cce co cpw wceq eqid 1cex ce2 ax-mp ncpw1c eqtri ) ABCDZ
|
|
EFZBGDZBDOOHPQHOIBOJKLMN $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
nclec.1 $e |- A e. _V $.
|
|
nclec.2 $e |- B e. _V $.
|
|
$( A relationship between cardinality, subset, and cardinal less than.
|
|
(Contributed by SF, 17-Mar-2015.) $)
|
|
nclec $p |- ( A C_ B -> Nc A <_c Nc B ) $=
|
|
( vx vy wss cnc wrex clec wbr wcel ncid sseq1 sseq2 rspc2ev mp3an12 brlec
|
|
cv ncex sylibr ) ABGZESZFSZGZFBHZIEAHZIZUGUFJKAUGLBUFLUBUHACMBDMUEUBAUDGE
|
|
FABUGUFUCAUDNUDBAOPQEFUGUFATBTRUA $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( A non-empty set is less than or equal to itself. Theorem XI.2.14 of
|
|
[Rosser] p. 375. (Contributed by SF, 4-Mar-2015.) $)
|
|
lecidg $p |- ( ( A e. V /\ A =/= (/) ) -> A <_c A ) $=
|
|
( vx vy wcel c0 wne wa clec wbr cv wss wrex ssid sseq2 rspcev mpan2 ancli
|
|
wex eximi n0 df-rex 3imtr4i adantl wb brlecg anidms adantr mpbird ) ABEZA
|
|
FGZHAAIJZCKZDKZLZDAMZCAMZUKUQUJUMAEZCSURUPHZCSUKUQURUSCURUPURUMUMLZUPUMNU
|
|
OUTDUMAUNUMUMOPQRTCAUAUPCAUBUCUDUJULUQUEZUKUJVACDAABBUFUGUHUI $.
|
|
$}
|
|
|
|
$( A cardinal is less than or equal to itself. Corollary 1 of theorem
|
|
XI.2.14 of [Rosser] p. 376. (Contributed by SF, 4-Mar-2015.) $)
|
|
nclecid $p |- ( A e. NC -> A <_c A ) $=
|
|
( cncs wcel c0 wne clec wbr wceq nulnnc eleq1 mtbiri necon2ai lecidg mpdan
|
|
) ABCZADEAAFGOADADHODBCIADBJKLABMN $.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d x y $.
|
|
$( Cardinal zero is a minimal element of cardinal less than or equal.
|
|
Theorem XI.2.15 of [Rosser] p. 376. (Contributed by SF, 4-Mar-2015.) $)
|
|
lec0cg $p |- ( ( A e. V /\ A =/= (/) ) -> 0c <_c A ) $=
|
|
( vx vy wcel c0 wne wa c0c clec wbr cv wss wrex wex 0ss jctr wceq wb cvv
|
|
eximi df-rex 3imtr4i csn df-0c rexeq ax-mp 0ex sseq1 rexbidv rexsn sylibr
|
|
n0 bitri adantl 0cex brlecg mpan adantr mpbird ) ABEZAFGZHIAJKZCLZDLZMZDA
|
|
NZCINZVBVHVAVBFVEMZDANZVHVEAEZDOVKVIHZDOVBVJVKVLDVKVIVEPQUADAUMVIDAUBUCVH
|
|
VGCFUDZNZVJIVMRVHVNSUEVGCIVMUFUGVGVJCFUHVDFRVFVIDAVDFVEUIUJUKUNULUOVAVCVH
|
|
SZVBITEVAVOUPCDIATBUQURUSUT $.
|
|
|
|
$( The cardinality of ` _V ` is a maximal element of cardinal less than or
|
|
equal. Theorem XI.2.16 of [Rosser] p. 376. (Contributed by SF,
|
|
4-Mar-2015.) $)
|
|
lecncvg $p |- ( ( A e. V /\ A =/= (/) ) -> A <_c Nc _V ) $=
|
|
( vx vy wcel c0 wne wa cvv cnc clec wbr wss wrex wex vvex ncid ssv sseq2
|
|
cv rspcev mp2an jctr eximi n0 df-rex 3imtr4i adantl wb ncex brlecg adantr
|
|
mpan2 mpbird ) ABEZAFGZHAIJZKLZCTZDTZMZDUQNZCANZUPVCUOUSAEZCOVDVBHZCOUPVC
|
|
VDVECVDVBIUQEUSIMZVBIPQUSRVAVFDIUQUTIUSSUAUBUCUDCAUEVBCAUFUGUHUOURVCUIZUP
|
|
UOUQIEVGIUJCDAUQBIUKUMULUN $.
|
|
$}
|
|
|
|
$( Cardinal zero is a minimal element of cardinal less than or equal. Lemma
|
|
1 of theorem XI.2.15 of [Rosser] p. 376. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
le0nc $p |- ( A e. NC -> 0c <_c A ) $=
|
|
( cncs wcel wne c0c clec wbr wceq nulnnc eleq1 mtbiri necon2ai lec0cg mpdan
|
|
c0 ) ABCZAODEAFGPAOAOHPOBCIAOBJKLABMN $.
|
|
|
|
$( The cardinality of a unit power class is not equal to the cardinality of
|
|
the power class. Theorem XI.2.4 of [Rosser] p. 372. (Contributed by SF,
|
|
10-Mar-2015.) $)
|
|
ncpw1pwneg $p |- ( A e. V -> Nc ~P1 A =/= Nc ~P A ) $=
|
|
( wcel cpw1 cnc cpw wne cen wbr wn nenpw1pw wceq wb pw1exg eqncg necon3abid
|
|
cvv syl mpbiri ) ABCZADZEZAFZEZGUAUCHIZJAKTUEUBUDTUAQCUBUDLUEMABNUAUCQORPS
|
|
$.
|
|
|
|
${
|
|
$d A x $. $d A y $. $d x y $.
|
|
$( The cardinality of a unit power class is strictly less than the
|
|
cardinality of the power class. Theorem XI.2.17 of [Rosser] p. 376.
|
|
(Contributed by SF, 10-Mar-2015.) $)
|
|
ltcpw1pwg $p |- ( A e. V -> Nc ~P1 A <c Nc ~P A ) $=
|
|
( vx vy wcel cpw1 cnc cpw clec wbr wne cltc wss wrex cvv pw1exg ncidg syl
|
|
cv ncex pwexg pw1sspw sseq1 sseq2 rspc2ev mp3an3 syl2anc brlec ncpw1pwneg
|
|
sylibr brltc sylanbrc ) ABEZAFZGZAHZGZIJZUOUQKUOUQLJUMCSZDSZMZDUQNCUONZUR
|
|
UMUNUOEZUPUQEZVBUMUNOEVCABPUNOQRUMUPOEVDABUAUPOQRVCVDUNUPMZVBAUBVAVEUNUTM
|
|
CDUNUPUOUQUSUNUTUCUTUPUNUDUEUFUGCDUOUQUNTUPTUHUJABUIUOUQUKUL $.
|
|
$}
|
|
|
|
${
|
|
sbthlem1.1 $e |- R e. _V $.
|
|
sbthlem1.2 $e |- X e. _V $.
|
|
sbthlem1.3 $e |- G = Clos1 ( ( X \ ran R ) , R ) $.
|
|
sbthlem1.4 $e |- A = ( X i^i G ) $.
|
|
sbthlem1.5 $e |- B = ( X \ G ) $.
|
|
sbthlem1.6 $e |- C = ( ran R i^i G ) $.
|
|
sbthlem1.7 $e |- D = ( ran R \ G ) $.
|
|
$( Lemma for ~ sbth . Set up similarity with a range. Theorem XI.1.14 of
|
|
[Rosser] p. 350. (Contributed by SF, 11-Mar-2015.) $)
|
|
sbthlem1 $p |- ( ( ( Fun R /\ Fun `' R ) /\
|
|
( X C_ dom R /\ ran R C_ X ) ) -> ran R ~~ X ) $=
|
|
( wa wss cun cen cin c0 wfun ccnv cdm crn wbr cima cres wf1o wf1 wf df-f1
|
|
wfn ssid df-f mpbiran2 funfn bitr4i anbi1i bitri inss1 sstr mpan syl5eqss
|
|
biimpri adantr f1ores syl2an cdif cclos1 rnex difex clos1ex eqeltri resex
|
|
cvv inex f1oen clos1baseima ineq2i indi disjdif uneq1i uncom eqtri 3eqtri
|
|
syl un0 inss2 wceq imassrn simprr syl5ss difss jctil sylib sseqin2 syl5eq
|
|
a1i unss imaeq2d sseqtr4d ssun2 sseqtr4i sseq1d mpbiri syl6sseqr breqtrrd
|
|
ssin eqssd enrflx ccompl difsscompl clos1base sscon34 mpbi compleqi iinun
|
|
df-dif dblcompl uneq2i sseqtri sslin ax-mp eqsstri incompl sstri syl5breq
|
|
3sstr4g ssdif ineq12i inindif unen mpanr12 syl2anc ensym uneq12i inundif
|
|
3brtr3g ) EUAZEUBUAZOZGEUCZPZEUDZGPZOZOZCDQZABQZUUDGRUUGUUIUUHRUEZUUHUUIR
|
|
UEUUGACRUEZBDRUEZUUJUUGAEAUFZCRUUGAUUMEAUGZUHZAUUMRUEUUAUUBUUDEUIZAUUBPZU
|
|
UOUUFUUPUUAUUPUUBUUDEUJZYTOUUAUUBUUDEUKUURYSYTUUREUUBULZYSUURUUSUUDUUDPUU
|
|
DUMUUBUUDEUNUOEUPUQURUSVDUUCUUQUUEUUCAGFSZUUBKUUTGPUUCUUTUUBPGFUTUUTGUUBV
|
|
AVBVCVEUUBUUDAEVFVGAUUMUUNEAHAUUTVOKGFIFGUUDVHZEVIVOJEUVAGUUDIEHVJVKZHVLV
|
|
MZVPVMVNVQWFUUGCUUMUUGCEFUFZUUMUUGCUUDUVDSZUVDCUUDFSZUUDUVAUVDQZSZUVEMFUV
|
|
GUUDFEUVAUVBHJVRZVSUVHUUDUVASZUVEQTUVEQZUVEUUDUVAUVDVTUVJTUVEUUDGWAWBUVKU
|
|
VETQUVETUVEWCUVEWGWDWEWEUVEUVDPUUGUUDUVDWHWRVCUUGAFEUUGAUUTFKUUGFGPUUTFWI
|
|
UUGFUVGGUVIUUGUVAGPZUVDGPZOUVGGPUUGUVMUVLUUGUVDUUDGEFWJUUAUUCUUEWKZWLGUUD
|
|
WMWNUVAUVDGWSWOVCFGWPWOWQWTZXAUUGUUMUVFCUUGUUMUUDPZUUMFPZOUUMUVFPUUGUVQUV
|
|
PUUGUVQUVDFPUVDUVGFUVDUVAXBUVIXCUUGUUMUVDFUVOXDXEEAWJWNUUMUUDFXHWOMXFXIXG
|
|
UUGBBDRBBGFVHZVOLGFIUVCVKVMXJUUGBDUUGUVRUUDFXKZSZBDUUGUVRUUDPZUVRUVSPZOUV
|
|
RUVTPUUGUWBUWAUWBUUGGFXLWRUVRGGXKZUUDQZSZUUDUVRGUVSSZUWEGFXRUVSUWDPUWFUWE
|
|
PUVSUVAXKZUWDUVAFPUVSUWGPFEUVAJXMUVAFXNXOUWGGUUDXKZSZXKUWCUWHXKZQUWDUVAUW
|
|
IGUUDXRXPGUWHXQUWJUUDUWCUUDXSXTWEYAUVSUWDGYBYCYDUWEGUUDSZUUDUWEGUWCSZUWKQ
|
|
TUWKQZUWKGUWCUUDVTUWLTUWKGYEWBUWMUWKTQUWKTUWKWCUWKWGWDWEGUUDWHYDYFWNUVRUU
|
|
DUVSXHWOLDUUDFVHZUVTNUUDFXRWDYHUUGUWNUVRDBUUGUUEUWNUVRPUVNUUDGFYIWFNLYHXI
|
|
YGUUKUULOABSZTWICDSZTWIUUJUWOUUTUVRSTAUUTBUVRKLYJGFYKWDUWPUVFUWNSTCUVFDUW
|
|
NMNYJUUDFYKWDACBDYLYMYNUUIUUHYOWOUUHUVFUWNQUUDCUVFDUWNMNYPUUDFYQWDUUIUUTU
|
|
VRQGAUUTBUVRKLYPGFYQWDYR $.
|
|
$}
|
|
|
|
${
|
|
$d B b $. $d R b $.
|
|
sbthlem2.1 $e |- R e. _V $.
|
|
$( Lemma for ~ sbth . Eliminate hypotheses from ~ sbthlem1 . Theorem
|
|
XI.1.14 of [Rosser] p. 350. (Contributed by SF, 10-Mar-2015.) $)
|
|
sbthlem2 $p |- ( ( ( Fun R /\ Fun `' R ) /\
|
|
( B e. V /\ B C_ dom R /\ ran R C_ B ) ) -> ran R ~~ B ) $=
|
|
( vb wcel cdm wss crn w3a wfun ccnv wa cen wbr wi cv cdif cin eqid imbi2d
|
|
wceq sseq1 sseq2 anbi12d imbi12d cclos1 vex sbthlem1 expcom vtoclg 3impib
|
|
breq2 impcom ) ACFZABGZHZBIZAHZJBKBLKMZURANOZUOUQUSUTVAPZEQZUPHZURVCHZMZU
|
|
TURVCNOZPZPUQUSMZVBPEACVCAUBZVFVIVHVBVJVDUQVEUSVCAUPUCVCAURUDUEVJVGVAUTVC
|
|
AURNUMUAUFUTVFVGVCVCURRBUGZSZVCVKRZURVKSZURVKRZBVKVCDEUHVKTVLTVMTVNTVOTUI
|
|
UJUKULUN $.
|
|
$}
|
|
|
|
${
|
|
$d A r $. $d A s $. $d B r $. $d B s $. $d C r $. $d C s $. $d D r $.
|
|
$d D s $. $d r s $.
|
|
$( Lemma for ~ sbth . If ` A ` is equinumerous with a subset of ` B ` and
|
|
vice-versa, then ` A ` is equinumerous with ` B ` . Theorem XI.1.15 of
|
|
[Rosser] p. 353. (Contributed by SF, 10-Mar-2015.) $)
|
|
sbthlem3 $p |- ( ( ( A ~~ C /\ C C_ B ) /\ ( B ~~ D /\ D C_ A ) ) ->
|
|
A ~~ B ) $=
|
|
( vr vs cen wbr wss wf1o wex bitr4i crn cdm wceq syl ad2antlr wfun ccnv
|
|
wa cv wi bren anbi12i eeanv ccom simprl f1ofo forn ad2antrr f1odm 3sstr4d
|
|
wfo dmcosseq eqtrd f1ofun syl2anr wfn dff1o2 simp2bi syl2an funeqi sylibr
|
|
funco cnvco jca adantr w3a funfn anbi1i eqid df-3an mpbiran2 vex eqbrtrrd
|
|
coex f1oen wcel rnex syl6eqelr simprr sseqtr4d syl5sseq sbthlem2 syl13anc
|
|
cvv rncoss entr syl2anc ex exlimivv sylbi imp an4s ensymi ad2antrl ) ACGH
|
|
ZCBIZTZBDGHZDAIZTTADGHZDBGHZABGHWQWTWRXAXBWQWTTZWRXATZXBXDACEUAZJZBDFUAZJ
|
|
ZTZFKEKZXEXBUBZXDXGEKZXIFKZTXKWQXMWTXNACEUCBDFUCUDXGXIEFUELXJXLEFXJXEXBXJ
|
|
XETZAXHXFUFZMZGHXQDGHZXBXOXPNZAXQGXOXSXFNZAXOXFMZXHNZIXSXTOXOCBYAYBXJWRXA
|
|
UGXGYACOZXIXEXGACXFUMYCACXFUHACXFUIPUJXIYBBOXGXEBDXHUKQULXHXFUNPXGXTAOXIX
|
|
EACXFUKUJUOZXOXSXQXPJZXSXQGHXOXPRZXPSZRZTZYEXJYIXEXJYFYHXIXHRXFRYFXGBDXHU
|
|
PACXFUPXHXFVDUQXJXFSZXHSZUFZRZYHXGYJRZYKRZYMXIXGXFAURYNYCACXFUSUTXIXHBURY
|
|
OXHMZDOZBDXHUSUTYJYKVDVAYGYLXHXFVEVBVCVFVGZYEXPXSURZYHXQXQOZVHZYIXSXQXPUS
|
|
YIYSYHTZUUAYFYSYHXPVIVJUUAUUBYTXQVKYSYHYTVLVMLLVCXSXQXPXHXFFVNZEVNVPZVQPV
|
|
OXOYIDWFVRZDXSIXQDIXRYRXIUUEXGXEXIDYPWFXIBDXHUMYQBDXHUHBDXHUIPZXHUUCVSVTQ
|
|
XODAXSXJWRXAWAYDWBXOYPXQDXHXFWGXIYQXGXEUUFQWCDXPWFUUDWDWEAXQDWHWIWJWKWLWM
|
|
WNWTXCWSXABDWOWPADBWHWI $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a d $. $d A d $. $d a g $. $d A g $.
|
|
$d B a $. $d B b $. $d b d $. $d B d $. $d b g $. $d B g $. $d d g $.
|
|
$( The Schroder-Bernstein Theorem. This theorem gives the antisymmetry law
|
|
for cardinal less than or equal. Translated out, it means that, if
|
|
` A ` is no larger than ` B ` and ` B ` is no larger than ` A ` , then
|
|
` Nc A = Nc B ` . Theorem XI.2.20 of [Rosser] p. 376. (Contributed by
|
|
SF, 11-Mar-2015.) $)
|
|
sbth $p |- ( ( A e. NC /\ B e. NC ) ->
|
|
( ( A <_c B /\ B <_c A ) -> A = B ) ) $=
|
|
( vg vb vd va cncs wcel wa clec wbr cv wss wrex wceq brlecg reeanv cnc wi
|
|
cen wb ancoms anbi12d 2rexbii bitri syl6bbr ncseqnc bi2anan9 simplr ensym
|
|
biimpar sylib simprl simpll simprr sbthlem3 syl22anc entr syl2anc ex elnc
|
|
anbi12i vex eqnc imbi2i 3imtr4i rexlimivv rexeq rexbidv sylan9bbr imbi12d
|
|
eqeq12 mpbiri syl rexlimdvva sylbid ) AGHZBGHZIZABJKZBAJKZIZCLZDLZMZELZFL
|
|
ZMZIZFANZDBNZEBNCANZABOZVSWBWEDBNZCANZWHFANZEBNZIZWLVSVTWOWAWQCDABGGPVRVQ
|
|
WAWQUAEFBAGGPUBUCWLWNWPIZEBNCANWRWKWSCEABWEWHDFBAQUDWNWPCEABQUEUFVSWKWMCE
|
|
ABVSWCAHZWFBHZIZIAWCRZOZBWFRZOZIZWKWMSZVSXGXBVQXDWTVRXFXAAWCUGBWFUGUHUKXG
|
|
XHWIFXCNZDXENZXCXEOZSWIXKDFXEXCWDWFTKZWGWCTKZIZWIWCWFTKZSWDXEHZWGXCHZIWIX
|
|
KSXNWIXOXNWIIZWCWDTKZXLXOXRWCWGTKZWGWDTKZXSXRXMXTXLXMWIUIZWGWCUJULXRXMWEX
|
|
LWHYAYBXNWEWHUMXLXMWIUNZXNWEWHUOWGWDWCWFUPUQWCWGWDURUSYCWCWDWFURUSUTXPXLX
|
|
QXMWDWFVAWGWCVAVBXKXOWIWCWFCVCVDVEVFVGXGWKXJWMXKXFWKWJDXENXDXJWJDBXEVHXDW
|
|
JXIDXEWIFAXCVHVIVJAXCBXEVLVKVMVNVOVP $.
|
|
$}
|
|
|
|
$( Cardinal less than is equivalent to one-way cardinal less than or equal.
|
|
Theorem XI.2.21 of [Rosser] p. 377. (Contributed by SF, 11-Mar-2015.) $)
|
|
ltlenlec $p |- ( ( M e. NC /\ N e. NC ) ->
|
|
( M <c N <-> ( M <_c N /\ -. N <_c M ) ) ) $=
|
|
( cltc wbr clec wa cncs wcel wn brltc wceq nclecid breq1 syl5ibcom ad2antrr
|
|
wne wi sbth expdimp impbid necon3abid pm5.32da syl5bb ) ABCDABEDZABPZFAGHZB
|
|
GHZFZUDBAEDZIZFABJUHUDUEUJUHUDFZUIABUKABKZUIUFULUIQUGUDUFAAEDULUIALABAEMNOU
|
|
HUDUIULABRSTUAUBUC $.
|
|
|
|
${
|
|
$d M x $. $d M y $. $d M z $. $d N x $. $d N y $. $d N z $. $d x y $.
|
|
$d x z $. $d y z $.
|
|
$( For non-empty sets, cardinal sum always increases cardinal less than or
|
|
equal. Theorem XI.2.19 of [Rosser] p. 376. (Contributed by SF,
|
|
11-Mar-2015.) $)
|
|
addlec $p |- ( ( M e. V /\ N e. W /\ ( M +c N ) =/= (/) ) ->
|
|
M <_c ( M +c N ) ) $=
|
|
( vx vz vy wcel cplc c0 wne w3a clec cv wss wrex wex wa wceq cvv ssun1 n0
|
|
wbr cin cun eladdc sseq2 mpbiri adantl rexlimivw reximi sylbi ancli eximi
|
|
rexcom df-rex bitri 3imtr4i 3ad2ant3 addcexg brlecg syldan 3adant3 mpbird
|
|
wb ) ACHZBDHZABIZJKZLAVHMUCZENZFNZOZFVHPEAPZVIVFVNVGVLVHHZFQVOVMEAPZRZFQZ
|
|
VIVNVOVQFVOVPVOVKGNZUDJSZVLVKVSUEZSZRZGBPZEAPVPVLABEGUFWDVMEAWCVMGBWBVMVT
|
|
WBVMVKWAOVKVSUAVLWAVKUGUHUIUJUKULUMUNFVHUBVNVPFVHPVRVMEFAVHUOVPFVHUPUQURU
|
|
SVFVGVJVNVEZVIVFVGVHTHWEABCDUTEFAVHCTVAVBVCVD $.
|
|
$}
|
|
|
|
$( For cardinals, cardinal sum always increases cardinal less than or equal.
|
|
Corollary of theorem XI.2.19 of [Rosser] p. 376. (Contributed by SF,
|
|
11-Mar-2015.) $)
|
|
addlecncs $p |- ( ( M e. NC /\ N e. NC ) -> M <_c ( M +c N ) ) $=
|
|
( cncs wcel cplc c0 wne clec wbr ncaddccl wceq nulnnc eleq1 mtbiri necon2ai
|
|
wa syl addlec mpd3an3 ) ACDZBCDZABEZFGZAUBHITUAPUBCDZUCABJUDUBFUBFKUDFCDLUB
|
|
FCMNOQABCCRS $.
|
|
|
|
${
|
|
$d a b $. $d a p $. $d b p $. $d M a $. $d M b $. $d M p $. $d N a $.
|
|
$d N b $. $d N p $.
|
|
$( Cardinal less than or equal in terms of cardinal addition. Theorem
|
|
XI.2.22 of [Rosser] p. 377. (Contributed by SF, 11-Mar-2015.) $)
|
|
dflec2 $p |- ( ( M e. NC /\ N e. NC ) ->
|
|
( M <_c N <-> E. p e. NC N = ( M +c p ) ) ) $=
|
|
( va vb cncs wcel wa clec wbr cv cplc wceq wrex wss cnc ncseqnc vex cun
|
|
wi brlecg bi2anan9 biimpar cdif difex ncelncsi cin disjdif ncdisjun ax-mp
|
|
undif2 ssequn1 biimpi syl5eq nceqd syl5reqr addceq2 eqeq2d rspcev sylancr
|
|
c0 wb id addceq1 eqeqan12d rexbidv ancoms syl rexlimdvva sylbid addlecncs
|
|
syl5ibr breq2 syl5ibrcom adantlr rexlimdva impbid ) AFGZBFGZHZABIJZBACKZL
|
|
ZMZCFNZVTWADKZEKZOZEBNDANWEDEABFFUAVTWHWEDEABVTWFAGZWGBGZHZHAWFPZMZBWGPZM
|
|
ZHZWHWETVTWPWKVRWMWIVSWOWJAWFQBWGQUBUCWHWEWPWNWLWBLZMZCFNZWHWGWFUDZPZFGWN
|
|
WLXALZMZWSWTWGWFERDRZUEZUFWHXBWFWTSZPZWNWFWTUGVAMXGXBMWFWGUHWFWTXDXEUIUJW
|
|
HXFWGWHXFWFWGSZWGWFWGUKWHXHWGMWFWGULUMUNUOUPWRXCCXAFWBXAMWQXBWNWBXAWLUQUR
|
|
USUTWOWMWEWSVBWOWMHWDWRCFWOWMBWNWCWQWOVCAWLWBVDVEVFVGVLVHVIVJVTWDWACFVRWB
|
|
FGZWDWATVSVRXIHWAWDAWCIJAWBVKBWCAIVMVNVOVPVQ $.
|
|
$}
|
|
|
|
${
|
|
$d A x $. $d A y $. $d B x $. $d B y $. $d C x $. $d C y $. $d x y $.
|
|
$( Cardinal less than or equal is transitive. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
lectr $p |- ( ( A e. NC /\ B e. NC /\ C e. NC ) ->
|
|
( ( A <_c B /\ B <_c C ) -> A <_c C ) ) $=
|
|
( vx vy cncs wcel w3a clec wa cv cplc wceq wrex wb dflec2 3adant3 3adant1
|
|
wbr anbi12d reeanv syl6bbr addceq1 addcass syl6eq eqeq2d biimpa addlecncs
|
|
simp1 ncaddccl syl2an breq2 syl5ibrcom syl5 rexlimdvva sylbid ) AFGZBFGZC
|
|
FGZHZABISZBCISZJZBADKZLZMZCBEKZLZMZJZEFNDFNZACISZUTVCVFDFNZVIEFNZJVKUTVAV
|
|
MVBVNUQURVAVMOUSABDPQURUSVBVNOUQBCEPRTVFVIDEFFUAUBUTVJVLDEFFVJCAVDVGLZLZM
|
|
ZUTVDFGVGFGJZJZVLVFVIVQVFVHVPCVFVHVEVGLVPBVEVGUCAVDVGUDUEUFUGVSVLVQAVPISZ
|
|
UTUQVOFGVTVRUQURUSUIVDVGUJAVOUHUKCVPAIULUMUNUOUP $.
|
|
$}
|
|
|
|
$( Transitivity law for cardinal less than or equal and less than.
|
|
(Contributed by SF, 16-Mar-2015.) $)
|
|
leltctr $p |- ( ( A e. NC /\ B e. NC /\ C e. NC ) ->
|
|
( ( A <_c B /\ B <c C ) -> A <c C ) ) $=
|
|
( cncs wcel w3a clec wbr cltc wa lectr expdimp adantrd wi wceq breq1 anbi1d
|
|
wne biimpac brltc sbth ancoms 3adant1 eqcom syl6ibr syl5 necon3d expr imp3a
|
|
jcad 3imtr4g expimpd ) ADEZBDEZCDEZFZABGHZBCIHZACIHZUPUQJZBCGHZBCRZJZACGHZA
|
|
CRZJURUSUTVCVDVEUTVAVDVBUPUQVAVDABCKLMUTVAVBVEUPUQVAVBVENUPUQVAJZJACBCUPVFA
|
|
COZBCOZVFVGJCBGHZVAJZUPVHVGVFVJVGUQVIVAACBGPQSUPVJCBOZVHUNUOVJVKNZUMUOUNVLC
|
|
BUAUBUCBCUDUEUFLUGUHUIUJBCTACTUKUL $.
|
|
|
|
${
|
|
$d x y z $.
|
|
$( Cardinal less than or equal partially orders the cardinals.
|
|
(Contributed by SF, 12-Mar-2015.) $)
|
|
lecponc $p |- <_c Po NC $=
|
|
( vx vy vz clec cncs cpartial wbr wtru cvv wcel lecex a1i ncsex cv adantl
|
|
nclecid w3a wa imp 3adant1 lectr weq sbth pod trud ) DEFGHABCEDIIDIJHKLEI
|
|
JHMLANZEJZUFUFDGHUFPOUGBNZEJZCNZEJQZUFUHDGZUHUJDGRZUFUJDGZHUKUMUNUFUHUJUA
|
|
STUGUIRZULUHUFDGRZABUBZHUOUPUQUFUHUCSTUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d M x $. $d N x $. $d P x $.
|
|
$( Addition law for cardinal less than. Theorem XI.2.23 of [Rosser]
|
|
p. 377. (Contributed by SF, 12-Mar-2015.) $)
|
|
leaddc1 $p |- ( ( ( M e. NC /\ N e. NC /\ P e. NC ) /\ M <_c N ) ->
|
|
( M +c P ) <_c ( N +c P ) ) $=
|
|
( vx cncs wcel w3a clec wbr cplc cv wceq wb dflec2 3adant3 wi wa ncaddccl
|
|
wrex addlecncs sylan addc32 syl6eq breq2d syl5ibrcom rexlimdva sylbid imp
|
|
addceq1 3adant2 ) BEFZCEFZAEFZGZBCHIZBAJZCAJZHIZUNUOCBDKZJZLZDESZURUKULUO
|
|
VBMUMBCDNOUKUMVBURPULUKUMQZVAURDEVCUSEFZQURVAUPUPUSJZHIZVCUPEFVDVFBARUPUS
|
|
TUAVAUQVEUPHVAUQUTAJVECUTAUIBUSAUBUCUDUEUFUJUGUH $.
|
|
$}
|
|
|
|
$( Addition law for cardinal less than. Theorem XI.2.23 of [Rosser] p. 377.
|
|
(Contributed by SF, 12-Mar-2015.) $)
|
|
leaddc2 $p |- ( ( ( M e. NC /\ N e. NC /\ P e. NC ) /\ N <_c P ) ->
|
|
( M +c N ) <_c ( M +c P ) ) $=
|
|
( cncs wcel w3a clec wbr wa cplc 3anrot leaddc1 sylanb addccom 3brtr4g ) BD
|
|
EZCDEZADEZFZCAGHZICBJZABJZBCJBAJGSQRPFTUAUBGHPQRKBCALMBCNBANO $.
|
|
|
|
${
|
|
$d a p $. $d a q $. $d a x $. $d N a $. $d p q $. $d p x $. $d q x $.
|
|
$( Any cardinal is either zero or no greater than one. Theorem XI.2.24 of
|
|
[Rosser] p. 377. (Contributed by SF, 12-Mar-2015.) $)
|
|
nc0le1 $p |- ( N e. NC -> ( N = 0c \/ 1c <_c N ) ) $=
|
|
( va vx vq vp wcel cv cnc wceq wex c0c c1c clec wbr wo wss wrex vex sylbi
|
|
c0 cncs elncs nceq df0c2 syl6eqr orcd wne wel csn snss snel1c sseq2 sseq1
|
|
ncid rspc2ev mp3an12 exlimiv n0 1cex ncex brlec rexcom bitri 3imtr4i olcd
|
|
pm2.61ine eqeq1 breq2 orbi12d mpbiri ) AUAFABGZHZIZBJAKIZLAMNZOZBAUBVMVPB
|
|
VMVPVLKIZLVLMNZOZVSVKTVKTIZVQVRVTVLTHKVKTUCUDUEUFVKTUGZVRVQCBUHZCJDGZEGZP
|
|
ZDLQEVLQZWAVRWBWFCWBCGZUIZVKPZWFWGVKCRZUJVKVLFWHLFWIWFVKBRUNWGWJUKWEWIWCV
|
|
KPEDVKWHVLLWDVKWCULWCWHVKUMUOUPSUQCVKURVRWEEVLQDLQWFDELVLUSVKUTVAWEDELVLV
|
|
BVCVDVEVFVMVNVQVOVRAVLKVGAVLLMVHVIVJUQS $.
|
|
$}
|
|
|
|
${
|
|
$d N m $.
|
|
$( Any cardinal is either zero or the successor of a cardinal. Corollary
|
|
of theorem XI.2.24 of [Rosser] p. 377. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
nc0suc $p |- ( N e. NC -> ( N = 0c \/ E. m e. NC N = ( m +c 1c ) ) ) $=
|
|
( cncs wcel c0c wceq c1c clec wbr wo cplc wrex nc0le1 1cnc dflec2 addccom
|
|
cv wb wa eqeq2i rexbii syl6bb mpan orbi2d mpbid ) BCDZBEFZGBHIZJUGBAQZGKZ
|
|
FZACLZJBMUFUHULUGGCDZUFUHULRNUMUFSUHBGUIKZFZACLULGBAOUOUKACUNUJBGUIPTUAUB
|
|
UCUDUE $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a m $. $d a n $. $d A n $. $d B n $. $d m n $. $d m p $.
|
|
$d n p $.
|
|
$( Cardinal less than or equal is total over the naturals. (Contributed by
|
|
SF, 12-Mar-2015.) $)
|
|
leconnnc $p |- ( ( A e. Nn /\ B e. Nn ) -> ( A <_c B \/ B <_c A ) ) $=
|
|
( va vp cnnc wcel clec wbr wo cv wceq breq2 breq1 orbi12d imbi2d c0c cncs
|
|
wi c1c wa vn vm cplc wn cab ccnv csn cima cun cvv elun eliniseg cop df-br
|
|
elimasn bitr4i orbi12i bitri abbi2i uneq2i eqtri abbii eqtr4i abexv lecex
|
|
unab imor cnvex snex imaex unex eqeltrri weq nnnc le0nc syl dflec2 nc0le1
|
|
orc wrex 1cnc ax-mp mpbiri orim1i a1i orcomd adantl simpll simplr leaddc2
|
|
simpr syl31anc ex orim12d biimprd com12 rexlimdva adantr sylbid addlecncs
|
|
mpd mpan2 peano2nc lectr mpd3an3 mpan2d ancoms olc syl6 jaod syl2an finds
|
|
a2d vtoclga imp ) AEFZBEFZABGHZBAGHZIZXQXPXTXPAUAJZGHZYAAGHZIZRXPXTRUABEY
|
|
ABKZYDXTXPYEYBXRYCXSYABAGLYABAGMNOXPYAEFZYDYFCJZYAGHZYAYGGHZIZRZYFPYAGHZY
|
|
APGHZIZRYFUBJZYAGHZYAYOGHZIZRYFYOSUCZYAGHZYAYSGHZIZRYFYDRCUBAYFUDZCUEZGUF
|
|
ZYAUGZUHZGUUFUHZUIZUIZYKCUEZUJUUJUUCYJIZCUEZUUKUUJUUDYJCUEZUIUUMUUIUUNUUD
|
|
YJCUUIYGUUIFYGUUGFZYGUUHFZIYJYGUUGUUHUKUUOYHUUPYIGYAYGULUUPYAYGUMGFYIGYAY
|
|
GUOYAYGGUNUPUQURUSUTUUCYJCVFVAYKUULCYFYJVGVBVCUUDUUIUUCCVDUUGUUHUUEUUFGVE
|
|
VHYAVIZVJGUUFVEUUQVJVKVKVLYGPKZYJYNYFUURYHYLYIYMYGPYAGMYGPYAGLNOCUBVMZYJY
|
|
RYFUUSYHYPYIYQYGYOYAGMYGYOYAGLNOYGYSKZYJUUBYFUUTYHYTYIUUAYGYSYAGMYGYSYAGL
|
|
NOYGAKZYJYDYFUVAYHYBYIYCYGAYAGMYGAYAGLNOYFYLYNYFYAQFZYLYAVNZYAVOVPYLYMVSV
|
|
PYOEFZYFYRUUBUVDYFYRUUBRZUVDYOQFZUVBUVEYFYOVNUVCUVFUVBTZYPUUBYQUVGYPYAYOD
|
|
JZUCZKZDQVTZUUBYOYADVQUVFUVKUUBRUVBUVFUVJUUBDQUVFUVHQFZTZYSUVIGHZUVIYSGHZ
|
|
IZUVJUUBRUVMSUVHGHZUVHSGHZIZUVPUVLUVSUVFUVLUVRUVQUVLUVHPKZUVQIZUVRUVQIZUV
|
|
HVRUWAUWBRUVLUVTUVRUVQUVTUVRPSGHZSQFZUWCWASVOWBUVHPSGMWCWDWEXAWFWGUVMUVQU
|
|
VNUVRUVOUVMUVQUVNUVMUVQTZUVFUWDUVLUVQUVNUVFUVLUVQWHUWDUWEWAWEUVFUVLUVQWIU
|
|
VMUVQWKUVHYOSWJWLWMUVMUVRUVOUVMUVRTZUVFUVLUWDUVRUVOUVFUVLUVRWHUVFUVLUVRWI
|
|
UWDUWFWAWEUVMUVRWKSYOUVHWJWLWMWNXAUVJUVPUUBUVJUUBUVPUVJYTUVNUUAUVOYAUVIYS
|
|
GLYAUVIYSGMNWOWPVPWQWRWSUVGYQUUAUUBUVBUVFYQUUARUVBUVFTYQYOYSGHZUUAUVFUWGU
|
|
VBUVFUWDUWGWAYOSWTXBWGUVBUVFYSQFZYQUWGTUUARUVFUWHUVBYOXCWGYAYOYSXDXEXFXGU
|
|
UAYTXHXIXJXKWMXMXLWPXNWPXO $.
|
|
$}
|
|
|
|
${
|
|
$d A p $. $d B p $.
|
|
$( The sum of two cardinals is zero iff both addends are zero.
|
|
(Contributed by SF, 12-Mar-2015.) $)
|
|
addceq0 $p |- ( ( A e. NC /\ B e. NC ) -> ( ( A +c B ) = 0c <->
|
|
( A = 0c /\ B = 0c ) ) ) $=
|
|
( vp cncs wcel wa cplc c0c wceq c1c wrex nc0suc ord wne 0cnsuc necon3bbid
|
|
wn wi eqeq1d mpbiri wo ianor adantr addc32 eqnetri addceq1 rexlimivw syl6
|
|
adantl addcass eqnetrri addceq2 jaod syl5bi con4d addceq12 addcid2 syl6eq
|
|
cv impbid1 ) ADEZBDEZFZABGZHIZAHIZBHIZFZVCVHVEVHQVFQZVGQZUAVCVEQZVFVGUBVC
|
|
VIVKVJVCVIACUSZJGZIZCDKZVKVAVIVORVBVAVFVOCALMUCVNVKCDVNVKVMBGZHNVPVLBGZJG
|
|
HVLJBUDVQOUEVNVEVPHVNVDVPHAVMBUFSPTUGUHVCVJBVMIZCDKZVKVBVJVSRVAVBVGVSCBLM
|
|
UIVRVKCDVRVKAVMGZHNAVLGZJGVTHAVLJUJWAOUKVRVEVTHVRVDVTHBVMAULSPTUGUHUMUNUO
|
|
VHVDHHGHABHHUPHUQURUT $.
|
|
$}
|
|
|
|
${
|
|
$d M x $.
|
|
$( Ordering law for cardinal exponentiation to two. Theorem XI.2.71 of
|
|
[Rosser] p. 390. (Contributed by SF, 13-Mar-2015.) $)
|
|
ce2lt $p |- ( ( M e. NC /\ ( M ^c 0c ) e. NC ) -> M <c ( 2c ^c M ) ) $=
|
|
( vx cncs wcel c0c cce co wa cv cpw1 cnc wceq wex c2c wbr ce0ncpw1 cpw id
|
|
cltc cvv vex ltcpw1pwg ax-mp syl6eqbr ce2 breqtrrd exlimiv syl ) ACDAEFGC
|
|
DHABIZJKZLZBMANAFGZSOZABPUKUMBUKAUIQKZULSUKAUJUNSUKRUITDUJUNSOBUAZUITUBUC
|
|
UDUIAUOUEUFUGUH $.
|
|
$}
|
|
|
|
${
|
|
$d a x $. $d a y $. $d b x $. $d b y $. $d f x $. $d f y $. $d M a $.
|
|
$d M x $. $d M y $. $d N a $. $d N b $. $d N x $. $d N y $. $d x y $.
|
|
$d a b $. $d a c $. $d a f $. $d b c $. $d b f $. $d c f $. $d c x $.
|
|
$d c y $.
|
|
$( Another potential definition of cardinal inequality. (Contributed by
|
|
SF, 23-Mar-2015.) $)
|
|
dflec3 $p |- ( ( M e. NC /\ N e. NC ) -> ( M <_c N <->
|
|
E. a e. M E. b e. N E. f f : a -1-1-> b ) ) $=
|
|
( vx vy vc wcel wa cv cnc wceq wex clec wbr wf1 wrex wb vex elncs anbi12i
|
|
cncs eeanv bitr4i wss ncex brlec cid cres wf1o f1oi f1of1 ax-mp f1ss mpan
|
|
rexcom resex f1eq1 spcev syl weq f1eq2 exbidv rspcev sylan2 rexlimiva crn
|
|
idex wi cen eqnc f1f1orn f1oen ensym sylib sylibr eleq2 syl5ib sylbir imp
|
|
elnc wf f1f frn adantl sseq1 syl2anc exlimdv rexlimiv impbii rexbii bitri
|
|
ex 3bitri breq12 simpl rexeq rexeqbidv bibi12d mpbiri exlimivv sylbi ) BU
|
|
CIZCUCIZJZBFKZLZMZCGKZLZMZJZGNFNZBCOPZDKZEKZAKZQZANZECRZDBRZSZXFXIFNZXLGN
|
|
ZJXNXDYDXEYEFBUAGCUAUBXIXLFGUDUEXMYCFGXMYCXHXKOPZXTEXKRZDXHRZSYFHKZXQUFZE
|
|
XKRHXHRZXTDXHRZEXKRZYHHEXHXKXGUGXJUGUHYKYJHXHRZEXKRYMYJHEXHXKUQYNYLEXKYNY
|
|
LYJYLHXHYJYIXHIYIXQXRQZANZYLYJYIXQUIYIUJZQZYPYIYIYQQZYJYRYIYIYQUKYSYIULYI
|
|
YIYQUMUNYIYIXQYQUOUPYOYRAYQUIYIVIHTURYIXQXRYQUSUTVAXTYPDYIXHDHVBXSYOAXPYI
|
|
XQXRVCVDVEVFVGXTYNDXHXPXHIZXSYNAYTXSYNYTXSJXRVHZXHIZUUAXQUFZYNYTXSUUBYTXP
|
|
LZXHMZXSUUBVJUUEXPXGVKPYTXPXGDTVLXPXGWBUEXSUUAUUDIZUUEUUBXSUUAXPVKPZUUFXS
|
|
XPUUAVKPZUUGXSXPUUAXRUKUUHXPXQXRVMXPUUAXRATVNVAXPUUAVOVPUUAXPWBVQUUDXHUUA
|
|
VRVSVTWAXSUUCYTXSXPXQXRWCUUCXPXQXRWDXPXQXRWEVAWFYJUUCHUUAXHYIUUAXQWGVEWHW
|
|
NWIWJWKWLWMXTEDXKXHUQWOXMXOYFYBYHBXHCXKOWPXMYAYGDBXHXIXLWQXLYAYGSXIXTECXK
|
|
WRWFWSWTXAXBXC $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a b $. $d A b $. $d a f $. $d A f $. $d A g $. $d A h $.
|
|
$d A i $. $d A p $. $d A q $. $d B a $. $d B b $. $d b f $. $d B f $.
|
|
$d B g $. $d B h $. $d B i $. $d B p $. $d B q $. $d f g $. $d f h $.
|
|
$d f i $. $d f p $. $d f q $. $d g h $. $d g i $. $d g p $. $d g q $.
|
|
$d h i $. $d h p $. $d h q $. $d i p $. $d i q $. $d p q $.
|
|
nclenc.1 $e |- A e. _V $.
|
|
nclenc.2 $e |- B e. _V $.
|
|
$( Comparison rule for cardinalities. (Contributed by SF, 24-Mar-2015.) $)
|
|
nclenc $p |- ( Nc A <_c Nc B <-> E. f f : A -1-1-> B ) $=
|
|
( vp vq vg vh vi va vb wbr cv wf1 wex wrex wcel wa wf1o cnc clec ncelncsi
|
|
cncs wb dflec3 mp2an wi cen elnc bren bitri anbi12i eeanv bitr4i w3a ccom
|
|
ccnv f1of1 3ad2ant2 simp3 f1co syl2anc f1ocnv syl 3ad2ant1 vex coex cnvex
|
|
f1eq1 spcev 3expia exlimivv sylbi exlimdv rexlimivv ncid wceq f1eq2 f1eq3
|
|
exbidv rspc2ev mp3an12 sylibr impbii ) AUAZBUAZUBMZABCNZOZCPZWHFNZGNZHNZO
|
|
ZHPZGWGQFWFQZWKWFUDRZWGUDRZWHWQUEADUCZBEUCZHWFWGFGUFUGWPWKFGWFWGWLWFRZWMW
|
|
GRZSZWOWKHXDWLAINZTZWMBJNZTZSZJPIPZWOWKUHZXDXFIPZXHJPZSXJXBXLXCXMXBWLAUIM
|
|
XLWLAUJWLAIUKULXCWMBUIMXMWMBUJWMBJUKULUMXFXHIJUNUOXIXKIJXFXHWOWKXFXHWOUPZ
|
|
ABXGWNUQZXEURZUQZOZWKXNWLBXOOZAWLXPOZXRXNWMBXGOZWOXSXHXFYAWOWMBXGUSUTXFXH
|
|
WOVAWLWMBXGWNVBVCXFXHXTWOXFAWLXPTXTWLAXEVDAWLXPUSVEVFAWLBXOXPVBVCWJXRCXQX
|
|
OXPXGWNJVGHVGVHXEIVGVIVHABWIXQVJVKVEVLVMVNVOVPVNWKKNZLNZWIOZCPZLWGQKWFQZW
|
|
HAWFRBWGRWKYFADVQBEVQYEWKAYCWIOZCPKLABWFWGYBAVRYDYGCYBAYCWIVSWAYCBVRYGWJC
|
|
YCBAWIVTWAWBWCWRWSWHYFUEWTXACWFWGKLUFUGWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d M x $. $d A x $. $d A f $. $d A g $. $d A p $. $d A q $. $d A y $.
|
|
$d f g $. $d f p $. $d f q $. $d f x $. $d f y $. $d g p $. $d g q $.
|
|
$d g x $. $d g y $. $d M y $. $d p q $. $d p x $. $d p y $. $d q x $.
|
|
$d q y $. $d x y $.
|
|
lenc.1 $e |- A e. _V $.
|
|
$( Less than or equal condition for the cardinality of a number.
|
|
(Contributed by SF, 18-Mar-2015.) $)
|
|
lenc $p |- ( M e. NC -> ( M <_c Nc A <-> E. x e. M x C_ A ) ) $=
|
|
( vy vp vq vf vg wcel cv wex clec wbr wss wrex wf1o cen elnc vex cncs cnc
|
|
wceq wb elncs ncex brlec wa bren bitri anbi12i eeanv bitr4i w3a cima cres
|
|
wi ccnv ccom wf1 f1of1 3ad2ant2 simp3 f1ores syl2anc 3ad2ant1 f1oco resex
|
|
f1ocnv cnvex coex f1oen 3syl sylibr imass2 3ad2ant3 wfo f1ofo syl sseqtrd
|
|
foima sseq1 rspcev 3expia exlimivv sylbi nclec eqnc breq1 sylbir rexlimiv
|
|
rexlimivv syl5ib impbii rexeq bibi12d mpbiri exlimiv ) CUAJCEKZUBZUCZELCB
|
|
UBZMNZAKZBOZACPZUDZECUEXAXGEXAXGWTXBMNZXEAWTPZUDXHXIXHFKZGKZOZGXBPFWTPXIF
|
|
GWTXBWSUFBUFUGXLXIFGWTXBXJWTJZXKXBJZUHZXJWSHKZQZXKBIKZQZUHZILHLZXLXIUQZXO
|
|
XQHLZXSILZUHYAXMYCXNYDXMXJWSRNYCXJWSSXJWSHUIUJXNXKBRNYDXKBSXKBIUIUJUKXQXS
|
|
HIULUMXTYBHIXQXSXLXIXQXSXLUNZXRXJUOZWTJZYFBOZXIYEYFWSRNZYGYEWSYFXRXJUPZXP
|
|
URZUSZQZYFWSYLURZQYIYEXJYFYJQZWSXJYKQZYMYEXKBXRUTZXLYOXSXQYQXLXKBXRVAVBXQ
|
|
XSXLVCXKBXJXRVDVEXQXSYPXLXJWSXPVIVFWSXJYFYJYKVGVEWSYFYLVIYFWSYNYLYJYKXRXJ
|
|
ITFTVHXPHTVJVKVJVLVMYFWSSVNYEYFXRXKUOZBXLXQYFYROXSXJXKXRVOVPXSXQYRBUCZXLX
|
|
SXKBXRVQYSXKBXRVRXKBXRWAVSVBVTXEYHAYFWTXDYFBWBWCVEWDWEWFWLWFXEXHAWTXEXDUB
|
|
ZXBMNZXDWTJZXHXDBATZDWGUUBYTWTUCZUUAXHUDUUDXDWSRNUUBXDWSUUCWHXDWSSUMYTWTX
|
|
BMWIWJWMWKWNXAXCXHXFXICWTXBMWIXEACWTWOWPWQWRWF $.
|
|
$}
|
|
|
|
$( Compute the T-raising of a cardinality. (Contributed by SF,
|
|
23-Apr-2021.) $)
|
|
tcncg $p |- ( A e. V -> T_c Nc A = Nc ~P1 A ) $=
|
|
( wcel cnc ctc cncs cpw1 wceq ncelncs tccl syl pw1exg ncidg pw1eltc syl2anc
|
|
cvv nceleq syl22anc ) ABCZADZEZFCZAGZDZFCZUCUACZUCUDCZUAUDHSTFCZUBABIZTJKSU
|
|
CPCZUEABLZUCPIKSUHATCUFUIABMTANOSUJUGUKUCPMKUAUDUCQR $.
|
|
|
|
${
|
|
tcnc.1 $e |- A e. _V $.
|
|
$( Compute the T-raising of a cardinality. (Contributed by SF,
|
|
4-Mar-2015.) $)
|
|
tcnc $p |- T_c Nc A = Nc ~P1 A $=
|
|
( cvv wcel cnc ctc cpw1 wceq tcncg ax-mp ) ACDAEFAGEHBACIJ $.
|
|
$}
|
|
|
|
$( Compute the T-raising of the cardinality of the universe. Part of Theorem
|
|
5.2 of [Specker] p. 973. (Contributed by SF, 4-Mar-2015.) $)
|
|
tcncv $p |- T_c Nc _V = Nc 1c $=
|
|
( cvv cnc ctc cpw1 c1c vvex tcnc df1c2 nceqi eqtr4i ) ABCADZBEBAFGEKHIJ $.
|
|
|
|
$( Compute the T-raising of the cardinality of one. Part of Theorem 5.2 of
|
|
[Specker] p. 973. (Contributed by SF, 4-Mar-2015.) $)
|
|
tcnc1c $p |- T_c Nc 1c = Nc ~P1 1c $=
|
|
( c1c 1cex tcnc ) ABC $.
|
|
|
|
${
|
|
$d M x $. $d M y $. $d N x $. $d N y $. $d x y $.
|
|
$( Cardinal T is one-to-one. Based on theorem 2.4 of [Specker] p. 972.
|
|
(Contributed by SF, 10-Mar-2015.) $)
|
|
tc11 $p |- ( ( M e. NC /\ N e. NC ) -> ( T_c M = T_c N <-> M = N ) ) $=
|
|
( vx vy cncs wcel wa cv cnc wceq wex ctc elncs cpw1 vex tcnc cen wbr eqnc
|
|
wb anbi12i eeanv bitr4i eqeq12i enpw1 pw1ex 3bitr4ri bitri tceq eqeqan12d
|
|
eqeq12 bibi12d mpbiri exlimivv sylbi ) AEFZBEFZGZACHZIZJZBDHZIZJZGZDKCKZA
|
|
LZBLZJZABJZTZURVACKZVDDKZGVFUPVLUQVMCAMDBMUAVAVDCDUBUCVEVKCDVEVKUTLZVCLZJ
|
|
ZUTVCJZTVPUSNZIZVBNZIZJZVQVNVSVOWAUSCOZPVBDOPUDUSVBQRVRVTQRVQWBUSVBUEUSVB
|
|
WCSVRVTUSWCUFSUGUHVEVIVPVJVQVAVDVGVNVHVOAUTUIBVCUIUJAUTBVCUKULUMUNUO $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a c $. $d a w $. $d a x $. $d A x $. $d a y $. $d A y $.
|
|
$d a z $. $d A z $. $d b c $. $d b w $. $d b x $. $d B x $. $d b y $.
|
|
$d B y $. $d b z $. $d B z $. $d c w $. $d c x $. $d c z $. $d w x $.
|
|
$d w y $. $d w z $. $d X c $. $d X w $. $d X x $. $d x y $. $d X y $.
|
|
$d x z $. $d X z $. $d y z $.
|
|
$( T raising rule for cardinal sum. (Contributed by SF, 11-Mar-2015.) $)
|
|
taddc $p |- ( ( ( A e. NC /\ B e. NC /\ X e. NC ) /\
|
|
T_c A = ( T_c B +c X ) ) -> E. c e. NC X = T_c c ) $=
|
|
( vw vx vy vz va vb cncs wcel ctc wceq wa cv cnc wex vex sylbi cplc elncs
|
|
w3a cpw1 wrex 3anbi123i eeeanv bitr4i cen wbr tcnc addceq1i eqeq12i eqcom
|
|
wi wb pw1ex ncelncsi ncaddccl mp2an ncseqnc 3bitri cin cun eladdc pw1equn
|
|
ax-mp simp3 elnc ensym breq2 biimpcd syl5 eximdv exlimdv adantld rexlimiv
|
|
c0 syl5bi rexlimivw 3ad2ant1 adantr simpr addceq12d 3adant1 eqeq12d eqeq1
|
|
tceq syl6bb exbidv 3ad2ant3 imbi12d mpbiri exlimiv exlimivv df-rex anbi1i
|
|
eqnc imp 19.41v exbii excom ncex syl6eq eqeq2d ceqsexv bitri sylibr ) AKL
|
|
ZBKLZCKLZUCZAMZBMZCUAZNZOCEPZUDZQZNZERZCDPZMZNZDKUEZXLXPYAXLAFPZQZNZBGPZQ
|
|
ZNZCHPZQZNZUCZHRZGRFRZXPYAUOZXLYHFRZYKGRZYNHRZUCYQXIYSXJYTXKUUAFAUBGBUBHC
|
|
UBUFYHYKYNFGHUGUHYPYRFGYOYRHYOYRYGMZYJMZYMUAZNZYLXRUIUJZERZUOUUEYFUDZYIUD
|
|
ZQZYMUAZLZUUGUUEUUHQZUUKNUUKUUMNZUULUUBUUMUUDUUKYFFSUKUUCUUJYMYIGSZUKULUM
|
|
UUMUUKUNUUKKLZUUNUULUPUUJKLYMKLUUPUUIYIUUOUQURYLHSZURUUJYMUSUTUUKUUHVAVGV
|
|
BUULIPZJPZVCVRNZUUHUURUUSVDNZOZJYMUEZIUUJUEUUGUUHUUJYMIJVEUVCUUGIUUJUVBUU
|
|
GJYMUUSYMLZUVAUUGUUTUVAYFYBXQVDNZUURYBUDNZUUSXRNZUCZERZDRUVDUUGDEUURUUSYF
|
|
ISJSVFUVDUVIUUGDUVDUVHUUFEUVHUVGUVDUUFUVEUVFUVGVHUVDUUSYLUIUJZUVGUUFUOZUU
|
|
SYLVIUVJYLUUSUIUJZUVKUUSYLVJUVGUVLUUFUUSXRYLUIVKVLTTVMVNVOVSVPVQVTTTYOXPU
|
|
UEYAUUGYOXMUUBXOUUDYHYKXMUUBNYNAYGWHWAYKYNXOUUDNYHYKYNOXNUUCCYMYKXNUUCNYN
|
|
BYJWHWBYKYNWCWDWEWFYNYHYAUUGUPYKYNXTUUFEYNXTYMXSNUUFCYMXSWGYLXRUUQWRWIWJW
|
|
KWLWMWNWOTWSYEYBKLZYDOZDRYBXQQZNZYDOZERZDRZYAYDDKWPUVNUVRDUVNUVPERZYDOUVR
|
|
UVMUVTYDEYBUBWQUVPYDEWTUHXAUVSUVQDRZERYAUVQDEXBUWAXTEYDXTDUVOXQXCUVPYCXSC
|
|
UVPYCUVOMXSYBUVOWHXQESUKXDXEXFXAXGVBXH $.
|
|
$}
|
|
|
|
${
|
|
$d M p q $. $d N p q $.
|
|
$( T-raising perserves order for cardinals. Theorem 5.5 of [Specker]
|
|
p. 973. (Contributed by SF, 11-Mar-2015.) $)
|
|
tlecg $p |- ( ( M e. NC /\ N e. NC ) -> ( M <_c N <-> T_c M <_c T_c N ) )
|
|
$=
|
|
( vp vq cncs wcel wa clec wbr ctc cv cplc wceq wrex dflec2 tccl rexlimdva
|
|
wi sylbid adantlr addlecncs syl2an tcdi breqtrrd breq2d syl5ibrcom adantr
|
|
tceq wb simplr simpll simprl simprr taddc syl31anc addceq2 eqeq2d biimpac
|
|
ncaddccl tc11 syl2anc breq2 sylbird syl5 expdimp an32s adantrl mpd impbid
|
|
expr ) AEFZBEFZGZABHIZAJZBJZHIZVMVNBACKZLZMZCENZVQABCOVKWAVQRVLVKVTVQCEVK
|
|
VREFZGZVQVTVOVSJZHIWCVOVOVRJZLZWDHVKVOEFZWEEFVOWFHIWBAPZVRPVOWEUAUBAVRUCU
|
|
DVTVPWDVOHBVSUHUEUFQUGSVMVQVPVOVRLZMZCENZVNVKWGVPEFVQWKUIVLWHBPVOVPCOUBVM
|
|
WJVNCEVMWBWJVNVMWBWJGZGZVRDKZJZMZDENZVNWMVLVKWBWJWQVKVLWLUJVKVLWLUKVMWBWJ
|
|
ULVMWBWJUMBAVRDUNUOVMWJWQVNRWBVMWJGWPVNDEVMWNEFZWJWPVNRVMWRGZWJWPVNWJWPGV
|
|
PVOWOLZMZWSVNWPWJXAWPWIWTVPVRWOVOUPUQURWSXAVPAWNLZJZMZVNWSXCWTVPVKWRXCWTM
|
|
VLAWNUCTUQWSXDBXBMZVNWSVLXBEFZXDXEUIVKVLWRUJVKWRXFVLAWNUSTBXBUTVAVKWRXEVN
|
|
RVLVKWRGVNXEAXBHIAWNUABXBAHVBUFTSVCVDVEVFQVGVHVJQSVI $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a c $. $d a m $. $d a n $. $d a p $. $d a q $. $d a x $.
|
|
$d a y $. $d b c $. $d b m $. $d b n $. $d b p $. $d b q $. $d b x $.
|
|
$d b y $. $d c m $. $d c n $. $d c p $. $d c q $. $d c x $. $d c y $.
|
|
$d M a $. $d M b $. $d M c $. $d m n $. $d m p $. $d M p $. $d M q $.
|
|
$d m x $. $d m y $. $d N a $. $d N b $. $d N c $. $d n p $. $d N q $.
|
|
$d n x $. $d n y $. $d p q $. $d p x $. $d p y $. $d x y $.
|
|
$( If a cardinal is less than or equal to a T-raising, then it is also a
|
|
T-raising. Theorem 5.6 of [Specker] p. 973. (Contributed by SF,
|
|
11-Mar-2015.) $)
|
|
letc $p |- ( ( M e. NC /\ N e. NC /\ M <_c T_c N ) ->
|
|
E. p e. NC M = T_c p ) $=
|
|
( vq va vb vc vx vy vn vm cncs wcel cv wceq wrex wa cnc wex vex clec cplc
|
|
ctc wbr wb tccl dflec2 sylan2 wi elncs 3anbi123i eeeanv bitr4i cpw1 eqcom
|
|
w3a ncelncsi ncaddccl mp2an ncseqnc ax-mp bitri cin c0 cun eladdc pw1equn
|
|
bi2anan9 ineq12 eqeq1d anbi12d tceq tcnc syl6eq eqeq2d rspcev mpan sylbir
|
|
eleq1 ad2antrr syl6bi 3adant1 exlimivv syl5bi rexlimivv 3ad2ant2 addceq12
|
|
com12 expimpd sylbi 3adant2 eqeq12d eqeq1 rexbidv 3ad2ant1 imbi12d mpbiri
|
|
exlimiv 3expa rexlimdva sylbid 3impia ) ALMZBLMZABUCZUAUDZACNZUCZOZCLPZXC
|
|
XDQZXFXEADNZUBZOZDLPZXJXDXCXELMXFXOUEBUFAXEDUGUHXKXNXJDLXCXDXLLMZXNXJUIZX
|
|
CXDXPUPZAENZRZOZBFNZRZOZXLGNZRZOZUPZGSZFSESZXQXRYAESZYDFSZYGGSZUPYJXCYKXD
|
|
YLXPYMEAUJFBUJGXLUJUKYAYDYGEFGULUMYIXQEFYHXQGYHXQYBUNZRZXTYFUBZOZXTXHOZCL
|
|
PZUIYQYNYPMZYSYQYPYOOZYTYOYPUOYPLMZUUAYTUEXTLMZYFLMUUBXSETUQZYEGTUQXTYFUR
|
|
USYPYNUTVAVBYTHNZINZVCZVDOZYNUUEUUFVEOZQZIYFPHXTPYSYNXTYFHIVFUUJYSHIXTYFU
|
|
UEXTMZUUFYFMZQZUUHUUIYSUUIYBJNZKNZVEOZUUEUUNUNZOZUUFUUOUNZOZUPZKSJSZUUMUU
|
|
HQZYSJKUUEUUFYBHTITVGUVBUVCYSUVAUVCYSUIZJKUURUUTUVDUUPUURUUTQZUVCUUQXTMZU
|
|
USYFMZQZUUQUUSVCZVDOZQYSUVEUUMUVHUUHUVJUURUUKUVFUUTUULUVGUUEUUQXTVSUUFUUS
|
|
YFVSVHUVEUUGUVIVDUUEUUQUUFUUSVIVJVKUVFYSUVGUVJUVFXTUUQRZOZYSUUCUVLUVFUEUU
|
|
DXTUUQUTVAUUNRZLMUVLYSUUNJTZUQYRUVLCUVMLXGUVMOZXHUVKXTUVOXHUVMUCUVKXGUVMV
|
|
LUUNUVNVMVNVOVPVQVRVTWAWBWCWHWDWIWEWJWJYHXNYQXJYSYHXEYOXMYPYDYAXEYOOYGYDX
|
|
EYCUCYOBYCVLYBFTVMVNWFYAYGXMYPOYDAXLXTYFWGWKWLYAYDXJYSUEYGYAXIYRCLAXTXHWM
|
|
WNWOWPWQWRWCWJWSWTXAXB $.
|
|
$}
|
|
|
|
${
|
|
$d M n $. $d M x $. $d n x $.
|
|
$( If ` ( M ^c 0c ) ` is a cardinal, then ` M ` is a T-raising of some
|
|
cardinal. (Contributed by SF, 17-Mar-2015.) $)
|
|
ce0t $p |- ( ( M e. NC /\ ( M ^c 0c ) e. NC ) -> E. n e. NC M = T_c n ) $=
|
|
( vx cncs wcel c0c cce co wa cpw1 cnc wceq wex ctc wrex ce0ncpw1 ncelncsi
|
|
cv vex tcnc eqcomi tceq eqeq2d rspcev mp2an eqeq1 rexbidv mpbiri exlimiv
|
|
syl ) BDEBFGHDEIBCRZJKZLZCMBARZNZLZADOZBCPUMUQCUMUQULUOLZADOZUKKZDEULUTNZ
|
|
LZUSUKCSZQVAULUKVCTUAURVBAUTDUNUTLUOVAULUNUTUBUCUDUEUMUPURADBULUOUFUGUHUI
|
|
UJ $.
|
|
$}
|
|
|
|
${
|
|
$d M y $. $d N y $. $d M p $. $d M q $. $d N p $. $d p q $. $d p r $.
|
|
$d p s $. $d p x $. $d p y $. $d q r $. $d q s $. $d q x $. $d q y $.
|
|
$d r s $. $d r x $. $d r y $. $d s y $. $d x y $.
|
|
$( Partial ordering law for base two cardinal exponentiation. Theorem 4.8
|
|
of [Specker] p. 973. (Contributed by SF, 16-Mar-2015.) $)
|
|
ce2le $p |- ( ( ( M e. NC /\ N e. NC /\ ( N ^c 0c ) e. NC ) /\ M <_c N ) ->
|
|
( 2c ^c M ) <_c ( 2c ^c N ) ) $=
|
|
( vp vq vx vy vr cncs wcel cce co clec wbr wa cv wceq wrex c2c wi cnc c0c
|
|
vs w3a ctc ce0t 3adant1 adantr letc tlecg ancoms wex elncs anbi12i bitr4i
|
|
wb eeanv cpw wss cen enpw 3imtr4i adantl sspwb biimpi sseq1 sseq2 rspc2ev
|
|
elnc syl3anc ex rexlimivv ncex brlec cpw1 vex tcnc ce2 ax-mp 3brtr4g tceq
|
|
breq12 oveq2d breqan12d imbi12d mpbiri exlimivv sylbi sylbird an32s breq1
|
|
imp anbi2d anbi1d oveq2 breq1d com12 rexlimdva mpd 3expa breq2d 3ad2antl1
|
|
breq2 ) AHIZBHIZBUAJKHIZUCZABLMZNBCOZUDZPZCHQZRAJKZRBJKZLMZXFXKXGXDXEXKXC
|
|
CBUEUFUGXCXDXGXKXNSXEXCXGNZXJXNCHXJXOXHHIZNZXNXJXQXNSXCAXILMZNZXPNZXLRXIJ
|
|
KZLMZSXCXPXRYBXCXPXRYBXCXPXRUCADOZUDZPZDHQZYBAXHDUHXPXRYFYBSXCXPXRNZYEYBD
|
|
HYEYGYCHIZNZYBYEYIYBSXPYDXILMZNZYHNZRYDJKZYALMZSXPYHYJYNXPYHNZYJYNYOYJYCX
|
|
HLMZYNYHXPYPYJUOYCXHUIUJYHXPYPYNSZYHXPNZYCEOZTZPZXHFOZTZPZNZFUKEUKZYQYRUU
|
|
AEUKZUUDFUKZNUUFYHUUGXPUUHEYCULFXHULUMUUAUUDEFUPUNUUEYQEFUUEYQYTUUCLMZRYT
|
|
UDZJKZRUUCUDZJKZLMZSUUIYSUQZTZUUBUQZTZUUKUUMLXHYCURZDUUCQCYTQGOZUBOZURZUB
|
|
UURQGUUPQZUUIUUPUURLMUUSUVCCDYTUUCXHYTIZYCUUCIZNZUUSUVCUVFUUSNXHUQZUUPIZY
|
|
CUQZUURIZUVGUVIURZUVCUVFUVHUUSUVDUVHUVEXHYSUSMUVGUUOUSMUVDUVHXHYSUTXHYSVH
|
|
UVGUUOVHVAUGUGUVFUVJUUSUVEUVJUVDYCUUBUSMUVIUUQUSMUVEUVJYCUUBUTYCUUBVHUVIU
|
|
UQVHVAVBUGUUSUVKUVFUUSUVKXHYCVCVDVBUVBUVKUVGUVAURGUBUVGUVIUUPUURUUTUVGUVA
|
|
VEUVAUVIUVGVFVGVIVJVKCDYTUUCYSVLUUBVLVMGUBUUPUURUUOVLUUQVLVMVAUUJYSVNTPUU
|
|
KUUPPYSEVOZVPYSUUJUVLVQVRUULUUBVNTPUUMUURPUUBFVOZVPUUBUULUVMVQVRVSUUEYPUU
|
|
IYNUUNYCYTXHUUCLWAUUAUUDYMUUKYAUUMLUUAYDUUJRJYCYTVTWBUUDXIUULRJXHUUCVTWBW
|
|
CWDWEWFWGUJWHWKWIYEYIYLYBYNYEYGYKYHYEXRYJXPAYDXILWJWLWMYEXLYMYALAYDRJWNWO
|
|
WDWEWPWQUFWRWSWIXJXQXTXNYBXJXOXSXPXJXGXRXCBXIALXBWLWMXJXMYAXLLBXIRJWNWTWD
|
|
WEWPWQXAWR $.
|
|
$}
|
|
|
|
${
|
|
$d M x $. $d N x $.
|
|
$( The exponent of a T-raising to a T-raising is always a cardinal.
|
|
(Contributed by SF, 13-Mar-2015.) $)
|
|
cet $p |- ( ( M e. NC /\ N e. NC ) -> ( T_c M ^c T_c N ) e. NC ) $=
|
|
( vx cncs wcel ctc c0c cce co tccl adantr adantl cnc wceq wex tceq oveq1d
|
|
elncs ax-mp syl6eqel wa cv wne cpw1 vex ncelncsi pw1ex ncid tcnc eleqtrri
|
|
c0 ce0nnuli mp2an wb ce0nulnc mpbi exlimiv sylbi cecl syl22anc ) ADEZBDEZ
|
|
UAAFZDEZBFZDEZVCGHIZDEZVEGHIZDEZVCVEHIDEVAVDVBAJKVBVFVABJLVAVHVBVAACUBZMZ
|
|
NZCOVHCARVMVHCVMVGVLFZGHIZDVMVCVNGHAVLPQVOUKUCZVODEZVNDEZVKUDZVNEVPVLDEVR
|
|
VKCUEZUFVLJSZVSVSMVNVSVKVTUGUHVKVTUIUJVKVNULUMVRVPVQUNWAVNUOSUPZTUQURKVBV
|
|
JVAVBBVLNZCOVJCBRWCVJCWCVIVODWCVEVNGHBVLPQWBTUQURLVCVEUSUT $.
|
|
$}
|
|
|
|
$( The exponent of two to a T-raising is always a cardinal. Theorem 5.8 of
|
|
[Specker] p. 973. (Contributed by SF, 13-Mar-2015.) $)
|
|
ce2t $p |- ( M e. NC -> ( 2c ^c T_c M ) e. NC ) $=
|
|
( cncs wcel c2c ctc cce co tc2c oveq1i 2nc cet mpan syl5eqelr ) ABCZDAEZFGD
|
|
EZOFGZBPDOFHIDBCNQBCJDAKLM $.
|
|
|
|
${
|
|
$d M x $.
|
|
$( Distributive law for T-raising and cardinal exponentiation to two.
|
|
(Contributed by SF, 13-Mar-2015.) $)
|
|
tce2 $p |- ( ( M e. NC /\ ( M ^c 0c ) e. NC ) ->
|
|
T_c ( 2c ^c M ) = ( 2c ^c T_c M ) ) $=
|
|
( vx cncs wcel c0c cce co wa cpw1 cnc wceq c2c ctc cpw tcnc eqid ce2 tceq
|
|
ax-mp syl wex ce0ncpw1 vex pwex ncpwpw1 3eqtr4i pw1ex eqtr4i oveq2 syl6eq
|
|
cv oveq2d 3eqtr4a exlimiv ) ACDAEFGCDHABUKZIZJZKZBUALAFGZMZLAMZFGZKZABUBU
|
|
RVCBURLUQFGZMZLUPIJZFGZUTVBVEUPNJZVGUONZJZMZVIIJVEVHVIUOBUCZUDOVDVJKZVEVK
|
|
KUQUQKVMUQPUOUQVLQSVDVJRSUOVLUEUFVFVFKVGVHKVFPUPVFUOVLUGZQSUHURUSVDKUTVEK
|
|
AUQLFUIUSVDRTURVAVFLFURVAUQMVFAUQRUPVNOUJULUMUNT $.
|
|
$}
|
|
|
|
${
|
|
$d M x $.
|
|
$( A T-raising raised to zero is always a cardinal. (Contributed by SF,
|
|
16-Mar-2015.) $)
|
|
te0c $p |- ( M e. NC -> ( T_c M ^c 0c ) e. NC ) $=
|
|
( vx cncs wcel cv cpw1 ctc wex c0c cce cnc wceq elncs vex pw1ex ncid tceq
|
|
co wb syl tcnc syl6eq syl5eleqr eximi c0 wne tccl ce0nnul ce0nulnc bitr3d
|
|
sylbi mpbid ) ACDZBEZFZAGZDZBHZUPIJRZCDZUMAUNKZLZBHURBAMVBUQBVBUOUOKZUPUO
|
|
UNBNZOPVBUPVAGVCAVAQUNVDUAUBUCUDUKUMUSUEUFZURUTUMUPCDZVEURSAUGZUPBUHTUMVF
|
|
VEUTSVGUPUITUJUL $.
|
|
$}
|
|
|
|
${
|
|
$d M n $.
|
|
$( ` ( M ^c 0c ) ` is a cardinal iff ` M ` is a T-raising of some
|
|
cardinal. (Contributed by SF, 17-Mar-2015.) $)
|
|
ce0tb $p |- ( M e. NC -> ( ( M ^c 0c ) e. NC <-> E. n e. NC M = T_c n ) )
|
|
$=
|
|
( cncs wcel c0c cce co cv ctc wceq wrex ce0t te0c oveq1 eleq1d syl5ibrcom
|
|
ex rexlimiv impbid1 ) BCDZBEFGZCDZBAHZIZJZACKZTUBUFABLQUEUBACUCCDUBUEUDEF
|
|
GZCDUCMUEUAUGCBUDEFNOPRS $.
|
|
$}
|
|
|
|
${
|
|
$d M n $. $d M x $. $d n x $. $d n y $. $d x y $.
|
|
$( Cardinal exponentiation to zero is a cardinal iff the number is less
|
|
than the size of cardinal one. (Contributed by SF, 18-Mar-2015.) $)
|
|
ce0lenc1 $p |- ( M e. NC -> ( ( M ^c 0c ) e. NC <-> M <_c Nc 1c ) ) $=
|
|
( vn vx vy cncs wcel cv ctc wceq wrex c1c cnc clec wbr wex cpw1 tceq tcnc
|
|
vex wss c0c co ce0tb elncs syl6eq pw1ss1c pw1ex 1cex nclec ax-mp syl6eqbr
|
|
exlimiv sylbi breq1 syl5ibrcom rexlimiv lenc wa wi ncseqnc biimpar sspw12
|
|
cce ncelncsi eqeq1d rspcev mp2an nceq eqcom syl6bb rexbidv mpbiri syl5ibr
|
|
eqeq1 syl rexlimdva sylbid impbid2 bitrd ) AEFZAUAVCUBEFABGZHZIZBEJZAKLZM
|
|
NZBAUCVTWDWFWCWFBEWAEFZWFWCWBWEMNZWGWACGZLZIZCOWHCWAUDWKWHCWKWBWIPZLZWEMW
|
|
KWBWJHWMWAWJQWICSZRUEWLKTWMWEMNWIUFWLKWIWNUGUHUIUJUKULUMAWBWEMUNUOUPVTWFW
|
|
IKTZCAJWDCKAUHUQVTWOWDCAVTWIAFZURAWJIZWOWDUSVTWQWPAWIUTVAWOWDWQWJWBIZBEJZ
|
|
WOWIDGZPZIZDOWSDWIWNVBXBWSDXBWSWBXALZIZBEJZWTLZEFXFHZXCIZXEWTDSZVDWTXIRXD
|
|
XHBXFEWAXFIWBXGXCWAXFQVEVFVGXBWRXDBEXBWRXCWBIXDXBWJXCWBWIXAVHVEXCWBVIVJVK
|
|
VLULUMWQWCWRBEAWJWBVNVKVMVOVPVQVRVS $.
|
|
$}
|
|
|
|
${
|
|
$d M x y z $.
|
|
$( A T-raising is less than or equal to the cardinality of cardinal one.
|
|
(Contributed by SF, 16-Mar-2015.) $)
|
|
tlenc1c $p |- ( M e. NC -> T_c M <_c Nc 1c ) $=
|
|
( vx vy vz cncs wcel cv cnc wceq wex ctc c1c clec wbr elncs cpw1 wss wrex
|
|
ncid ncex tceq vex tcnc syl6eq pw1ex 1cex pw1ss1c sseq1 sseq2 mp3an brlec
|
|
rspc2ev mpbir syl6eqbr exlimiv sylbi ) AEFABGZHZIZBJAKZLHZMNZBAOUSVBBUSUT
|
|
UQPZHZVAMUSUTURKVDAURUAUQBUBZUCUDVDVAMNCGZDGZQZDVARCVDRZVCVDFLVAFVCLQZVIV
|
|
CUQVEUESLUFSUQUGVHVJVCVGQCDVCLVDVAVFVCVGUHVGLVCUIULUJCDVDVAVCTLTUKUMUNUOU
|
|
P $.
|
|
$}
|
|
|
|
$( Cardinal one is not zero. (Contributed by SF, 4-Mar-2015.) $)
|
|
1ne0c $p |- 1c =/= 0c $=
|
|
( c0c c1c cplc addcid2 0cnsuc eqnetrri ) ABCBABDAEF $.
|
|
|
|
$( Cardinal two is not zero. (Contributed by SF, 4-Mar-2015.) $)
|
|
2ne0c $p |- 2c =/= 0c $=
|
|
( c1c cplc c2c c0c 1p1e2c 0cnsuc eqnetrri ) AABCDEAFG $.
|
|
|
|
${
|
|
$d A x $.
|
|
$( A set is finite iff its cardinality is a natural. (Contributed by SF,
|
|
18-Mar-2015.) $)
|
|
finnc $p |- ( A e. Fin <-> Nc A e. Nn ) $=
|
|
( vx cv cnc wceq cnnc wrex wcel cfin cncs nnnc ncseqnc syl rexbiia risset
|
|
wb elfin 3bitr4ri ) BCZADZEZBFGASHZBFGTFHAIHUAUBBFSFHSJHUAUBPSKSALMNBTFOB
|
|
AQR $.
|
|
$}
|
|
|
|
${
|
|
$d p q $. $d p x $. $d p z $. $d q x $. $d q z $. $d x z $. $d y z $.
|
|
$d p t $. $d p u $. $d q t $. $d q u $. $d t u $. $d t x $. $d x y $.
|
|
$( The stratified T raising function is a set. (Contributed by SF,
|
|
18-Mar-2015.) $)
|
|
tcfnex $p |- TcFn e. _V $=
|
|
( vx vz vp vq vu vt c1c cvv csset cid cv cop wcel wex wbr vex bitri exbii
|
|
wa 3bitri vy ctcfn cxp cins3 ccnv cncs cpw1fn csi ccom ctxp cima cpw1 cin
|
|
csymdif crn ccompl cuni ctc cmpt df-tcfn csn wel cnc wceq wrex weq wb wal
|
|
cins2 oteltxp df-br brcnv brssetsn bitr3i wn opex elrn2 elsymdif otelins2
|
|
elcompl elin opelxp mpbiran2 anbi1i ncseqnc rexbidv brsnsi1 19.41v bitr4i
|
|
snex excom anass anbi2d ceqsexv brpw1fn anbi12i opelco df-clel opelssetsn
|
|
breq1 3bitr4i brsnsi elima1c eluni ancom df-rex syl6rbbr pm5.32i otelins3
|
|
elimapw11c ideq bibi12i xchbinx exnal 3bitrri con1bii cab cio df-tc eqtri
|
|
dfiota2 eleq2i eluniab releqmpt eqtr4i 1cex cnvex ncsex vvex xpex pw1fnex
|
|
ssetex siex coex txpex imaex pw1ex inex ins2ex rnex idex symdifex complex
|
|
ins3ex mptexlem eqeltri ) UBGHUCIUDIUEZUFHUCZIUGUHZUIZUUGUHZIUJZGUKZUJZGU
|
|
LZUKZUMZVIZJUDZUNZUOZUPZUJZUOZVIUNGUKUPUEUMZHUBAGAKZUQZURZUSUVEAUTAUAGUVD
|
|
UVHBKZUAKZVAZUVFLZLUVCMZBNUABVBZCKZUFMZUVODKZULZVCVDZDUVGVEZSZCBVFZVGZCVH
|
|
ZSZBNZUVLUVDMUVJUVHMZUVMUWEBUVMUVIUVKLUUGMZUVIUVFLZUVBMZSUWEUVIUVKUVFUUGU
|
|
VBVJUWHUVNUWJUWDUWHUVIUVKUUGOZUVNUVIUVKUUGVKUWKUVKUVIIOUVNUVIUVKIVLUVJUVI
|
|
UAPBPZVMQVNUWJUWIUVAMZVOUWDUWIUVAUVIUVFUWLAPZVPVTUWDUWMUWMUVOUWILZUUTMZCN
|
|
UWCVOZCNUWDVOCUWIUUTVQUWPUWQCUWPUWOUURMZUWOUUSMZVGUWCUWOUURUUSVRUWRUWAUWS
|
|
UWBUWRUVOUVFLZUUQMUWTUUHMZUWTUUPMZSZUWAUVOUVIUVFUUQUWLVSUWTUUHUUPWAUXCUVP
|
|
UXBSUWAUXAUVPUXBUXAUVPUVFHMUWNUVOUVFUFHWBWCWDUVPUXBUVTUVPUVTUVRUVOMZDUVGV
|
|
EZUXBUVPUVSUXDDUVGUVOUVRWEWFUVQVAZVAZUWTLUUNMZDNUVQUVGMZUXDSZDNUXBUXEUXHU
|
|
XJDUXHUXGUVOLUUJMZUXGUVFLZUUMMZSUXDUXISUXJUXGUVOUVFUUJUUMVJUXKUXDUXMUXIUX
|
|
GEKZUUIOZUXNUVOIOZSZENZFKZUVRVDZFCVBZSZFNZUXKUXDUXRUXNUXSVAZVDZUXFUXSUGOZ
|
|
SZUXPSZFNZENUYHENZFNUYCUXQUYIEUXQUYGFNZUXPSUYIUXOUYKUXPFUXFUXNUGUVQWJZWGW
|
|
DUYGUXPFWHWIRUYHEFWKUYJUYBFUYJUYEUYFUXPSZSZENUYFUYDUVOIOZSZUYBUYHUYNEUYEU
|
|
YFUXPWLRUYMUYPEUYDUXSWJUYEUXPUYOUYFUXNUYDUVOIWTWMWNUYFUXTUYOUYAUVQUXSDPZW
|
|
OUXSUVOFPZCPVMWPTRTEUXGUVOIUUIWQFUVRUVOWRXAUYDUXLLUULMZFNDFVBZFAVBZSZFNUX
|
|
MUXIUYSVUBFUYSUYDUXGLUUKMZUYDUVFLIMZSVUBUYDUXGUVFUUKIVJVUCUYTVUDVUAVUCUYD
|
|
UXGUUKOZUYTUYDUXGUUKVKVUEUXSUXFUUGOUXFUXSIOUYTUXSUXFUUGUYRUYLXBUXSUXFIVLU
|
|
VQUXSUYQUYRVMTVNUXSUVFUYRUWNWSWPQRFUXLUULXCFUVQUVFXDXAWPUXDUXIXETRDUWTUUN
|
|
XJUXDDUVGXFXAXGXHQTUWSUVOUVILJMZUWBUVOUVIUVFJUWNXIVUFUVOUVIJOUWBUVOUVIJVK
|
|
UVOUVIUWLXKVNQXLXMRUWCCXNXOXPQWPQRBUVLUVCVQUWGUVJUWDBXQUQZMUWFUVHVUGUVJUV
|
|
HUWACXRVUGDUVGCXSUWACBYAXTYBUWDBUVJYCQXAYDYEGUVDYFUVCUUGUVBIYLYGZUVAUUTUU
|
|
RUUSUUQUUHUUPUFHYHYIYJUUNUUOUUJUUMIUUIYLUGYKYMYNUULGUUKIUUGVUHYMYLYOYFYPY
|
|
OGYFYQYPYRYSJUUAUUDUUBYTUUCYOYTUUEUUF $.
|
|
$}
|
|
|
|
$( Functionhood statement for the stratified T-raising function.
|
|
(Contributed by SF, 18-Mar-2015.) $)
|
|
fntcfn $p |- TcFn Fn 1c $=
|
|
( vx cv cuni ctc cvv wcel ctcfn c1c wfn df-tcfn fnmpt tcex a1i mprg ) ABZCZ
|
|
DZEFZGHIAHAHQGEAJKROHFPLMN $.
|
|
|
|
${
|
|
$d A x $.
|
|
brtcfn.1 $e |- A e. _V $.
|
|
$( Binary relationship form of the stratified T-raising function.
|
|
(Contributed by SF, 18-Mar-2015.) $)
|
|
brtcfn $p |- ( { A } TcFn B <-> B = T_c A ) $=
|
|
( vx csn ctcfn cfv wceq ctc wbr c1c wcel snel1c cv cuni unieq syl6eq tceq
|
|
unisn syl df-tcfn tcex fvmpt ax-mp eqeq1i wb fntcfn fnbrfvb mp2an 3bitr3i
|
|
wfn eqcom ) AEZFGZBHZAIZBHUMBFJZBUPHUNUPBUMKLZUNUPHACMZDUMDNZOZIZUPKFUTUM
|
|
HZVAAHVBUPHVCVAUMOAUTUMPACSQVAARTDUAAUBUCUDUEFKUKURUOUQUFUGUSKUMBFUHUIUPB
|
|
ULUJ $.
|
|
$}
|
|
|
|
${
|
|
$d A n $.
|
|
ncfin.1 $e |- A e. _V $.
|
|
$( The cardinality of a set is a natural iff the set is finite.
|
|
(Contributed by SF, 19-Mar-2015.) $)
|
|
ncfin $p |- ( Nc A e. Nn <-> A e. Fin ) $=
|
|
( vn cnnc wcel cv wrex cfin ncid eleq2 rspcev mpan2 wa wceq eqcom cncs wb
|
|
cnc nnnc ncseqnc syl syl5bb biimpar eleq1d exbiri pm2.43a rexlimiv impbii
|
|
elfin bitr4i ) ARZDEZACFZEZCDGZAHEULUOULAUKEZUOABIUNUPCUKDUMUKAJKLUNULCDU
|
|
NUMDEZULUQUNULUQUQUNMUKUMDUQUKUMNZUNURUMUKNZUQUNUKUMOUQUMPEUSUNQUMSUMATUA
|
|
UBUCUDUEUFUGUHCAUIUJ $.
|
|
$}
|
|
|
|
${
|
|
$d n x $.
|
|
$( Lemma for ~ nclenn . Set up stratification for induction. (Contributed
|
|
by SF, 19-Mar-2015.) $)
|
|
nclennlem1 $p |- { x | A. n e. NC ( n <_c x -> n e. Nn ) } e. _V $=
|
|
( clec cnnc ccompl cres cncs cima cv wbr wcel wi wral wn vex elcompl wrex
|
|
wa bitri complex cab cvv elima brres anbi2i rexbii 3bitrri con1bii abbi2i
|
|
rexanali lecex nncex resex ncsex imaex eqeltrri ) CDEZFZGHZEZBIZAIZCJZVAD
|
|
KZLBGMZAUAUBVEAUTVBUTKVBUSKZNVEVBUSAOPVEVFVFVAVBURJZBGQVCVDNZRZBGQVENBVBU
|
|
RGUCVGVIBGVGVCVAUQKZRVIVAVBCUQUDVJVHVCVADBOPUESUFVCVDBGUJUGUHSUIUSURGCUQU
|
|
KDULTUMUNUOTUP $.
|
|
$}
|
|
|
|
${
|
|
$d m n $. $d M n $. $d m p $. $d m q $. $d m x $. $d N n $. $d n p $.
|
|
$d n q $. $d n x $. $d N x $. $d p q $.
|
|
$( A cardinal that is less than or equal to a natural is a natural.
|
|
Theorem XI.3.3 of [Rosser] p. 391. (Contributed by SF, 19-Mar-2015.) $)
|
|
nclenn $p |- ( ( M e. NC /\ N e. Nn /\ M <_c N ) -> M e. Nn ) $=
|
|
( vn vx vp vq cncs wcel cnnc clec wbr wi cv wral c0c c1c cplc wceq breq2
|
|
wa vm nclennlem1 imbi1d ralbidv weq le0nc 0cnc sbth mpan2 peano1 syl6eqel
|
|
imp ex mpan2d rgen w3a wrex wb peano2 syl dflec2 sylan2 ancoms 3adant3 wo
|
|
nnnc nc0suc addceq2 addcid1 syl6eq eqeq2d biimpa eleq1 biimpcd syl5 exp3a
|
|
3ad2ant1 addcass syl6eqr ncaddccl 3ad2antl2 peano4nc addlecncs syl5ibrcom
|
|
adantr syl2anc com23 adantl pm2.27 com24 3impia sylbid rexlimdva rexlimdv
|
|
syl8 jaod 3expia ralimdva finds breq1 imbi12d rspccv com12 3imp ) AGHZBIH
|
|
ZABJKZAIHZXFXEXGXHLZXFCMZBJKZXJIHZLZCGNZXEXILXJDMZJKZXLLZCGNXJOJKZXLLZCGN
|
|
XJUAMZJKZXLLZCGNXJXTPQZJKZXLLZCGNXNDUABDCUBXOORZXQXSCGYFXPXRXLXOOXJJSUCUD
|
|
DUAUEZXQYBCGYGXPYAXLXOXTXJJSUCUDXOYCRZXQYECGYHXPYDXLXOYCXJJSUCUDXOBRZXQXM
|
|
CGYIXPXKXLXOBXJJSUCUDXSCGXJGHZXROXJJKZXLXJUFYJXRYKTZXLYJYLTXJOIYJYLXJORZY
|
|
JOGHYLYMLUGXJOUHUIULUJUKUMUNUOXTIHZYBYECGYNYJYBYEYNYJYBUPZYDYCXJEMZQZRZEG
|
|
UQZXLYNYJYDYSURZYBYJYNYTYNYJYCGHZYTYNYCIHZUUAXTUSZYCVFUTXJYCEVAVBVCVDYOYR
|
|
XLEGYPGHYPORZYPFMZPQZRZFGUQZVEYOYRXLLZFYPVGYOUUDUUIUUHYNYJUUDUUILZYBYNUUB
|
|
UUJUUCUUBUUDYRXLUUDYRTYCXJRZUUBXLUUDYRUUKUUDYQXJYCUUDYQXJOQXJYPOXJVHXJVIV
|
|
JVKVLUUKUUBXLYCXJIVMVNVOVPUTVQYOUUGUUIFGYOUUEGHZTZUUGYRXLUUGYRTYCXJUUEQZP
|
|
QZRZUUMXLUUGYRUUPUUGYQUUOYCUUGYQXJUUFQUUOYPUUFXJVHXJUUEPVRVSVKVLUUMUUPXTU
|
|
UNRZXLUUMXTGHZUUNGHZUUPUUQURYOUURUULYNYJUURYBXTVFVQWEYJYNUULUUSYBXJUUEVTW
|
|
AXTUUNWBWFYOUULUUQXLLZYNYJYBUULUUTLYNYJTZUUQUULYBXLUVAUUQUULYAYBXLLYJUUQU
|
|
ULYALLYNYJUULUUQYAYJUULUUQYALYJUULTYAUUQXJUUNJKXJUUEWCXTUUNXJJSWDUMWGWHYA
|
|
XLWIWOWJWKULWLVOVPWMWPVOWNWLWQWRWSXMXICAGXJARXKXGXLXHXJABJWTXJAIVMXAXBUTX
|
|
CXD $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $. $d B x y z $. $d C x y z $.
|
|
$( Distributivity law for cardinal addition and multiplication. Theorem
|
|
XI.2.31 of [Rosser] p. 379. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
addcdi $p |- ( ( A e. NC /\ B e. NC /\ C e. NC ) ->
|
|
( A .c ( B +c C ) ) = ( ( A .c B ) +c ( A .c C ) ) ) $=
|
|
( vx vy vz cncs wcel cplc cmuc co wceq cv cnc vex wa wi cxp mucnc oveq1
|
|
w3a ncaddccl 3adant1 wex elncs ncid mpbiri cin c0 cun wrex eladdc ncseqnc
|
|
eleq2 wb bi2anan9 ncdisjun oveq2d xpdisj2 xpex syl xpundi nceqi addceq12i
|
|
unex eqtri 3eqtr4g eqtr3d addceq12d eqeq12d syl5ibr exlimiv sylbi adantrd
|
|
addceq12 oveq2 adantr adantl imbi2d syl5ibrcom 3ad2ant1 sylbird rexlimdvv
|
|
syl5bi syl5 exlimdv mpd ) AGHZBGHZCGHZUAZBCIZGHZAWLJKZABJKZACJKZIZLZWIWJW
|
|
MWHBCUBUCWMWLDMZNZLZDUDWKWRDWLUEWKXAWRDXAWSWLHZWKWRXAXBWSWTHWSDOZUFWLWTWS
|
|
UNUGXBEMZFMZUHUILZWSXDXEUJZLZPZFCUKEBUKWKWRWSBCEFULWKXIWREFBCWKXDBHZXECHZ
|
|
PZBXDNZLZCXENZLZPZXIWRQZWIWJXQXLUOWHWIXNXJWJXPXKBXDUMCXEUMUPUCWHWIXQXRQWJ
|
|
WHXRXQXIAXMXOIZJKZAXMJKZAXOJKZIZLZQWHXFYDXHWHAWTLZDUDXFYDQZDAUEYEYFDXFYDY
|
|
EWTXSJKZWTXMJKZWTXOJKZIZLXFWTXGNZJKZYGYJXFYKXSWTJXDXEEOZFOZUQURXFWSXDRZWS
|
|
XERZUJZNZYONZYPNZIZYLYJXFYOYPUHUILYRUUALXDXEWSWSUSYOYPWSXDXCYMUTWSXEXCYNU
|
|
TUQVAYLWSXGRZNYRWSXGXCXDXEYMYNVESUUBYQWSXDXEVBVCVFYHYSYIYTWSXDXCYMSWSXEXC
|
|
YNSVDVGVHYEXTYGYCYJAWTXSJTYEYAYHYBYIAWTXMJTAWTXOJTVIVJVKVLVMVNXQWRYDXIXQW
|
|
NXTWQYCXQWLXSAJBCXMXOVOURXQWOYAWPYBXNWOYALXPBXMAJVPVQXPWPYBLXNCXOAJVPVRVI
|
|
VJVSVTWAWBWCWDWEWFWDWG $.
|
|
$}
|
|
|
|
$( Distributivity law for cardinal addition and multiplication. Theorem
|
|
XI.2.30 of [Rosser] p. 379. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
addcdir $p |- ( ( A e. NC /\ B e. NC /\ C e. NC ) ->
|
|
( ( A +c B ) .c C ) = ( ( A .c C ) +c ( B .c C ) ) ) $=
|
|
( cncs wcel w3a cplc cmuc wceq addcdi 3coml ncaddccl 3adant3 muccom syl2anc
|
|
co simp3 3adant2 3adant1 addceq12d 3eqtr4d ) ADEZBDEZCDEZFZCABGZHPZCAHPZCBH
|
|
PZGZUFCHPZACHPZBCHPZGUDUBUCUGUJICABJKUEUFDEZUDUKUGIUBUCUNUDABLMUBUCUDQUFCNO
|
|
UEULUHUMUIUBUDULUHIUCACNRUCUDUMUIIUBBCNSTUA $.
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
$( The cardinal product of two cardinal numbers is zero iff one of the
|
|
numbers is zero. Biconditional form of theorem XI.2.34 of [Rosser]
|
|
p. 380. (Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
muc0or $p |- ( ( A e. NC /\ B e. NC ) -> ( ( A .c B ) = 0c <->
|
|
( A = 0c \/ B = 0c ) ) ) $=
|
|
( vx vy cncs wcel wa cmuc co c0c wceq wo cv cnc wex wi df0c2 sylbi eqeq1d
|
|
c0 elncs anbi12i eeanv bitr4i cxp vex mucnc eqeq12i cen wbr xpex eqnc en0
|
|
bitri xpeq0 nceq orim12i oveq12 eqeq1 adantr eqeq2i syl6bb adantl orbi12d
|
|
imbi12d mpbiri exlimivv 0cnc muccom mpan muc0 eqtrd oveq1 syl5ibrcom jaod
|
|
wb oveq2 impbid ) AEFZBEFZGZABHIZJKZAJKZBJKZLZWAACMZNZKZBDMZNZKZGZDOCOZWC
|
|
WFPZWAWICOZWLDOZGWNVSWPVTWQCAUADBUAUBWIWLCDUCUDWMWOCDWMWOWHWKHIZJKZWHTNZK
|
|
ZWKWTKZLZPWSWGWJUEZNZWTKZXCWRXEJWTWGWJCUFZDUFZUGQUHXFXDTKZXCXFXDTUIUJXIXD
|
|
TWGWJXGXHUKULXDUMUNXIWGTKZWJTKZLXCWGWJUOXJXAXKXBWGTUPWJTUPUQRRRWMWCWSWFXC
|
|
WMWBWRJAWHBWKHURSWMWDXAWEXBWMWDWHJKZXAWIWDXLVPWLAWHJUSUTJWTWHQVAVBWLWEXBV
|
|
PWIWLWEWKJKXBBWKJUSJWTWKQVAVBVCVDVEVFVGRWAWDWCWEVTWDWCPVSVTWCWDJBHIZJKVTX
|
|
MBJHIZJJEFVTXMXNKVHJBVIVJBVKVLWDWBXMJAJBHVMSVNVCVSWEWCPVTVSWCWEAJHIZJKAVK
|
|
WEWBXOJBJAHVQSVNUTVOVR $.
|
|
$}
|
|
|
|
${
|
|
$d A q $. $d B q $. $d C q $.
|
|
$( Multiplication law for cardinal less than. Theorem XI.2.35 of [Rosser]
|
|
p. 380. (Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
lemuc1 $p |- ( ( ( A e. NC /\ B e. NC /\ C e. NC ) /\ A <_c B ) ->
|
|
( A .c C ) <_c ( B .c C ) ) $=
|
|
( vq cncs wcel w3a clec cmuc co cv cplc wceq wrex wb dflec2 3adant3 muccl
|
|
wbr wa wi adantr ancoms adantll addlecncs syl2anc simpll addcdir breqtrrd
|
|
simpr simplr syl3anc oveq1 breq2d syl5ibrcom rexlimdva 3adant2 sylbid imp
|
|
) AEFZBEFZCEFZGZABHSZACIJZBCIJZHSZVCVDBADKZLZMZDENZVGUTVAVDVKOVBABDPQUTVB
|
|
VKVGUAVAUTVBTZVJVGDEVLVHEFZTZVGVJVEVICIJZHSVNVEVEVHCIJZLZVOHVNVEEFZVPEFZV
|
|
EVQHSVLVRVMACRUBVBVMVSUTVMVBVSVHCRUCUDVEVPUEUFVNUTVMVBVOVQMUTVBVMUGVLVMUJ
|
|
UTVBVMUKAVHCUHULUIVJVFVOVEHBVICIUMUNUOUPUQURUS $.
|
|
$}
|
|
|
|
$( Multiplication law for cardinal less than. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
lemuc2 $p |- ( ( ( A e. NC /\ B e. NC /\ C e. NC ) /\ B <_c C ) ->
|
|
( A .c B ) <_c ( A .c C ) ) $=
|
|
( cncs wcel w3a clec wbr wa cmuc co 3anrot lemuc1 sylanb wceq simpl1 simpl2
|
|
muccom syl2anc simpl3 3brtr4d ) ADEZBDEZCDEZFZBCGHZIZBAJKZCAJKZABJKZACJKZGU
|
|
EUCUDUBFUFUHUIGHUBUCUDLBCAMNUGUBUCUJUHOUBUCUDUFPZUBUCUDUFQABRSUGUBUDUKUIOUL
|
|
UBUCUDUFTACRSUA $.
|
|
|
|
$( A cardinal is less than or equal to its product with another. Theorem
|
|
XI.2.36 of [Rosser] p. 381. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
ncslemuc $p |- ( ( M e. NC /\ N e. NC /\ N =/= 0c ) ->
|
|
M <_c ( M .c N ) ) $=
|
|
( cncs wcel c0c wne w3a c1c clec wbr cmuc co wn df-ne nc0le1 ord syl5bi imp
|
|
wceq wa 3adant1 wi mucid1 ad2antrr 1cnc lemuc2 mp3anl2 eqbrtrrd 3adant3 mpd
|
|
ex ) ACDZBCDZBEFZGHBIJZAABKLZIJZUMUNUOULUMUNUOUNBESZMUMUOBENUMURUOBOPQRUAUL
|
|
UMUOUQUBUNULUMTZUOUQUSUOTAHKLZAUPIULUTASUMUOAUCUDULHCDUMUOUTUPIJUEAHBUFUGUH
|
|
UKUIUJ $.
|
|
|
|
$( The product of the cardinality of ` _V ` squared is just the cardinality
|
|
of ` _V ` . Theorem XI.2.37 of [Rosser] p. 381. (Contributed by Scott
|
|
Fenton, 31-Jul-2019.) $)
|
|
ncvsq $p |- ( Nc _V .c Nc _V ) = Nc _V $=
|
|
( cvv cnc cmuc co clec wbr wceq wcel c0 ovex wn cncs vvex mp2an mpbii df-ne
|
|
wne mto mpbir c0c nulnnc ncelncsi muccl eleq1 lecncvg vn0 el0c nemtbir ncid
|
|
eleq2 ncslemuc mp3an wa wi sbth ) ABZUPCDZUPEFZUPUQEFZUQUPGZUQAHUQIQZURUPUP
|
|
CJVAUQIGZKVBILHZUAVBUQLHZVCUPLHZVEVDAMUBZVFUPUPUCNZUQILUDORUQIPSUQAUENVEVEU
|
|
PTQZUSVFVFVHUPTGZKVIATHZVJAIUFAUGUHVIAUPHVJAMUIUPTAUJORUPTPSUPUPUKULVDVEURU
|
|
SUMUTUNVGVFUQUPUONN $.
|
|
|
|
$( There are exactly as many ordered pairs as there are sets. Corollary to
|
|
theorem XI.2.37 of [Rosser] p. 381. (Contributed by Scott Fenton,
|
|
31-Jul-2019.) $)
|
|
vvsqenvv $p |- ( _V X. _V ) ~~ _V $=
|
|
( cvv cxp cen xpvv vvex enrflx eqbrtri ) AABAACDAEFG $.
|
|
|
|
$( Cardinal one is strictly greater than cardinal zero. (Contributed by
|
|
Scott Fenton, 1-Aug-2019.) $)
|
|
0lt1c $p |- 0c <c 1c $=
|
|
( vx c0c cv csn cnc c1c cltc wbr clec wne c0 df0c2 wss 0ss 0ex snex wceq wn
|
|
df-ne wcel nclec ax-mp eqbrtri snnz mpbi ncid eleq2 mpbiri el0c sylib mpbir
|
|
vex mto brltc mpbir2an df1c3 breqtrri ) BACZDZEZFGBUTGHBUTIHBUTJZBKEZUTILKU
|
|
SMVBUTIHUSNKUSOURPZUAUBUCVABUTQZRVDUSKQZUSKJVERURAULZUDUSKSUEVDUSBTZVEVDVGU
|
|
SUTTUSVCUFBUTUSUGUHUSUIUJUMBUTSUKBUTUNUOURVFUPUQ $.
|
|
|
|
${
|
|
$d x y z w $.
|
|
$( The function mapping ` x ` to its cardinal successor exists.
|
|
(Contributed by Scott Fenton, 30-Jul-2019.) $)
|
|
csucex $p |- ( x e. _V |-> ( x +c 1c ) ) e. _V $=
|
|
( vy vz vw cvv cv c1c cplc caddcfn c1st wbr wex wceq cop wcel bitri exbii
|
|
wa vex syl6bb cmpt csn cxp cres ccnv ccom brcnv br1st anbi1i 19.41v excom
|
|
bitr4i opex breq1 brres braddcfn opelxp mpbiran elsn ceqsexv 1cex addceq2
|
|
anbi12ci eqeq1d opelco copab mptv eleq2i weq addceq1 eqeq1 eqcom opelopab
|
|
eqeq2d 3bitr4ri eqrelriv addcfnex vvex snex xpex resex 1stex coex eqeltri
|
|
cnvex ) AEAFZGHZUAZIEGUBZUCZUDZJUEZUFZEBCWHWMBFZDFZWLKZWOCFZWKKZRZDLZWNGH
|
|
ZWQMZWNWQNZWMOXCWHOZWTWFGMZWNWFHZWQMZRZALZXBWTWOWNWFNZMZWRRZALZDLZXIWSXMD
|
|
WSXKALZWRRXMWPXOWRWPWOWNJKXOWNWOJUGAWOWNBSZUHPUIXKWRAUJULQXNXLDLZALXIXLDA
|
|
UKXQXHAWRXHDXJWNWFXPASZUMXKWRXJWQWKKZXHWOXJWQWKUNXSXJWQIKZXJWJOZRXHXJWQIW
|
|
JUOXTXGYAXEWNWFWQXPXRUPYAWFWIOZXEYAWNEOYBXPWNWFEWIUQURAGUSPVCPTUTQPPXGXBA
|
|
GVAXEXFXAWQWFGWNVBVDUTPDWNWQWKWLVEXDXCWOWGMZADVFZOXBWHYDXCADWGVGVHYCWOXAM
|
|
ZXBADWNWQXPCSABVIWGXAWOWFWNGVJVNDCVIYEWQXAMXBWOWQXAVKWQXAVLTVMPVOVPWKWLIW
|
|
JVQEWIVRGVSVTWAJWBWEWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $. $d B x y $.
|
|
brcsuc.1 $e |- A e. _V $.
|
|
brcsuc.2 $e |- B e. _V $.
|
|
$( Binary relationship form of the successor mapping function.
|
|
(Contributed by Scott Fenton, 2-Aug-2019.) $)
|
|
brcsuc $p |- ( A ( x e. _V |-> ( x +c 1c ) ) B <-> B = ( A +c 1c ) ) $=
|
|
( vy cv c1c cplc wceq cvv cmpt addceq1 eqeq2d eqeq1 mptv brab ) FGZAGZHIZ
|
|
JRBHIZJCUAJAFBCAKTLDESBJTUARSBHMNRCUAOAFTPQ $.
|
|
$}
|
|
|
|
${
|
|
$d x y w $.
|
|
$( Lemma for ~ nnltp1c . Set up stratification. (Contributed by SF,
|
|
25-Mar-2015.) $)
|
|
nnltp1clem1 $p |- { x | x <c ( x +c 1c ) } e. _V $=
|
|
( vw vy cltc ccnv cvv cv c1c cplc cmpt ccom cfix wbr cab elfix wa wex vex
|
|
wcel bitri brco wceq brcsuc brcnv anbi12i exbii 1cex addcex breq2 ceqsexv
|
|
abbi2i ltcex cnvex csucex coex fixex eqeltrri ) DEZBFBGHIJZKZLZAGZVBHIZDM
|
|
ZANFVDAVAVBVASVBVBUTMZVDVBUTOVEVBCGZUSMZVFVBURMZPZCQZVDCVBVBURUSUAVJVFVCU
|
|
BZVBVFDMZPZCQVDVIVMCVGVKVHVLBVBVFARZCRUCVFVBDUDUEUFVLVDCVCVBHVNUGUHVFVCVB
|
|
DUIUJTTTUKUTURUSDULUMBUNUOUPUQ $.
|
|
$}
|
|
|
|
${
|
|
$d n x $. $d N x $.
|
|
$( Any natural is less than one plus itself. (Contributed by SF,
|
|
25-Mar-2015.) $)
|
|
nnltp1c $p |- ( N e. Nn -> N <c ( N +c 1c ) ) $=
|
|
( vx vn cv c1c cplc cltc wbr c0c wceq addceq1 breq12d clec cncs wcel 1cnc
|
|
id wne brltc cnnc nnltp1clem1 0cnc addlecncs mp2an 0cnsuc necomi mpbir2an
|
|
weq wa wi nnnc peano2nc syl w3a leaddc1 ex mp3an3 syl2anc peano2 suc11nnc
|
|
wb mpdan biimpd necon3d anim12d 3imtr4g finds ) BDZVHEFZGHIIEFZGHZCDZVLEF
|
|
ZGHZVMVMEFZGHZAAEFZGHBCABUAVHIJZVHIVIVJGVRQVHIEKLBCUHZVHVLVIVMGVSQVHVLEKL
|
|
VHVMJZVHVMVIVOGVTQVHVMEKLVHAJZVHAVIVQGWAQVHAEKLVKIVJMHZIVJRINOENOZWBUBPIE
|
|
UCUDVJIIUEUFIVJSUGVLTOZVLVMMHZVLVMRZUIVMVOMHZVMVORZUIVNVPWDWEWGWFWHWDVLNO
|
|
ZVMNOZWEWGUJZVLUKZWDWIWJWLVLULUMWIWJWCWKPWIWJWCUNWEWGEVLVMUOUPUQURWDVMVOV
|
|
LVMWDVMVOJZVLVMJZWDVMTOWMWNVAVLUSVLVMUTVBVCVDVEVLVMSVMVOSVFVG $.
|
|
$}
|
|
|
|
${
|
|
$d n p q x y z $.
|
|
$( Lemma for ~ addccan2nc . Stratification helper theorem. (Contributed
|
|
by Scott Fenton, 2-Aug-2019.) $)
|
|
addccan2nclem1 $p |- ( x ( AddC o. `' ( 1st |` ( _V X. { n } ) ) ) y
|
|
<-> y = ( x +c n ) ) $=
|
|
( vz vp vq cv caddcfn c1st cvv wbr cop wceq wa opeq2 eqeq2d 3bitri anbi1i
|
|
wex weq csn cxp cres ccnv ccom cplc brco wcel brcnv brres w3a ancom elxp2
|
|
wrex rexv rexsn exbii bitri exancom 19.41v bitr4i br1st eqeq1 opth syl6bb
|
|
pm5.32ri equcom anbi2i equcoms adantl pm5.32i df-3an 2exbii ceqsex2v opex
|
|
vex opeq1 breq1 ceqsexv braddcfn eqcom ) AGZBGZHIJCGZUAZUBZUCZUDZUEKZWBWD
|
|
LZWCHKZWBWDUFZWCMWCWLMWIWBDGZWHKZWMWCHKZNZDSWMWJMZWONZDSWKDWBWCHWHUGWPWRD
|
|
WNWQWOWNWMWBWGKWMWBIKZWMWFUHZNZWQWBWMWGUIWMWBIWFUJXAWSWMEGZWDLZMZNZESZEAT
|
|
ZFCTZWMXBFGZLZMZUKZFSESZWQXAXDESZWSNZXFXAWTWSNXOWSWTULWTXNWSWTXKFWEUNZEJU
|
|
NXPESXNEFWMJWEUMXPEUOXPXDEXKXDFWDCVPZXHXJXCWMXIWDXBOPUPUQQRURXFXDWSNESXOW
|
|
SXDEUSXDWSEUTURVAXFWMWBXILZMZXDNZFSZESXMXEYAEXEXSFSZXDNYAWSYBXDFWMWBAVPZV
|
|
BRXSXDFUTVAUQXTXLEFXTXGXHNZXKNZXLXTXGCFTZNZXDNYDXDNYEXDXSYGXDXSXCXRMYGWMX
|
|
CXRVCXBWDWBXIVDVEVFYGYDXDYFXHXGCFVGVHRYDXDXKYDXCXJWMXHXCXJMZXGYHCFWDXIXBO
|
|
VIVJPVKQXGXHXKVLVAVMURXKXSWQEFWBWDYCXQXGXJXRWMXBWBXIVQPXHXRWJWMXIWDWBOPVN
|
|
QQRUQWOWKDWJWBWDYCXQVOWMWJWCHVRVSQWBWDWCYCXQVTWLWCWAQ $.
|
|
$}
|
|
|
|
${
|
|
$d N n p x y $. $d P p x y $.
|
|
$( Lemma for ~ addccan2nc . Establish stratification for induction.
|
|
(Contributed by Scott Fenton, 2-Aug-2019.) $)
|
|
addccan2nclem2 $p |- ( ( N e. V /\ P e. W ) ->
|
|
{ x | ( ( x +c N ) = ( x +c P ) -> N = P ) } e. _V ) $=
|
|
( vn vp vy wcel wa cv cplc wceq cab cvv caddcfn c1st ccnv ccom wbr ccompl
|
|
wi cun wn wo unab complab uneq1i imor abbii 3eqtr4i addceq2 eqeq1d abbidv
|
|
eleq1d eqeq2d csn cxp cres cfix wex elfix brco addccan2nclem1 brcnv bitri
|
|
anbi12i exbii addcex eqeq1 ceqsexv 3bitri abbi2i addcfnex 1stex vvex snex
|
|
vex xpex resex cnvex coex fixex eqeltrri vtocl2g complexg syl abexv unexg
|
|
sylancl syl5eqelr ) CDIBEIJZAKZCLZWMBLZMZCBMZUBZANZWPANZUAZWQANZUCZOWPUDZ
|
|
ANZXBUCXDWQUEZANXCWSXDWQAUFXAXEXBWPAUGUHWRXFAWPWQUIUJUKWLXAOIZXBOIXCOIWLW
|
|
TOIZXGWMFKZLZWMGKZLZMZANZOIWNXLMZANZOIXHFGCBDEXICMZXNXPOXQXMXOAXQXJWNXLXI
|
|
CWMULUMUNUOXKBMZXPWTOXRXOWPAXRXLWOWNXKBWMULUPUNUOPQOXKUQZURZUSZRZSZRZPQOX
|
|
IUQZURZUSZRZSZSZUTZXNOXMAYKWMYKIWMWMYJTZHKZXJMZYMXLMZJZHVAZXMWMYJVBYLWMYM
|
|
YITZYMWMYDTZJZHVAYQHWMWMYDYIVCYTYPHYRYNYSYOAHFVDYSWMYMYCTYOYMWMYCVEAHGVDV
|
|
FVGVHVFYOXMHXJWMXIAVRFVRVIYMXJXLVJVKVLVMYJYDYIYCPYBVNYAQXTVOOXSVPXKVQVSVT
|
|
WAWBWAPYHVNYGQYFVOOYEVPXIVQVSVTWAWBWBWCWDWEWTOWFWGWQAWHXAXBOOWIWJWK $.
|
|
$}
|
|
|
|
${
|
|
$d M x m $. $d N x m $. $d P x m $.
|
|
$( Cancellation law for addition over the cardinal numbers. Biconditional
|
|
form of theorem XI.3.2 of [Rosser] p. 391. (Contributed by Scott
|
|
Fenton, 2-Aug-2019.) $)
|
|
addccan2nc $p |- ( ( M e. Nn /\ N e. NC /\ P e. NC ) ->
|
|
( ( M +c N ) = ( M +c P ) <-> N = P ) ) $=
|
|
( vx vm cnnc wcel cncs cplc wceq wi cv c0c c1c wa addceq1 eqeq12d addcid2
|
|
imbi1d eqeq12i w3a cvv addccan2nclem2 weq biimpi a1i addc32 nnnc ncaddccl
|
|
sylan adantrr adantr adantrl peano4nc biimpd syl2anc simpr syld syl5bi ex
|
|
findsd 3impb addceq2 impbid1 ) BFGZCHGZAHGZUABCIZBAIZJZCAJZVEVFVGVJVKKZDL
|
|
ZCIZVMAIZJZVKKMCIZMAIZJZVKKZELZCIZWAAIZJZVKKZWANIZCIZWFAIZJZVKKZVLVFVGOZD
|
|
EBUBDACHHUCVMMJZVPVSVKWLVNVQVOVRVMMCPVMMAPQSDEUDZVPWDVKWMVNWBVOWCVMWACPVM
|
|
WAAPQSVMWFJZVPWIVKWNVNWGVOWHVMWFCPVMWFAPQSVMBJZVPVJVKWOVNVHVOVIVMBCPVMBAP
|
|
QSVTWKVSVKVQCVRACRARTUEUFWAFGZWKOZWEWJWIWBNIZWCNIZJZWQWEOZVKWGWRWHWSWANCU
|
|
GWANAUGTXAWTWDVKXAWBHGZWCHGZWTWDKWQXBWEWPVFXBVGWPWAHGZVFXBWAUHZWACUIUJUKU
|
|
LWQXCWEWPVGXCVFWPXDVGXCXEWAAUIUJUMULXBXCOWTWDWBWCUNUOUPWQWEUQURUSUTVAVBCA
|
|
BVCVD $.
|
|
$}
|
|
|
|
${
|
|
$d M q $. $d N q $. $d P q $.
|
|
$( Cardinal addition preserves cardinal less than. Biconditional form of
|
|
corollary 4 of theorem XI.3.2 of [Rosser] p. 391. (Contributed by Scott
|
|
Fenton, 2-Aug-2019.) $)
|
|
lecadd2 $p |- ( ( M e. Nn /\ N e. NC /\ P e. NC ) ->
|
|
( ( M +c N ) <_c ( M +c P ) <-> N <_c P ) ) $=
|
|
( vq cnnc wcel cncs w3a cplc clec wbr cv wceq wrex wb nnnc ncaddccl sylan
|
|
3ad2antl2 sylbid 3adant3 3adant2 dflec2 syl2anc addcass eqeq2i addccan2nc
|
|
simpl1 simpl3 syl3anc addlecncs breq2 syl5ibrcom syl5bi rexlimdva leaddc2
|
|
wa wi ex syl3an1 impbid ) BEFZCGFZAGFZHZBCIZBAIZJKZCAJKZVEVHVGVFDLZIZMZDG
|
|
NZVIVEVFGFZVGGFZVHVMOVBVCVNVDVBBGFZVCVNBPZBCQRUAVBVDVOVCVBVPVDVOVQBAQRUBV
|
|
FVGDUCUDVEVLVIDGVLVGBCVJIZIZMZVEVJGFZUQZVIVKVSVGBCVJUEUFWBVTAVRMZVIWBVBVD
|
|
VRGFZVTWCOVBVCVDWAUHVBVCVDWAUIVCVBWAWDVDCVJQSVRBAUGUJWBVIWCCVRJKZVCVBWAWE
|
|
VDCVJUKSAVRCJULUMTUNUOTVBVPVCVDVIVHURVQVPVCVDHVIVHABCUPUSUTVA $.
|
|
$}
|
|
|
|
${
|
|
$d M p q $. $d N p q $.
|
|
$( Relationship between successor and cardinal less than or equal.
|
|
(Contributed by Scott Fenton, 3-Aug-2019.) $)
|
|
ncslesuc $p |- ( ( M e. NC /\ N e. NC ) -> ( M <_c ( N +c 1c ) <->
|
|
( M <_c N \/ M = ( N +c 1c ) ) ) ) $=
|
|
( vp vq cncs wcel wa c1c cplc clec wbr wceq wo cv wrex c0c addceq2 adantl
|
|
wb wi peano2nc dflec2 sylan2 nc0suc addcid1 syl6eq eqeq2d olc a1i addcass
|
|
eqcoms syl6bi syl6eqr biimpa ncaddccl adantlr peano4nc syl2anc syl5ibrcom
|
|
simplr addlecncs breq2 orc syl6 sylbid syl5 exp3a rexlimdva jaod rexlimdv
|
|
1cnc mpan2 lectr mpd3an3 mpan2d nclecid syl breq1 impbid ) AEFZBEFZGZABHI
|
|
ZJKZABJKZAWCLZMZWBWDWCACNZIZLZCEOZWGWAVTWCEFZWDWKSBUAZAWCCUBUCWBWJWGCEWHE
|
|
FWHPLZWHDNZHIZLZDEOZMWBWJWGTZDWHUDWBWNWSWRWNWSTWBWNWJWCALWGWNWIAWCWNWIAPI
|
|
AWHPAQAUEUFUGWGAWCWFWEUHUKULUIWBWQWSDEWBWOEFZGZWQWJWGWQWJGWCAWOIZHIZLZXAW
|
|
GWQWJXDWQWIXCWCWQWIAWPIXCWHWPAQAWOHUJUMUGUNXAXDBXBLZWGXAWAXBEFZXDXESVTWAW
|
|
TUTVTWTXFWAAWOUOUPBXBUQURXAXEWEWGXAWEXEAXBJKZVTWTXGWAAWOVAUPBXBAJVBUSWEWF
|
|
VCVDVEVFVGVHVIVFVJVEWBWEWDWFWBWEBWCJKZWDWAXHVTWAHEFXHVKBHVAVLRVTWAWLWEXHG
|
|
WDTWAWLVTWMRABWCVMVNVOWBWDWFWCWCJKZWAXIVTWAWLXIWMWCVPVQRAWCWCJVRUSVIVS $.
|
|
$}
|
|
|
|
${
|
|
$d m p $. $d m q $. $d m x $. $d m y $. $d p q $. $d p t $. $d p x $.
|
|
$d p y $. $d q t $. $d q x $. $d t x $. $d x y $.
|
|
$( Lemma for ~ nmembers1 . Set up stratification. (Contributed by SF,
|
|
25-Mar-2015.) $)
|
|
nmembers1lem1 $p |- { x | { m e. Nn | ( 1c <_c m /\ m <_c x ) } e. T_c
|
|
T_c x } e. _V $=
|
|
( vp vq vt clec c1c cnnc cvv ctcfn cin cv wbr wa wcel wex cop exbii bitri
|
|
3bitri vy csset cins3 csn cima cres csi cins2 csymdif ccompl ccnv cxp crn
|
|
ccom cuni1 crab ctc cab vex snex wceq wel w3a elrn2 elima1c elin otelins2
|
|
eluni1 opsnelsi opelres ancom df-br brres bitr4i anbi1i bitr3i anbi2i weq
|
|
elimasn breq2 breq1 anbi12d 3bitr4i releqel opelxp mpbiran opelco brsnsi2
|
|
elrab 19.42v excom an12 anbi1d ceqsexv brcnv brtcfn anbi12i tcex otelins3
|
|
tceq eqeq2d opelssetsn df-3an imasn iniseg inab eqtri ineq1i dfrab2 lecex
|
|
eqtr4i imaex cnvex inex nncex eqeltrri eleq1 eleq2 ceqsex2v abbi2i ssetex
|
|
ineq12i ins3ex resex siex ins2ex symdifex 1cex vvex tcfnex coex xpex rnex
|
|
complex uni1ex ) UBUCZFFGUDZUEZUFZHUFZUGZUHZUIZGUEZUJZUGZUHZIJUKZUGZUUHUN
|
|
ZULZKZYPKZGUEZUMZUOZUOZGBLZFMZUURALZFMZNZBHUPZUUTUQZUQZOZAURIUVFAUUQUUTUU
|
|
QOUUTUDZUUPOUVGUDZUUOOZUVFUUTUUPAUSZVHUVGUUOUUTUTZVHUVICLZUVCVAZDLZUVEVAZ
|
|
CDVBZVCZCPZDPZUVQDPCPUVFUVIUVNUVHQZUUNOZDPUVSDUVHUUNVDUWAUVRDUWAUVLUDZUVT
|
|
QZUUMOZCPUVRCUVTUUMVEUWDUVQCUWCUULOZUWCYPOZNUVMUVONZUVPNUWDUVQUWEUWGUWFUV
|
|
PUWEUWCUUGOZUWCUUKOZNUWGUWCUUGUUKVFUWHUVMUWIUVOUWHUWBUVHQUUFOUVLUVGQUUEOU
|
|
VMUWBUVNUVHUUFDUSZVGUVLUVGUUECUSZUVKVICUAUVCUUAUVGUVKUALZUUTQZYTOZUWLHOZG
|
|
UWLFMZUWLUUTFMZNZNZUWLUDUVGQUUAOUWLUVCOUWNUWMYSOZUWONUWOUWTNUWSUWLUUTYSHV
|
|
JUWTUWOVKUWTUWRUWOUWTUWLUUTYSMZUWRUWLUUTYSVLUXAUWQUWLYROZNUXBUWQNUWRUWLUU
|
|
TFYRVMUWQUXBVKUXBUWPUWQUXBGUWLQFOUWPFGUWLVSGUWLFVLVNVOTVPVQTUWLUUTYTUAUSU
|
|
VJVIUVBUWRBUWLHBUAVRUUSUWPUVAUWQUURUWLGFVTUURUWLUUTFWAWBWIWCWDTUWIUVTUUJO
|
|
ZUVNELZUUHMZUXDUVHUUIMZNZEPZUVOUWIUWBIOUXCUVLUTZUWBUVTIUUJWEWFEUVNUVHUUIU
|
|
UHWGUXHUXEUXDUWBVAZUVLUVGUUHMZNZNZCPZEPUXMEPZCPZUVOUXGUXNEUXGUXEUXLCPZNUX
|
|
NUXFUXQUXECUVGUXDUUHUVKWHVQUXEUXLCWJVNRUXMECWKUXPUVNUWBUUHMZUXKNZCPUVLUVD
|
|
VAZUVNUVLUQZVAZNZCPUVOUXOUXSCUXOUXJUXEUXKNZNZEPUXSUXMUYEEUXEUXJUXKWLRUYDU
|
|
XSEUWBUXIUXJUXEUXRUXKUXDUWBUVNUUHVTWMWNSRUXSUYCCUXSUYBUXTNUYCUXRUYBUXKUXT
|
|
UXRUWBUVNJMUYBUVNUWBJWOUVLUVNUWKWPSUXKUVGUVLJMUXTUVLUVGJWOUUTUVLUVJWPSWQU
|
|
YBUXTVKSRUYBUVOCUVDUUTWRUXTUYAUVEUVNUVLUVDWTXAWNTTTWQSUWFUWBUVNQUBOUVPUWB
|
|
UVNUVHUBUVGUTWSUVLUVNUWKUWJXBSWQUWCUULYPVFUVMUVOUVPXCWCRSRSUVQDCWKUVPUVCU
|
|
VNOUVFCDUVCUVEYRFUKZUVGUEZKZHKZUVCIUYIUVBBURZHKUVCUYHUYJHUYHUUSBURZUVABUR
|
|
ZKUYJYRUYKUYGUYLBGFXDBFUUTXEYBUUSUVABXFXGXHUVBBHXIXKUYHHYRUYGFYQXJGUTXLZU
|
|
YFUVGFXJXMUVKXLXNXOXNXPUVDWRUVLUVCUVNXQUVNUVEUVCXRXSTTXTUUPUUOUUNUUMGUULY
|
|
PUUGUUKUUFUUEUUDUUCGYPUUBUBYAYCZUUAYTYSHFYRXJUYMYDXOYDYEYFYGYHXLYNYEYFIUU
|
|
JYIUUIUUHUUHJYJXMZYEUYOYKYLXNUYNXNYHXLYMYOYOXP $.
|
|
$}
|
|
|
|
$( Lemma for ~ nmembers1 . The set of all elements between one and zero is
|
|
empty. (Contributed by Scott Fenton, 1-Aug-2019.) $)
|
|
nmembers1lem2 $p |- { m e. Nn | ( 1c <_c m /\ m <_c 0c ) } e. 0c $=
|
|
( c1c cv clec wbr wa cnnc crab wcel wn wral wi cltc 0lt1c cncs wb 0cnc 1cnc
|
|
c0c ltlenlec mp2an mpbi simpri nnnc lectr mp3an13 syl exp3a imp imnan sylib
|
|
mtoi ex rgen c0 wceq el0c rabeq0 bitri mpbir ) BACZDEZVASDEZFZAGHZSIZVDJZAG
|
|
KZVGAGVAGIZVBVCJZLVGVIVBVJVIVBFVCBSDEZSBDEZVKJZSBMEZVLVMFZNSOIZBOIZVNVOPQRS
|
|
BTUAUBUCVIVBVCVKLVIVBVCVKVIVAOIZVDVKLZVAUDVQVRVPVSRQBVASUEUFUGUHUIULUMVBVCU
|
|
JUKUNVFVEUOUPVHVEUQVDAGURUSUT $.
|
|
|
|
${
|
|
$d A m x y $. $d B m x y $.
|
|
$( Lemma for ~ nmembers1 . If the interval from one to a natural is in a
|
|
given natural, extending it by one puts it in the next natural.
|
|
(Contributed by Scott Fenton, 3-Aug-2019.) $)
|
|
nmembers1lem3 $p |- ( ( A e. Nn /\ B e. Nn ) ->
|
|
( { m e. Nn | ( 1c <_c m /\ m <_c A ) } e. B ->
|
|
{ m e. Nn | ( 1c <_c m /\ m <_c ( A +c 1c ) ) } e. ( B +c 1c ) ) ) $=
|
|
( vx vy c1c cv clec wbr wa cnnc wcel cplc cun wceq wn cncs adantr anbi12d
|
|
wo crab csn ccompl wrex wi cltc nnltp1c wb nnnc peano2 syl ltlenlec mpbid
|
|
syl2anc simprd intnand a1d breq2 breq1 elrab notbii imnan bitr4i elcomplg
|
|
sylibr mpbird ncslesuc syl2an expcom adantrd pm5.32d anass orbi1i 3bitr3g
|
|
andi bitri 1cnc addlecncs sylancr syl6breqr jca eleq1 syl5ibrcom pm4.71rd
|
|
addccom bicomd orbi2d bitrd elun elsn orbi12i 3bitr4g eqrdv uneq2d eqeq2d
|
|
weq sneq rspcev compleq uneq1 rexeqbidv sylan2 elsuc ) FCGZHIZXDAHIZJZCKU
|
|
AZBLZAKLZBKLZJZXEXDAFMZHIZJZCKUAZBFMLZXIXLJXPDGZEGZUBZNZOZEXRUCZUDZDBUDZX
|
|
QXLXIXPXHXTNZOZEXHUCZUDZYEXLXMYHLZXPXHXMUBZNZOZYIXLYJXMXHLZPZXLXMKLZFXMHI
|
|
ZXMAHIZJZPZUEZYOXLYTYPXLYRYQXJYRPZXKXJAXMHIZUUBXJAXMUFIZUUCUUBJZAUGXJAQLZ
|
|
XMQLZUUDUUEUHAUIZXJYPUUGAUJZXMUIUKAXMULUNUMUORUPUQYOYPYSJZPUUAYNUUJXGYSCX
|
|
MKXDXMOXEYQXFYRXDXMFHURXDXMAHUSSUTVAYPYSVBVCVEXLYPYJYOUHXJYPXKUUIRXMXHKVD
|
|
UKVFXLDXPYLXLXRKLZFXRHIZXRXMHIZJZJZUUKUULXRAHIZJZJZXRXMOZTZXRXPLXRYLLZXLU
|
|
UOUURUUKUULJZUUSJZTZUUTXLUVBUUMJUVBUUPUUSTZJZUUOUVDXLUVBUUMUVEXJUVBUUMUVE
|
|
UHZUEXKXJUUKUVGUULUUKXJUVGUUKXRQLUUFUVGXJXRUIUUHXRAVGVHVIVJRVKUUKUULUUMVL
|
|
UVFUVBUUPJZUVCTUVDUVBUUPUUSVOUVHUURUVCUUKUULUUPVLVMVPVNXLUVCUUSUURXLUUSUV
|
|
CXLUUSUVBXJUUSUVBUEXKXJUVBUUSYPYQJXJYPYQUUIXJFFAMZXMHXJFQLUUFFUVIHIVQUUHF
|
|
AVRVSAFWEVTWAUUSUUKYPUULYQXRXMKWBXRXMFHURSWCRWDWFWGWHXOUUNCXRKCDWPZXEUULX
|
|
NUUMXDXRFHURZXDXRXMHUSSUTUVAXRXHLZXRYKLZTUUTXRXHYKWIUVLUURUVMUUSXGUUQCXRK
|
|
UVJXEUULXFUUPUVKXDXRAHUSSUTDXMWJWKVPWLWMYGYMEXMYHXSXMOZYFYLXPUVNXTYKXHXSX
|
|
MWQWNWOWRUNYDYIDXHBXRXHOZYBYGEYCYHXRXHWSUVOYAYFXPXRXHXTWTWOXAWRXBEXPBDXCV
|
|
EVI $.
|
|
$}
|
|
|
|
${
|
|
$d N m n a $.
|
|
$( Count the number of elements in a natural interval. From
|
|
~ nmembers1lem2 and ~ nmembers1lem3 , we would expect to arrive at
|
|
` { m e. Nn | ( 1c <_c m /\ m <_c N ) } e. N ` , but this proposition is
|
|
not stratifiable. Instead, we arrive at the weaker conclusion below.
|
|
We can arrive at the earlier proposition once we add the Axiom of
|
|
Counting, which we will do later. (Contributed by Scott Fenton,
|
|
3-Aug-2019.) $)
|
|
nmembers1 $p |- ( N e. Nn ->
|
|
{ m e. Nn | ( 1c <_c m /\ m <_c N ) } e. T_c T_c N ) $=
|
|
( vn c1c clec wbr wa cnnc crab ctc wcel c0c cplc wceq anbi2d rabbidv tceq
|
|
breq2 syl eleq12d va nmembers1lem1 weq nmembers1lem2 ax-mp eqtri eleqtrri
|
|
cv tc0c wi nntccl nmembers1lem3 cncs nnnc 1cnc tcdi mpan2 addceq2i syl6eq
|
|
wb tc1c tccl eqtrd eleq2d adantr sylibrd mpdan finds ) DAUHZEFZVICUHZEFZG
|
|
ZAHIZVKJZJZKVJVILEFZGZAHIZLJZJZKVJVIUAUHZEFZGZAHIZWBJZJZKZVJVIWBDMZEFZGZA
|
|
HIZWIJZJZKZVJVIBEFZGZAHIZBJZJZKCUABCAUBVKLNZVNVSVPWAXAVMVRAHXAVLVQVJVKLVI
|
|
EROPXAVOVTNVPWANVKLQVOVTQSTCUAUCZVNWEVPWGXBVMWDAHXBVLWCVJVKWBVIEROPXBVOWF
|
|
NVPWGNVKWBQVOWFQSTVKWINZVNWLVPWNXCVMWKAHXCVLWJVJVKWIVIEROPXCVOWMNVPWNNVKW
|
|
IQVOWMQSTVKBNZVNWRVPWTXDVMWQAHXDVLWPVJVKBVIEROPXDVOWSNVPWTNVKBQVOWSQSTVSL
|
|
WAAUDWAVTLVTLNWAVTNUIVTLQUEUIUFUGWBHKZWGHKZWHWOUJXEWFHKXFWBUKWFUKSXEXFGWH
|
|
WLWGDMZKZWOWBWGAULXEWOXHUTXFXEWNXGWLXEWBUMKZWNXGNWBUNXIWNWFDMZJZXGXIWMXJN
|
|
WNXKNXIWMWFDJZMZXJXIDUMKZWMXMNUOWBDUPUQXLDWFVAURUSWMXJQSXIXKWGXLMZXGXIWFU
|
|
MKZXKXONZWBVBXPXNXQUOWFDUPUQSXLDWGVAURUSVCSVDVEVFVGVH $.
|
|
$}
|
|
|
|
$( Cardinal less than is irreflexive. (Contributed by Scott Fenton,
|
|
12-Dec-2021.) $)
|
|
ltcirr $p |- -. A <c A $=
|
|
( clec wbr wn wne wo cltc neirr olci wa brltc anor bitri con2bii mpbi ) AAB
|
|
CZDZAAEZDZFZAAGCZDSQAHIUATUAPRJTDAAKPRLMNO $.
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Specker's disproof of the axiom of choice
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
$c Sp[ac] $. $( Special set generator for axiom of choice. $)
|
|
|
|
$( Extend the definition of a class to include the special set generator for
|
|
the axiom of choice. $)
|
|
cspac $a class Sp[ac] $.
|
|
|
|
${
|
|
$d m x y $.
|
|
$( Define the special class generator for the disproof of the axiom of
|
|
choice. Definition 6.1 of [Specker] p. 973. (Contributed by SF,
|
|
3-Mar-2015.) $)
|
|
df-spac $a |- Sp[ac] = ( m e. NC |->
|
|
Clos1 ( { m } ,
|
|
{ <. x , y >. | ( x e. NC /\ y e. NC /\ y = ( 2c ^c x ) ) } ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d b m $. $d m n $. $d m t $. $d n t $. $d b t $.
|
|
$( Lemma for ~ nncdiv3 . Set up a helper for stratification. (Contributed
|
|
by SF, 3-Mar-2015.) $)
|
|
nncdiv3lem1 $p |- ( <. n , b >. e. ran ( Ins3 `' ( ( ran ( `' 1st (x) ( 1st
|
|
i^i 2nd ) ) (x) 2nd ) " AddC ) i^i ( ( ( 1st o. 1st ) (x) ( ( 2nd o.
|
|
1st ) (x) 2nd ) ) " AddC ) ) <-> b = ( ( n +c n ) +c n ) ) $=
|
|
( vm vt cv cop c1st c2nd ctxp caddcfn wcel wex wceq wa vex wbr wrex df-br
|
|
bitri 3bitri ccnv cin crn cima cins3 ccom cplc elrn2 elin otelins3 cproj1
|
|
opelcnv proj1ex eqvinc cproj2 opeq breq1i proj2ex opbr1st oteltxp 3bitr2i
|
|
trtxp eqcom anbi12i op1st2nd exbii 3bitr4ri anbi1i rexbii risset braddcfn
|
|
opex elima 3bitr4i anbi2i anass brco ceqsexv bitr2i addcex addceq1 eqeq2d
|
|
breq1 ) AEZBEZFZGUAZGHUBZIZUCZHIZJUDZUAZUEZGGUFZHGUFZHIZIZJUDZUBZUCKCEZWF
|
|
FZWTKZCLXAWDWDUGZMZWEXAWDUGZMZNZCLWEXDWDUGZMZCWFWTUHXCXHCXCXBWNKZXBWSKZNX
|
|
HXBWNWSUIXKXEXLXGXKXAWDFZWMKWDXAFZWLKZXEXAWDWEWMBOZUJXAWDWLULXOWDWDFZXAJP
|
|
ZXDXAMXEDEZXNWKPZDJQXSXQXAFZMZDJQZXOXRXTYBDJXTXSWDWJPZXSXAHPZNXSXQGPZYENY
|
|
BXSWDXAWJHVBYDYFYEYDXSWDFZWJKXAYGFWIKZCLZYFXSWDWJRCYGWIUHXSUKZXQMZXAYJMZX
|
|
AXQMZNZCLYFYICYJXQXSDOZUMZUNYFYJXSUOZFZXQGPYKXSYRXQGXSUPZUQYJYQXQYPXSYOUR
|
|
ZUSSYHYNCYHXAXSFWGKZXMWHKZNYNXAXSWDWGWHUTUUAYLUUBYMUUAXSXAFGKXSXAGPZYLXAX
|
|
SGULXSXAGRUUCYRXAGPYJXAMYLXSYRXAGYSUQYJYQXAYPYTUSYJXAVCTZVAUUBXMGKZXMHKZN
|
|
XAWDGPZXAWDHPZNYMXMGHUIUUGUUEUUHUUFXAWDGRXAWDHRVDWDWDXAAOZUUIVEVAVDSVFVGT
|
|
VHXQXAXSWDWDUUIUUIVLCOZVETVIDXNWKJVMXRYAJKYCXQXAJRDYAJVJSVNWDWDXAUUIUUIVK
|
|
XDXAVCTTXLXSXMWEFZMZDJQZXMWEJPZXGXLXSXBWRPZDJQUUMDXBWRJVMUUOUULDJUUOXSXAW
|
|
OPZXSWFWQPZNZXSXMGPZXSWEHPZNZUULXSXAWFWOWQVBUURUUPXSWDWPPZUUTNZNUUPUVBNZU
|
|
UTNUVAUUQUVCUUPXSWDWEWPHVBVOUUPUVBUUTVPUVDUUSUUTYJXAGPZYJWDHPZNYJXMMZUVDU
|
|
USXAWDYJUUJUUIVEUUPUVEUVBUVFUUPXSWDGPZWDXAGPZNZALWDYJMZUVINZALUVEAXSXAGGV
|
|
QUVJUVLAUVHUVKUVIUVHYRWDGPYJWDMUVKXSYRWDGYSUQYJYQWDYPYTUSYJWDVCTVHVFUVIUV
|
|
EAYJYPWDYJXAGWCVRTUVBUUCUUHNZCLYLUUHNZCLUVFCXSWDHGVQUVMUVNCUUCYLUUHUUDVHV
|
|
FUUHUVFCYJYPXAYJWDHWCVRTVDUUSYRXMGPUVGXSYRXMGYSUQYJYQXMYPYTUSSVNVHVAXMWEX
|
|
SXAWDUUJUUIVLXPVETVISUUNUUKJKUUMXMWEJRDUUKJVJVSUUNXFWEMXGXAWDWEUUJUUIVKXF
|
|
WEVCSTVDSVFXGXJCXDWDWDUUIUUIVTXEXFXIWEXAXDWDWAWBVRT $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a n $. $d b n $.
|
|
$( Lemma for ~ nncdiv3 . Set up stratification for induction.
|
|
(Contributed by SF, 2-Mar-2015.) $)
|
|
nncdiv3lem2 $p |- { a | E. n e. Nn ( a = ( ( n +c n ) +c n ) \/
|
|
a = ( ( ( n +c n ) +c n ) +c 1c ) \/ a = ( ( ( n +c n ) +c n ) +c 2c ) ) }
|
|
e. _V $=
|
|
( vb c1st c2nd ctxp caddcfn c1c cvv c2c wceq wcel wbr cop wa bitri 3bitri
|
|
wex 1stex txpex ccnv cin crn cima cins3 ccom csn cxp cun cnnc cv cplc w3o
|
|
wrex cab elima df-br wo elun nncdiv3lem1 elrn2 oteltxp opelcnv bicomi vex
|
|
elin opelxp mpbiran2 eliniseg anbi12i 1cex op1st2nd opex ceqsexv braddcfn
|
|
exbii breq1 eqcom addcex addceq1 eqeq2d orbi12i c0 cpr df-2c ncex eqeltri
|
|
cnc df-3or rexbii abbi2i cnvex 2ndex inex rnex addcfnex imaex ins3ex coex
|
|
3bitr4i snex vvex xpex unex nncex eqeltrri ) DUAZDEUBZFZUCZEFZGUDZUAZUEZD
|
|
DUFZEDUFZEFZFZGUDZUBZUCZYAUAZDEUAZHUGZUDZIUHZUBZGFZUCZFZUCZUIZYBDYCJUGZUD
|
|
ZIUHZUBZGFZUCZFZUCZUIZUJUDZBUKZAUKZUUDULZUUDULZKZUUCUUFHULZKZUUCUUFJULZKZ
|
|
UMZAUJUNZBUOIUUMBUUBUUCUUBLUUDUUCUUAMZAUJUNUUMAUUCUUAUJUPUUNUULAUJUUNUUDU
|
|
UCNZUUALZUULUUDUUCUUAUQUUOYLLZUUOYTLZURUUGUUIURZUUKURUUPUULUUQUUSUURUUKUU
|
|
QUUOYALZUUOYKLZURUUSUUOYAYKUSUUTUUGUVAUUIABUTUVACUKZUUONZYJLZCRUVBUUFKZUU
|
|
CUVBHULZKZOZCRUUICUUOYJVAUVDUVHCUVDUVBUUDNYBLZUVBUUCNZYILZOUVHUVBUUDUUCYB
|
|
YIVBUVIUVEUVKUVGUVIUUDUVBNZYALUVEUVBUUDYAVCACUTPZUVKUUDUVBHNZKZUUDUUCGMZO
|
|
ZARZUVNUUCGMZUVGUVKUUDUVJNZYHLZARUVRAUVJYHVAUWAUVQAUWAUVLYGLZUUOGLZOUVQUU
|
|
DUVBUUCYGGVBUWBUVOUWCUVPUWBUVLDLZUVLYFLZOUUDUVBDMZUUDHEMZOUVOUVLDYFVFUWDU
|
|
WFUWEUWGUWFUWDUUDUVBDUQVDZUWEUUDYELZUWGUWEUWIUVBILZCVEZUUDUVBYEIVGVHEHUUD
|
|
VIPVJUVBHUUDUWKVKVLQUVPUWCUUDUUCGUQVDZVJPVPPUVPUVSAUVNUVBHUWKVKVMUUDUVNUU
|
|
CGVQVNUVSUVFUUCKUVGUVBHUUCUWKVKVOUVFUUCVRPQVJPVPUVGUUICUUFUUEUUDUUDUUDAVE
|
|
ZUWMVSUWMVSZUVEUVFUUHUUCUVBUUFHVTWAVNQWBPUURUVCYSLZCRUVEUUCUVBJULZKZOZCRU
|
|
UKCUUOYSVAUWOUWRCUWOUVIUVJYRLZOUWRUVBUUDUUCYBYRVBUVIUVEUWSUWQUVMUWSUVBJNZ
|
|
UUCGMZUWPUUCKUWQUWSUVTYQLZARUUDUWTKZUVPOZARUXAAUVJYQVAUXBUXDAUXBUVLYPLZUW
|
|
COUXDUUDUVBUUCYPGVBUXEUXCUWCUVPUXEUWDUVLYOLZOUWFUUDJEMZOUXCUVLDYOVFUWDUWF
|
|
UXFUXGUWHUXFUUDYNLZUXGUXFUXHUWJUWKUUDUVBYNIVGVHEJUUDVIPVJUVBJUUDUWKJWCIWD
|
|
ZWHIWEUXIWFWGZVLQUWLVJPVPUVPUXAAUWTUVBJUWKUXJVMUUDUWTUUCGVQVNQUVBJUUCUWKU
|
|
XJVOUWPUUCVRQVJPVPUWQUUKCUUFUWNUVEUWPUUJUUCUVBUUFJVTWAVNQWBUUOYLYTUSUUGUU
|
|
IUUKWIWTPWJPWKUUAUJYLYTYAYKXTXNXSXMXLXKGXJEXIXGXHDSWLDESWMWNTWOWMTWPWQWLW
|
|
RXRGXOXQDDSSWSXPEEDWMSWSWMTTWPWQWNWOZYJYBYIYAUXKWLZYHYGGDYFSYEIYCYDEWMWLZ
|
|
HXAWQXBXCWNWPTWOTWOXDYSYBYRUXLYQYPGDYOSYNIYCYMUXMJXAWQXBXCWNWPTWOTWOXDXEW
|
|
QXF $.
|
|
$}
|
|
|
|
${
|
|
$d A n m a $.
|
|
$( Divisibility by three rule for finite cardinals. Part of Theorem 3.4 of
|
|
[Specker] p. 973. (Contributed by SF, 2-Mar-2015.) $)
|
|
nncdiv3 $p |- ( A e. Nn ->
|
|
E. n e. Nn ( A = ( ( n +c n ) +c n ) \/
|
|
A = ( ( ( n +c n ) +c n ) +c 1c ) \/
|
|
A = ( ( ( n +c n ) +c n ) +c 2c ) ) ) $=
|
|
( va vm cplc wceq c1c c2c w3o cnnc wrex c0c 3orbi123d rexbidv wcel eqeq2d
|
|
eqeq1 wo df-3or r19.43 cv nncdiv3lem2 peano1 addcid1 addcid2 eqtr2i 3mix1
|
|
weq ax-mp addceq12 anidms id addceq12d addceq1d rspcev addceq1 reximi a1i
|
|
mp2an wi 1p1e2c addceq2i eqtri syl6eq peano2 addc32 addc4 eqtr3i addceq1i
|
|
addcass sylancl eqeq1d syl5ibrcom rexlimiv cbvrexv sylib 3orim123d rexbii
|
|
orbi1i 3bitr4i bitri 3orrot 3bitr2i 3imtr4g finds ) CUAZBUAZWGEZWGEZFZWFW
|
|
IGEZFZWFWIHEZFZIZBJKLWIFZLWKFZLWMFZIZBJKZDUAZWIFZXAWKFZXAWMFZIZBJKZXAGEZW
|
|
IFZXGWKFZXGWMFZIZBJKZAWIFZAWKFZAWMFZIZBJKCDABCUBWFLFZWOWSBJXQWJWPWLWQWNWR
|
|
WFLWIQWFLWKQWFLWMQMNCDUHZWOXEBJXRWJXBWLXCWNXDWFXAWIQWFXAWKQWFXAWMQMNWFXGF
|
|
ZWOXKBJXSWJXHWLXIWNXJWFXGWIQWFXGWKQWFXGWMQMNWFAFZWOXPBJXTWJXMWLXNWNXOWFAW
|
|
IQWFAWKQWFAWMQMNLJOLLLEZLEZFZLYBGEZFZLYBHEZFZIZWTUCYCYHYBYALYAUDLUEUFYCYE
|
|
YGUGUIWSYHBLJWGLFZWPYCWQYEWRYGYIWIYBLYIWHYAWGLYIWHYAFWGWGLLUJUKYIULUMZPYI
|
|
WKYDLYIWIYBGYJUNPYIWMYFLYIWIYBHYJUNPMUOUSXAJOZXBBJKZXCBJKZXDBJKZIZXIBJKZX
|
|
JBJKZXHBJKZIZXFXLYKYLYPYMYQYNYRYLYPUTYKXBXIBJXAWIGUPUQURYMYQUTYKXCXJBJXCX
|
|
GWKGEZWMXAWKGUPYTWIGGEZEWMWIGGVJUUAHWIVAVBVCVDUQURYNYRUTYKYNXGWFWFEZWFEZF
|
|
ZCJKZYRXDUUEBJWGJOZUUEXDWMGEZUUCFZCJKZUUFWGGEZJOUUGUUJUUJEZUUJEZFZUUIWGVE
|
|
UUGUUKWGEZGEUULWMUUNGWMWHHEZWGEUUNWHWGHVFUUOUUKWGWHUUAEUUOUUKUUAHWHVAVBWG
|
|
WGGGVGVHVIVCVIUUKWGGVJVCUUHUUMCUUJJWFUUJFZUUCUULUUGUUPUUBUUKWFUUJUUPUUBUU
|
|
KFWFWFUUJUUJUJUKUUPULUMPUOVKXDUUDUUHCJXDXGUUGUUCXAWMGUPVLNVMVNUUDXHCBJCBU
|
|
HZUUCWIXGUUQUUBWHWFWGUUQUUBWHFWFWFWGWGUJUKUUQULUMPVOVPURVQXFXBXCRZXDRZBJK
|
|
ZYOXEUUSBJXBXCXDSVRUURBJKZYNRYLYMRZYNRUUTYOUVAUVBYNXBXCBJTVSUURXDBJTYLYMY
|
|
NSVTWAXLXHXIRZXJRZBJKZYSXKUVDBJXHXIXJSVRUVEUVCBJKZYQRZYSUVCXJBJTUVGYRYPRZ
|
|
YQRYRYPYQIYSUVFUVHYQXHXIBJTVSYRYPYQSYRYPYQWBWCWAWAWDWE $.
|
|
$}
|
|
|
|
${
|
|
$d A a $. $d a m $. $d a n $. $d A n $. $d a p $. $d B n $. $d m n $.
|
|
$d m p $. $d n p $. $d n q $. $d p q $. $d p x $. $d q x $. $d m q $.
|
|
$( Three times a natural is not one more than three times a natural.
|
|
Another part of Theorem 3.4 of [Specker] p. 973. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
nnc3n3p1 $p |- ( ( A e. Nn /\ B e. Nn ) ->
|
|
-. ( ( A +c A ) +c A ) = ( ( ( B +c B ) +c B ) +c 1c ) ) $=
|
|
( vn vp vq cnnc wcel cplc c1c wceq wn c0c caddcfn c1st c2nd cop wa anidms
|
|
addceq12 nncaddccl va vm vx cv wral wi ccnv csn cima cres ccom ctxp cins3
|
|
cin crn ccompl cab cvv vex elcompl wbr wrex elima df-br wex oteltxp bitri
|
|
elrn elrn2 opelco brcnv brres eliniseg anbi2i 1cex op1st2nd 3bitri anbi1i
|
|
exbii opex breq1 ceqsexv braddcfn eqcom opelcnv nncdiv3lem1 anbi12i ancom
|
|
addcex addceq1 eqeq2d eqeq1 rexbii dfrex2 3bitrri con1bii abbi2i addcfnex
|
|
1stex 2ndex cnvex snex imaex resex coex inex rnex ins3ex complex eqeltrri
|
|
txpex nncex addceq12d eqtri syl6eq eqeq1d notbid ralbidv weq addceq1d wne
|
|
df-ne mpbi cncs mpancom syl sylancl syl5bb addcass addceq1i addc32 eqeq1i
|
|
id wb peano2 syl5ibrcom addc6 1cnnc mp2an imp addcid1 syl6bb 1ne0c intnan
|
|
addcid2 cbvralv nnnc 1cnc addceq0 mtbiri wo nnc0suc 0cnsuc eqtr3i sylnibr
|
|
rgen a1i peano1 suc11nnc mtbird sylnib adantr rspcv adantl eqeq12i mp3an3
|
|
addccan1 syl2an biimpd nsyld an32s rexlimdva syl5bi ralrimiv finds rspccv
|
|
jaod ex ) AFGZBFGZAAHZAHZBBHZBHZIHZJZKZUVSUWBCUDZUWHHZUWHHZIHZJZKZCFUEZUV
|
|
TUWGUFUAUDZUWOHZUWOHZUWKJZKZCFUEZLUWKJZKZCFUEUBUDZUXCHZUXCHZUWKJZKZCFUEZU
|
|
XCIHZUXIHZUXIHZDUDZUXLHZUXLHZIHZJZKZDFUEZUWNUAUBAMNOUGZIUHZUIZUJZUGZUKZNU
|
|
GZNOUNZULZUOZOULZMUIZUGZUMZNNUKZONUKZOULZULZMUIZUNZUOZUGZULZUOZUYTULZUOZF
|
|
UIZUPZUWTUAUQURUWTUAVUFUWOVUFGUWOVUEGZKUWTUWOVUEUAUSZUTUWTVUGVUGUWHUWOVUD
|
|
VAZCFVBUWRCFVBUWTKCUWOVUDFVCVUIUWRCFVUIUWHUWOPZVUDGZUWRUWHUWOVUDVDVUKUXLV
|
|
UJVUCVAZDVEUXLUWQJZUXLUWKJZQZDVEUWRDVUJVUCVHVULVUODVULUXLUWHPZVUBGZUXLUWO
|
|
PUYTGZQZVUNVUMQVUOVULUXLVUJPVUCGVUSUXLVUJVUCVDUXLUWHUWOVUBUYTVFVGVUQVUNVU
|
|
RVUMVUQEUDZVUPPVUAGZEVEVUTUWJJZUXLVUTIHZJZQZEVEVUNEVUPVUAVIVVAVVEEVVAVUTU
|
|
XLPUYDGZVUTUWHPUYTGZQVVDVVBQVVEVUTUXLUWHUYDUYTVFVVFVVDVVGVVBVVFVUTUCUDZUY
|
|
CVAZVVHUXLMVAZQZUCVEZVUTIPZUXLMVAZVVDUCVUTUXLMUYCVJVVLVVHVVMJZVVJQZUCVEVV
|
|
NVVKVVPUCVVIVVOVVJVVIVVHVUTNVAZVVHUYAGZQZVVQVVHIOVAZQVVOVVIVVHVUTUYBVAVVS
|
|
VUTVVHUYBVKVVHVUTNUYAVLVGVVRVVTVVQOIVVHVMVNVUTIVVHEUSZVOVPVQVRVSVVJVVNUCV
|
|
VMVUTIVWAVOVTVVHVVMUXLMWAWBVGVVNVVCUXLJVVDVUTIUXLVWAVOWCVVCUXLWDVGVQVVGUW
|
|
HVUTPUYSGVVBVUTUWHUYSWECEWFVGWGVVDVVBWHVQVSVVDVUNEUWJUWIUWHUWHUWHCUSZVWBW
|
|
IVWBWIVVBVVCUWKUXLVUTUWJIWJWKWBVQVURUWOUXLPUYSGVUMUXLUWOUYSWEUADWFVGWGVUN
|
|
VUMWHVQVSVUNUWRDUWQUWPUWOUWOUWOVUHVUHWIVUHWIUXLUWQUWKWLWBVQVGWMUWRCFWNWOW
|
|
PVGWQVUEVUDFVUCVUBUYTVUAUYDUYTMUYCWRUYBNUYAWSUXSUXTOWTXAIXBXCXDXAXEUYSUYR
|
|
UYLUYQUYKUYJUYIMUYHOUYGUYEUYFNWSXANOWSWTXFXKXGWTXKWRXCXAXHUYPMUYMUYONNWSW
|
|
SXEUYNOONWTWSXEWTXKXKWRXCXFXGXAZXKXGVWCXKXGXLXCXIXJUWOLJZUWSUXBCFVWDUWRUX
|
|
AVWDUWQLUWKVWDUWQLLHZLHZLVWDUWPVWEUWOLVWDUWPVWEJUWOUWOLLSRVWDYMXMVWFVWELV
|
|
WEUUALUUEXNZXOXPXQXRUAUBXSZUWSUXGCFVWHUWRUXFVWHUWQUXEUWKVWHUWPUXDUWOUXCVW
|
|
HUWPUXDJUWOUWOUXCUXCSRVWHYMXMXPXQXRUWOUXIJZUWTUXKUWKJZKZCFUEUXRVWIUWSVWKC
|
|
FVWIUWRVWJVWIUWQUXKUWKVWIUWPUXJUWOUXIVWIUWPUXJJUWOUWOUXIUXISRVWIYMXMXPXQX
|
|
RVWKUXQCDFCDXSZVWJUXPVWLUWKUXOUXKVWLUWJUXNIVWLUWIUXMUWHUXLVWLUWIUXMJUWHUW
|
|
HUXLUXLSRVWLYMXMXTWKXQUUFUUBUWOAJZUWSUWMCFVWMUWRUWLVWMUWQUWBUWKVWMUWPUWAU
|
|
WOAVWMUWPUWAJUWOUWOAASRVWMYMXMXPXQXRUXBCFUWHFGZUXAUWJLJZILJZQZVWPVWOILYAV
|
|
WPKUUCILYBYCUUDUXAUWKLJZVWNVWQLUWKWDVWNUWJYDGZIYDGVWRVWQYNVWNUWJFGZVWSUWI
|
|
FGZVWNVWTVWNVXAUWHUWHTRUWIUWHTYEUWJUUGYFUUHUWJIUUIYGYHUUJUUPUXCFGZUXHUXRV
|
|
XBUXHQZUXQDFUXLFGUXLLJZVVDEFVBZUUKVXCUXQEUXLUULVXCVXDUXQVXEVXBVXDUXQUFUXH
|
|
VXBUXQVXDUXKLIHZJZKVXBUXJUXCHZIHZVXFJZVXGVXBVXJVXHLJZVXBUXIUXCHZUXCHZIHZL
|
|
JZVXKVXOKZVXBVXNLYAVXPVXMUUMVXNLYBYCUUQVXHVXNLVXLIHZUXCHVXHVXNVXQUXJUXCUX
|
|
IUXCIYIYJVXLIUXCYKUUNYLUUOVXBVXHFGZLFGVXJVXKYNUXJFGZVXBVXRVXBUXIFGZVXSUXC
|
|
YOVXTVXSUXIUXITRYFUXJUXCTYEUURVXHLUUSYGUUTVXIUXKVXFUXJUXCIYIYLUVAVXDUXPVX
|
|
GVXDUXOVXFUXKVXDUXNLIVXDUXNVWFLVXDUXMVWEUXLLVXDUXMVWEJUXLUXLLLSRVXDYMXMVW
|
|
GXOXTWKXQYPUVBVXCVVDUXQEFVXCVUTFGZQUXQVVDUXKVVCVVCHZVVCHZIHZJZKZVXBVYAUXH
|
|
VYFVXBVYAQZUXHVYFVYGUXHUXEVUTVUTHZVUTHZIHZJZVYEVYAUXHVYKKZUFVXBUXGVYLCVUT
|
|
FCEXSZUXFVYKVYMUWKVYJUXEVYMUWJVYIIVYMUWIVYHUWHVUTVYMUWIVYHJUWHUWHVUTVUTSR
|
|
VYMYMXMXTWKXQUVCUVDVYGVYEVYKVYEUXEIIHZIHZHZVYJVYOHZJZVYGVYKUXKVYPVYDVYQUX
|
|
CIUXCIUXCIYQVYDVYIVYOHZIHVYQVYCVYSIVUTIVUTIVUTIYQYJVYIVYOIYKXNUVEVXBUXEFG
|
|
ZVYJFGZVYRVYKYNZVYAUXDFGZVXBVYTVXBWUCUXCUXCTRUXDUXCTYEVYAVYIFGZWUAVYHFGZV
|
|
YAWUDVYAWUEVUTVUTTRVYHVUTTYEVYIYOYFVYTWUAVYOFGZWUBVYNFGZIFGZWUFWUHWUHWUGY
|
|
RYRIITYSYRVYNITYSVYOUXEVYJUVGUVFUVHYHUVIUVJYTUVKVVDUXPVYEVVDUXOVYDUXKVVDU
|
|
XNVYCIVVDUXMVYBUXLVVCVVDUXMVYBJUXLUXLVVCVVCSRVVDYMXMXTWKXQYPUVLUVQUVMUVNU
|
|
VRUVOUWMUWGCBFUWHBJZUWLUWFWUIUWKUWEUWBWUIUWJUWDIWUIUWIUWCUWHBWUIUWIUWCJUW
|
|
HUWHBBSRWUIYMXMXTWKXQUVPYFYT $.
|
|
$}
|
|
|
|
$( Three times a natural is not two more than three times a natural. Another
|
|
part of Theorem 3.4 of [Specker] p. 973. (Contributed by SF,
|
|
12-Mar-2015.) $)
|
|
nnc3n3p2 $p |- ( ( A e. Nn /\ B e. Nn ) ->
|
|
-. ( ( A +c A ) +c A ) = ( ( ( B +c B ) +c B ) +c 2c ) ) $=
|
|
( cnnc wcel wa cplc c1c wceq wn peano2 nnc3n3p1 sylan ancoms eqcom addceq1i
|
|
c2c addc4 nncaddccl anidms mpancom addc32 1p1e2c 3eqtrri eqtri eqeq1i bitri
|
|
addceq2i addcass sylnibr wb 2nnc sylancl suc11nnc syl2an mtbid ) ACDZBCDZEZ
|
|
AAFZAFZGFZBBFZBFZPFZGFZHZUTVDHZURBGFZVHFZVHFZVAHZVFUQUPVKIZUQVHCDUPVLBJVHAK
|
|
LMVFVEVAHVKVAVENVEVJVAVEVIBFZGFVJVDVMGVMVBGGFZFZBFVCVNFVDVIVOBBGBGQOVBVNBUA
|
|
VNPVCUBUGUCOVIBGUHUDUEUFUIUPUTCDZVDCDZVFVGUJUQUSCDZUPVPUPVRAARSUSARTUQVCCDZ
|
|
PCDVQVBCDZUQVSUQVTBBRSVBBRTUKVCPRULUTVDUMUNUO $.
|
|
|
|
$( One more than three times a natural is not two more than three times a
|
|
natural. Final part of Theorem 3.4 of [Specker] p. 973. (Contributed by
|
|
SF, 12-Mar-2015.) $)
|
|
nnc3p1n3p2 $p |- ( ( A e. Nn /\ B e. Nn ) ->
|
|
-. ( ( ( A +c A ) +c A ) +c 1c ) =
|
|
( ( ( B +c B ) +c B ) +c 2c ) ) $=
|
|
( cnnc wcel wa cplc c1c c2c nnc3n3p1 wb nncaddccl anidms mpancom peano2 syl
|
|
wceq suc11nnc syl2an mtbird addcass 1p1e2c addceq2i eqtr2i eqeq2i sylnibr )
|
|
ACDZBCDZEZAAFZAFZGFZBBFZBFZGFZGFZPZUKUMHFZPUHUPUJUNPZABIUFUJCDZUNCDZUPURJUG
|
|
UICDZUFUSUFVAAAKLUIAKMUGUMCDZUTULCDZUGVBUGVCBBKLULBKMUMNOUJUNQRSUQUOUKUOUMG
|
|
GFZFUQUMGGTVDHUMUAUBUCUDUE $.
|
|
|
|
${
|
|
$d x y t $.
|
|
$( Lemma for ~ spacval . Set up stratification for the recursive
|
|
relationship. (Contributed by SF, 6-Mar-2015.) $)
|
|
spacvallem1 $p |- { <. x , y >. | ( x e. NC /\ y e. NC /\ y = ( 2c ^c x
|
|
) ) } e. _V $=
|
|
( vt cncs cce c2nd c1st ccnv c2c cv wcel wceq cop wa wbr wex anbi1i bitri
|
|
3bitri ncsex cxp cfullfun csn cima cres ccom cin co w3a cvv opelxp opelco
|
|
copab brcnv brres ancom eliniseg 2nc elexi vex op1st2nd exbii brfullfunop
|
|
opex breq1 ceqsexv eqcom anbi12i elin df-3an opabbi2i xpex ceex fullfunex
|
|
3bitr4i 2ndex 1stex cnvex snex imaex resex coex inex eqeltrri ) DDUAZEUBZ
|
|
FGHZIUCZUDZUEZHZUFZUGZAJZDKZBJZDKZWPIWNEUHZLZUIZABUMUJWTABWMWNWPMZWEKZXAW
|
|
LKZNWOWQNZWSNXAWMKWTXBXDXCWSWNWPDDUKXCCJZIWNMZLZXEWPWFOZNZCPZXFWPWFOZWSXC
|
|
WNXEWKOZXHNZCPXJCWNWPWFWKULXMXICXLXGXHXLXEWNWJOZXEIGOZXEWNFOZNZXGWNXEWJUN
|
|
XNXPXEWIKZNXRXPNXQXEWNFWIUOXPXRUPXRXOXPGIXEUQQSIWNXEIDURUSZAUTZVASQVBRXHX
|
|
KCXFIWNXSXTVDXEXFWPWFVEVFXKWRWPLWSIWNWPEXSXTVCWRWPVGRSVHXAWEWLVIWOWQWSVJV
|
|
OVKWEWLDDTTVLWFWKEVMVNWJFWIVPWGWHGVQVRIVSVTWAVRWBWCWD $.
|
|
$}
|
|
|
|
${
|
|
$d N n $. $d n x $. $d n y $. $d x y $.
|
|
$( The value of the special set generator. (Contributed by SF,
|
|
4-Mar-2015.) $)
|
|
spacval $p |- ( N e. NC -> ( Sp[ac] ` N ) =
|
|
Clos1 ( { N } , { <. x , y >. | ( x e. NC /\ y e. NC /\ y = ( 2c ^c x
|
|
) ) } ) ) $=
|
|
( vn cv csn cncs wcel c2c cce co wceq w3a copab cclos1 cspac clos1eq1 syl
|
|
sneq df-spac snex spacvallem1 clos1ex fvmpt ) DCDEZFZAEZGHBEZGHUHIUGJKLMA
|
|
BNZOZCFZUIOZGPUECLUFUKLUJULLUECSUIUFUKQRABDTUIUKCUAABUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d n x y $.
|
|
$( The special set generator is a function over the cardinals.
|
|
(Contributed by SF, 18-Mar-2015.) $)
|
|
fnspac $p |- Sp[ac] Fn NC $=
|
|
( vn vx vy cv csn cncs wcel c2c cce co w3a copab cclos1 cvv cspac df-spac
|
|
wceq wfn fnmpt snex spacvallem1 clos1ex a1i mprg ) ADZEZBDZFGCDZFGUHHUGIJ
|
|
QKBCLZMZNGZOFRAFAFUJONBCAPSUKUEFGUIUFUETBCUAUBUCUD $.
|
|
$}
|
|
|
|
${
|
|
$d x y $.
|
|
$( The special set generator generates a set of cardinals. (Contributed by
|
|
SF, 13-Mar-2015.) $)
|
|
spacssnc $p |- ( N e. NC -> ( Sp[ac] ` N ) C_ NC ) $=
|
|
( vx vy cncs wcel cspac cfv csn cv c2c cce co wceq w3a copab spacval cima
|
|
cclos1 cun wss snex spacvallem1 clos1baseima wa snssi crn imassrn wex cab
|
|
eqid rnopab simp2 exlimiv abssi eqsstri sstri jctir unss syl5eqss eqsstrd
|
|
sylib ) ADEZAFGAHZBIZDEZCIZDEZVFJVDKLMZNZBCOZRZDBCAPVBVKVCVJVKQZSZDVKVJVC
|
|
AUABCUBVKUJUCVBVCDTZVLDTZUDVMDTVBVNVOADUEVLVJUFZDVJVKUGVPVIBUHZCUIDVIBCUK
|
|
VQCDVIVGBVEVGVHULUMUNUOUPUQVCVLDURVAUSUT $.
|
|
$}
|
|
|
|
${
|
|
$d M x y $.
|
|
$( The initial value of the special set generator is an element.
|
|
(Contributed by SF, 13-Mar-2015.) $)
|
|
spacid $p |- ( M e. NC -> M e. ( Sp[ac] ` M ) ) $=
|
|
( vx vy cncs wcel csn cv c2c cce wceq w3a copab cclos1 cspac cfv wss eqid
|
|
co clos1base snssg mpbiri spacval eleqtrrd ) ADEZAAFZBGZDECGZDEUGHUFIRJKB
|
|
CLZMZANOUDAUIEUEUIPUIUHUEUIQSAUIDTUABCAUBUC $.
|
|
$}
|
|
|
|
${
|
|
$d N x $. $d N y $. $d x y $.
|
|
$( Closure law for the special set generator. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
spaccl $p |- ( ( M e. NC /\ N e. ( Sp[ac] ` M ) /\ ( N ^c 0c ) e. NC ) ->
|
|
( 2c ^c N ) e. ( Sp[ac] ` M ) ) $=
|
|
( vx vy cncs wcel cspac cfv c0c cce co w3a c2c csn wceq syl2anc cvv eleq1
|
|
cv eqid copab cclos1 wbr spacval 3ad2ant1 eleqtrd spacssnc sselda 3adant3
|
|
simp2 simp3 cnnc 2nnc ceclnn1 mp3an1 eqidd wb oveq2 eqeq2d 3anbi13d eqeq1
|
|
ovex 3anbi23d brabg mpan2 3ad2ant2 mpbir3and clos1conn eleqtrrd ) AEFZBAG
|
|
HZFZBIJKEFZLZMBJKZANZCSZEFZDSZEFZVSMVQJKZOZLZCDUAZUBZVKVNBWEFBVOWDUCZVOWE
|
|
FVNBVKWEVJVLVMUJVJVLVKWEOVMCDAUDUEZUFVNWFBEFZVOEFZVOVOOZVJVLWHVMVJVKEBAUG
|
|
UHUIZVNWHVMWIWKVJVLVMUKMULFWHVMWIUMMBUNUOPVNVOUPVLVJWFWHWIWJLZUQZVMVLVOQF
|
|
WMMBJVBWCWHVTVSVOOZLWLCDBVOVKQWDVQBOZVRWHWBWNVTVQBERWOWAVOVSVQBMJURUSUTWN
|
|
VTWIWNWJWHVSVOERVSVOVOVAVCWDTVDVEVFVGBVOWEWDVPWETVHPWGVI $.
|
|
$}
|
|
|
|
${
|
|
$d M x $. $d M z $. $d p q $. $d p x $. $d p z $. $d q x $. $d q z $.
|
|
$d S x $. $d S z $. $d x z $.
|
|
$( Inductive law for the special set generator. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
spacind $p |- ( ( ( M e. NC /\ S e. V ) /\
|
|
( M e. S /\ A. x e. ( Sp[ac] ` M ) ( ( x e. S /\ ( x ^c 0c ) e. NC ) ->
|
|
( 2c ^c x ) e. S ) ) ) -> ( Sp[ac] ` M ) C_ S ) $=
|
|
( vp vq vz wcel cncs cv cce co wa c2c wi wceq w3a wal impexp bitri cclos1
|
|
cvv c0c cspac cfv wral wss elex csn copab spacval adantr wbr simplr snssi
|
|
adantl spacssnc sseld 2nc ceclr simprd mp3an1 ex imim1d a1dd 3anass albii
|
|
imbi1i 19.21v bi2.04 ovex eleq1 imbi12d ceqsalv imbi2i syl6ibr vex eqeq2d
|
|
weq oveq2 3anbi13d 3anbi23d eqid brab imim2d 3imtr4g syld imp ralimdva wb
|
|
eqeq1 raleq sylibd ad2ant2rl snex spacvallem1 clos1induct syl3anc eqsstrd
|
|
syl sylanl2 ) BDHCIHZBUBHZCBHZAJZBHZXEUCKLIHZMNXEKLZBHZOZACUDUEZUFZMZXKBU
|
|
GBDUHXBXCMZXMMZXKCUIZEJZIHZFJZIHZXSNXQKLZPZQZEFUJZUAZBXNXKYEPZXMXBYFXCEFC
|
|
UKZULULXOXCXPBUGZXFXEGJZYDUMZMYIBHZOZGRZAYEUFZYEBUGXBXCXMUNXMYHXNXDYHXLCB
|
|
UOULUPXBXLYNXCXDXBXLYNXBXLYMAXKUFZYNXBXJYMAXKXBXEXKHZXJYMOZXBYPXEIHZYQXBX
|
|
KIXECUQURXBYRYQXBYRMZXFXGXIOZOXFYJYKOZGRZOZXJYMYSYTUUBXFYSYTYRYIIHZYIXHPZ
|
|
QZYKOZGRZUUBYSYTYRXHIHZXIOZOZUUHYRYTUUKOXBYRYTUUJYRYRUUIXGXIYRUUIXGNIHZYR
|
|
UUIXGUSUULYRUUIQNUCKLIHXGNXEUTVAVBVCVDVEUPUUHYRUUDUUEMZYKOZOZGRZUUKUUGUUO
|
|
GUUGYRUUMMZYKOUUOUUFUUQYKYRUUDUUEVFVHYRUUMYKSTVGUUPYRUUNGRZOUUKYRUUNGVIUU
|
|
RUUJYRUURUUEUUDYKOZOZGRUUJUUNUUTGUUNUUDUUEYKOOUUTUUDUUEYKSUUDUUEYKVJTVGUU
|
|
SUUJGXHNXEKVKUUEUUDUUIYKXIYIXHIVLYIXHBVLVMVNTVOTTVPUUAUUGGYJUUFYKYCYRXTXS
|
|
XHPZQUUFEFXEYIYDAVQGVQEAVSZXRYRYBUVAXTXQXEIVLUVBYAXHXSXQXENKVTVRWAFGVSXTU
|
|
UDUVAUUEYRXSYIIVLXSYIXHWKWBYDWCWDVHVGVPWEXFXGXISYMXFUUAOZGRUUCYLUVCGXFYJY
|
|
KSVGXFUUAGVITWFVCWGWHWIXBYFYOYNWJYGYMAXKYEWLWTWMWHWNAGYEYDXPUBBCWOEFWPYEW
|
|
CWQWRWSXA $.
|
|
$}
|
|
|
|
${
|
|
$d ch x $. $d M x $. $d M y $. $d N x $. $d ph y $. $d ps x $.
|
|
$d ta x $. $d th x $. $d x y $.
|
|
spacis.1 $e |- { x | ph } e. _V $.
|
|
spacis.2 $e |- ( x = M -> ( ph <-> ps ) ) $.
|
|
spacis.3 $e |- ( x = y -> ( ph <-> ch ) ) $.
|
|
spacis.4 $e |- ( x = ( 2c ^c y ) -> ( ph <-> th ) ) $.
|
|
spacis.5 $e |- ( x = N -> ( ph <-> ta ) ) $.
|
|
spacis.6 $e |- ( M e. NC -> ps ) $.
|
|
spacis.7 $e |-
|
|
( ( ( M e. NC /\ y e. ( Sp[ac] ` M ) ) /\ ( ( y ^c 0c ) e. NC /\ ch ) )
|
|
-> th ) $.
|
|
$( Induction scheme for the special set generator. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
spacis $p |- ( ( M e. NC /\ N e. ( Sp[ac] ` M ) ) -> ta ) $=
|
|
( cncs wcel wa cce cspac cfv cab cvv cv c0c co c2c wi wral wss a1i mpbird
|
|
id elabg ancom vex elab anbi2i bitri ovex sylibr syl5bi ralrimiva spacind
|
|
ex syl22anc sselda wb adantl mpbid ) HQRZIHUAUBZRZSIAFUCZRZEVLVMVOIVLVLVO
|
|
UDRZHVORZGUEZVORZVSUFTUGQRZSZUHVSTUGZVORZUIZGVMUJVMVOUKVLUNVQVLJULVLVRBOA
|
|
BFHQKUOUMVLWEGVMWBWACSZVLVSVMRSZWDWBWAVTSWFVTWAUPVTCWAACFVSGUQLURUSUTWGWF
|
|
WDWGWFSDWDPADFWCUHVSTVAMURVBVFVCVDGVOHUDVEVGVHVNVPEVIVLAEFIVMNUOVJVK $.
|
|
$}
|
|
|
|
${
|
|
$d A n $.
|
|
$( Lemma for ~ nchoice . A finite cardinal is not one more than its
|
|
T-raising. (Contributed by SF, 3-Mar-2015.) $)
|
|
nchoicelem1 $p |- ( A e. Nn -> -. A = ( T_c A +c 1c ) ) $=
|
|
( vn cnnc wcel cplc wceq c1c c2c ctc wn id syl2anc cncs nnnc syl addceq1d
|
|
tcdi eqtrd mtbird addcass cv w3o nncdiv3 nntccl nnc3n3p1 nncaddccl anidms
|
|
wrex eqeq2d tceq eqeq12d notbid syl5ibrcom 1cnc sylancl tc1c addceq12d wb
|
|
a1i peano2 suc11nnc nnc3n3p2 2nnc eqcomd addceq2i addccom 1p1e2c addceq1i
|
|
eqtr2i 3eqtr3i 3eqtr4i addc4 ax-mp 3eqtr3g ncaddccl eqeq1d eqcom rexlimiv
|
|
sylnib 3jaod ) ACDABUAZWAEZWAEZFZAWCGEZFZAWCHEZFZUBZBCUHAAIZGEZFZJZABUCWI
|
|
WMBCWACDZWDWMWFWHWNWMWDWCWCIZGEZFZJWNWQWCWAIZWREZWREZGEZFZWNWNWRCDXBJWNKZ
|
|
WAUDWAWRUELZWNWPXAWCWNWOWTGWNWOWBIZWREZWTWNWBMDZWAMDZWOXFFWNWBCDZXGWNXIWA
|
|
WAUFUGZWBNOWANZWBWAQLWNXEWSWRWNXHXHXEWSFXKXKWAWAQLPRZPUISWDWLWQWDAWCWKWPW
|
|
DKWDWJWOGAWCUJPUKULUMWNWMWFWEWEIZGEZFZJWNXOWCXMFZWNXPXBXDWNXMXAWCWNXMWOGI
|
|
ZEZXAWNWCMDZGMDZXMXRFWNWCCDZXSWNXIWNYAXJXCWBWAUFLZWCNOUNWCGQUOWNWOWTXQGXL
|
|
XQGFWNUPUSUQRUISWNYAXMCDZXOXPURYBWNWECDZYCWNYAYDYBWCUTOWEUDOWCXMVALSWFWLX
|
|
OWFAWEWKXNWFKWFWJXMGAWEUJPUKULUMWNWMWHWGWGIZGEZFZJWNYFWGFZYGWNYHWAGEZIZYJ
|
|
EZYJEZWGFZWNYJCDZWNYMJWNYICDZYNWAUTZYIUDOXCYJWAVBLWNYFYLWGWNYFYIYIEZYIEZI
|
|
ZYLWNYEXQEZWGGEZIZYFYSWNUUBYTWNWGMDZXTUUBYTFWNWGCDZUUCWNYAHCDUUDYBVCWCHUF
|
|
UOWGNOUNWGGQUOVDXQGYEUPVEUUAYRFUUBYSFWCHGEZEZWBGGEZEZYIEZUUAYRWBWAUUEEZEW
|
|
BUUGYIEZEUUFUUIUUJUUKWBWAHEZGEUUGWAEZGEUUJUUKUULUUMGUUMWAUUGEUULUUGWAVFUU
|
|
GHWAVGVEVIVHWAHGTUUGWAGTVJVEWBWAUUETWBUUGYITVKWCHGTYQUUHYIWAGWAGVLVHVKUUA
|
|
YRUJVMVNWNYSYQIZYJEZYLWNYQMDZYIMDZYSUUOFWNUUQUUQUUPWNYOUUQYPYINOZUURYIYIV
|
|
OLUURYQYIQLWNUUNYKYJWNUUQUUQUUNYKFUURUURYIYIQLPRRVPSYFWGVQVSWHWLYGWHAWGWK
|
|
YFWHKWHWJYEGAWGUJPUKULUMVTVRO $.
|
|
$}
|
|
|
|
${
|
|
$d A n $.
|
|
$( Lemma for ~ nchoice . A finite cardinal is not two more than its
|
|
T-raising. (Contributed by SF, 12-Mar-2015.) $)
|
|
nchoicelem2 $p |- ( A e. Nn -> -. A = ( T_c A +c 2c ) ) $=
|
|
( vn cnnc wcel cplc wceq c1c c2c ctc wn nntccl syl2anc cncs tcdi addceq1d
|
|
id syl mtbird sylancl addceq2i w3o wrex nncdiv3 nnc3n3p2 nncaddccl anidms
|
|
cv nnnc eqtrd eqeq2d tceq eqeq12d notbid syl5ibrcom wb 2nnc suc11nnc tc1c
|
|
addc32 eqtri eqeq2i sylnibr 1cnc peano2 nnc3p1n3p2 eqcom addcass addceq1i
|
|
1p1e2c addc4 tc2c 3eqtr4i addceq12i 3eqtr3ri 2nc syl6eq addceq12d 3eqtr4a
|
|
3jaod rexlimiv ) ACDABUGZWAEZWAEZFZAWCGEZFZAWCHEZFZUAZBCUBAAIZHEZFZJZABUC
|
|
WIWMBCWACDZWDWMWFWHWNWMWDWCWCIZHEZFZJWNWQWCWAIZWREZWREZHEZFZWNWNWRCDXBJWN
|
|
PZWAKWAWRUDLWNWPXAWCWNWOWTHWNWOWBIZWREZWTWNWBMDZWAMDZWOXEFWNWBCDZXFWNXHWA
|
|
WAUEUFZWBUHQWAUHZWBWANLWNXDWSWRWNXGXGXDWSFXJXJWAWANLOUIZOUJRZWDWLWQWDAWCW
|
|
KWPWDPWDWJWOHAWCUKOULUMUNWNWMWFWEWEIZHEZFZJWNXOWEWOGIZEZHEZFZWNWEWPGEZFZX
|
|
SWNYAWQXLWNWCCDZWPCDZYAWQUOWNXHWNYBXIXCWBWAUELZWNWOCDZHCDYCWNYBYEYDWCKQUP
|
|
WOHUESWCWPUQLRXRXTWEXRWPXPEXTWOXPHUSXPGWPURTUTVAVBWNXNXRWEWNXMXQHWNWCMDZG
|
|
MDZXMXQFWNYBYFYDWCUHQZVCWCGNSOUJRWFWLXOWFAWEWKXNWFPWFWJXMHAWEUKOULUMUNWNW
|
|
MWHWGWGIZHEZFZJWNYKWGWAGEZIZYMEZYMEZGEZFZWNYPWGFZYQWNYMCDZWNYRJWNYLCDYSWA
|
|
VDYLKQXCYMWAVELWGYPVFVBWNYJYPWGWNWTHIZEZHEZWRGEZUUCEZUUCEZGEZYJYPUUDWREZG
|
|
EZGEUUGGGEZEUUFUUBUUGGGVGUUHUUEGUUDWRGVGVHUUGUUAUUIHUUGWSYTEZWREUUAUUDUUJ
|
|
WRWSUUIEWSHEUUDUUJUUIHWSVITWRGWRGVJYTHWSVKTVLVHWSYTWRUSUTVIVMVNWNYIUUAHWN
|
|
YIWOYTEZUUAWNYFHMDYIUUKFYHVOWCHNSWNWOWTYTXKOUIOWNYOUUEGWNYNUUDYMUUCWNYMUU
|
|
CYMUUCWNYMWRXPEZUUCWNXGYGYMUULFXJVCWAGNSXPGWRURTVPZUUMVQUUMVQOVRUJRWHWLYK
|
|
WHAWGWKYJWHPWHWJYIHAWGUKOULUMUNVSVTQ $.
|
|
$}
|
|
|
|
${
|
|
$d M p $. $d M x $. $d M y $. $d p x $. $d p y $. $d x y $.
|
|
$( Lemma for ~ nchoice . Compute the value of ` Sp[ac] ` when the argument
|
|
is not exponentiable. Theorem 6.2 of [Specker] p. 973. (Contributed by
|
|
SF, 13-Mar-2015.) $)
|
|
nchoicelem3 $p |- ( ( M e. NC /\ -. ( M ^c 0c ) e. NC ) ->
|
|
( Sp[ac] ` M ) = { M } ) $=
|
|
( vx vy vp cncs wcel c0c cce co wn wa cspac cfv cv c2c wceq w3a cvv eleq1
|
|
eqid csn copab cclos1 spacval adantr cima c0 wbr cop elimasn df-br bitr4i
|
|
wb vex oveq2 eqeq2d 3anbi13d weq eqeq1 3anbi23d brabg mpan2 biimpac ceclr
|
|
2nc simprd mp3an1 sylan2 3impb syl6bi syl5bi con3d imp eq0rdv spacvallem1
|
|
snex clos1nrel syl eqtrd ) AEFZAGHIEFZJZKZALMZAUAZBNZEFZCNZEFZWHOWFHIZPZQ
|
|
ZBCUBZUCZWEVTWDWNPWBBCAUDUEWCWMWEUFZUGPWNWEPWCDWOVTWBDNZWOFZJVTWQWAWQAWPW
|
|
MUHZVTWAWQAWPUIWMFWRWMAWPUJAWPWMUKULVTWRVTWPEFZWPOAHIZPZQZWAVTWPRFWRXBUMD
|
|
UNWLVTWIWHWTPZQXBBCAWPERWMWFAPZWGVTWKXCWIWFAESXDWJWTWHWFAOHUOUPUQCDURWIWS
|
|
XCXAVTWHWPESWHWPWTUSUTWMTVAVBVTWSXAWAWSXAKVTWTEFZWAXAWSXEWPWTESVCOEFZVTXE
|
|
WAVEXFVTXEQOGHIEFWAOAVDVFVGVHVIVJVKVLVMVNWNWMWEAVPBCVOWNTVQVRVS $.
|
|
$}
|
|
|
|
${
|
|
$d M n $. $d M p $. $d n p $. $d N p $.
|
|
$( Lemma for ~ nchoice . The initial value of ` Sp[ac] ` is a minimum
|
|
value. Theorem 6.4 of [Specker] p. 973. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
nchoicelem4 $p |- ( ( M e. NC /\ N e. ( Sp[ac] ` M ) ) ->
|
|
M <_c N ) $=
|
|
( vp vn cv clec wbr c2c cce co csn cima cab cvv imasn breq2 cncs wa sylan
|
|
wcel lecex snex imaex eqeltrri nclecid cspac cfv c0c cltc spacssnc sselda
|
|
ce2lt wne brltc simplbi wi simpll adantr cnnc 2nnc ceclnn1 mp3an1 syl3anc
|
|
syl lectr mpan2d impr spacis ) ACEZFGZAAFGADEZFGZAHVKIJZFGZABFGCDABFAKZLV
|
|
JCMNCAFOFVOUAAUBUCUDVIAAFPVIVKAFPVIVMAFPVIBAFPAUEAQTZVKAUFUGZTZRZVKUHIJQT
|
|
ZVLVNVSVTRZVLVKVMFGZVNWAVKVMUIGZWBVSVKQTZVTWCVPVQQVKAUJUKZVKULSWCWBVKVMUM
|
|
VKVMUNUOVDWAVPWDVMQTZVLWBRVNUPVPVRVTUQVSWDVTWEURVSWDVTWFWEHUSTWDVTWFUTHVK
|
|
VAVBSAVKVMVEVCVFVGVH $.
|
|
$}
|
|
|
|
$( Lemma for ~ nchoice . A cardinal is not a member of the special set of
|
|
itself raised to two. Theorem 6.5 of [Specker] p. 973. (Contributed by
|
|
SF, 13-Mar-2015.) $)
|
|
nchoicelem5 $p |- ( ( M e. NC /\ ( M ^c 0c ) e. NC ) ->
|
|
-. M e. ( Sp[ac] ` ( 2c ^c M ) ) ) $=
|
|
( cncs wcel c0c cce co wa c2c cspac cfv clec wbr wn cltc ce2lt wb 2nc ax-mp
|
|
jctl c0 wne cnnc 2nnc ce0nn ce0nulnc mpbi cecl syl2an ltlenlec syldan mpbid
|
|
simprd nchoicelem4 sylan mtand ) ABCZADEFBCZGZAHAEFZIJCZUSAKLZURAUSKLZVAMZU
|
|
RAUSNLZVBVCGZAOUPUQUSBCZVDVEPUPHBCZUPGHDEFZBCZUQGVFUQUPVGQSUQVIVHTUAZVIHUBC
|
|
VJUCHUDRVGVJVIPQHUERUFSHAUGUHZAUSUIUJUKULURVFUTVAVKUSAUMUNUO $.
|
|
|
|
${
|
|
$d M x $.
|
|
$( Lemma for ~ nchoice . Split the special set generator into base and
|
|
inductive values. Theorem 6.6 of [Specker] p. 973. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
nchoicelem6 $p |- ( ( M e. NC /\ ( M ^c 0c ) e. NC ) ->
|
|
( Sp[ac] ` M ) = ( { M } u. ( Sp[ac] ` ( 2c ^c M ) ) ) ) $=
|
|
( vx cncs wcel c0c cce co wa cspac cfv c2c cvv wi wral wss a1i adantr syl
|
|
fvex spaccl csn cun cv simpl snex unex snidg elun1 wceq elun orbi1i bitri
|
|
wo elsn spacssnc spacid simpr syl3anc sseldd oveq2 eleq1d syl5ibrcom cnnc
|
|
2nnc ceclnn1 mp3an1 simprr simprl expr jaod syl5bi com23 imp3a elun2 syl6
|
|
ex ralrimivw spacind syl22anc snssd 3expib unssd eqssd ) ACDZAEFGCDZHZAIJ
|
|
ZAUAZKAFGZIJZUBZWFWDWKLDZAWKDZBUCZWKDZWNEFGCDZHZKWNFGZWKDZMZBWGNWGWKOWDWE
|
|
UDZWLWFWHWJAUEWIISUFPWFAWHDZWMWDXBWEACUGQAWHWJUHRWFWTBWGWFWQWRWJDZWSWFWOW
|
|
PXCWFWPWOXCWFWPWOXCMWOWNAUIZWNWJDZUMZWFWPHZXCWOWNWHDZXEUMXFWNWHWJUJXHXDXE
|
|
BAUNUKULXGXDXCXEWFXDXCMWPWFXCXDWIWJDZWFWICDZXIWFWGCWIWDWGCOWEAUOQWFWDAWGD
|
|
ZWEWIWGDZXAWDXKWEAUPZQWDWEUQAATURZUSZWIUPRXDWRWIWJWNAKFUTVAVBQWFWPXEXCWFW
|
|
PXEHZHXJXEWPXCWFXJXPKVCDWDWEXJVDKAVEVFQWFWPXEVGWFWPXEVHWIWNTURVIVJVKVPVLV
|
|
MWRWJWHVNVOVQBWKALVRVSWFWHWJWGWDWHWGOWEWDAWGXMVTQWFXJWGLDZXLWNWGDZWPHWRWG
|
|
DZMZBWJNWJWGOXOXQWFAISPXNWFXTBWJWDXTWEWDXRWPXSAWNTWAQVQBWGWILVRVSWBWC $.
|
|
$}
|
|
|
|
$( Lemma for ~ nchoice . Calculate the cardinality of a special set
|
|
generator. Theorem 6.7 of [Specker] p. 974. (Contributed by SF,
|
|
13-Mar-2015.) $)
|
|
nchoicelem7 $p |- ( ( M e. NC /\ ( M ^c 0c ) e. NC ) ->
|
|
Nc ( Sp[ac] ` M ) = ( Nc ( Sp[ac] ` ( 2c ^c M ) ) +c 1c ) ) $=
|
|
( cncs wcel c0c cce co cspac cfv cnc csn c2c cun c1c cplc nchoicelem6 nceqd
|
|
wa cin c0 wceq incom wn nchoicelem5 disjsn sylibr syl5eq snex fvex ncdisjun
|
|
syl df1c3g addceq1d adantr eqtr4d addccom syl6eq eqtrd ) ABCZADEFBCZQZAGHZI
|
|
AJZKAEFZGHZLZIZVDIZMNZUTVAVEAOPUTVFMVGNZVHUTVFVBIZVGNZVIUTVBVDRZSTVFVKTUTVL
|
|
VDVBRZSVBVDUAUTAVDCUBVMSTAUCVDAUDUEUFVBVDAUGVCGUHUIUJURVIVKTUSURMVJVGABUKUL
|
|
UMUNMVGUOUPUQ $.
|
|
|
|
$( Lemma for ~ nchoice . An anti-closure condition for cardinal
|
|
exponentiation to zero. Theorem 4.5 of [Specker] p. 973. (Contributed by
|
|
SF, 18-Mar-2015.) $)
|
|
nchoicelem8 $p |- ( ( <_c We NC /\ M e. NC ) ->
|
|
( -. ( M ^c 0c ) e. NC <-> Nc 1c <c M ) ) $=
|
|
( clec cncs cwe wbr wcel wa wn c1c cstrict cfound breqi brin bitri cpartial
|
|
cin simprbi adantr a1i cantisym c0c cce cnc cltc ce0lenc1 notbid adantl wne
|
|
co wb cconnex wo df-we sopc sylbi simpl simpr ncelncsi connexd sylan ord wi
|
|
1cex wceq id nclecid syl6eqbrr necon3bi jcad simplbi cref ctrans df-partial
|
|
ax-mp 3syl simplr simprl simprr antid necon3ad expimpd impbid brltc syl6bbr
|
|
expr bitrd ) BCDEZACFZGZAUAUBUICFZHZAIUCZBEZHZWLAUDEZWHWKWNUJWGWHWJWMAUEUFU
|
|
GWIWNWLABEZWLAUHZGZWOWIWNWRWIWNWPWQWIWMWPWGBCUKEZWHWMWPULWGBCJEZBCKEZGZWSWG
|
|
BCJKPZEXBBCDXCUMLBCJKMNZWTWSXAWTBCOEZWSCBUNZQRUOWSWHGZCBAWLWSWHUPWSWHUQWLCF
|
|
ZXGIVCURZSUSUTVAWNWQVBWIWMWLAWLAVDZAWLWLBXJVEXHWLWLBEXIWLVFVNVGVHSVIWIWPWQW
|
|
NWIWPGWMWLAWIWPWMXJWIWPWMGZGZCBWLAWIBCTEZXKWGXMWHWGWTXEXMWGWTXAXDVJWTXEWSXF
|
|
VJXEBCVKVLPZEZXMXEBCXNTPZEXOXMGBCOXPVMLBCXNTMNQVORRXHXLXISWGWHXKVPWIWPWMVQW
|
|
IWPWMVRVSWEVTWAWBWLAWCWDWF $.
|
|
|
|
$( Lemma for ~ nchoice . Calculate the cardinality of the special set
|
|
generator when near the end of raisability. Theorem 6.8 of [Specker]
|
|
p. 974. (Contributed by SF, 18-Mar-2015.) $)
|
|
nchoicelem9 $p |- ( ( <_c We NC /\ M e. NC /\ -. ( M ^c 0c ) e. NC ) ->
|
|
( Nc ( Sp[ac] ` T_c M ) = 2c \/ Nc ( Sp[ac] ` T_c M ) = 3c ) ) $=
|
|
( clec cncs wbr wcel c0c cce co wn cspac cfv cnc c2c wceq c3c wo wa cplc wb
|
|
c1c cwe ctc cltc wne brltc simplbi 1cex ncelncsi tlecg adantl tcnc1c breq1i
|
|
mpan cpw1 wi tccl te0c pw1ex w3a ce2le mp3an1 syl2anc ce2ncpw11c orc orbi1i
|
|
pm2.1 df-ne mpbir ordir mpbiran2 sylibr ce2t nchoicelem8 sylan2 nchoicelem3
|
|
bitri csn nceqd ovex df1c3 syl6eqr syl sylbird nclecid ax-mp ce0lenc1 ce2lt
|
|
ex mp2an cnnc ceclnn1 mp3an mpan2 mpbiri addceq1d nchoicelem7 1p1e2c eqcomi
|
|
2nnc 3eqtr4g fveq2 eqeq1d syl5ibcom adantr orim12d syl5 syl5bi syld addceq1
|
|
sylbid orim12i syl56 a1i eqeq12d 2p1e3c orbi12d 3imtr4d 3impia ) BCUADZACEZ
|
|
AFGHCEIZAUBZJKLZMNZYCONZPZXSXTQZTLZAUCDZMYBGHZJKZLZTRZTTRZNZYMMTRZNZPZYAYFY
|
|
IYHABDZYGYLTNZYLMNZPZYRYIYSYHAUDYHAUEUFYGYSYHUBZYBBDZUUBXTYSUUDSZXSYHCEZXTU
|
|
UETUGUHZYHAUIUMUJUUDTUNZLZYBBDZYGUUBUUCUUIYBBUKULYGUUJMUUIGHZYJBDZUUBXTUUJU
|
|
ULUOZXSXTYBCEZYBFGHCEZUUMAUPZAUQZUUICEZUUNUUOUUMUUHTUGURUHUURUUNUUOUSUUJUUL
|
|
UUIYBUTWHVAVBUJUULYHYJBDZYGUUBUUKYHYJBVCULUUSYHYJUCDZYHYJNZPZYGUUBUUSUUSUVA
|
|
PZUVBUUSUVAVDUVBUUSYHYJUDZQZUVAPZUVCUUTUVEUVAYHYJUEVEUVFUVCUVDUVAPZUVGUVAIZ
|
|
UVAPUVAVFUVDUVHUVAYHYJVGVEVHUUSUVDUVAVIVJVPVKYGUUTYTUVAUUAYGUUTYJFGHCEIZYTX
|
|
TXSYJCEZUVIUUTSAVLZYJVMVNXTUVIYTUOZXSXTUVJUVLUVKUVJUVIYTUVJUVIQZYLYJVQZLTUV
|
|
MYKUVNYJVOVRYJMYBGVSVTWAWHWBUJWCXSUVAUUAUOXTXSYHJKZLZMNUVAUUAXSMYHGHZJKZLZT
|
|
RZYNUVPMXSUVSTTXSUVQFGHCEIZUVSTNZXSUWAYHUVQUCDZUUFYHFGHCEZUWCUUGUWDYHYHBDZU
|
|
UFUWEUUGYHWDWEUUFUWDUWESUUGYHWFWEVHZYHWGWIXSUVQCEZUWAUWCSMWJEUUFUWDUWGWSUUG
|
|
UWFMYHWKWLZUVQVMWMWNUWGUWAUWBUWHUWGUWAQZUVSUVQVQZLTUWIUVRUWJUVQVOVRUVQMYHGV
|
|
SVTWAUMWBWOUUFUWDUVPUVTNUUGUWFYHWPWIYNMWQWRZWTUVAUVPYLMUVAUVOYKYHYJJXAVRXBX
|
|
CXDXEXFXGXHXGXJYTYOUUAYQYLTTXIYLMTXIXKXLAVMXTYFYRSXSXTYDYOYEYQXTYCYMMYNXTUU
|
|
NUUOYCYMNUUPUUQYBWPVBZMYNNXTUWKXMXNXTYCYMOYPUWLOYPNXTYPOXOWRXMXNXPUJXQXR $.
|
|
|
|
${
|
|
$d c y $. $d S t $. $d S y $. $d S z $. $d t z $. $d X y $. $d X z $.
|
|
$d y z $.
|
|
nchoicelem10.1 $e |- S e. _V $.
|
|
nchoicelem10.2 $e |- X e. _V $.
|
|
$( Lemma for ~ nchoice . Stratification helper lemma. (Contributed by SF,
|
|
18-Mar-2015.) $)
|
|
nchoicelem10 $p |- ( <. c , X >. e. ~ ( ( Ins3 _S (+) Ins2 ~ ran ( `' ~
|
|
_S (x) ( `' _S |` Fix ( _S o. Image S ) ) ) ) " 1c )
|
|
<-> c = Clos1 ( X , S ) ) $=
|
|
( vy vz vt csset cv wcel wn wss wa wbr wex vex 3bitri bitri anbi12i trtxp
|
|
cclos1 ccompl ccnv cimage ccom cfix cres ctxp crn csn cop cima wel wi wal
|
|
elrn brcnv df-br snex opex elcompl opelssetsn xchbinx brres elfix brimage
|
|
wceq brco exbii imaex sseq1 ceqsexv ancom annim exnal 3bitrri con1bii cab
|
|
brsset cint df-clos1 eleq2i elintab 3bitr4i releqel ) CFBAUBZIUCZUDZIUDZI
|
|
AUEZUFZUGZUHZUIZUJZUCZBEFJZUKZBULZWPKZLBGJZMZAXBUMZXBMZNZFGUNZUOZGUPZWTWQ
|
|
KWRWGKZXIXAXAXBWTWOOZGPXHLZGPXILGWTWOUQXKXLGXKXBWSWIOZXBBWNOZNZXFXGLZNZXL
|
|
XBWSBWIWNUAXOXPXFNXQXMXPXNXFXMWSXBWHOWSXBULZWHKZXPXBWSWHURWSXBWHUSXSXRIKX
|
|
GXRIWSXBWRUTZGQZVAVBWRXBFQZYAVCVDRXNXBBWJOZXBWMKZNXFXBBWJWMVEYCXCYDXEYCBX
|
|
BIOXCXBBIURBXBEYAVTSYDXBXBWLOZXEXBWLVFYEXBHJZWKOZYFXBIOZNZHPYFXDVHZYFXBMZ
|
|
NZHPXEHXBXBIWKVIYIYLHYGYJYHYKXBYFAYAHQZVGYFXBYMYAVTTVJYKXEHXDAXBDYAVKYFXD
|
|
XBVLVMRSTSTXPXFVNSXFXGVORVJXHGVPVQVRWTWPWSBXTEVAVBXJWRXFGVSWAZKXIWGYNWRAB
|
|
GWBWCXFGWRYBWDSWEWF $.
|
|
$}
|
|
|
|
${
|
|
$d a b $. $d a m $. $d a t $. $d a u $. $d a x $. $d a y $. $d b m $.
|
|
$d b t $. $d b u $. $d b x $. $d b y $. $d m t $. $d m u $. $d m x $.
|
|
$d m y $. $d t u $. $d t x $. $d t y $. $d u x $. $d u y $. $d x y $.
|
|
$( Lemma for ~ nchoice . Set up stratification for ~ nchoicelem12 .
|
|
(Contributed by SF, 18-Mar-2015.) $)
|
|
nchoicelem11 $p |- { t | A. m e. NC ( t = Nc ( Sp[ac] `
|
|
T_c m ) -> Nc ( Sp[ac] ` m ) e. Nn ) } e. _V $=
|
|
( vu va vb cncs cvv csset cv wcel wceq cfin wn csn wa wex wbr bitri exbii
|
|
snex vx vy cxp cins3 ccompl ccnv c2c cce co w3a copab ccom cfix cres ctxp
|
|
crn cins2 csymdif c1c cima csi cin ctcfn cpw1 cdif ctc cspac cfv cnc cnnc
|
|
cimage wi wral cab vex elcompl wrex cop elimapw13 cclos1 tccl spacval syl
|
|
nceqd eqeq2d finnc eleq1d syl5bbr notbid eldif opelco anbi1i 19.41v excom
|
|
imbi12d brsnsi1 bicomi anass breq1 anbi2d bitr4i sneq breq1d brtcfn df-br
|
|
ceqsexv opelcnv elin opelxp mpbiran2 wel ancom brsnsi2 breq2 nchoicelem10
|
|
spacvallem1 brcnv brssetsn anbi12i df-clel 3bitr4i clos1ex eqnc2 clos1eq1
|
|
tcex rexbii snelpw1 elima risset notbii ncsex vvex xpex ssetex cnvex coex
|
|
complex imaex siex pw1ex annim 3bitri rexbiia rexnal bitr2i abbi2i ins3ex
|
|
syl6rbbr con1bii imageex fixex resex txpex rnex ins2ex symdifex 1cex inex
|
|
tcfnex finex difex eqeltrri ) FGUCZHUDZHUEZUFZHUFZHUAIZFJUBIZFJUVIUGUVHUH
|
|
UIKUJUAUBUKZVKZULZUMZUNZUOZUPZUEZUQZURZUSUTZUEZVAZUVGULZVBZUFZVCVAZVAZULZ
|
|
UWALUTZVDZVDZGUCZVEZFVDZVDZVDZUTZUEZAIZBIZVFZVGVHZVIZKZUWTVGVHZVIVJJZVLZB
|
|
FVMZAVNGUXHAUWRUWSUWRJUWSUWQJZMUXHUWSUWQAVOZVPUXHUXIUXIUXGMZBFVQZUXHMUXIU
|
|
WTNZNZNZUWSVRZUWMJZBFVQUXLBUWSUWMFVSUXQUXKBFUWTFJZUXKUWSUXANZUVJVTZVIZKZU
|
|
XMUVJVTZLJZVLZMZUXQUXRUXGUYEUXRUXDUYBUXFUYDUXRUXCUYAUWSUXRUXBUXTUXRUXAFJU
|
|
XBUXTKUWTWAUAUBUXAWBWCWDWEUXFUXELJUXRUYDUXEWFUXRUXEUYCLUAUBUWTWBWGWHWOWIU
|
|
XQUXPUWHJZUXPUWLJZMZOUYBUYDMZOUYFUXPUWHUWLWJUYGUYBUYIUYJUYGCIZUXAKZUWSUYK
|
|
NZUVJVTZVIZKZOZCPZUYBUYGUXMUYKVCQZUYMNZUWSUWEQZOZCPZUYRUYGUXODIZUWGQZVUDU
|
|
WSUWEQZOZDPZVUCDUXOUWSUWEUWGWKVUHVUDEIZNZKZUXNVUIUWFQZOZVUFOZEPZDPZVUCVUG
|
|
VUODVUGVUMEPZVUFOZVUOVUEVUQVUFEUXNVUDUWFUXMTWPWLVUOVURVUMVUFEWMWQRSVUPVUN
|
|
DPZEPZVUCVUNDEWNVUTVULVUJUWSUWEQZOZEPZVUCVUSVVBEVUSVUKVULVUFOZOZDPVVBVUNV
|
|
VEDVUKVULVUFWRSVVDVVBDVUJVUITVUKVUFVVAVULVUDVUJUWSUWEWSWTXFRSVVCVUIUYMKZU
|
|
YSOZVVAOZCPZEPZVUCVVBVVIEVVBVVGCPZVVAOVVIVULVVKVVACUXMVUIVCUWTTZWPWLVVGVV
|
|
ACWMXASVVJVVHEPZCPVUCVVHECWNVVMVUBCVVMVVFUYSVVAOZOZEPVUBVVHVVOEVVFUYSVVAW
|
|
RSVVNVUBEUYMUYKTZVVFVVAVUAUYSVVFVUJUYTUWSUWEVUIUYMXBXCWTXFRSRRRRRRVUBUYQC
|
|
UYSUYLVUAUYPUWTUYKBVOXDVUAUWSFJZUYNUWSJZOZUYPVUAUYTUWSVRUWEJZVVSUYTUWSUWE
|
|
XEVVTUWSUYTVRZUWDJZVVSUYTUWSUWDXGVWBVWAUVCJZVWAUWCJZOVVSVWAUVCUWCXHVWCVVQ
|
|
VWDVVRVWCVVQUYTGJUYMTUWSUYTFGXIXJUWSVUIUVGQZVUIUYTUWBQZOZEPZVUDUYNKZDAXKZ
|
|
OZDPZVWDVVRVWHVUIVUDNZKZVUDUYMUWAQZOZVWEOZEPZDPZVWLVWHVWQDPZEPVWSVWGVWTEV
|
|
WGVWPDPZVWEOZVWTVWGVWFVWEOVXBVWEVWFXLVWFVXAVWEDUYMVUIUWAVVPXMWLRVWPVWEDWM
|
|
XASVWQEDWNRVWRVWKDVWRVWOUWSVWMUVGQZOZVWKVWRVWNVWOVWEOZOZEPVXDVWQVXFEVWNVW
|
|
OVWEWRSVXEVXDEVWMVUDTVWNVWEVXCVWOVUIVWMUWSUVGXNWTXFRVWOVWIVXCVWJVWOVUDUYM
|
|
VRUWAJVWIVUDUYMUWAXEUVJUYMDUAUBXPZVVPXORVXCVWMUWSHQVWJUWSVWMHXQVUDUWSDVOU
|
|
XJXRRXSRSREUWSUYTUWBUVGWKDUYNUWSXTYAXSRRRUWSUYNUVJUYMVVPVXGYBYCXAXSSRUYPU
|
|
YBCUXAUWTYEUYLUYOUYAUWSUYLUYNUXTUYLUYMUXSKUYNUXTKUYKUXAXBUVJUYMUXSYDWCWDW
|
|
EXFRUYHUYDVUDUXMUWAQZDLVQZVUDUYCKZDLVQUYHUYDVXHVXJDLVXHVUDUXMVRUWAJVXJVUD
|
|
UXMUWAXEUVJUXMDVXGVVLXORYFUYHUXOUWKJZVXIUYHVXKUWSGJUXJUXOUWSUWKGXIXJVXKUX
|
|
NUWJJZVXIUXNUWJYGVXLUXMUWIJVXIUXMUWIYGDUXMUWALYHRRRDUYCLYIYAYJXSUYBUYDUUA
|
|
UUBUUHUUCRUXGBFUUDUUEUUIRUUFUWQUWMUWPUWHUWLUWEUWGUWDUVCUWCFGYKYLYMUWBUVGU
|
|
WAUVTUVSUSUVDUVRHYNUUGUVQUVPUVOUVFUVNUVEHYNYQYOUVGUVMHYNYOZUVLHUVKYNUVJVX
|
|
GUUJYPUUKUULUUMUUNYQUUOUUPUUQYRYQZYSVXMYPUURYOUWFVCUUSYSYSYPUWKGUWJUWIUWA
|
|
LVXNUUTYRYTYTYLYMUVAUWOUWNFYKYTYTYTYRYQUVB $.
|
|
$}
|
|
|
|
${
|
|
$d M m $. $d m t $. $d m x $. $d M x $. $d k m $. $d k n $. $d k t $.
|
|
$d m n $. $d n t $. $d t x $.
|
|
$( Lemma for ~ nchoice . If the T-raising of a cardinal yields a finite
|
|
special set, then so does the initial set. Theorem 7.1 of [Specker]
|
|
p. 974. (Contributed by SF, 18-Mar-2015.) $)
|
|
nchoicelem12 $p |- ( ( M e. NC /\ ( Sp[ac] ` T_c M ) e. Fin ) ->
|
|
( Sp[ac] ` M ) e. Fin ) $=
|
|
( vx vm vt vk cspac cfv wcel cncs cv cnc wceq cnnc wi c0c nceqd cce co wa
|
|
wral ctc cfin wrex finnc risset bitri c1c cplc nchoicelem11 eqeq1 ralbidv
|
|
imbi1d weq tceq fveq2d eqeq2d fveq2 eleq1d imbi12d cbvralv syl6bb wne c2c
|
|
vn wn tccl te0c nchoicelem7 syl2anc 0cnsuc a1i eqnetrd necomd df-ne sylib
|
|
pm2.21d rgen 2nnc ceclnn1 mp3an1 syl ancoms adantrl adantl wb nnnc adantr
|
|
fvex ncelncsi peano4nc sylancl tce2 biimprd sylbid imim1d imp peano2 syl6
|
|
rspcv sylibrd ex syld expimpd csn nchoicelem3 vex df1c3 eqeltrri syl6eqel
|
|
1cnnc expcom adantld adantrd pm2.61i an32s ralrimiva finds syl6bbr rspccv
|
|
a1d com23 rexlimiv sylbi impcom ) AUAZFGZUBHZAIHZAFGZUBHZYGBJZYFKZLZBMUCZ
|
|
YHYJNZYGYLMHYNYFUDBYLMUEUFYMYOBMYKMHZYHYMYJYPYKCJZUAZFGZKZLZYQFGZKZMHZNZC
|
|
ITZYHYMYJNZNDJZYTLZUUDNZCITZOYTLZUUDNZCITVDJZYTLZUUDNZCITZUUNUGUHZEJZUAZF
|
|
GZKZLZUUSFGZKZMHZNZEITZUUFDVDYKDCUIUUHOLZUUJUUMCIUVIUUIUULUUDUUHOYTUJULUK
|
|
DVDUMZUUJUUPCIUVJUUIUUOUUDUUHUUNYTUJULUKUUHUURLZUUKUURYTLZUUDNZCITUVHUVKU
|
|
UJUVMCIUVKUUIUVLUUDUUHUURYTUJULUKUVMUVGCEICEUMZUVLUVCUUDUVFUVNYTUVBUURUVN
|
|
YSUVAUVNYRUUTFYQUUSUNUOPUPUVNUUCUVEMUVNUUBUVDYQUUSFUQPURUSUTVADBUMZUUJUUE
|
|
CIUVOUUIUUAUUDUUHYKYTUJULUKUUMCIYQIHZUULUUDUVPOYTVBUULVEUVPYTOUVPYTVCYRQR
|
|
FGKZUGUHZOUVPYRIHYROQRIHYTUVRLYQVFYQVGYRVHVIUVROVBUVPUVQVJVKVLVMOYTVNVOVP
|
|
VQUUNMHZUUQUVHUVSUUQSUVGEIUVSUUSIHZUUQUVGUUSOQRIHZUVSUVTSZUUQSUVGNUWAUWBU
|
|
UQUVGUWAUWBSZUUQUUNVCUUSQRZUAZFGZKZLZUWDFGZKZMHZNZUVGUWAUVTUUQUWLNZUVSUVT
|
|
UWAUWMUVTUWASUWDIHZUWMVCMHUVTUWAUWNVRVCUUSVSVTUUPUWLCUWDIYQUWDLZUUOUWHUUD
|
|
UWKUWOYTUWGUUNUWOYSUWFUWOYRUWEFYQUWDUNUOPUPUWOUUCUWJMUWOUUBUWIYQUWDFUQPUR
|
|
USWSWAWBWCUWCUWLUVGUWCUWLSZUVCUWJUGUHZMHZUVFUWPUVCUWKUWRUWCUWLUVCUWKNUWCU
|
|
VCUWHUWKUWCUVCUURVCUUTQRZFGZKZUGUHZLZUWHUWCUVBUXBUURUWBUVBUXBLZUWAUVTUXDU
|
|
VSUVTUUTIHUUTOQRIHUXDUUSVFUUSVGUUTVHVIWDWDUPUWCUXCUUNUXALZUWHUWCUUNIHZUXA
|
|
IHUXCUXEWEUWBUXFUWAUVSUXFUVTUUNWFWGWDUWTUWSFWHWIUUNUXAWJWKUWCUWHUXEUWCUWG
|
|
UXAUUNUWCUWFUWTUWCUWEUWSFUWAUVTUWEUWSLZUVSUVTUWAUXGUUSWLWBWCUOPUPWMWNWNWO
|
|
WPUWJWQWRUWPUVEUWQMUWCUVEUWQLZUWLUWAUVTUXHUVSUVTUWAUXHUUSVHWBWCWGURWTXAXB
|
|
XCUWAVEZUWBUVGUUQUXIUVTUVGUVSUVTUXIUVGUVTUXISZUVFUVCUXJUVEUUSXDZKZMUXJUVD
|
|
UXKUUSXEPUGUXLMUUSEXFXGXJXHXIXTXKXLXMXNXOXPXAXQUUEUUGCAIYQALZUUAYMUUDYJUX
|
|
MYTYLYKUXMYSYFUXMYRYEFYQAUNUOPUPUXMUUDYIKZMHYJUXMUUCUXNMUXMUUBYIYQAFUQPUR
|
|
YIUDXRUSXSWAYAYBYCYD $.
|
|
$}
|
|
|
|
${
|
|
$d M x $.
|
|
$( Lemma for ~ nchoice . The cardinality of a special set is at least
|
|
one. (Contributed by SF, 18-Mar-2015.) $)
|
|
nchoicelem13 $p |- ( M e. NC -> 1c <_c Nc ( Sp[ac] ` M ) ) $=
|
|
( vx cncs wcel cv cspac cfv wss c1c wrex cnc wbr csn snel1cg spacid snssi
|
|
clec syl sseq1 rspcev syl2anc wb 1cnc fvex lenc ax-mp sylibr ) ACDZBEZAFG
|
|
ZHZBIJZIUJKQLZUHAMZIDUNUJHZULACNUHAUJDUOAOAUJPRUKUOBUNIUIUNUJSTUAICDUMULU
|
|
BUCBUJIAFUDUEUFUG $.
|
|
$}
|
|
|
|
${
|
|
$d M k $.
|
|
$( Lemma for ~ nchoice . When the special set generator yields a
|
|
singleton, then the cardinal is not raisable. (Contributed by SF,
|
|
19-Mar-2015.) $)
|
|
nchoicelem14 $p |- ( ( M e. NC /\ Nc ( Sp[ac] ` M ) = 1c ) ->
|
|
-. ( M ^c 0c ) e. NC ) $=
|
|
( vk cncs wcel cspac cfv cnc c1c wceq c0c cce co wn c2c cplc wne c0 bitri
|
|
cin sylibr csn cun nchoicelem5 incom eqeq1i disjsn snex fvex ncdisjun syl
|
|
df1c3g adantr addceq2d addccom syl6reqr clec wbr cnnc 2nnc ceclnn1 mp3an1
|
|
wa nchoicelem13 cv wrex wb 1cnc ncelncsi dflec2 mp2an 0cnsuc neeq1 mpbiri
|
|
eqnetri rexlimivw sylbi 3syl 0cnc peano4nc necon3bii eqnetrd neeq2i df-ne
|
|
addcid2 sylib nchoicelem6 nceqd eqeq1d mtbird ex con2d imp ) ACDZAEFZGZHI
|
|
ZAJKLCDZMWMWQWPWMWQWPMWMWQVBZWPAUAZNAKLZEFZUBZGZHIZWRXCJHOZPZXDMZWRXCWSGZ
|
|
XAGZOZXEWRWSXASZQIZXCXJIWRAXADMZXLAUCXLXAWSSZQIXMXKXNQWSXAUDUEXAAUFRTWSXA
|
|
AUGWTEUHZUIUJWRXJXIHOZXEWRXPXIXHOXJWRHXHXIWMHXHIWQACUKULUMXHXIUNUOWRXIJPZ
|
|
XPXEPWRWTCDZHXIUPUQZXQNURDWMWQXRUSNAUTVAWTVCXSXIHBVDZOZIZBCVEZXQHCDXICDZX
|
|
SYCVFVGXAXOVHZHXIBVIVJYBXQBCYBXQYAJPYAXTHOJHXTUNXTVKVNXIYAJVLVMVOVPVQXPXE
|
|
XIJYDJCDXPXEIXIJIVFYEVRXIJVSVJVTTWAWAXFXCHPXGXEHXCHWDWBXCHWCRWEWRWOXCHWRW
|
|
NXBAWFWGWHWIWJWKWL $.
|
|
$}
|
|
|
|
$( Lemma for ~ nchoice . When the special set generator does not yield a
|
|
singleton, then the cardinal is raisable. (Contributed by SF,
|
|
19-Mar-2015.) $)
|
|
nchoicelem15 $p |- ( ( M e. NC /\ 1c <c Nc ( Sp[ac] ` M ) ) ->
|
|
( M ^c 0c ) e. NC ) $=
|
|
( cncs wcel c1c cspac cfv cnc cltc wbr c0c cce co wne clec brltc simprbi wn
|
|
wceq wa csn df1c3g adantr nchoicelem3 nceqd eqtr4d ex necon1ad syl5 imp ) A
|
|
BCZDAEFZGZHIZAJKLBCZUMDULMZUJUNUMDULNIUODULOPUJUNDULUJUNQZDULRUJUPSZDATZGZU
|
|
LUJDUSRUPABUAUBUQUKURAUCUDUEUFUGUHUI $.
|
|
|
|
${
|
|
$d m n $. $d m t $. $d m u $. $d m v $. $d m x $. $d m y $. $d n t $.
|
|
$d n u $. $d n v $. $d n x $. $d n y $. $d t u $. $d t v $. $d t x $.
|
|
$d t y $. $d u v $. $d u x $. $d u y $. $d v x $. $d v y $. $d x y $.
|
|
$( Lemma for ~ nchoice . Set up stratification for ~ nchoicelem17 .
|
|
(Contributed by SF, 19-Mar-2015.) $)
|
|
nchoicelem16 $p |- { t |
|
|
( <_c We NC -> A. m e. NC ( Nc ( Sp[ac] ` m ) = ( 1c +c t ) -> ( ( Sp[ac] ` T_c
|
|
m ) e. Fin /\ ( Nc ( Sp[ac] ` T_c m ) = ( T_c Nc ( Sp[ac] ` m ) +c 1c ) \/ Nc (
|
|
Sp[ac] ` T_c m ) = ( T_c Nc ( Sp[ac] ` m ) +c 2c ) ) ) ) ) } e. _V
|
|
$=
|
|
( vu vv cncs wbr ccnv c1c caddcfn csset wcel c2c wceq cop wex exbii bitri
|
|
wa 3bitri cnvex vx vy vn clec cwe cab c2nd c1st csn cima cres ccom ccompl
|
|
wn cins3 cv cce co w3a copab cimage cfix ctxp csi ctcfn cun cnnc cpw1 cfv
|
|
cspac cnc cplc ctc cfin wo wral vex snex wrex cclos1 spacval nceqd eqeq1d
|
|
wi syl eleq1d orbi12d anbi12d notbid opelcnv opelco brsnsi1 anbi1i 19.41v
|
|
opsnelsi bitr4i excom anass breq1 anbi2d ceqsexv brcnv df-br nchoicelem10
|
|
wel brssetsn anbi12i clos1ex eleq1 ancom eqcom eqnc2 brtcfn brco eliniseg
|
|
brres anbi2i 1cex op1st2nd braddcfn addceq1 eqeq2d syl6bbr eqeq1 addceq1d
|
|
opex tcex 2ndex 1stex imaex resex addcfnex coex ssetex complex txpex siex
|
|
ncsex unex pw1ex crn cins2 csymdif cdif cuni1 cvv elima3 eluni1 elimapw13
|
|
unab elcompl eldif opelres spacvallem1 elimapw11c oteltxp 3bitr2i df-clel
|
|
tccl brun 2nc elexi orbi12i ncex finnc sneq clos1eq1 notbii syl6rbbr tceq
|
|
pm5.32i rexbiia 3bitrri con1bii addcex eqeq2 imbi1d ralbidv abbi2i uneq2i
|
|
rexanali imor abbii 3eqtr4i abexv ins3ex imageex fixex rnex ins2ex tcfnex
|
|
symdifex nncex difex uni1ex eqeltrri ) UDEUEFZUNZAUFZUGUHGZHUIZUJZUKZIGZU
|
|
LZJJUOZJUMZGZJGZJUAUPZEKUBUPZEKUXKLUXJUQURMUSUAUBUTZVAZULZVBZUKZVCZUUAZUM
|
|
ZUUBZUUCZHUJZUMZGZVDZULZGZEUKZGZVDZVEGZVDZVDZUYKIUHUGGZUXAUJZUKZGZULZIUHU
|
|
YNLUIZUJZUKZGZULZVFZGZULZVGUKZUYIULZVCZHVHZUJZUUDZEVHZVHZVHZUJZUMZUUEZUJZ
|
|
VFZUWQBUPZVJVIZVKZHAUPZVLZMZVVAVMZVJVIZVNKZVVHVKZVVCVMZHVLZMZVVJVVKLVLZMZ
|
|
VOZRZWDZBEVPZWDZAUFZUUFUWSVVSAUFZVFUWRVVSVOZAUFVUTVWAUWRVVSAUUJVUSVWBUWSV
|
|
VSAVUSVVDVUSKCUPZVURKZVWDVVDNUXEKZRZCOVWDVVEMZVVCVWDMZVVQWDZBEVPZRZCOVVSC
|
|
VVDUXEVURUUGVWGVWLCVWGVWKVWHRVWLVWEVWKVWFVWHVWEVWDUIZVUQKVWMVUPKZUNVWKVWD
|
|
VUQCVQZUUHVWMVUPVWDVRUUKVWKVWNVWNVVAUIZUIZUIZVWMNZVULKZBEVSVWIVVQUNZRZBEV
|
|
SVWKUNBVWMVULEUUIVWTVXBBEVVAEKZVWTVWIVVIVVJVWDVMZHVLZMZVVJVXDLVLZMZVOZRZU
|
|
NZRZVXBVXCVXLVWPUXLVTZVKZVWDMZVVGUIZUXLVTZVNKZVXQVKZVXEMZVXSVXGMZVOZRZUNZ
|
|
RZVWTVXCVWIVXOVXKVYDVXCVVCVXNVWDVXCVVBVXMUAUBVVAWAWBWCVXCVXJVYCVXCVVIVXRV
|
|
XIVYBVXCVVHVXQVNVXCVVGEKVVHVXQMVVAUUSUAUBVVGWAWEZWFVXCVXFVXTVXHVYAVXCVVJV
|
|
XSVXEVXCVVHVXQVYFWBZWCVXCVVJVXSVXGVYGWCWGWHWIWHVWTVWSUYJKZVWSVUKKZUNZRVYE
|
|
VWSUYJVUKUULVYHVXOVYJVYDVYHVWDEKZVXMVWDKZRZVXOVYHVWQVWDNZUYIKVWDVWQNZUYHK
|
|
ZVYMVWQVWDUYIVWPVRZVWOWOVWQVWDUYHWJVYPVYOUYGKZVYKRVYLVYKRVYMVWDVWQUYGEUUM
|
|
VYRVYLVYKVYRVYNUYFKZDUPZVXMMZDCXEZRZDOZVYLVWDVWQUYFWJVYSVWQVVDUYEFZVVDVWD
|
|
JFZRZAOZVVDVYTUIZMZVWPVYTUYDFZRZWUFRZAOZDOZWUDAVWQVWDJUYEWKWUHWUMDOZAOWUO
|
|
WUGWUPAWUGWULDOZWUFRWUPWUEWUQWUFDVWPVVDUYDVVAVRZWLWMWULWUFDWNWPPWUMADWQQW
|
|
UNWUCDWUNWUJWUKWUFRZRZAOWUKWUIVWDJFZRZWUCWUMWUTAWUJWUKWUFWRPWUSWVBAWUIVYT
|
|
VRWUJWUFWVAWUKVVDWUIVWDJWSWTXAWUKWUAWVAWUBWUKVYTVWPUYCFVYTVWPNUYCKWUAVWPV
|
|
YTUYCXBVYTVWPUYCXCUXLVWPDUAUBUUNZWURXDSVYTVWDDVQZVWOXFXGSPSWUBVYLDVXMUXLV
|
|
WPWURWVCXHZVYTVXMVWDXIXASWMVYLVYKXJSSVXOVWDVXNMVYMVXNVWDXKVWDVXMWVEXLQWPV
|
|
YIVYCVYIUCUPZUIZUIZVWSNVUIKZUCOWVFVVGMZWVGUXLVTZVNKZWVKVKZVXEMZWVMVXGMZVO
|
|
ZRZRZUCOVYCUCVWSVUIUUOWVIWVRUCWVIWVHVWRNUYMKZWVHVWMNVUHKZRWVRWVHVWRVWMUYM
|
|
VUHUUPWVSWVJWVTWVQWVSWVGVWQNUYLKZVWPWVFVEFZWVJWVGVWQUYLWVFVRZVYQWOWWAWVFV
|
|
WPNUYKKWVFVWPUYKFWWBWVFVWPUYKUCVQWURWOWVFVWPUYKXCWVFVWPVEXBUUQVVAWVFBVQXM
|
|
SWVTWVHVYTUYIFZVYTVWMVUGFZRZDOVYTWVMMZVYTVGKZVYTVXEMZVYTVXGMZVOZRZRZDOWVQ
|
|
DWVHVWMVUGUYIWKWWFWWMDWWDWWGWWEWWLWWDVYTWVHUYHFZWWGWVHVYTUYHXBWWNVYTEKZWV
|
|
KVYTKZRZWWGWWNVYTWVHUYGFZWWORWWPWWORWWQVYTWVHUYGEXPWWRWWPWWOWWRWVHVYTUYFF
|
|
WVHVWDUYEFZVWDVYTJFZRZCOZWWPVYTWVHUYFXBCWVHVYTJUYEXNWXBVVDWVKMZADXEZRZAOZ
|
|
WWPWXBVWDVVDUIZMZWVGVVDUYDFZRZWWTRZAOZCOWXKCOZAOWXFWXAWXLCWXAWXJAOZWWTRWX
|
|
LWWSWXNWWTAWVGVWDUYDWWCWLWMWXJWWTAWNWPPWXKCAWQWXMWXEAWXMWXHWXIWWTRZRZCOWX
|
|
IWXGVYTJFZRZWXEWXKWXPCWXHWXIWWTWRPWXOWXRCWXGVVDVRWXHWWTWXQWXIVWDWXGVYTJWS
|
|
WTXAWXIWXCWXQWXDWXIVVDWVGUYCFVVDWVGNUYCKWXCWVGVVDUYCXBVVDWVGUYCXCUXLWVGAW
|
|
VCWWCXDSVVDVYTAVQZWVDXFXGSPSAWVKVYTUURWPSWMWWPWWOXJSVYTWVKUXLWVGWWCWVCXHX
|
|
LWPQWWEVYTVWMVUFFZWWHRWWKWWHRWWLVYTVWMVUFVGXPWXTWWKWWHWXTVYTVVDVUEFZVVDVW
|
|
MUYKFZRZAOVVDVXDMZVYTVVDHVLZMZVYTVVDLVLZMZVOZRZAOWWKAVYTVWMUYKVUEXNWYCWYJ
|
|
AWYCWYIWYDRWYJWYAWYIWYBWYDWYAVVDVYTVUDFVVDVYTUYRFZVVDVYTVUCFZVOWYIVYTVVDV
|
|
UDXBVVDVYTUYRVUCUUTWYKWYFWYLWYHWYKVVDHNZVYTIFZWYEVYTMWYFWYKVVDVWDUYQFZVWD
|
|
VYTIFZRZCOVWDWYMMZWYPRZCOWYNCVVDVYTIUYQXNWYQWYSCWYOWYRWYPWYOVWDVVDUYPFZVW
|
|
DVVDUHFZVWDHUGFZRZWYRVVDVWDUYPXBWYTXUAVWDUYOKZRXUCVWDVVDUHUYOXPXUDXUBXUAU
|
|
GHVWDXOXQQVVDHVWDWXSXRXSSWMPWYPWYNCWYMVVDHWXSXRYFVWDWYMVYTIWSXASVVDHVYTWX
|
|
SXRXTWYEVYTXKSWYLVVDLNZVYTIFZWYGVYTMWYHWYLVVDVWDVUBFZWYPRZCOVWDXUEMZWYPRZ
|
|
COXUFCVVDVYTIVUBXNXUHXUJCXUGXUIWYPXUGVWDVVDVUAFZXUAVWDLUGFZRZXUIVVDVWDVUA
|
|
XBXUKXUAVWDUYTKZRXUMVWDVVDUHUYTXPXUNXULXUAUGLVWDXOXQQVVDLVWDWXSLEUVAUVBZX
|
|
SSWMPWYPXUFCXUEVVDLWXSXUOYFVWDXUEVYTIWSXASVVDLVYTWXSXUOXTWYGVYTXKSUVCSWYB
|
|
VWMVVDVEFWYDVVDVWMVEXBVWDVVDVWOXMQXGWYIWYDXJQPWYIWWKAVXDVWDYGWYDWYFWWIWYH
|
|
WWJWYDWYEVXEVYTVVDVXDHYAYBWYDWYGVXGVYTVVDVXDLYAYBWGXASWMWWKWWHXJSXGPWWLWV
|
|
QDWVMWVKUVDWWGWWHWVLWWKWVPWWGWWHWVMVGKWVLVYTWVMVGXIWVKUVEYCWWGWWIWVNWWJWV
|
|
OVYTWVMVXEYDVYTWVMVXGYDWGWHXASXGQPWVQVYCUCVVGVVAYGWVJWVLVXRWVPVYBWVJWVKVX
|
|
QVNWVJWVGVXPMWVKVXQMWVFVVGUVFUXLWVGVXPUVGWEZWFWVJWVNVXTWVOVYAWVJWVMVXSVXE
|
|
WVJWVKVXQXUPWBZWCWVJWVMVXSVXGXUQWCWGWHXASUVHXGQUVIVWIVXAVXKVWIVVQVXJVWIVV
|
|
PVXIVVIVWIVVMVXFVVOVXHVWIVVLVXEVVJVWIVVKVXDHVVCVWDUVJZYEYBVWIVVNVXGVVJVWI
|
|
VVKVXDLXURYEYBWGWTWIUVKYCUVLVWIVVQBEUWAUVMUVNSVWFHVVDNZVWDIFZVVEVWDMVWHVW
|
|
FVWDVYTUXDFZVYTVVDUXCFZRZDOVYTXUSMZVYTVWDIFZRZDOXUTDVWDVVDUXCUXDWKXVCXVFD
|
|
XVCXVEXVDRXVFXVAXVEXVBXVDVWDVYTIXBXVBVYTVVDUGFZVYTUXBKZRZVYTHUHFZXVGRZXVD
|
|
VYTVVDUGUXBXPXVIXVGXVJRXVKXVHXVJXVGUHHVYTXOXQXVGXVJXJQHVVDVYTXRWXSXSSXGXV
|
|
EXVDXJQPXVEXUTDXUSHVVDXRWXSYFVYTXUSVWDIWSXASHVVDVWDXRWXSXTVVEVWDXKSXGVWKV
|
|
WHXJQPVWKVVSCVVEHVVDXRWXSUVOVWHVWJVVRBEVWHVWIVVFVVQVWDVVEVVCUVPUVQUVRXASU
|
|
VSUVTVVTVWCAUWQVVSUWBUWCUWDUWSVUSUWRAUWEUXEVURUXCUXDUGUXBYHUWTUXAUHYITHVR
|
|
ZYJYKIYLTYMVUQVUPVULVUOUYJVUKUYIUYHUYGEUYFJUYEYNUYDUYCUYBUYAHUXFUXTJYNUWF
|
|
UXSUXRUXQUXHUXPUXGJYNYOTUXIUXOJYNTUXNJUXMYNUXLWVCUWGYMUWHYKYPUWIYOUWJUWLX
|
|
RYJYOTYQYMTYRYKTZYQVUIVUJUYMVUHUYLUYKVEUWKTZYQYQVUGUYIVUFVGUYKVUEXVNVUDUY
|
|
RVUCIUYQYLUYPUHUYOYIUYNUXAUGYHTZXVLYJYKTYMIVUBYLVUAUHUYTYIUYNUYSXVOLVRYJY
|
|
KTYMYSTYMUWMYKXVMYMYPHXRYTYJUWNVUNVUMEYRYTYTYTYJYOUWOYJYSUWP $.
|
|
$}
|
|
|
|
${
|
|
$d k m $. $d k n $. $d M m $. $d m n $. $d m t $. $d m x $. $d n t $.
|
|
$d k t $. $d k x $. $d M t $. $d M x $. $d t x $.
|
|
$( Lemma for ~ nchoice . If the special set of a cardinal is finite, then
|
|
so is the special set of its T-raising, and there is a calculable
|
|
relationship between their sizes. Theorem 7.2 of [Specker] p. 974.
|
|
(Contributed by SF, 19-Mar-2015.) $)
|
|
nchoicelem17 $p |- ( ( <_c We NC /\ M e. NC /\ ( Sp[ac] ` M ) e. Fin ) ->
|
|
( ( Sp[ac] ` T_c M ) e. Fin /\
|
|
( Nc ( Sp[ac] ` T_c M ) = ( T_c Nc ( Sp[ac] ` M ) +c 1c ) \/
|
|
Nc ( Sp[ac] ` T_c M ) = ( T_c Nc ( Sp[ac] ` M ) +c 2c ) ) ) ) $=
|
|
( vt vk vm cncs wcel cspac cfv cfin ctc cnc c1c cplc wceq c2c wa cnnc c0c
|
|
wi addceq1d vx vn clec cwe wbr wo finnc wrex risset nchoicelem13 ad2antlr
|
|
cv wb 1cnc fvex ncelncsi dflec2 mp2an eqtr ancoms eqtr2 ex adantl addceq2
|
|
jcai addccom syl6eq eqeq2d rspcev mpan nnnc sylan2 syl5ibr adantll nclenn
|
|
3expia wral nchoicelem16 addcid1 imbi1d ralbidv imbi2d fveq2 nceqd eqeq1d
|
|
weq tceq fveq2d eleq1d syl eqeq12d orbi12d anbi12d imbi12d cbvralv syl6bb
|
|
cce co wn nchoicelem14 w3a c3c nchoicelem9 id 2nnc syl6eqel 2p1e3c peano2
|
|
ax-mp eqeltrri jaoi sylibr 1p1e2c eqeq2i eqtri orbi12i nchoicelem3 df1c3g
|
|
csn adantr eqtr4d tc1c 3adant1 ralrimiva wne ad2antrl simp2 simpr 3adant2
|
|
syld nchoicelem7 addceq1 syl5 com23 3impia imp mpd expr syl5bi rexlimdva
|
|
mpbird cltc 0cnsuc peano1 1cnnc addccan1 mp3an23 necon3bid mpbiri addcid2
|
|
jca neeq2i sylib eqnetrd syl2an necomd brltc sylanbrc nchoicelem15 df-3an
|
|
ceclnn1 mp3an1 rspccva tce2 eqtr3d peano4nc 3syl mpbid tccl te0c 3ad2ant1
|
|
syl2anc 3imtr4g tcdi addceq2i addc32 orim12d anim12d embantd sylbid exp4b
|
|
sylan2br 3exp com12 a2d finds rspccv syl6com 3syld imp3a exp3a ) UCEUDUEZ
|
|
AEFZAGHZIFZAJZGHZIFZUWQKZUWNKZJZLMZNZUWSUXAOMZNZUFZPZUWOUWTQFZUWLUWMPZUXG
|
|
UWNUGUXHBULZUWTNZBQUHUXIUXGBUWTQUIUXIUXKUXGBQUXIUXJQFZPZLUWTUCUEZUXKUXGSZ
|
|
UWMUXNUWLUXLAUJUKUXNUWTLUAULZMZNZUAEUHZUXMUXOLEFZUWTEFUXNUXSUMUNUWNAGUOUP
|
|
LUWTUAUQURUXMUXRUXOUAEUXMUXPEFZPZUXRUXKUXGUXRUXKPZUXJUXQNZUXRPUYBUXGUYCUY
|
|
DUXRUXKUXRUYDUXJUWTUXQUSUTUXKUYDUXRSUXRUXKUYDUXRUXJUWTUXQVAVBVCVEUYBUYDUX
|
|
RUXGUYBUYDUXPUXJUCUEZUXPQFZUXRUXGSZUXLUYAUYDUYESUXIUYDUYEUXLUYAPUXJUXPCUL
|
|
ZMZNZCEUHZUXTUYDUYKUNUYJUYDCLEUYHLNZUYIUXQUXJUYLUYIUXPLMUXQUYHLUXPVDUXPLV
|
|
FVGVHVIVJUYAUXLUYEUYKUMZUXLUYAUXJEFUYMUXJVKUXPUXJCUQVLUTVMVNUXLUYAUYEUYFS
|
|
ZUXIUYAUXLUYNUYAUXLUYEUYFUXPUXJVOVPUTVNUXMUYFUYGSZUYAUXIUYOUXLUWLUWMUYOUW
|
|
LUYFUWMUYGUYFUWLDULZGHZKZUXQNZUYPJZGHZIFZVUAKZUYRJZLMZNZVUCVUDOMZNZUFZPZS
|
|
ZDEVQZUWMUYGSUWLUYRLUXJMZNZVUJSZDEVQZSUWLUYRLNZVUJSZDEVQZSUWLUYRLUBULZMZN
|
|
ZVUJSZDEVQZSUWLUYHGHZKZLVUTLMZMZNZUYHJZGHZIFZVVKKZVVFJZLMZNZVVMVVNOMZNZUF
|
|
ZPZSZCEVQZSUWLVULSBUBUXPBDVRUXJRNZVUPVUSUWLVWCVUOVURDEVWCVUNVUQVUJVWCVUML
|
|
UYRVWCVUMLRMLUXJRLVDLVSVGVHVTWAWBBUBWFZVUPVVDUWLVWDVUOVVCDEVWDVUNVVBVUJVW
|
|
DVUMVVAUYRUXJVUTLVDVHVTWAWBUXJVVGNZVUPVWBUWLVWEVUPUYRVVHNZVUJSZDEVQVWBVWE
|
|
VUOVWGDEVWEVUNVWFVUJVWEVUMVVHUYRUXJVVGLVDVHVTWAVWGVWADCEDCWFZVWFVVIVUJVVT
|
|
VWHUYRVVFVVHVWHUYQVVEUYPUYHGWCWDZWEVWHVUBVVLVUIVVSVWHVUAVVKIVWHUYTVVJGUYP
|
|
UYHWGWHZWIVWHVUFVVPVUHVVRVWHVUCVVMVUEVVOVWHVUAVVKVWJWDZVWHVUDVVNLVWHUYRVV
|
|
FNVUDVVNNVWIUYRVVFWGWJZTWKVWHVUCVVMVUGVVQVWKVWHVUDVVNOVWLTWKWLWMWNWOWPWBB
|
|
UAWFZVUPVULUWLVWMVUOVUKDEVWMVUNUYSVUJVWMVUMUXQUYRUXJUXPLVDVHVTWAWBUWLVURD
|
|
EUWLUYPEFZPVUQUYPRWQWREFWSZVUJVWNVUQVWOSUWLVWNVUQVWOUYPWTVBVCUWLVWNVWOVUJ
|
|
UWLVWNVWOXAZVUBVUIVWPVUCONZVUCXBNZUFZVUBUYPXCZVWSVUCQFZVUBVWQVXAVWRVWQVUC
|
|
OQVWQXDXEXFVWRVUCXBQVWRXDOLMZXBQXGOQFZVXBQFXEOXHXIXJXFXKVUAUGXLWJVWPVUIVU
|
|
CLLMZNZVUCLOMZNZUFZVWPVWSVXHVWTVXEVWQVXGVWRVXDOVUCXMXNVXFXBVUCVXFVXBXBLOV
|
|
FXGXOXNXPXLVWNVWOVUIVXHUMUWLVWNVWOPZVUFVXEVUHVXGVXIVUEVXDVUCVXIVUDLLVXIVU
|
|
DLJZLVXIVUQVUDVXJNVXIUYRUYPXSZKZLVXIUYQVXKUYPXQWDVWNLVXLNVWOUYPEXRXTYAUYR
|
|
LWGWJYBVGZTVHVXIVUGVXFVUCVXIVUDLOVXMTVHWLYCUUAUUKVPYJYDVUTQFZUWLVVDVWBUWL
|
|
VXNVVDVWBSUWLVXNVVDVWBUWLVXNVVDXAZVWACEVVIVVFVVGLMZNZVXOUYHEFZPVVTVVHVXPV
|
|
VFLVVGVFXNVXOVXRVXQVVTVXOVXRVXQPZPZLVVFUUBUEZVVTVXTLVVFUCUEZLVVFYEVYAVXRV
|
|
YBVXOVXQUYHUJYFVXTVVFLVXOVXNVXQVVFLYEVXSUWLVXNVVDYGVXRVXQYHVXNVXQPVVFVXPL
|
|
VXNVXQYHVXNVXPLYEZVXQVXNVXPRLMZYEZVYCVXNVYEVVGRYEVUTUUCVXNVXPVYDVVGRVXNVV
|
|
GQFZVXPVYDNVVGRNUMZVUTXHZVYFRQFLQFVYGUUDUUELVVGRUUFUUGWJUUHUUIVYDLVXPLUUJ
|
|
UULUUMXTUUNUUOUUPLVVFUUQUURVXTVYAUYHRWQWREFZVVTVXRVYAVYISVXOVXQVXRVYAVYIU
|
|
YHUUSVBYFVXOVXSVYIVVTVXSVYIPVXOVXRVXQVYIXAZVVTVXRVXQVYIUUTVXOVYJPOUYHWQWR
|
|
ZEFZVVTVYJVYLVXOVXRVYIVYLVXQVXCVXRVYIVYLXEOUYHUVAUVBYIVCVXOVYJVYLVVTSZUWL
|
|
VXNVVDVYJVYMSUWLVXNPZVYJVVDVYMVYNVYJVVDVYLVVTVVDVYLPVYKGHZKZVVANZVYKJZGHZ
|
|
IFZVYSKZVYPJZLMZNZWUAWUBOMZNZUFZPZSZVYNVYJPZVVTVVCWUIDVYKEUYPVYKNZVVBVYQV
|
|
UJWUHWUKUYRVYPVVAWUKUYQVYOUYPVYKGWCWDZWEWUKVUBVYTVUIWUGWUKVUAVYSIWUKUYTVY
|
|
RGUYPVYKWGWHZWIWUKVUFWUDVUHWUFWUKVUCWUAVUEWUCWUKVUAVYSWUMWDZWUKVUDWUBLWUK
|
|
UYRVYPNVUDWUBNWULUYRVYPWGWJZTWKWUKVUCWUAVUGWUEWUNWUKVUDWUBOWUOTWKWLWMWNUV
|
|
CWUJWUIVYQOVVJWQWRZGHZIFZWUQKZWUCNZWUSWUENZUFZPZSZVVTVYJWUIWVDUMZVYNVXRVY
|
|
IWVEVXQVXRVYIPZWUHWVCVYQWVFVYTWURWUGWVBWVFVYSWUQIWVFVYRWUPGUYHUVDWHZWIWVF
|
|
WUDWUTWUFWVAWVFWUAWUSWUCWVFVYSWUQWVGWDZWEWVFWUAWUSWUEWVHWEWLWMWBYIVCWUJVY
|
|
QWVCVVTWUJVYPVVGVVAWUJVYPLMZVXPNZVYPVVGNZVYJWVJVYNVYJVVFWVIVXPVXRVYIVVFWV
|
|
INZVXQUYHYKZYIVXRVXQVYIYGUVEVCVXNWVJWVKUMZUWLVYJVXNVYFVVGEFZWVNVYHVVGVKVY
|
|
PEFZWVOWVNVYOVYKGUOUPZVYPVVGUVFVJUVGUKUVHVUTLVFVGWUJWURVVLWVBVVSWUJWUSQFZ
|
|
VVMQFZWURVVLWVRWVSWUJWUSLMZQFWUSXHWUJVVMWVTQVYJVVMWVTNZVYNVXRVXQWWAVYIVXR
|
|
VVJEFVVJRWQWREFWWAUYHUVIUYHUVJVVJYKUVLUVKVCZWIVMWUQUGVVKUGUVMWUJWUTVVPWVA
|
|
VVRWUTVVPWUJWVTWUCLMZNWUSWUCLYLWUJVVMWVTVVOWWCWWBWUJVVNWUCLWUJVVNWVIJZWUC
|
|
VYJVVNWWDNZVYNVXRVYIWWEVXQWVFWVLWWEWVMVVFWVIWGWJYIVCWWDWUBVXJMZWUCWVPUXTW
|
|
WDWWFNWVQUNVYPLUVNURVXJLWUBYBUVOXOVGZTWKVMWVAVVRWUJWVTWUELMZNWUSWUELYLWUJ
|
|
VVMWVTVVQWWHWWBWUJVVQWUCOMWWHWUJVVNWUCOWWGTWUBLOUVPVGWKVMUVQUVRUVSUVTYMUW
|
|
AYNYOYPYQUWBYRYJYQYRYSYDUWCUWDUWEUWFVUKUYGDAEUYPANZUYSUXRVUJUXGWWIUYRUWTU
|
|
XQWWIUYQUWNUYPAGWCWDZWEWWIVUBUWRVUIUXFWWIVUAUWQIWWIUYTUWPGUYPAWGWHZWIWWIV
|
|
UFUXCVUHUXEWWIVUCUWSVUEUXBWWIVUAUWQWWKWDZWWIVUDUXALWWIUYRUWTNVUDUXANWWJUY
|
|
RUWTWGWJZTWKWWIVUCUWSVUGUXDWWLWWIVUDUXAOWWMTWKWLWMWNUWGUWHYNYPXTXTUWIUWJY
|
|
MUWKYTYSYQYTYSYSYO $.
|
|
$}
|
|
|
|
${
|
|
$d c x $. $d c p $. $d c q $. $d p q $. $d p x $. $d q x $.
|
|
$( Lemma for ~ nchoice . Set up stratification for ~ nchoicelem19 .
|
|
(Contributed by SF, 20-Mar-2015.) $)
|
|
nchoicelem18 $p |- { x | ( Sp[ac] ` x ) e. Fin } e. _V $=
|
|
( vp vq vc cncs ccompl csset ccnv cv wcel wceq c1c cima cspac wo wa bitri
|
|
cfin complex ssetex cins3 c2c cce co w3a copab cimage ccom cfix cres ctxp
|
|
crn cins2 csymdif cuni1 cin cun cfv cab cvv wn pm2.1 cdm wfn fnspac ax-mp
|
|
c0 fndm eleq2i sylnbir 0fin syl6eqel pm4.71i orbi1i elun vex elcompl elin
|
|
ndmfv csn cclos1 spacval eleq1d eluni1 wbr df-br spacvallem1 nchoicelem10
|
|
wrex cop snex rexbii elima 3bitr4i syl6rbbr pm5.32i orbi12i andir mpbiran
|
|
risset abbi2i ncsex ins3ex cnvex imageex coex fixex resex ins2ex symdifex
|
|
txpex rnex 1cex imaex finex uni1ex inex unex eqeltrri ) EFZEGUAZGFZHZGHZG
|
|
BIZEJCIZEJYFUBYEUCUDKUEBCUFZUGZUHZUIZUJZUKZULZFZUMZUNZLMZFZRMZUOZUPZUQZAI
|
|
ZNURZRJZAUSUTUUEAUUBUUCUUBJZUUCEJZVAZUUGOZUUEUUGVBUUHUUGUUEPZOZUUHUUEPZUU
|
|
JOUUFUUIUUEPUUHUULUUJUUHUUEUUHUUDVGRUUGUUCNVCZJUUDVGKUUMEUUCNEVDUUMEKVEEN
|
|
VHVFVIUUCNVSVJVKVLVMVNUUFUUCXTJZUUCUUAJZOUUKUUCXTUUAVOUUNUUHUUOUUJUUCEAVP
|
|
ZVQUUOUUGUUCYTJZPUUJUUCEYTVRUUGUUQUUEUUGUUEUUCVTZYGWAZRJZUUQUUGUUDUUSRBCU
|
|
UCWBWCUUQUURYSJZUUTUUCYSUUPWDDIZUURYRWEZDRWIUVBUUSKZDRWIUVAUUTUVCUVDDRUVC
|
|
UVBUURWJYRJUVDUVBUURYRWFYGUURDBCWGZUUCWKWHQWLDUURYRRWMDUUSRWTWNQWOWPQWQQU
|
|
UHUUGUUEWRWNWSXAXTUUAEXBSEYTXBYSYRRYQYPLYAYOGTXCYNYMYLYCYKYBGTSXDYDYJGTXD
|
|
YIGYHTYGUVEXEXFXGXHXKXLSXIXJXMXNSXOXNXPXQXRXS $.
|
|
$}
|
|
|
|
${
|
|
$d m n $. $d m x $. $d n x $. $d m p $. $d n p $.
|
|
$( Lemma for ~ nchoice . Assuming well-ordering, there is a cardinal with
|
|
a finite special set that is its own T-raising. Theorem 7.3 of
|
|
[Specker] p. 974. (Contributed by SF, 20-Mar-2015.) $)
|
|
nchoicelem19 $p |- ( <_c We NC ->
|
|
E. m e. NC ( ( Sp[ac] ` m ) e. Fin /\ T_c m = m ) ) $=
|
|
( vn vx vp clec cncs wbr cv cspac cfv cfin wcel wi wa wceq eleq1d cvv cnc
|
|
fveq2 syl cwe wral wrex ctc nchoicelem18 weq id vvex ncelncsi csn c0c cce
|
|
co wn c1c cltc cpw1 cpw ltcpw1pwg ax-mp df1c2 nceqi pwv eqcomi 3brtr4i wb
|
|
nchoicelem8 mpbiri nchoicelem3 sylancr snfi syl6eqel rspcev weds cantisym
|
|
mpan2 simpll cstrict cfound cin df-we breqi brin simplbi cpartial cconnex
|
|
bitri sopc cref ctrans porta simp3bi simplr tccl simprr cplc nchoicelem17
|
|
c2c wo simprl syl3anc simpld breq2 imbi12d rspcv letc 3expia nchoicelem12
|
|
syl3c syl2anc ad2ant2lr imp ad2ant2l mpd tlecg mpbid breq1 imbi2d ralbidv
|
|
anbi12d anbi2d tceq breq12d com12 an32s rexlimdva adantlr antid imdistand
|
|
syld exp32 reximdva ) EFUAGZAHZIJZKLZBHZIJZKLZYNYQEGZMZBFUBZNZAFUCYPYNUDZ
|
|
YNOZNZAFUCYMCHZIJZKLZYPYSCABFECUECAUFUUHYOKUUGYNISPCBUFUUHYRKUUGYQISPYMUG
|
|
YMQRZFLZUUJIJZKLZUUICFUCQUHUIZYMUULUUJUJZKYMUUKUUJUKULUMFLUNZUULUUOOUUNYM
|
|
UUPUORZUUJUPGZQUQZRZQURZRZUUQUUJUPQQLUUTUVBUPGUHQQUSUTUOUUSVAVBUVBUUJUVAQ
|
|
VCVBVDVEYMUUKUUPUURVFUUNUUJVGVPVHUUJVIVJUUJVKVLUUIUUMCUUJFUUGUUJOUUHUULKU
|
|
UGUUJISPVMVJVNYMUUCUUFAFYMYNFLZNZYPUUBUUEUVDYPUUBUUEUVDUUCNZFEUUDYNUVEYME
|
|
FVOGZYMUVCUUCVQZYMEFVRGZUVFYMUVHEFVSGZYMEFVRVSVTZGUVHUVINEFUAUVJWAWBEFVRV
|
|
SWCWGWDUVHEFWEGZUVFUVHUVKEFWFGFEWHWDUVKEFWIGEFWJGUVFFEWKWLTTTUVEUVCUUDFLZ
|
|
YMUVCUUCWMZYNWNTZUVMUVEYNUUDEGZUUDYNEGZUVEUVLUUBUUDIJZKLZUVOUVNUVDYPUUBWO
|
|
UVEUVRUVQRZYORUDZUOWPOUVSUVTWRWPOWSZUVEYMUVCYPUVRUWANUVGUVMUVDYPUUBWTYNWQ
|
|
XAXBUUAUVRUVOMBUUDFYQUUDOZYSUVRYTUVOUWBYRUVQKYQUUDISPYQUUDYNEXCXDXEXIZUVE
|
|
UVOYNDHZUDZOZDFUCZUVPUVEUVCUVCUVOUWGMUVMUVMUVCUVCUVOUWGYNYNDXFXGXJYMUUCUW
|
|
GUVPMUVCYMUUCNUWFUVPDFYMUWDFLZUUCUWFUVPMUWFYMUWHNZUUCNZUVPUWFUWJUVPMUWIUW
|
|
EIJZKLZYSUWEYQEGZMZBFUBZNZNZUWEUDZUWEEGZMUWQUWEUWDEGZUWSUWQUWDIJZKLZUWTUW
|
|
HUWLUXBYMUWOUWDXHXKUWHUWOUXBUWTMZYMUWLUWHUWOUXCUWNUXCBUWDFBDUFZYSUXBUWMUW
|
|
TUXDYRUXAKYQUWDISPYQUWDUWEEXCXDXEXLXMXNUWQUWEFLZUWHUWTUWSVFUWQUWHUXEYMUWH
|
|
UWPWMZUWDWNTUXFUWEUWDXOXJXPUWFUWJUWQUVPUWSUWFUUCUWPUWIUWFYPUWLUUBUWOUWFYO
|
|
UWKKYNUWEISPUWFUUAUWNBFUWFYTUWMYSYNUWEYQEXQXRXSXTYAUWFUUDUWRYNUWEEYNUWEYB
|
|
UWFUGYCXDVHYDYEYFYGYJXNUWCYHYKYIYLXN $.
|
|
$}
|
|
|
|
${
|
|
$d m n $.
|
|
$( Cardinal less than or equal does not well-order the cardinals. This is
|
|
equivalent to saying that the axiom of choice from ZFC is false in NF.
|
|
Theorem 7.5 of [Specker] p. 974. (Contributed by SF, 20-Mar-2015.) $)
|
|
nchoice $p |- -. <_c We NC $=
|
|
( vn vm cncs cv ctc c1c cplc wceq c2c wo cnnc wrex wcel wn cspac cfv cfin
|
|
wa cnc eqeq1d cwe wbr nchoicelem1 nchoicelem2 ioran sylanbrc nchoicelem19
|
|
clec nrex finnc biimpi ad2antrl simpll simplr simprl nchoicelem17 syl3anc
|
|
simprd simprr fveq2d nceqd orbi12d mpbid id tceq addceq1d eqeq12d syl2anc
|
|
rspcev ex rexlimdva mpd mto ) UHCUAUBZADZVOEZFGZHZVOVPIGZHZJZAKLZWAAKVOKM
|
|
VRNVTNWANVOUCVOUDVRVTUEUFUIVNBDZOPZQMZWCEZWCHZRZBCLWBBUGVNWHWBBCVNWCCMZRZ
|
|
WHWBWJWHRZWDSZKMZWLWLEZFGZHZWLWNIGZHZJZWBWEWMWJWGWEWMWDUJUKULWKWFOPZSZWOH
|
|
ZXAWQHZJZWSWKWTQMZXDWKVNWIWEXEXDRVNWIWHUMVNWIWHUNWJWEWGUOWCUPUQURWKXBWPXC
|
|
WRWKXAWLWOWKWTWDWKWFWCOWJWEWGUSUTVAZTWKXAWLWQXFTVBVCWAWSAWLKVOWLHZVRWPVTW
|
|
RXGVOWLVQWOXGVDZXGVPWNFVOWLVEZVFVGXGVOWLVSWQXHXGVPWNIXIVFVGVBVIVHVJVKVLVM
|
|
$.
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Finite recursion
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Declare new constants $)
|
|
$c FRec $. $( Finite recursion generator $)
|
|
|
|
$( Extend the definition of a class to include the finite recursive function
|
|
generator. $)
|
|
cfrec $a class FRec ( F , I ) $.
|
|
|
|
${
|
|
$d F x $. $d I x $.
|
|
$( Define the finite recursive function generator. This is a function over
|
|
` Nn ` that obeys the standard recursion relationship. Definition
|
|
adapted from theorem XI.3.24 of [Rosser] p. 412. (Contributed by Scott
|
|
Fenton, 30-Jul-2019.) $)
|
|
df-frec $a |- FRec ( F , I ) =
|
|
Clos1 ( { <. 0c , I >. } , PProd ( ( x e. _V |-> ( x +c 1c ) ) , F ) ) $.
|
|
$}
|
|
|
|
${
|
|
$d F x $. $d G x $. $d I x $. $d J x $.
|
|
$( Equality theorem for finite recursive function generator. (Contributed
|
|
by Scott Fenton, 31-Jul-2019.) $)
|
|
freceq12 $p |- ( ( F = G /\ I = J ) -> FRec ( F , I ) = FRec ( G , J ) ) $=
|
|
( vx wceq wa c0c cop csn cvv cv c1c cplc cmpt cpprod cclos1 cfrec df-frec
|
|
syl opeq2 sneqd clos1eq1 pprodeq2 clos1eq2 sylan9eqr 3eqtr4g ) ABFZCDFZGH
|
|
CIZJZEKELMNOZAPZQZHDIZJZULBPZQZACRBDRUIUHUNUPUMQZURUIUKUPFUNUSFUIUJUOCDHU
|
|
AUBUMUKUPUCTUHUMUQFUSURFABULUDUMUPUQUETUFEACSEBDSUG $.
|
|
$}
|
|
|
|
${
|
|
$d G x $. $d I x $.
|
|
frecex.1 $e |- F = FRec ( G , I ) $.
|
|
$( The finite recursive function generator preserves sethood. (Contributed
|
|
by Scott Fenton, 30-Jul-2019.) $)
|
|
frecexg $p |- ( G e. V -> F e. _V ) $=
|
|
( vx wcel c0c cop csn cvv c1c cplc cmpt cpprod cclos1 cfrec df-frec eqtri
|
|
cv snex csucex pprodexg mpan clos1exg sylancr syl5eqel ) BDGZAHCIZJZFKFTL
|
|
MNZBOZPZKABCQUMEFBCRSUHUJKGULKGZUMKGUIUAUKKGUHUNFUBUKBKDUCUDULUJKKUEUFUG
|
|
$.
|
|
|
|
frecex.2 $e |- G e. _V $.
|
|
$( The finite recursive function generator preserves sethood. (Contributed
|
|
by Scott Fenton, 30-Jul-2019.) $)
|
|
frecex $p |- F e. _V $=
|
|
( cvv wcel frecexg ax-mp ) BFGAFGEABCFDHI $.
|
|
$}
|
|
|
|
${
|
|
$d F x y z a b c d i $. $d G x y z a b c d i $. $d I i x $.
|
|
frecxp.1 $e |- F = FRec ( G , I ) $.
|
|
frecxp.2 $e |- G e. _V $.
|
|
$( Subset relationship for the finite recursive function generator.
|
|
(Contributed by Scott Fenton, 30-Jul-2019.) $)
|
|
frecxp $p |- F C_ ( Nn X. ( ran G u. { I } ) ) $=
|
|
( vy vz vx va vb vc vd cnnc cvv wcel cv wceq c0c wa c0 vi crn csn cun cxp
|
|
cfrec wss eqid freceq12 mpan sneq uneq2d xpeq2d sseq12d cop c1c cplc cmpt
|
|
cpprod wbr wal wral nncex rnex snex unex xpex peano1 vex snid elun2 ax-mp
|
|
0cex opex snss opelxp bitr3i mpbir2an w3a wex brpprod brcsuc brelrn elun1
|
|
wi syl peano2 anim12ci adantrr eleq1 anbi1d syl5ibr exp3a sylbi wb syl6bb
|
|
adantr adantl imbi12d 3impia exlimivv impcom ax-gen rgenw pprodex df-frec
|
|
csucex clos1induct mp3an vtoclg wn cclos1 opexb simprbi con3i snprc sylib
|
|
imp clos1eq1 clos10 syl6eq 0ss syl6eqss syl5eqss pm2.61i eqsstri ) ABCUFZ
|
|
MBUBZCUCZUDZUEZDCNOZYGYKUGZBUAPZUFZMYHYNUCZUDZUEZUGZYMUACNYNCQZYOYGYRYKBB
|
|
QYTYOYGQBUHBBYNCUIUJYTYQYJMYTYPYIYHYNCUKULUMUNYRNORYNUOZUCZYRUGZFPZYROZUU
|
|
DGPZHNHPUPUQURZBUSZUTZSUUFYROZWEZGVAZFYOVBYSMYQVCYHYPBEVDYNVEVFVGUUCRMOZY
|
|
NYQOZVHYNYPOUUNYNUAVIZVJYNYPYHVKVLUUCUUAYROUUMUUNSUUAYRRYNVMUUOVNVORYNMYQ
|
|
VPVQVRUULFYOUUKGUUIUUEUUJUUIUUDIPZJPZUOZQZUUFKPZLPZUOZQZUUPUUTUUGUTZUUQUV
|
|
ABUTZSZVSZLVTKVTZJVTIVTUUEUUJWEZIJKLUUDUUFUUGBWAUVHUVIIJUVGUVIKLUUSUVCUVF
|
|
UVIUVFUVIUUSUVCSZUUPMOZUUQYQOZSZUUTMOZUVAYQOZSZWEZUVDUVEUVQUVDUUTUUPUPUQZ
|
|
QZUVEUVQWEHUUPUUTIVIKVIWBUVSUVEUVMUVPUVEUVMSUVPUVSUVRMOZUVOSZUVEUVKUWAUVL
|
|
UVEUVOUVKUVTUVEUVAYHOUVOUUQUVABWCUVAYHYPWDWFUUPWGWHWIUVSUVNUVTUVOUUTUVRMW
|
|
JWKWLWMWNXRUVJUUEUVMUUJUVPUUSUUEUVMWOUVCUUSUUEUURYROUVMUUDUURYRWJUUPUUQMY
|
|
QVPWPWQUVCUUJUVPWOUUSUVCUUJUVBYROUVPUUFUVBYRWJUUTUVAMYQVPWPWRWSWLWTXAXAWN
|
|
XBXCXDFGYOUUHUUBNYRUUAVEUUGBHXGEXEZHBYNXFXHXIXJYLXKZYGRCUOZUCZUUHXLZYKHBC
|
|
XFUWCUWFTYKUWCUWFTUUHXLZTUWCUWETQZUWFUWGQUWCUWDNOZXKUWHUWIYLUWIRNOYLRCXMX
|
|
NXOUWDXPXQUUHUWETXSWFUWGUUHUWBUWGUHXTYAYKYBYCYDYEYF $.
|
|
$}
|
|
|
|
${
|
|
$d G g $. $d I g $.
|
|
frecxpg.1 $e |- F = FRec ( G , I ) $.
|
|
$( Subset relationship for the finite recursive function generator.
|
|
(Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
frecxpg $p |- ( G e. V -> F C_ ( Nn X. ( ran G u. { I } ) ) ) $=
|
|
( vg wcel cfrec cnnc crn csn cun cxp cv wss wceq eqid freceq12 mpan2 rneq
|
|
uneq1d xpeq2d sseq12d vex frecxp vtoclg syl5eqss ) BDGABCHZIBJZCKZLZMZEFN
|
|
ZCHZIUMJZUJLZMZOUHULOFBDUMBPZUNUHUQULURCCPUNUHPCQUMBCCRSURUPUKIURUOUIUJUM
|
|
BTUAUBUCUNUMCUNQFUDUEUFUG $.
|
|
|
|
$}
|
|
|
|
${
|
|
$d F w x y z t $. $d G w x y z t $. $d I w x y z t $. $d ph w x y z t $.
|
|
dmfrec.1 $e |- F = FRec ( G , I ) $.
|
|
dmfrec.2 $e |- ( ph -> G e. V ) $.
|
|
dmfrec.3 $e |- ( ph -> I e. dom G ) $.
|
|
dmfrec.4 $e |- ( ph -> ran G C_ dom G ) $.
|
|
$( The domain of the finite recursive function generator is the naturals.
|
|
(Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
dmfrec $p |- ( ph -> dom F = Nn ) $=
|
|
( vx vt vw vy vz cnnc wcel cvv c0c syl wbr cdm crn csn cun cxp wss dmxpss
|
|
frecxpg dmss 3syl syl6ss cv c1c cplc wi wral frecexg cop cmpt cpprod wrex
|
|
dmexg wo 0cex opexg mpan snidg orcd wb snex csucex pprodexg cfrec df-frec
|
|
cclos1 eqtri clos1basesucg sylancr mpbird opeldm wex eldm2 wceq vex elsnc
|
|
wa opex opth bitri simprbi eleq1 biimprcd syl5 cproj2 cproj1 opeq qrpprod
|
|
breq1i brelrn sseld adantr rexlimdva jaod ancld clos1conn eximi eldm eqid
|
|
sylbid 1cex addcex brcsuc mpbir mpbiran bitr4i anbi2i 19.42v 3imtr4i syl6
|
|
exbii exlimdv syl5bi ralrimivw peano5 syl3anc eqssd ) ABUAZOAYGOCUBZDUCUD
|
|
ZUEZUAZOACEPZBYJUFYGYKUFGBCDEFUHBYJUIUJOYIUGUKAYGQPZRYGPZJULZYGPZYOUMUNZY
|
|
GPZUOZJOUPOYGUFABQPZYMAYLYTGBCDEFUQSBQVBSARDURZBPZYNAUUBUUAUUAUCZPZKULZUU
|
|
ALQLULUMUNUSZCUTZTKBVAZVCZAUUDUUHAUUAQPZUUDADCUAZPZUUJHRQPUULUUJVDRDQUUKV
|
|
EVFSUUAQVGSVHAUUCQPZUUGQPZUUBUUIVIUUAVJZAYLUUNGUUFQPYLUUNLVKUUFCQEVLVFSZK
|
|
UUABUUGUUCQQBCDVMUUCUUGVOFLCDVNVPZVQVRVSRDBVTSAYSJOYPYOMULZURZBPZMWAAYRMY
|
|
OBWBAUUTYRMAUUTUUTUURUUKPZWFZYRAUUTUVAAUUTUUSUUCPZUUEUUSUUGTZKBVAZVCZUVAA
|
|
UUMUUNUUTUVFVIUUOUUPKUUSBUUGUUCQQUUQVQVRAUVCUVAUVEUVCUURDWCZAUVAUVCYORWCZ
|
|
UVGUVCUUSUUAWCUVHUVGWFUUSUUAYOUURJWDZMWDWGWEYOUURRDWHWIWJAUULUVGUVAUOHUVG
|
|
UVAUULUURDUUKWKWLSWMAUVDUVAKBAUVDUVAUOUUEBPUVDUURYHPZAUVAUVDUUEWNZUURCTZU
|
|
VJUVDUUEWOZYOUUFTZUVLUVDUVMUVKURZUUSUUGTUVNUVLWFUUEUVOUUSUUGUUEWPWRUVMUVK
|
|
YOUURUUFCWQWIWJUVKUURCWSSAYHUUKUURIWTWMXAXBXCXIXDUUTUUSYQNULZURZUUGTZWFZN
|
|
WAZUVQBPZNWAUVBYRUVSUWANUUSUVQBUUGUUCUUQXEXFUVBUUTUVRNWAZWFUVTUVAUWBUUTUV
|
|
AUURUVPCTZNWAUWBNUURCXGUVRUWCNUVRYOYQUUFTZUWCUWDYQYQWCYQXHLYOYQUVIYOUMUVI
|
|
XJXKXLXMYOUURYQUVPUUFCWQXNXTXOXPUUTUVRNXQXONYQBWBXRXSYAYBYCJYGQYDYEYF $.
|
|
$}
|
|
|
|
${
|
|
$d F w y z $.
|
|
$( Lemma for ~ fnfrec . Establish stratification for induction.
|
|
(Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
fnfreclem1 $p |- ( F e. V ->
|
|
{ w | A. y A. z ( ( w F y /\ w F z ) -> y = z ) } e. _V ) $=
|
|
( wcel cv wbr wa weq wal cvv cid wn vex cop wex df-br bitri exbii wi ccnv
|
|
cab cxp cins2 cin cins3 cdif crn ccompl elcompl elrn2 elin opelcnv opelxp
|
|
eldif mpbiran 3bitr4i otelins2 anbi12i ideq equcom 3bitr2i notbii exanali
|
|
otelins3 exnal 3bitrri con1bii abbi2i vvex cnvexg xpexg sylancr syl inexg
|
|
ins2exg syl2anc idex ins3ex difexg mpan2 rnexg 3syl complexg syl5eqelr )
|
|
DEFZCGZAGZDHZWHBGZDHZIZABJZUABKZAKZCUCLDUBZUDZWQUEZUFZMUGZUHZUIZUIZUJZLWP
|
|
CXEWHXEFWHXDFZNWPWHXDCOZUKWPXFXFWIWHPZXCFZAQWMWNNZIZBQZAQZWPNZAWHXCULXIXL
|
|
AXIWKXHPZXBFZBQXLBXHXBULXPXKBXPXOWTFZXOXAFZNZIXKXOWTXAUPXQWMXSXJXQXOWRFZX
|
|
OWSFZIWMXOWRWSUMXTWJYAWLXHWQFZWHWIPDFXTWJWIWHDUNXTWKLFYBBOWKXHLWQUOUQWHWI
|
|
DRURWKWHPWQFWHWKPDFYAWLWKWHDUNWKWIWHWQAOZUSWHWKDRURUTSXRWNXRWKWIPMFWKWIMH
|
|
ZWNWKWIWHMXGVFWKWIMRYDBAJWNWKWIYCVABAVBSVCVDUTSTSTXMWONZAQXNXLYEAWMWNBVET
|
|
WOAVGSVHVISVJWGXCLFZXDLFXELFWGWTLFZXBLFZYFWGWRLFZWSLFZYGWGLLFWQLFZYIVKDEV
|
|
LZLWQLLVMVNWGYKYJYLWQLVQVOWRWSLLVPVRYGXALFYHMVSVTWTXALLWAWBXBLWCWDXCLWCXD
|
|
LWEWDWF $.
|
|
$}
|
|
|
|
${
|
|
$d G w z a $. $d I w z a $. $d X z a $. $d ph z $. $d F z $.
|
|
fnfreclem2.1 $e |- F = FRec ( G , I ) $.
|
|
fnfreclem2.2 $e |- ( ph -> G e. V ) $.
|
|
fnfreclem2.3 $e |- ( ph -> I e. dom G ) $.
|
|
fnfreclem2.4 $e |- ( ph -> ran G C_ dom G ) $.
|
|
$( Lemma for ~ fnfrec . Calculate the unique value of ` F ` at zero.
|
|
(Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
fnfreclem2 $p |- ( ph -> ( 0c F X -> X = I ) ) $=
|
|
( vz vw c0c wbr cop wcel wceq cvv c1c wb df-br csn cv cplc cmpt cpprod wo
|
|
wrex csucex pprodexg sylancr cfrec cclos1 df-frec eqtri clos1basesucg cdm
|
|
snex 0cex opexg elsnc2g opth simprbi syl6bi wi wn cproj1 cproj2 wa 0cnsuc
|
|
syl wne df-ne mpbi intnanr qrpprod opeq cfv vex proj1ex addceq1 eqid 1cex
|
|
breq1i addcex fvmpt ax-mp eqeq1i wfn fnmpti fnbrfvb bitr3i anbi1i 3bitr4i
|
|
mp2an mtbir a1i nrex pm2.21i jaod sylbid syl5bi ) MFBNMFOZBPZAFDQZMFBUAAX
|
|
DXCMDOZUBZPZKUCZXCLRLUCZSUDZUEZCUFZNZKBUHZUGZXEAXGRPXMRPZXDXPTXFURAXLRPCE
|
|
PXQLUIHXLCREUJUKKXCBXMXGRRBCDULXGXMUMGLCDUNUOUPUKAXHXEXOAXHXCXFQZXEAXFRPZ
|
|
XHXRTAMRPDCUQZPXSUSIMDRXTUTUKXCXFRVAVKXRMMQXEMFMDVBVCVDXOXEVEAXOXEXNKBXNV
|
|
FXIBPXNXIVGZSUDZMQZXIVHZFCNZVIZYCYEYBMVLYCVFYAVJYBMVMVNVOYAYDOZXCXMNYAMXL
|
|
NZYEVIXNYFYAYDMFXLCVPXIYGXCXMXIVQWDYCYHYEYCYAXLVRZMQZYHYIYBMYARPZYIYBQXIK
|
|
VSVTZLYAXKYBRXLXJYASWAXLWBZYASYLWCWEWFWGWHXLRWIYKYJYHTLRXKXLXJSLVSWCWEYMW
|
|
JYLRYAMXLWKWOWLWMWNWPWQWRWSWQWTXAXB $.
|
|
|
|
$d ph a t $. $d F a t $. $d X t $. $d G a t w z $. $d Y z t a $.
|
|
fnfreclem3.5 $e |- ( ph -> X e. Nn ) $.
|
|
fnfreclem3.6 $e |- ( ph -> ( X +c 1c ) F Y ) $.
|
|
$( Lemma for ~ fnfrec . The value of ` F ` at a successor is ` G ` related
|
|
to a previous element. (Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
fnfreclem3 $p |- ( ph -> E. z ( X F z /\ z G Y ) ) $=
|
|
( vw vt c1c wcel cvv wceq va cplc cop c0c csn cv wbr wex cmpt cpprod wrex
|
|
wa wb cdm 0cex opexg sylancr elsnc2g syl opth simplbi wi wne 0cnsuc df-ne
|
|
wn mpbi pm2.21i a1i syl5 sylbid vex opeqex ax-mp excom eleq1 df-br anbi2d
|
|
syl6bbr breq1 qrpprod cfv addceq1 eqid 1cex addcex fvmpt eqeq1i wfn fnmpt
|
|
addcexg mpan2 mprg fnbrfvb mp2an bitr3i anbi1i bitr4i syl6bb anbi12d cnnc
|
|
breldm adantl dmfrec adantr eleqtrd peano4 3expia syl2anc syld anim1d imp
|
|
biimpcd syl6bi com12 exlimdv eximdv mpi ex rexlimdva snex csucex pprodexg
|
|
wo cfrec cclos1 df-frec eqtri clos1basesucg syl5bb mpbid mpjaod ) AGQUBZH
|
|
UCZUDEUCZUEZRZGBUFZCUGZYRHDUGZULZBUHZUAUFZYNOSOUFZQUBZUIZDUJZUGZUACUKZAYQ
|
|
YNYOTZUUBAYOSRZYQUUJUMAUDSREDUNZRUUKUOKUDESUULUPUQYNYOSURUSUUJYMUDTZAUUBU
|
|
UJUUMHETYMHUDEUTVAUUMUUBVBAUUMUUBYMUDVCUUMVFGVDYMUDVEVGVHVIVJVKAUUHUUBUAC
|
|
AUUCCRZULZUUHUUBUUOUUHULZUUCPUFZYRUCZTZPUHZBUHZUUBUUSBUHPUHZUVAUUCSRUVBUA
|
|
VLPBUUCSVMVNUUSPBVOVGUUPUUTUUABUUPUUSUUAPUUSUUPUUAUUSUUPAUUQYRCUGZULZUUQQ
|
|
UBZYMTZYTULZULUUAUUSUUOUVDUUHUVGUUSUUNUVCAUUSUUNUURCRUVCUUCUURCVPUUQYRCVQ
|
|
VSVRUUSUUHUURYNUUGUGZUVGUUCUURYNUUGVTUVHUUQYMUUFUGZYTULUVGUUQYRYMHUUFDWAU
|
|
VFUVIYTUVFUUQUUFWBZYMTZUVIUVJUVEYMUUQSRZUVJUVETPVLZOUUQUUEUVESUUFUUDUUQQW
|
|
CUUFWDZUUQQUVMWEWFWGVNWHUUFSWIZUVLUVKUVIUMUUESRZUVOOSOSUUEUUFSUVNWJUUDSRQ
|
|
SRUVPWEUUDQSSWKWLWMUVMSUUQYMUUFWNWOWPWQWRWSWTUVDUVGUUAUVDUVFYSYTUVDUVFUUQ
|
|
GTZYSUVDUUQXARZGXARZUVFUVQVBUVDUUQCUNZXAUVCUUQUVTRAUUQYRCXBXCAUVTXATUVCAC
|
|
DEFIJKLXDXEXFAUVSUVCMXEUVRUVSUVFUVQUUQGXGXHXIUVCUVQYSVBAUVQUVCYSUUQGYRCVT
|
|
XMXCXJXKXLXNXOXPXQXRXSXTAYMHCUGZYQUUIYDZNUWAYNCRZAUWBYMHCVQAYPSRUUGSRZUWC
|
|
UWBUMYOYAAUUFSRDFRUWDOYBJUUFDSFYCUQUAYNCUUGYPSSCDEYEYPUUGYFIODEYGYHYIUQYJ
|
|
YKYL $.
|
|
$}
|
|
|
|
${
|
|
$d ph x y z w t a b $. $d F x y z w t a b $. $d G x y z w t a b $.
|
|
$d I x y z w t a b $.
|
|
fnfrec.1 $e |- F = FRec ( G , I ) $.
|
|
fnfrec.2 $e |- ( ph -> G e. Funs ) $.
|
|
fnfrec.3 $e |- ( ph -> I e. dom G ) $.
|
|
fnfrec.4 $e |- ( ph -> ran G C_ dom G ) $.
|
|
$( The recursive function generator is a function over the finite
|
|
cardinals. (Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
fnfrec $p |- ( ph -> F Fn Nn ) $=
|
|
( vy vz wceq cv wbr wa wi wal wcel cfuns breq1 wex vx vw vt wfun cdm cnnc
|
|
va vb wfn weq breldm adantl dmfrec adantr eleqtrd adantrr c0c c1c cvv cab
|
|
cplc frecexg fnfreclem1 3syl anbi12d imbi1d 2albidv breq2 bi2anan9 eqeq12
|
|
imbi12d cbval2v syl6bb fnfreclem2 imp adantrl eqtr4d ex alrimivv ad2antrr
|
|
crn wss simplr simpr fnfreclem3 adantlrr anim12d eeanv 19.29 eximi pm3.35
|
|
syl6ibr syl cfv anbi1d elfunsi funbrfv eqtr2 syl56 exp3a syl5 com34 imp3a
|
|
biimpa com12 an4s com3l exlimdvv impr syld ancoms findsd 19.21bbi alrimiv
|
|
expr mpcom dffun2 sylibr df-fn sylanbrc ) ABUDZBUEZUFKZBUFUIAUALZILZBMZYD
|
|
JLZBMZNZIJUJZOZJPIPZUAPYAAYLUAAYKIJAYIYJYDUFQZAYINYJAYFYMYHAYFNYDYBUFYFYD
|
|
YBQAYDYEBUKULAYCYFABCDREFGHUMZUNUOUPYMAYIYJYMAYKYMANYKIJUBLZYEBMZYOYGBMZN
|
|
ZYJOZJPIPZUQYEBMZUQYGBMZNZYJOZJPIPUCLZYEBMZUUEYGBMZNZYJOZJPZIPZUUEURVAZUG
|
|
LZBMZUULUHLZBMZNZUGUHUJZOZUHPUGPZYLAUBUCYDUSACRQZBUSQYTUBUTUSQFBCDREVBIJU
|
|
BBUSVCVDYOUQKZYSUUDIJUVBYRUUCYJUVBYPUUAYQUUBYOUQYEBSYOUQYGBSVEVFVGUBUCUJZ
|
|
YSUUIIJUVCYRUUHYJUVCYPUUFYQUUGYOUUEYEBSYOUUEYGBSVEVFVGYOUULKZYTUULYEBMZUU
|
|
LYGBMZNZYJOZJPIPUUTUVDYSUVHIJUVDYRUVGYJUVDYPUVEYQUVFYOUULYEBSYOUULYGBSVEV
|
|
FVGUVHUUSIJUGUHIUGUJZJUHUJZNUVGUUQYJUURUVIUVEUUNUVJUVFUUPYEUUMUULBVHYGUUO
|
|
UULBVHVIYEUUMYGUUOVJVKVLVMUBUAUJZYSYKIJUVKYRYIYJUVKYPYFYQYHYOYDYEBSYOYDYG
|
|
BSVEVFVGAUUDIJAUUCYJAUUCNYEDYGAUUAYEDKZUUBAUUAUVLABCDRYEEFGHVNVOUPAUUBYGD
|
|
KZUUAAUUBUVMABCDRYGEFGHVNVOVPVQVRVSAUUEUFQZUUKUUTOAUVNUUKUUTAUVNUUKNNZUUS
|
|
UGUHUVOUUQUUFYEUUMCMZNZUUGYGUUOCMZNZNZJTZITZUURUVOUUQUVQITZUVSJTZNUWBUVOU
|
|
UNUWCUUPUWDUVOUUNUWCAUVNUUNUWCUUKAUVNNZUUNNIBCDRUUEUUMEAUVAUVNUUNFVTADCUE
|
|
ZQZUVNUUNGVTACWAUWFWBZUVNUUNHVTAUVNUUNWCUWEUUNWDWEWFVRUVOUUPUWDAUVNUUPUWD
|
|
UUKUWEUUPNJBCDRUUEUUOEAUVAUVNUUPFVTAUWGUVNUUPGVTAUWHUVNUUPHVTAUVNUUPWCUWE
|
|
UUPWDWEWFVRWGUVQUVSIJWHWLAUVNUUKUWBUUROUWEUUKUWBUURUUKUWBNZUUIUVTNZJTZITZ
|
|
UWEUURUWIUUJUWANZITUWLUUJUWAIWIUWMUWKIUUIUVTJWIWJWMAUWLUUROUVNAUWJUURIJAU
|
|
UIUVTUURUVTAUUIUURUUFUUGUVPUVRAUUIUUROZOAUUHUVPUVRNZNUWNAUUHUWOUWNAUUHUUI
|
|
UWOUURAUUHUUIUWOUUROZUUHUUINYJAUWPUUHYJWKAYJUWOUURYJUWONYGUUMCMZUVRNZAYGC
|
|
WNZUUMKZUWSUUOKZNUURYJUWOUWRYJUVPUWQUVRYEYGUUMCSWOXDAUWQUWTUVRUXAAUVACUDZ
|
|
UWQUWTOFCWPZYGUUMCWQVDAUVAUXBUVRUXAOFUXCYGUUOCWQVDWGUWSUUMUUOWRWSWTXAWTXB
|
|
XCXEXFXGXCXHUNXAWTXIXJVSXOXKXLXMVRXCXPVRVSXNUAIJBXQXRYNBUFXSXT $.
|
|
$}
|
|
|
|
${
|
|
$d F x y $. $d G x y $. $d I x y $.
|
|
frec0.1 $e |- F = FRec ( G , I ) $.
|
|
frec0.2 $e |- ( ph -> G e. Funs ) $.
|
|
frec0.3 $e |- ( ph -> I e. dom G ) $.
|
|
frec0.4 $e |- ( ph -> ran G C_ dom G ) $.
|
|
$( Calculate the value of the finite recursive function generator at zero.
|
|
(Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
frec0 $p |- ( ph -> ( F ` 0c ) = I ) $=
|
|
( vy vx c0c wcel cv cvv cnnc peano1 sylancr wb cfuns mpbird cfv wceq cplc
|
|
cop csn c1c cmpt cpprod wbr wrex cdm opexg snidg syl orcd csucex pprodexg
|
|
snex cfrec cclos1 df-frec eqtri clos1basesucg wfn fnfrec fnopfvb sylancl
|
|
wo ) AKBUADUBZKDUDZBLZAVKVJVJUEZLZIMVJJNJMUFUCUGZCUHZUIIBUJZVHZAVMVPAVJNL
|
|
ZVMAKOLZDCUKZLVRPGKDOVTULQVJNUMUNUOAVLNLVONLZVKVQRVJURAVNNLCSLWAJUPFVNCNS
|
|
UQQIVJBVOVLNNBCDUSVLVOUTEJCDVAVBVCQTABOVDVSVIVKRABCDEFGHVEPOKDBVFVGT $.
|
|
$}
|
|
|
|
${
|
|
$d F w y $. $d G w y $. $d I w y $. $d X w y $.
|
|
frecsuc.1 $e |- F = FRec ( G , I ) $.
|
|
frecsuc.2 $e |- ( ph -> G e. Funs ) $.
|
|
frecsuc.3 $e |- ( ph -> I e. dom G ) $.
|
|
frecsuc.4 $e |- ( ph -> ran G C_ dom G ) $.
|
|
frecsuc.5 $e |- ( ph -> X e. Nn ) $.
|
|
$( Calculate the value of the finite recursive function generator at a
|
|
successor. (Contributed by Scott Fenton, 31-Jul-2019.) $)
|
|
frecsuc $p |- ( ph -> ( F ` ( X +c 1c ) ) = ( G ` ( F ` X ) ) ) $=
|
|
( vy vw wceq cop wcel cvv wbr cnnc syl syl2anc c1c cplc cfv c0c cv cpprod
|
|
csn cmpt wrex wo wfun cdm fnfrec fnfun cfuns dmfrec eleqtrrd funfvop eqid
|
|
wfn peano2 addceq1 eqeq2d eqeq1 mptv brabg mpbiri elfunsi crn snssd unssd
|
|
wb cun cxp frecxpg rnss rnxpss syl6ss fvelrn sseldd df-br sylibr wa breq1
|
|
wss qrpprod syl6bb rspcev syl12anc olcd snex csucex pprodexg cfrec cclos1
|
|
sylancr df-frec eqtri clos1basesucg mpbird fnopfvb ) AEUAUBZBUCEBUCZCUCZM
|
|
ZXBXDNZBOZAXGXFUDDNZUGZOZKUEZXFLPLUEZUAUBZUHZCUFZQZKBUIZUJZAXQXJAEXCNZBOZ
|
|
EXBXNQZXCXDCQZXQABUKZEBULZOZXTABRUTZYCABCDFGHIUMZRBUNSZAERYDJABCDUOFGHIUP
|
|
UQZEBURTAYAXBXBMZXBUSAEROZXBROZYAYJVLJAYKYLJEVASZXKXMMXKXBMYJLKEXBRRXNXLE
|
|
MXMXBXKXLEUAVBVCXKXBXBVDLKXMVEVFTVGAXCXDNCOZYBACUKZXCCULZOYNACUOOZYOGCVHS
|
|
ACVIZDUGZVMZYPXCAYRYSYPIADYPHVJVKABVIZYTXCAUUARYTVNZVIZYTABUUBWEZUUAUUCWE
|
|
AYQUUDGBCDUOFVOSBUUBVPSRYTVQVRAYCYEXCUUAOYHYIEBVSTVTVTXCCURTXCXDCWAWBXPYA
|
|
YBWCZKXSBXKXSMXPXSXFXOQUUEXKXSXFXOWDEXCXBXDXNCWFWGWHWIWJAXIPOXOPOZXGXRVLX
|
|
HWKAXNPOYQUUFLWLGXNCPUOWMWPKXFBXOXIPPBCDWNXIXOWOFLCDWQWRWSWPWTAYFYLXEXGVL
|
|
YGYMRXBXDBXATWT $.
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Cantorian and Strongly Cantorian Sets
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$c Can $. $( Cantorian class $)
|
|
$c SCan $. $( Strongly Cantorian class $)
|
|
|
|
$( Extend the definition of class to include the class of all Cantorian
|
|
sets. $)
|
|
ccan $a class Can $.
|
|
|
|
$( Define the class of all Cantorian sets. These are so-called because
|
|
Cantor's Theorem ` Nc A <c Nc ~P A ` holds for these sets. Definition
|
|
from [Rosser] p. 347 and [Holmes] p. 134. (Contributed by Scott Fenton,
|
|
19-Apr-2021.) $)
|
|
df-can $a |- Can = { x | ~P1 x ~~ x } $.
|
|
|
|
$( Extend the definition of class to include the class of all strongly
|
|
Cantorian sets. $)
|
|
cscan $a class SCan $.
|
|
|
|
${
|
|
$d x y $.
|
|
$( Define the class of strongly Cantorian sets. Unlike general Cantorian
|
|
sets, this fixes a specific mapping between ` x ` and ` ~P1 x ` .
|
|
Definition from [Holmes] p. 134.
|
|
(Contributed by Scott Fenton, 19-Apr-2021.) $)
|
|
df-scan $a |- SCan = { x | ( y e. x |-> { y } ) e. _V } $.
|
|
$}
|
|
|
|
${
|
|
$d x A $.
|
|
$( The domain of the singleton function. (Contributed by Scott Fenton,
|
|
20-Apr-2021.) $)
|
|
dmsnfn $p |- dom ( x e. A |-> { x } ) = A $=
|
|
( cv csn cvv wcel crab wa cab cmpt df-rab eqid dmmpt snex biantru abbi2i
|
|
cdm 3eqtr4i ) ACZDZEFZABGSBFZUAHZAIABTJZQBUAABKABTUDUDLMUCABUAUBSNOPR $.
|
|
$}
|
|
|
|
${
|
|
epelcres.1 $e |- Y e. _V $.
|
|
$( Version of ~ epelc with a restriction in place. (Contributed by
|
|
Scott Fenton, 20-Apr-2021.) $)
|
|
epelcres $p |- ( X e. A -> ( X ( _E |` A ) Y <-> X e. Y ) ) $=
|
|
( wcel cep wbr wa cres iba bicomd brres epelc bicomi 3bitr4g ) BAEZBCFGZ
|
|
PHZQBCFAIGBCEZPQRPQJKBCFALQSBCDMNO $.
|
|
$}
|
|
|
|
${
|
|
$d A x y $.
|
|
$( Membership in the class of Cantorian sets. (Contributed by Scott
|
|
Fenton, 19-Apr-2021.) $)
|
|
elcan $p |- ( A e. Can <-> ~P1 A ~~ A ) $=
|
|
( vx ccan wcel cvv cpw1 cen wbr elex brrelrnex wceq pw1eq breq12d df-can
|
|
cv id elab2g pm5.21nii ) ACDAEDAFZAGHZACISAGJBOZFZUAGHTBACEUAAKZUBSUAAGU
|
|
AALUCPMBNQR $.
|
|
|
|
$( Membership in the class of strongly Cantorian sets. (Contributed by
|
|
Scott Fenton, 19-Apr-2021.) $)
|
|
elscan $p |- ( A e. SCan <-> ( x e. A |-> { x } ) e. _V ) $=
|
|
( vy cscan wcel cvv csn cmpt elex cdm dmsnfn dmexg syl5eqelr wceq mpteq1
|
|
cv eleq1d df-scan elab2g pm5.21nii ) BDEBFEABAPGZHZFEZBDIUCBUBJFABKUBFLM
|
|
ACPZUAHZFEUCCBDFUDBNUEUBFAUDBUAOQCARST $.
|
|
$}
|
|
|
|
${
|
|
$d A x y z $.
|
|
$( Strongly Cantorian implies Cantorian. Observation from [Holmes],
|
|
p. 134. (Contributed by Scott Fenton, 19-Apr-2021.) $)
|
|
scancan $p |- ( A e. SCan -> A e. Can ) $=
|
|
( vx vy vz cv csn cmpt cvv wcel cpw1 cen wbr cscan wfn ccnv wceq weu wa
|
|
weq copab ccan wf1o snex eqid fnmpti wrex elpw1 euequ1 eqeq1 vex equcom
|
|
sneqb bitri syl6bb eubidv mpbiri rexlimivw sylbi df-mpt cnvopab snelpw1
|
|
cnveqi eleq1 syl6rbb pm5.32ri opabbii 3eqtri fnopab dff1o4 f1oeng mpan2
|
|
mpbir2an ensymi syl elscan elcan 3imtr4i ) BABEZFZGZHIZAJZAKLZAMIAUAIWA
|
|
AWBKLZWCWAAWBVTUBZWDWEVTANVTOZWBNBAVSVTVRUCVTUDUECEZVSPZCBWBWFWGWBIZWGD
|
|
EZFZPZDAUFWHBQZDWGAUGWLWMDAWLWMBDSZBQBDUHWLWHWNBWLWHWKVSPZWNWGWKVSUIWOD
|
|
BSWNWJVRDUJULDBUKUMUNUOUPUQURWFVRAIZWHRZBCTZOWQCBTWIWHRZCBTVTWRBCAVSUSV
|
|
BWQBCUTWQWSCBWHWPWIWHWIVSWBIWPWGVSWBVCVRAVAVDVEVFVGVHAWBVTVIVLAWBHVTVJV
|
|
KAWBVMVNBAVOAVPVQ $.
|
|
|
|
$}
|
|
|
|
$( The cardinality of a Cantorian set is equal to the cardinality
|
|
of its unit power set. (Contributed by Scott Fenton, 23-Apr-2021.) $)
|
|
canncb $p |- ( A e. V -> ( A e. Can <-> Nc ~P1 A = Nc A ) ) $=
|
|
( wcel cpw1 cnc wceq cen wbr ccan cvv wb pw1exg eqncg syl elcan syl6rbbr )
|
|
ABCZADZEAEFZRAGHZAICQRJCSTKABLRAJMNAOP $.
|
|
|
|
$( The cardinality of a Cantorian set is equal to the cardinality
|
|
of its unit power set. (Contributed by Scott Fenton, 21-Apr-2021.) $)
|
|
cannc $p |- ( A e. Can -> Nc ~P1 A = Nc A ) $=
|
|
( ccan wcel cpw1 cnc wceq canncb ibi ) ABCADEAEFABGH $.
|
|
|
|
$( The cardinality of a Cantorian set is strictly less than the cardinality
|
|
of its power set. (Contributed by Scott Fenton, 21-Apr-2021.) $)
|
|
canltpw $p |- ( A e. Can -> Nc A <c Nc ~P A ) $=
|
|
( ccan wcel cpw1 cnc cpw cltc cannc ltcpw1pwg eqbrtrrd ) ABCADEAEAFEGAHABIJ
|
|
$.
|
|
|
|
$( The cardinality of a Cantorian set is equal to the ` T_c ` raising
|
|
of that cardinal. (Contributed by Scott Fenton, 23-Apr-2021.) $)
|
|
cantcb $p |- ( A e. V -> ( A e. Can <-> T_c Nc A = Nc A ) ) $=
|
|
( wcel ccan cpw1 cnc wceq ctc canncb tcncg eqeq1d bitr4d ) ABCZADCAEFZAFZGO
|
|
HZOGABIMPNOABJKL $.
|
|
|
|
$( The cardinality of a Cantorian set is equal to the ` T_c ` raising
|
|
of that cardinal. (Contributed by Scott Fenton, 22-Apr-2021.) $)
|
|
cantc $p |- ( A e. Can -> T_c Nc A = Nc A ) $=
|
|
( ccan wcel cnc ctc wceq cantcb ibi ) ABCADZEIFABGH $.
|
|
|
|
$( The universe is not Cantorian. Theorem XI.1.8 of [Rosser] p. 348.
|
|
(Contributed by Scott Fenton, 22-Apr-2021.) $)
|
|
vncan $p |- -. _V e. Can $=
|
|
( cvv ccan wcel cnc cltc wbr ltcirr cpw canltpw pwv nceqi syl6breq mto ) AB
|
|
CZADZOEFOGNOAHZDOEAIPAJKLM $.
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Appendix: Typesetting definitions for the tokens in this file
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
|
|
$( $t
|
|
|
|
/* The '$ t' (no space between '$' and 't') token indicates the beginning
|
|
of the typesetting definition section, embedded in a Metamath
|
|
comment. There may only be one per source file, and the typesetting
|
|
section ends with the end of the Metamath comment. The typesetting
|
|
section uses C-style comment delimiters. Todo: Allow multiple
|
|
typesetting comments */
|
|
|
|
/* These are the LaTeX and HTML definitions in the order the tokens are
|
|
introduced in $c or $v statements. See HELP TEX or HELP HTML in the
|
|
Metamath program. */
|
|
|
|
|
|
/******* Web page format settings *******/
|
|
|
|
/* Page title, home page link */
|
|
htmltitle "New Foundations Explorer";
|
|
htmlhome '<A HREF="mmnf.html"><FONT SIZE=-2 FACE=sans-serif>' +
|
|
'<IMG SRC="nf.gif" BORDER=0 ALT=' +
|
|
'"Home" HEIGHT=32 WIDTH=32 ALIGN=MIDDLE STYLE="margin-bottom:0px">' +
|
|
'Home</FONT></A>';
|
|
/* Optional file where bibliographic references are kept */
|
|
/* If specified, e.g. "mmnf.html", Metamath will hyperlink all strings of the
|
|
form "[rrr]" (where "rrr" has no whitespace) to "mmnf.html#rrr" */
|
|
/* A warning will be given if the file "mmnf.html" with the bibliographical
|
|
references is not present. It is read in order to check correctness of
|
|
the references. */
|
|
/* Note: this is also used to determine the home page (rather than
|
|
extracting it from htmlhome) */
|
|
htmlbibliography "mmnf.html";
|
|
|
|
/* Variable color key at the bottom of each proof table */
|
|
htmlvarcolor '<FONT COLOR="#0000FF">wff</FONT> '
|
|
+ '<FONT COLOR="#FF0000">setvar</FONT> '
|
|
+ '<FONT COLOR="#CC33CC">class</FONT>';
|
|
|
|
/* GIF and Unicode HTML directories - these are used for the GIF version to
|
|
crosslink to the Unicode version and vice-versa */
|
|
htmldir "../nfegif/";
|
|
althtmldir "../nfeuni/";
|
|
|
|
|
|
/******* Symbol definitions *******/
|
|
|
|
htmldef "(" as "<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 TITLE='(' ALIGN=TOP>";
|
|
althtmldef "(" as "(";
|
|
latexdef "(" as "(";
|
|
htmldef ")" as "<IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 TITLE=')' ALIGN=TOP>";
|
|
althtmldef ")" as ")";
|
|
latexdef ")" as ")";
|
|
htmldef "->" as
|
|
" <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 TITLE='->' ALIGN=TOP> ";
|
|
althtmldef "->" as ' → ';
|
|
latexdef "->" as "\rightarrow";
|
|
htmldef "-." as
|
|
"<IMG SRC='lnot.gif' WIDTH=10 HEIGHT=19 TITLE='-.' ALIGN=TOP> ";
|
|
althtmldef "-." as '¬ ';
|
|
latexdef "-." as "\lnot";
|
|
htmldef "wff" as
|
|
"<IMG SRC='_wff.gif' WIDTH=24 HEIGHT=19 TITLE='wff' ALIGN=TOP> ";
|
|
althtmldef "wff" as '<FONT COLOR="#808080">wff </FONT>'; /* was #00CC00 */
|
|
latexdef "wff" as "{\rm wff}";
|
|
htmldef "|-" as
|
|
"<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 TITLE='|-' ALIGN=TOP> ";
|
|
althtmldef "|-" as
|
|
'<FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>'; /* ⊢ */
|
|
/* Without sans-serif, way too big in FF3 */
|
|
latexdef "|-" as "\vdash";
|
|
htmldef "ph" as
|
|
"<IMG SRC='_varphi.gif' WIDTH=11 HEIGHT=19 TITLE='ph' ALIGN=TOP>";
|
|
althtmldef "ph" as '<FONT COLOR="#0000FF"><I>φ</I></FONT>';
|
|
latexdef "ph" as "\varphi";
|
|
htmldef "ps" as "<IMG SRC='_psi.gif' WIDTH=12 HEIGHT=19 TITLE='ps' ALIGN=TOP>";
|
|
althtmldef "ps" as '<FONT COLOR="#0000FF"><I>ψ</I></FONT>';
|
|
latexdef "ps" as "\psi";
|
|
htmldef "ch" as "<IMG SRC='_chi.gif' WIDTH=12 HEIGHT=19 TITLE='ch' ALIGN=TOP>";
|
|
althtmldef "ch" as '<FONT COLOR="#0000FF"><I>χ</I></FONT>';
|
|
latexdef "ch" as "\chi";
|
|
htmldef "th" as
|
|
"<IMG SRC='_theta.gif' WIDTH=8 HEIGHT=19 TITLE='th' ALIGN=TOP>";
|
|
althtmldef "th" as '<FONT COLOR="#0000FF"><I>θ</I></FONT>';
|
|
latexdef "th" as "\theta";
|
|
htmldef "ta" as "<IMG SRC='_tau.gif' WIDTH=10 HEIGHT=19 TITLE='ta' ALIGN=TOP>";
|
|
althtmldef "ta" as '<FONT COLOR="#0000FF"><I>τ</I></FONT>';
|
|
latexdef "ta" as "\tau";
|
|
htmldef "et" as "<IMG SRC='_eta.gif' WIDTH=9 HEIGHT=19 TITLE='et' ALIGN=TOP>";
|
|
althtmldef "et" as '<FONT COLOR="#0000FF"><I>η</I></FONT>';
|
|
latexdef "et" as "\eta";
|
|
htmldef "ze" as "<IMG SRC='_zeta.gif' WIDTH=9 HEIGHT=19 TITLE='ze' ALIGN=TOP>";
|
|
althtmldef "ze" as '<FONT COLOR="#0000FF"><I>ζ</I></FONT>';
|
|
latexdef "ze" as "\zeta";
|
|
htmldef "si" as
|
|
"<IMG SRC='_sigma.gif' WIDTH=10 HEIGHT=19 TITLE='si' ALIGN=TOP>";
|
|
althtmldef "si" as '<FONT COLOR="#0000FF"><I>σ</I></FONT>';
|
|
latexdef "si" as "\sigma";
|
|
htmldef "rh" as "<IMG SRC='_rho.gif' WIDTH=9 HEIGHT=19 TITLE='rh' ALIGN=TOP>";
|
|
althtmldef "rh" as '<FONT COLOR="#0000FF"><I>ρ</I></FONT>';
|
|
latexdef "rh" as "\rho";
|
|
htmldef "mu" as "<IMG SRC='_mu.gif' WIDTH=10 HEIGHT=19 TITLE='mu' ALIGN=TOP>";
|
|
althtmldef "mu" as '<FONT COLOR="#0000FF"><I>μ</I></FONT>';
|
|
latexdef "mu" as "\rho";
|
|
htmldef "la" as
|
|
"<IMG SRC='_lambda.gif' WIDTH=9 HEIGHT=19 TITLE='la' ALIGN=TOP>";
|
|
althtmldef "la" as '<FONT COLOR="#0000FF"><I>λ</I></FONT>';
|
|
latexdef "la" as "\rho";
|
|
htmldef "ka" as
|
|
"<IMG SRC='_kappa.gif' WIDTH=9 HEIGHT=19 TITLE='ka' ALIGN=TOP>";
|
|
althtmldef "ka" as '<FONT COLOR="#0000FF"><I>κ</I></FONT>';
|
|
latexdef "ka" as "\rho";
|
|
htmldef "~P" as "<IMG SRC='scrp.gif' WIDTH=16 HEIGHT=19 TITLE='~P' ALIGN=TOP>";
|
|
althtmldef "~P" as '<FONT FACE=sans-serif>℘</FONT>';
|
|
latexdef "~P" as "{\cal P}";
|
|
htmldef "<->" as " <IMG SRC='leftrightarrow.gif' WIDTH=15 HEIGHT=19 " +
|
|
"TITLE='<->' ALIGN=TOP> ";
|
|
althtmldef "<->" as ' ↔ ';
|
|
latexdef "<->" as "\leftrightarrow";
|
|
htmldef "\/" as
|
|
" <IMG SRC='vee.gif' WIDTH=11 HEIGHT=19 TITLE='\/' ALIGN=TOP> ";
|
|
althtmldef "\/" as ' <FONT FACE=sans-serif> ∨</FONT> ' ;
|
|
/* althtmldef "\/" as ' <FONT FACE=sans-serif>⋁</FONT> ' ; */
|
|
/* was ∨ - changed to match font of ∧ replacement */
|
|
/* Changed back 6-Mar-2012 NM */
|
|
latexdef "\/" as "\vee";
|
|
htmldef "/\" as
|
|
" <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 TITLE='/\' ALIGN=TOP> ";
|
|
althtmldef "/\" as ' <FONT FACE=sans-serif>∧</FONT> ';
|
|
/* althtmldef "/\" as ' <FONT FACE=sans-serif>⋀</FONT> '; */
|
|
/* was ∧ which is circle in Mozilla on WinXP Pro (but not Home) */
|
|
/* Changed back 6-Mar-2012 NM */
|
|
latexdef "/\" as "\wedge";
|
|
htmldef "-/\" as
|
|
" <IMG SRC='barwedge.gif' WIDTH=9 HEIGHT=19 TITLE='-/\' ALIGN=TOP> ";
|
|
althtmldef "-/\" as ' <FONT FACE=sans-serif>⊼</FONT> ';
|
|
/*althtmldef "-/\" as " ⊼ "; */ /* too-high font bug in FF */
|
|
/* barwedge, U022BC, alias ISOAMSB barwed, ['nand'] */
|
|
latexdef "-/\" as "\barwedge";
|
|
htmldef "hadd" as "hadd";
|
|
althtmldef "hadd" as "hadd";
|
|
latexdef "hadd" as "\mbox{hadd}";
|
|
htmldef "cadd" as "cadd";
|
|
althtmldef "cadd" as "cadd";
|
|
latexdef "cadd" as "\mbox{cadd}";
|
|
htmldef "A." as
|
|
"<IMG SRC='forall.gif' WIDTH=10 HEIGHT=19 TITLE='A.' ALIGN=TOP>";
|
|
althtmldef "A." as '<FONT FACE=sans-serif>∀</FONT>'; /* ∀ */
|
|
latexdef "A." as "\forall";
|
|
htmldef "setvar" as
|
|
"<IMG SRC='_setvar.gif' WIDTH=20 HEIGHT=19 TITLE='setvar' ALIGN=TOP> ";
|
|
althtmldef "setvar" as '<FONT COLOR="#808080">setvar </FONT>';
|
|
latexdef "setvar" as "{\rm setvar}";
|
|
htmldef "x" as "<IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 TITLE='x' ALIGN=TOP>";
|
|
althtmldef "x" as '<I><FONT COLOR="#FF0000">x</FONT></I>';
|
|
latexdef "x" as "x";
|
|
htmldef "y" as "<IMG SRC='_y.gif' WIDTH=9 HEIGHT=19 TITLE='y' ALIGN=TOP>";
|
|
althtmldef "y" as '<I><FONT COLOR="#FF0000">y</FONT></I>';
|
|
latexdef "y" as "y";
|
|
htmldef "z" as "<IMG SRC='_z.gif' WIDTH=9 HEIGHT=19 TITLE='z' ALIGN=TOP>";
|
|
althtmldef "z" as '<I><FONT COLOR="#FF0000">z</FONT></I>';
|
|
latexdef "z" as "z";
|
|
htmldef "w" as "<IMG SRC='_w.gif' WIDTH=12 HEIGHT=19 TITLE='w' ALIGN=TOP>";
|
|
althtmldef "w" as '<I><FONT COLOR="#FF0000">w</FONT></I>';
|
|
latexdef "w" as "w";
|
|
htmldef "v" as "<IMG SRC='_v.gif' WIDTH=9 HEIGHT=19 TITLE='v' ALIGN=TOP>";
|
|
althtmldef "v" as '<I><FONT COLOR="#FF0000">v</FONT></I>';
|
|
latexdef "v" as "v";
|
|
htmldef "E." as
|
|
"<IMG SRC='exists.gif' WIDTH=9 HEIGHT=19 TITLE='E.' ALIGN=TOP>";
|
|
althtmldef "E." as '<FONT FACE=sans-serif>∃</FONT>'; /* ∃ */
|
|
/* Without sans-serif, bad in Opera and way too big in FF3 */
|
|
latexdef "E." as "\exists";
|
|
htmldef "=" as " <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 TITLE='=' ALIGN=TOP> ";
|
|
althtmldef "=" as ' = '; /* = */
|
|
latexdef "=" as "=";
|
|
htmldef "e." as " <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 TITLE='e.' ALIGN=TOP> ";
|
|
althtmldef "e." as ' <FONT FACE=sans-serif>∈</FONT> ';
|
|
latexdef "e." as "\in";
|
|
htmldef "[" as "<IMG SRC='lbrack.gif' WIDTH=5 HEIGHT=19 TITLE='[' ALIGN=TOP>";
|
|
althtmldef "[" as '['; /* [ */
|
|
latexdef "[" as "[";
|
|
htmldef "/" as
|
|
" <IMG SRC='solidus.gif' WIDTH=6 HEIGHT=19 TITLE='/' ALIGN=TOP> ";
|
|
althtmldef "/" as ' / '; /* / */
|
|
latexdef "/" as "/";
|
|
htmldef "]" as "<IMG SRC='rbrack.gif' WIDTH=5 HEIGHT=19 TITLE=']' ALIGN=TOP>";
|
|
althtmldef "]" as ']'; /* ] */
|
|
latexdef "]" as "]";
|
|
htmldef "u" as "<IMG SRC='_u.gif' WIDTH=10 HEIGHT=19 TITLE='u' ALIGN=TOP>";
|
|
althtmldef "u" as '<I><FONT COLOR="#FF0000">u</FONT></I>';
|
|
latexdef "u" as "u";
|
|
htmldef "f" as "<IMG SRC='_f.gif' WIDTH=9 HEIGHT=19 TITLE='f' ALIGN=TOP>";
|
|
althtmldef "f" as '<I><FONT COLOR="#FF0000">f</FONT></I>';
|
|
latexdef "f" as "f";
|
|
htmldef "g" as "<IMG SRC='_g.gif' WIDTH=9 HEIGHT=19 TITLE='g' ALIGN=TOP>";
|
|
althtmldef "g" as '<I><FONT COLOR="#FF0000">g</FONT></I>';
|
|
latexdef "g" as "g";
|
|
htmldef "E!" as "<IMG SRC='_e1.gif' WIDTH=12 HEIGHT=19 TITLE='E!' ALIGN=TOP>";
|
|
althtmldef "E!" as '<FONT FACE=sans-serif>∃!</FONT>';
|
|
latexdef "E!" as "\exists{!}";
|
|
htmldef "E*" as "<IMG SRC='_em1.gif' WIDTH=15 HEIGHT=19 TITLE='E*' ALIGN=TOP>";
|
|
althtmldef "E*" as '<FONT FACE=sans-serif>∃*</FONT>';
|
|
latexdef "E*" as "\exists^\ast";
|
|
htmldef "{" as "<IMG SRC='lbrace.gif' WIDTH=6 HEIGHT=19 TITLE='{' ALIGN=TOP>";
|
|
althtmldef "{" as '{'; /* { */
|
|
latexdef "{" as "\{";
|
|
htmldef "|" as " <IMG SRC='vert.gif' WIDTH=3 HEIGHT=19 TITLE='|' ALIGN=TOP> ";
|
|
althtmldef "|" as ' <FONT FACE=sans-serif>∣</FONT> '; /* &vertbar; */
|
|
latexdef "|" as "|";
|
|
htmldef "}" as "<IMG SRC='rbrace.gif' WIDTH=6 HEIGHT=19 TITLE='}' ALIGN=TOP>";
|
|
althtmldef "}" as '}'; /* } */
|
|
latexdef "}" as "\}";
|
|
htmldef "F/" as
|
|
"<IMG SRC='finv.gif' WIDTH=9 HEIGHT=19 ALT=' F/' TITLE='F/'>";
|
|
althtmldef "F/" as "Ⅎ";
|
|
latexdef "F/" as "\Finv";
|
|
htmldef "F/_" as
|
|
"<IMG SRC='_finvbar.gif' WIDTH=9 HEIGHT=19 ALT=' F/_' TITLE='F/_'>";
|
|
althtmldef "F/_" as "<U>Ⅎ</U>";
|
|
latexdef "F/_" as "\underline{\Finv}";
|
|
htmldef "class" as
|
|
"<IMG SRC='_class.gif' WIDTH=32 HEIGHT=19 TITLE='class' ALIGN=TOP> ";
|
|
althtmldef "class" as '<FONT COLOR="#808080">class </FONT>';
|
|
latexdef "class" as "{\rm class}";
|
|
htmldef "./\" as
|
|
" <IMG SRC='_.wedge.gif' WIDTH=11 HEIGHT=19 ALT=' ./\' TITLE='./\'> ";
|
|
althtmldef "./\" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∧</SPAN> ';
|
|
latexdef "./\" as "\wedge";
|
|
htmldef ".\/" as
|
|
" <IMG SRC='_.vee.gif' WIDTH=11 HEIGHT=19 ALT=' .\/' TITLE='.\/'> ";
|
|
althtmldef ".\/" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∨</SPAN> ';
|
|
latexdef ".\/" as "\vee";
|
|
htmldef ".<_" as
|
|
" <IMG SRC='_.le.gif' WIDTH=11 HEIGHT=19 ALT=' .<_' TITLE='.<_'> ";
|
|
althtmldef ".<_" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'≤</SPAN> ';
|
|
latexdef ".<_" as "\le";
|
|
htmldef ".<" as /* Symbol as variable */
|
|
" <IMG SRC='_.lt.gif' WIDTH=11 HEIGHT=19 ALT=' .<' TITLE='.<'> ";
|
|
althtmldef ".<" as
|
|
/* This is how to put a dotted box around the symbol: */
|
|
/* border means box around symbol; border-bottom underlines symbol */
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'<</SPAN> ';
|
|
/* Todo: can this STYLE sequence be done with a CLASS? */
|
|
/* Move the underline down 3px so it isn't too close to symbol */
|
|
/*
|
|
' <SPAN STYLE="vertical-align:-3px">' +
|
|
'<SPAN CLASS=symvar STYLE="text-decoration:underline dotted;color:#C3C">' +
|
|
'<SPAN STYLE="vertical-align:3px"><</SPAN></SPAN></SPAN> ';
|
|
*/
|
|
latexdef ".<" as "<";
|
|
htmldef ".+" as
|
|
" <IMG SRC='_.plus.gif' WIDTH=13 HEIGHT=19 ALT=' .+' TITLE='.+'> ";
|
|
althtmldef ".+" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'+</SPAN> ';
|
|
latexdef ".+" as "+";
|
|
htmldef ".-" as
|
|
" <IMG SRC='_.minus.gif' WIDTH=11 HEIGHT=19 ALT=' .-' TITLE='.-'> ";
|
|
althtmldef ".-" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'−</SPAN> ';
|
|
latexdef ".-" as "-";
|
|
htmldef ".X." as
|
|
" <IMG SRC='_.times.gif' WIDTH=9 HEIGHT=19 ALT=' .X.' TITLE='.X.'> ";
|
|
althtmldef ".X." as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'×</SPAN> ';
|
|
latexdef ".X." as "\times";
|
|
htmldef "./" as
|
|
" <IMG SRC='_.solidus.gif' WIDTH=8 HEIGHT=19 ALT=' ./' TITLE='./'> ";
|
|
althtmldef "./" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'/</SPAN> ';
|
|
latexdef "./" as "/";
|
|
htmldef ".^" as
|
|
" <IMG SRC='_.uparrow.gif' WIDTH=7 HEIGHT=19 ALT=' .^' TITLE='.^'> ";
|
|
althtmldef ".^" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'↑</SPAN> ';
|
|
latexdef ".^" as "\uparrow";
|
|
htmldef ".0." as
|
|
" <IMG SRC='_.0.gif' WIDTH=8 HEIGHT=19 ALT=' .0.' TITLE='.0.'> ";
|
|
althtmldef ".0." as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'0</SPAN> ';
|
|
latexdef ".0." as "0";
|
|
htmldef ".1." as
|
|
" <IMG SRC='_.1.gif' WIDTH=7 HEIGHT=19 ALT=' .1.' TITLE='.1.'> ";
|
|
althtmldef ".1." as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'1</SPAN> ';
|
|
latexdef ".1." as "1";
|
|
htmldef ".||" as
|
|
" <IMG SRC='_.parallel.gif' WIDTH=5 HEIGHT=19 ALT=' .||' TITLE='.||'> ";
|
|
althtmldef ".||" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∥</SPAN> ';
|
|
latexdef ".||" as "\parallel";
|
|
htmldef ".~" as
|
|
" <IMG SRC='_.sim.gif' WIDTH=13 HEIGHT=19 ALT=' .~' TITLE='.~'> ";
|
|
althtmldef ".~" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∼</SPAN> ';
|
|
latexdef ".~" as "\sim";
|
|
htmldef "._|_" as
|
|
" <IMG SRC='_.perp.gif' WIDTH=11 HEIGHT=19 ALT=' ._|_' TITLE='._|_'> ";
|
|
althtmldef "._|_" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'⊥</SPAN> ';
|
|
latexdef "._|_" as "\perp";
|
|
htmldef ".+^" as
|
|
" <IMG SRC='_.plushat.gif' WIDTH=11 HEIGHT=19 ALT=' .+^' TITLE='.+^'> ";
|
|
althtmldef ".+^" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'⨣</SPAN> '; /* ⨣ */
|
|
latexdef ".+^" as "\hat{+}";
|
|
htmldef ".+b" as
|
|
" <IMG SRC='_.plusb.gif' WIDTH=14 HEIGHT=19 ALT=' .+b' TITLE='.+b'> ";
|
|
althtmldef ".+b" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'✚</SPAN> ';
|
|
latexdef ".+b" as "\boldsymbol{+}";
|
|
htmldef ".(+)" as
|
|
" <IMG SRC='_.oplus.gif' WIDTH=13 HEIGHT=19 ALT=' .(+)' TITLE='.(+)'> ";
|
|
althtmldef ".(+)" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'⊕</SPAN> ';
|
|
latexdef ".(+)" as "\oplus";
|
|
htmldef ".*" as
|
|
" <IMG SRC='_.ast.gif' WIDTH=7 HEIGHT=19 ALT=' .*' TITLE='.*'> ";
|
|
althtmldef ".*" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∗</SPAN> ';
|
|
latexdef ".*" as "\ast";
|
|
htmldef ".x." as
|
|
" <IMG SRC='_.cdot.gif' WIDTH=4 HEIGHT=19 ALT=' .x.' TITLE='.x.'> ";
|
|
althtmldef ".x." as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'·</SPAN> ';
|
|
latexdef ".x." as "\cdot";
|
|
htmldef ".xb" as
|
|
" <IMG SRC='_.bullet.gif' WIDTH=8 HEIGHT=19 ALT=' .xb' TITLE='.xb'> ";
|
|
althtmldef ".xb" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'∙</SPAN> ';
|
|
latexdef ".xb" as "\bullet";
|
|
htmldef ".," as
|
|
" <IMG SRC='_.comma.gif' WIDTH=4 HEIGHT=19 ALT=' .,' TITLE='.,'> ";
|
|
althtmldef ".," as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
',</SPAN> ';
|
|
latexdef ".," as ",";
|
|
htmldef ".(x)" as
|
|
" <IMG SRC='_.otimes.gif' WIDTH=13 HEIGHT=19 ALT=' .(x)' TITLE='.(x)'> ";
|
|
althtmldef ".(x)" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'⊗</SPAN> ';
|
|
latexdef ".(x)" as "\otimes";
|
|
htmldef ".0b" as
|
|
" <IMG SRC='_.bf0.gif' WIDTH=9 HEIGHT=19 ALT=' .0b' TITLE='.0b'> ";
|
|
althtmldef ".0b" as
|
|
' <SPAN CLASS=symvar STYLE="border-bottom:1px dotted;color:#C3C">' +
|
|
'𝟎</SPAN> ';
|
|
latexdef ".0b" as "\mbox{\boldmath$0$}";
|
|
htmldef "A" as "<IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 TITLE='A' ALIGN=TOP>";
|
|
althtmldef "A" as '<I><FONT COLOR="#CC33CC">A</FONT></I>';
|
|
latexdef "A" as "A";
|
|
htmldef "B" as "<IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 TITLE='B' ALIGN=TOP>";
|
|
althtmldef "B" as '<I><FONT COLOR="#CC33CC">B</FONT></I>';
|
|
latexdef "B" as "B";
|
|
htmldef "C" as "<IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 TITLE='C' ALIGN=TOP>";
|
|
althtmldef "C" as '<I><FONT COLOR="#CC33CC">C</FONT></I>';
|
|
latexdef "C" as "C";
|
|
htmldef "D" as "<IMG SRC='_cd.gif' WIDTH=12 HEIGHT=19 TITLE='D' ALIGN=TOP>";
|
|
althtmldef "D" as '<I><FONT COLOR="#CC33CC">D</FONT></I>';
|
|
latexdef "D" as "D";
|
|
htmldef "P" as "<IMG SRC='_cp.gif' WIDTH=12 HEIGHT=19 TITLE='P' ALIGN=TOP>";
|
|
althtmldef "P" as '<I><FONT COLOR="#CC33CC">P</FONT></I>';
|
|
latexdef "P" as "P";
|
|
htmldef "R" as "<IMG SRC='_cr.gif' WIDTH=12 HEIGHT=19 TITLE='R' ALIGN=TOP>";
|
|
althtmldef "R" as '<I><FONT COLOR="#CC33CC">R</FONT></I>';
|
|
latexdef "R" as "R";
|
|
htmldef "S" as "<IMG SRC='_cs.gif' WIDTH=11 HEIGHT=19 TITLE='S' ALIGN=TOP>";
|
|
althtmldef "S" as '<I><FONT COLOR="#CC33CC">S</FONT></I>';
|
|
latexdef "S" as "S";
|
|
htmldef "T" as "<IMG SRC='_ct.gif' WIDTH=12 HEIGHT=19 TITLE='T' ALIGN=TOP>";
|
|
althtmldef "T" as '<I><FONT COLOR="#CC33CC">T</FONT></I>';
|
|
latexdef "T" as "T";
|
|
htmldef "=/=" as
|
|
" <IMG SRC='ne.gif' WIDTH=12 HEIGHT=19 TITLE='=/=' ALIGN=TOP> ";
|
|
althtmldef "=/=" as ' ≠ ';
|
|
latexdef "=/=" as "\ne";
|
|
htmldef "e/" as
|
|
" <IMG SRC='notin.gif' WIDTH=10 HEIGHT=19 TITLE='e/' ALIGN=TOP> ";
|
|
althtmldef "e/" as ' <FONT FACE=sans-serif>∉</FONT> ';
|
|
latexdef "e/" as "\notin";
|
|
htmldef "_V" as "<IMG SRC='rmcv.gif' WIDTH=10 HEIGHT=19 TITLE='_V' ALIGN=TOP>";
|
|
althtmldef "_V" as 'V';
|
|
latexdef "_V" as "{\rm V}";
|
|
htmldef "[." as
|
|
"<IMG SRC='_dlbrack.gif' WIDTH=6 HEIGHT=19 ALT=' [.' TITLE='[.'>";
|
|
/* althtmldef "[." as '⦏'; */ /* corner tick */
|
|
/* U+0323 COMBINING DOT BELOW (HTML ̣) */
|
|
althtmldef "[." as '[̣';
|
|
/* \underaccent is in accents package */
|
|
latexdef "[." as "\underaccent{\dot}{[}";
|
|
htmldef "]." as
|
|
"<IMG SRC='_drbrack.gif' WIDTH=5 HEIGHT=19 ALT=' ].' TITLE='].'>";
|
|
/* althtmldef "]." as '⦎'; */ /* corner tick */
|
|
althtmldef "]." as ']̣';
|
|
latexdef "]." as "\underaccent{\dot}{]}";
|
|
htmldef
|
|
"[_" as "<IMG SRC='_ulbrack.gif' WIDTH=6 HEIGHT=19 TITLE='[_' ALIGN=TOP>";
|
|
althtmldef "[_" as '<U>[</U>'; /* [ */
|
|
latexdef "[_" as "\underline{[}";
|
|
htmldef
|
|
"]_" as "<IMG SRC='_urbrack.gif' WIDTH=5 HEIGHT=19 TITLE=']_' ALIGN=TOP>";
|
|
althtmldef "]_" as '<U>]</U>'; /* ] */
|
|
latexdef "]_" as "\underline{]}";
|
|
htmldef "F" as "<IMG SRC='_cf.gif' WIDTH=13 HEIGHT=19 TITLE='F' ALIGN=TOP>";
|
|
althtmldef "F" as '<I><FONT COLOR="#CC33CC">F</FONT></I>';
|
|
latexdef "F" as "F";
|
|
htmldef "G" as "<IMG SRC='_cg.gif' WIDTH=12 HEIGHT=19 TITLE='G' ALIGN=TOP>";
|
|
althtmldef "G" as '<I><FONT COLOR="#CC33CC">G</FONT></I>';
|
|
latexdef "G" as "G";
|
|
htmldef "C_" as
|
|
" <IMG SRC='subseteq.gif' WIDTH=12 HEIGHT=19 TITLE='C_' ALIGN=TOP> ";
|
|
althtmldef "C_" as ' <FONT FACE=sans-serif>⊆</FONT> '; /* ⫅ */
|
|
latexdef "C_" as "\subseteq";
|
|
/* 7-Jun-2019 changed gif, unicode and latex def of "C." from subset to
|
|
subsetneq (BJ) */
|
|
htmldef "C." as
|
|
" <IMG SRC='subsetneq.gif' WIDTH=12 HEIGHT=19 TITLE='C.' ALIGN=TOP> ";
|
|
/* subset.gif */
|
|
althtmldef "C." as ' ⊊ '; /* <FONT FACE=sans-serif>⊂</FONT> */
|
|
latexdef "C." as "\subsetneq"; /* \subset */
|
|
htmldef "~" as " ∼ ";
|
|
althtmldef "~" as ' ∼ ';
|
|
latexdef "~" as "\sim";
|
|
htmldef "\" as
|
|
" <IMG SRC='setminus.gif' WIDTH=8 HEIGHT=19 TITLE='\' ALIGN=TOP> ";
|
|
althtmldef "\" as ' <FONT FACE=sans-serif>∖</FONT> '; /* ∖ */
|
|
latexdef "\" as "\setminus";
|
|
htmldef "u." as
|
|
" <IMG SRC='cup.gif' WIDTH=10 HEIGHT=19 TITLE='u.' ALIGN=TOP> ";
|
|
althtmldef "u." as ' ∪ ';
|
|
latexdef "u." as "\cup";
|
|
htmldef "-i^i" as
|
|
" &ncap ";
|
|
althtmldef "-i^i" as ' ⩃ ';
|
|
latexdef "-i^i" as "\overline{\cap}";
|
|
htmldef "i^i" as
|
|
" <IMG SRC='cap.gif' WIDTH=10 HEIGHT=19 TITLE='i^i' ALIGN=TOP> ";
|
|
althtmldef "i^i" as ' ∩ ';
|
|
latexdef "i^i" as "\cap";
|
|
htmldef "(/)" as
|
|
"<IMG SRC='varnothing.gif' WIDTH=11 HEIGHT=19 TITLE='(/)' ALIGN=TOP>";
|
|
althtmldef "(/)" as '<FONT FACE=sans-serif>∅</FONT>';
|
|
/*althtmldef "(/)" as '∅';*/ /* =∅ */ /* bad in Opera */
|
|
/*althtmldef "(/)" as '⌀';*/
|
|
latexdef "(/)" as "\varnothing";
|
|
htmldef "if" as "<IMG SRC='_if.gif' WIDTH=11 HEIGHT=19 TITLE='if' ALIGN=TOP>";
|
|
/*htmldef "ded" as
|
|
"<IMG SRC='_ded.gif' WIDTH=23 HEIGHT=19 TITLE='ded' ALIGN=TOP>";*/
|
|
althtmldef "if" as ' if';
|
|
/*althtmldef "ded" as 'ded';*/
|
|
latexdef "if" as "{\rm if}";
|
|
/*latexdef "ded" as "{\rm ded}";*/
|
|
htmldef "," as "<IMG SRC='comma.gif' WIDTH=4 HEIGHT=19 TITLE=',' ALIGN=TOP> ";
|
|
althtmldef "," as ', ';
|
|
latexdef "," as ",";
|
|
htmldef "<." as
|
|
"<IMG SRC='langle.gif' WIDTH=4 HEIGHT=19 TITLE='<.' ALIGN=TOP>";
|
|
althtmldef "<." as '<FONT FACE=sans-serif>⟨</FONT>'; /* 〈 */
|
|
latexdef "<." as "\langle";
|
|
htmldef ">." as
|
|
"<IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19 TITLE='>.' ALIGN=TOP>";
|
|
althtmldef ">." as '<FONT FACE=sans-serif>⟩</FONT>'; /* 〉 */
|
|
latexdef ">." as "\rangle";
|
|
htmldef "U." as
|
|
"<IMG SRC='bigcup.gif' WIDTH=13 HEIGHT=19 TITLE='U.' ALIGN=TOP>";
|
|
althtmldef "U." as '<FONT SIZE="+1">∪</FONT>'; /* ⋃ */
|
|
/* xcup does not render, and #8899 renders as a small bold cup, on
|
|
Mozilla 1.7.3 on Windows XP */
|
|
/*althtmldef "U." as '⋃';*/ /* ⋃ */
|
|
latexdef "U." as "\bigcup";
|
|
htmldef "|^|" as
|
|
"<IMG SRC='bigcap.gif' WIDTH=13 HEIGHT=19 TITLE='|^|' ALIGN=TOP>";
|
|
althtmldef "|^|" as '<FONT SIZE="+1">∩</FONT>'; /* ⋂ */
|
|
/*althtmldef "|^|" as '⋂';*/ /* ⋂ */
|
|
latexdef "|^|" as "\bigcap";
|
|
|
|
htmldef "Q" as "<IMG SRC='_cq.gif' WIDTH=12 HEIGHT=19 TITLE='Q' ALIGN=TOP>";
|
|
althtmldef "Q" as '<I><FONT COLOR="#CC33CC">Q</FONT></I>';
|
|
latexdef "Q" as "Q";
|
|
htmldef "t" as "<IMG SRC='_t.gif' WIDTH=7 HEIGHT=19 TITLE='t' ALIGN=TOP>";
|
|
althtmldef "t" as '<I><FONT COLOR="#FF0000">t</FONT></I>';
|
|
latexdef "t" as "t";
|
|
htmldef "s" as "<IMG SRC='_s.gif' WIDTH=7 HEIGHT=19 TITLE='s' ALIGN=TOP>";
|
|
althtmldef "s" as '<I><FONT COLOR="#FF0000">s</FONT></I>';
|
|
latexdef "s" as "s";
|
|
htmldef "r" as "<IMG SRC='_r.gif' WIDTH=8 HEIGHT=19 TITLE='r' ALIGN=TOP>";
|
|
althtmldef "r" as '<I><FONT COLOR="#FF0000">r</FONT></I>';
|
|
latexdef "r" as "r";
|
|
htmldef "a" as "<IMG SRC='_a.gif' WIDTH=9 HEIGHT=19 TITLE='a' ALIGN=TOP>";
|
|
althtmldef "a" as '<I><FONT COLOR="#FF0000">a</FONT></I>';
|
|
latexdef "a" as "a";
|
|
htmldef "b" as "<IMG SRC='_b.gif' WIDTH=8 HEIGHT=19 TITLE='b' ALIGN=TOP>";
|
|
althtmldef "b" as '<I><FONT COLOR="#FF0000">b</FONT></I>';
|
|
latexdef "b" as "b";
|
|
htmldef "c" as "<IMG SRC='_c.gif' WIDTH=7 HEIGHT=19 TITLE='c' ALIGN=TOP>";
|
|
althtmldef "c" as '<I><FONT COLOR="#FF0000">c</FONT></I>';
|
|
latexdef "c" as "c";
|
|
htmldef "d" as "<IMG SRC='_d.gif' WIDTH=9 HEIGHT=19 TITLE='d' ALIGN=TOP>";
|
|
althtmldef "d" as '<I><FONT COLOR="#FF0000">d</FONT></I>';
|
|
latexdef "d" as "d";
|
|
htmldef "e" as "<IMG SRC='_e.gif' WIDTH=8 HEIGHT=19 TITLE='e' ALIGN=TOP>";
|
|
althtmldef "e" as '<I><FONT COLOR="#FF0000">e</FONT></I>';
|
|
latexdef "e" as "e";
|
|
htmldef "i" as "<IMG SRC='_i.gif' WIDTH=6 HEIGHT=19 TITLE='i' ALIGN=TOP>";
|
|
althtmldef "i" as '<I><FONT COLOR="#FF0000">i</FONT></I>';
|
|
latexdef "i" as "i";
|
|
htmldef "j" as "<IMG SRC='_j.gif' WIDTH=7 HEIGHT=19 TITLE='j' ALIGN=TOP>";
|
|
althtmldef "j" as '<I><FONT COLOR="#FF0000">j</FONT></I>';
|
|
latexdef "j" as "j";
|
|
htmldef "k" as "<IMG SRC='_k.gif' WIDTH=9 HEIGHT=19 TITLE='k' ALIGN=TOP>";
|
|
althtmldef "k" as '<I><FONT COLOR="#FF0000">k</FONT></I>';
|
|
latexdef "k" as "k";
|
|
htmldef "m" as "<IMG SRC='_m.gif' WIDTH=14 HEIGHT=19 TITLE='m' ALIGN=TOP>";
|
|
althtmldef "m" as '<I><FONT COLOR="#FF0000">m</FONT></I>';
|
|
latexdef "m" as "m";
|
|
htmldef "n" as "<IMG SRC='_n.gif' WIDTH=10 HEIGHT=19 TITLE='n' ALIGN=TOP>";
|
|
althtmldef "n" as '<I><FONT COLOR="#FF0000">n</FONT></I>';
|
|
latexdef "n" as "n";
|
|
htmldef "o" as "<IMG SRC='_o.gif' WIDTH=8 HEIGHT=19 TITLE='o' ALIGN=TOP>";
|
|
althtmldef "o" as '<I><FONT COLOR="#FF0000">o</FONT></I>';
|
|
latexdef "o" as "o";
|
|
htmldef "p" as "<IMG SRC='_p.gif' WIDTH=10 HEIGHT=19 TITLE='p' ALIGN=TOP>";
|
|
althtmldef "p" as '<I><FONT COLOR="#FF0000">p</FONT></I>';
|
|
latexdef "p" as "p";
|
|
htmldef "q" as "<IMG SRC='_q.gif' WIDTH=8 HEIGHT=19 TITLE='q' ALIGN=TOP>";
|
|
althtmldef "q" as '<I><FONT COLOR="#FF0000">q</FONT></I>';
|
|
latexdef "q" as "q";
|
|
htmldef "E" as "<IMG SRC='_ce.gif' WIDTH=13 HEIGHT=19 TITLE='E' ALIGN=TOP>";
|
|
althtmldef "E" as '<I><FONT COLOR="#CC33CC">E</FONT></I>';
|
|
latexdef "E" as "E";
|
|
htmldef "I" as "<IMG SRC='_ci.gif' WIDTH=8 HEIGHT=19 TITLE='I' ALIGN=TOP>";
|
|
althtmldef "I" as '<I><FONT COLOR="#CC33CC">I</FONT></I>';
|
|
latexdef "I" as "I";
|
|
htmldef "J" as "<IMG SRC='_cj.gif' WIDTH=10 HEIGHT=19 TITLE='J' ALIGN=TOP>";
|
|
althtmldef "J" as '<I><FONT COLOR="#CC33CC">J</FONT></I>';
|
|
latexdef "J" as "J";
|
|
htmldef "K" as "<IMG SRC='_ck.gif' WIDTH=14 HEIGHT=19 TITLE='K' ALIGN=TOP>";
|
|
althtmldef "K" as '<I><FONT COLOR="#CC33CC">K</FONT></I>';
|
|
latexdef "K" as "K";
|
|
htmldef "L" as "<IMG SRC='_cl.gif' WIDTH=10 HEIGHT=19 TITLE='L' ALIGN=TOP>";
|
|
althtmldef "L" as '<I><FONT COLOR="#CC33CC">L</FONT></I>';
|
|
latexdef "L" as "L";
|
|
htmldef "M" as "<IMG SRC='_cm.gif' WIDTH=15 HEIGHT=19 TITLE='M' ALIGN=TOP>";
|
|
althtmldef "M" as '<I><FONT COLOR="#CC33CC">M</FONT></I>';
|
|
latexdef "M" as "M";
|
|
htmldef "N" as "<IMG SRC='_cn.gif' WIDTH=14 HEIGHT=19 TITLE='N' ALIGN=TOP>";
|
|
althtmldef "N" as '<I><FONT COLOR="#CC33CC">N</FONT></I>';
|
|
latexdef "N" as "N";
|
|
htmldef "O" as "<IMG SRC='_co.gif' WIDTH=12 HEIGHT=19 TITLE='O' ALIGN=TOP>";
|
|
althtmldef "O" as '<I><FONT COLOR="#CC33CC">O</FONT></I>';
|
|
latexdef "O" as "O";
|
|
htmldef "U" as "<IMG SRC='_cu.gif' WIDTH=12 HEIGHT=19 TITLE='U' ALIGN=TOP>";
|
|
althtmldef "U" as '<I><FONT COLOR="#CC33CC">U</FONT></I>';
|
|
latexdef "U" as "U";
|
|
htmldef "V" as "<IMG SRC='_cv.gif' WIDTH=12 HEIGHT=19 TITLE='V' ALIGN=TOP>";
|
|
althtmldef "V" as '<I><FONT COLOR="#CC33CC">V</FONT></I>';
|
|
latexdef "V" as "V";
|
|
htmldef "W" as "<IMG SRC='_cw.gif' WIDTH=16 HEIGHT=19 TITLE='W' ALIGN=TOP>";
|
|
althtmldef "W" as '<I><FONT COLOR="#CC33CC">W</FONT></I>';
|
|
latexdef "W" as "W";
|
|
htmldef "X" as "<IMG SRC='_cx.gif' WIDTH=13 HEIGHT=19 TITLE='X' ALIGN=TOP>";
|
|
althtmldef "X" as '<I><FONT COLOR="#CC33CC">X</FONT></I>';
|
|
latexdef "X" as "X";
|
|
htmldef "Y" as "<IMG SRC='_cy.gif' WIDTH=12 HEIGHT=19 TITLE='Y' ALIGN=TOP>";
|
|
althtmldef "Y" as '<I><FONT COLOR="#CC33CC">Y</FONT></I>';
|
|
latexdef "Y" as "Y";
|
|
htmldef "Z" as "<IMG SRC='_cz.gif' WIDTH=11 HEIGHT=19 TITLE='Z' ALIGN=TOP>";
|
|
althtmldef "Z" as '<I><FONT COLOR="#CC33CC">Z</FONT></I>';
|
|
latexdef "Z" as "Z";
|
|
|
|
htmldef "\/_" as
|
|
" <IMG SRC='veebar.gif' WIDTH=9 HEIGHT=19 ALT=' \/_' TITLE='\/_'> ";
|
|
althtmldef "\/_" as " ⊻ ";
|
|
/* 2-Jan-2016 reverted sans-serif */
|
|
latexdef "\/_" as "\veebar";
|
|
htmldef "T." as
|
|
" <IMG SRC='top.gif' WIDTH=11 HEIGHT=19 TITLE='T.' ALIGN=TOP> ";
|
|
althtmldef "T." as ' ⊤ ';
|
|
latexdef "T." as "\top";
|
|
htmldef "F." as
|
|
" <IMG SRC='perp.gif' WIDTH=11 HEIGHT=19 TITLE='F.' ALIGN=TOP> ";
|
|
althtmldef "F." as ' ⊥ ';
|
|
latexdef "F." as "\bot";
|
|
|
|
htmldef "iota" as
|
|
"<IMG SRC='riota.gif' WIDTH=6 HEIGHT=19 TITLE='iota' ALIGN=TOP>";
|
|
althtmldef "iota" as '℩';
|
|
latexdef "iota" as "\mathrm{\rotatebox[origin=C]{180}{$\iota$}}";
|
|
htmldef "h" as "<IMG SRC='_h.gif' WIDTH=10 HEIGHT=19 TITLE='h' ALIGN=TOP>";
|
|
althtmldef "h" as '<I><FONT COLOR="#FF0000">h</FONT></I>';
|
|
latexdef "h" as "h";
|
|
htmldef "H" as "<IMG SRC='_ch.gif' WIDTH=14 HEIGHT=19 TITLE='H' ALIGN=TOP>";
|
|
althtmldef "H" as '<I><FONT COLOR="#CC33CC">H</FONT></I>';
|
|
latexdef "H" as "H";
|
|
|
|
htmldef "X." as
|
|
" <IMG SRC='times.gif' WIDTH=9 HEIGHT=19 TITLE='X.' ALIGN=TOP> ";
|
|
althtmldef "X." as ' × ';
|
|
latexdef "X." as "\times";
|
|
htmldef "`'" as "<IMG SRC='_cnv.gif' WIDTH=10 HEIGHT=19 TITLE=" + '"' + "`'" +
|
|
'"' + " ALIGN=TOP>";
|
|
/*htmldef "`'" as
|
|
"<IMG SRC='smallsmile.gif' WIDTH=12 HEIGHT=19 TITLE=" +
|
|
'"' + "`'" + '"' + " ALIGN=TOP>";*/
|
|
althtmldef "`'" as '<FONT SIZE="-1"><SUP>◡</SUP></FONT>'; /* or 8995 */
|
|
latexdef "`'" as "{}^{\smallsmile}";
|
|
htmldef "dom" as
|
|
"<IMG SRC='_dom.gif' WIDTH=26 HEIGHT=19 TITLE='dom' ALIGN=TOP> ";
|
|
althtmldef "dom" as 'dom ';
|
|
latexdef "dom" as "{\rm dom}";
|
|
htmldef "ran" as
|
|
"<IMG SRC='_ran.gif' WIDTH=22 HEIGHT=19 TITLE='ran' ALIGN=TOP> ";
|
|
althtmldef "ran" as 'ran ';
|
|
latexdef "ran" as "{\rm ran}";
|
|
htmldef "|`" as " <IMG SRC='restriction.gif' WIDTH=5 HEIGHT=19 TITLE='|`'" +
|
|
" ALIGN=TOP> ";
|
|
althtmldef "|`" as ' <FONT FACE=sans-serif>↾</FONT> '; /* ↾ */
|
|
latexdef "|`" as "\restriction";
|
|
htmldef '"' as "<IMG SRC='backquote.gif' WIDTH=7 HEIGHT=19 TITLE='" + '"' +
|
|
"' ALIGN=TOP>";
|
|
althtmldef '"' as ' “ ';
|
|
latexdef '"' as "``";
|
|
htmldef "o." as
|
|
" <IMG SRC='circ.gif' WIDTH=8 HEIGHT=19 TITLE='o.' ALIGN=TOP> ";
|
|
althtmldef "o." as ' <FONT FACE=sans-serif>∘</FONT> ';
|
|
latexdef "o." as "\circ";
|
|
htmldef
|
|
"Fun" as "<IMG SRC='_fun.gif' WIDTH=25 HEIGHT=19 TITLE='Fun' ALIGN=TOP> ";
|
|
althtmldef "Fun" as 'Fun ';
|
|
latexdef "Fun" as "{\rm Fun}";
|
|
htmldef "Fn" as
|
|
" <IMG SRC='_fn.gif' WIDTH=17 HEIGHT=19 TITLE='Fn' ALIGN=TOP> ";
|
|
althtmldef "Fn" as ' Fn ';
|
|
latexdef "Fn" as "{\rm Fn}";
|
|
htmldef ":" as "<IMG SRC='colon.gif' WIDTH=4 HEIGHT=19 TITLE=':' ALIGN=TOP>";
|
|
althtmldef ":" as ':';
|
|
latexdef ":" as ":";
|
|
htmldef "-->" as
|
|
"<IMG SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 TITLE='-->' ALIGN=TOP>";
|
|
althtmldef "-->" as '–→';
|
|
/* ­‐–—− (possible symbols test) */
|
|
latexdef "-->" as "\longrightarrow";
|
|
htmldef "-1-1->" as
|
|
"<IMG SRC='onetoone.gif' WIDTH=23 HEIGHT=19 TITLE='-1-1->' ALIGN=TOP>";
|
|
althtmldef "-1-1->" as
|
|
'–<FONT SIZE=-2 FACE=sans-serif>1-1</FONT>→';
|
|
latexdef "-1-1->" as
|
|
"\raisebox{.5ex}{${\textstyle{\:}_{\mbox{\footnotesize\rm 1" +
|
|
"\tt -\rm 1}}}\atop{\textstyle{" +
|
|
"\longrightarrow}\atop{\textstyle{}^{\mbox{\footnotesize\rm {\ }}}}}$}";
|
|
htmldef "-onto->" as
|
|
"<IMG SRC='onto.gif' WIDTH=23 HEIGHT=19 TITLE='-onto->' ALIGN=TOP>";
|
|
althtmldef "-onto->" as
|
|
'–<FONT SIZE=-2 FACE=sans-serif>onto</FONT>→';
|
|
latexdef "-onto->" as
|
|
"\raisebox{.5ex}{${\textstyle{\:}_{\mbox{\footnotesize\rm {\ }}}}" +
|
|
"\atop{\textstyle{" +
|
|
"\longrightarrow}\atop{\textstyle{}^{\mbox{\footnotesize\rm onto}}}}$}";
|
|
htmldef "-1-1-onto->" as "<IMG SRC='onetooneonto.gif' WIDTH=23 HEIGHT=19 " +
|
|
"TITLE='-1-1-onto->' ALIGN=TOP>";
|
|
althtmldef "-1-1-onto->" as '–<FONT SIZE=-2 '
|
|
+ 'FACE=sans-serif>1-1</FONT>-<FONT SIZE=-2 '
|
|
+ 'FACE=sans-serif>onto</FONT>→';
|
|
latexdef "-1-1-onto->" as
|
|
"\raisebox{.5ex}{${\textstyle{\:}_{\mbox{\footnotesize\rm 1" +
|
|
"\tt -\rm 1}}}\atop{\textstyle{" +
|
|
"\longrightarrow}\atop{\textstyle{}^{\mbox{\footnotesize\rm onto}}}}$}";
|
|
htmldef "`" as
|
|
"<IMG SRC='backtick.gif' WIDTH=4 HEIGHT=19 TITLE='` ' ALIGN=TOP>";
|
|
/* Above, IE7 _printing_ is corrupted by '`'; use '` ' which works */
|
|
althtmldef "`" as ' ‘';
|
|
latexdef "`" as "`";
|
|
htmldef "Isom" as
|
|
" <IMG SRC='_isom.gif' WIDTH=30 HEIGHT=19 TITLE='Isom' ALIGN=TOP> ";
|
|
althtmldef "Isom" as ' Isom ';
|
|
latexdef "Isom" as "{\rm Isom}";
|
|
htmldef "|->" as " <IMG SRC='mapsto.gif' WIDTH=15 HEIGHT=19 TITLE='|->'" +
|
|
" ALIGN=TOP> ";
|
|
althtmldef "|->" as ' <FONT FACE=sans-serif>↦</FONT> ';
|
|
latexdef "|->" as "\mapsto";
|
|
htmldef "1st" as
|
|
"<IMG SRC='_1st.gif' WIDTH=15 HEIGHT=19 TITLE='1st' ALIGN=TOP>";
|
|
althtmldef "1st" as '1<SUP>st</SUP> ';
|
|
latexdef "1st" as "1^{\rm st}";
|
|
htmldef "2nd" as
|
|
"<IMG SRC='_2nd.gif' WIDTH=21 HEIGHT=19 TITLE='2nd' ALIGN=TOP>";
|
|
althtmldef "2nd" as '2<SUP>nd</SUP> ';
|
|
latexdef "2nd" as "2^{\rm nd}";
|
|
htmldef "Swap" as
|
|
"<FONT FACE=sans-serif> Swap </FONT>";
|
|
althtmldef "Swap" as '<FONT FACE=sans-serif> Swap </FONT>';
|
|
latexdef "Swap" as "{\rm Swap}";
|
|
|
|
htmldef "_E" as
|
|
" <IMG SRC='rmce.gif' WIDTH=9 HEIGHT=19 TITLE='_E' ALIGN=TOP> ";
|
|
althtmldef "_E" as ' E ';
|
|
latexdef "_E" as "{\rm E}";
|
|
htmldef "_I" as
|
|
" <IMG SRC='rmci.gif' WIDTH=4 HEIGHT=19 TITLE='_I' ALIGN=TOP> ";
|
|
althtmldef "_I" as ' I ';
|
|
latexdef "_I" as "{\rm I}";
|
|
|
|
htmldef "U_" as
|
|
"<IMG SRC='_cupbar.gif' WIDTH=13 HEIGHT=19 TITLE='U_' ALIGN=TOP>";
|
|
althtmldef "U_" as '<U><FONT SIZE="+1">∪</FONT></U>'; /* ⋃ */
|
|
latexdef "U_" as "\underline{\bigcup}";
|
|
htmldef "|^|_" as
|
|
"<IMG SRC='_capbar.gif' WIDTH=13 HEIGHT=19 TITLE='|^|_' ALIGN=TOP>";
|
|
althtmldef "|^|_" as '<U><FONT SIZE="+1">∩</FONT></U>'; /* ⋂ */
|
|
latexdef "|^|_" as "\underline{\bigcap}";
|
|
|
|
htmldef "(+)" as
|
|
" <IMG SRC='oplus.gif' WIDTH=13 HEIGHT=19 TITLE='(+)' ALIGN=TOP> ";
|
|
althtmldef "(+)" as " ⊕ ";
|
|
latexdef "(+)" as "\oplus";
|
|
|
|
htmldef "0c" as '0<SUB><I>c</I></SUB>';
|
|
althtmldef "0c" as '0<SUB><I>c</I></SUB>';
|
|
latexdef "0c" as "0_c";
|
|
htmldef "1c" as '1<SUB><I>c</I></SUB>';
|
|
althtmldef "1c" as '1<SUB><I>c</I></SUB>';
|
|
latexdef "1c" as "1_c";
|
|
htmldef "+c" as
|
|
" <IMG SRC='_plc.gif' WIDTH=18 HEIGHT=19 TITLE='+o' ALIGN=TOP> ";
|
|
althtmldef "+c" as ' +<SUB><I>c</I></SUB> ';
|
|
latexdef "+c" as "+_c";
|
|
|
|
htmldef "l" as "<IMG SRC='_l.gif' WIDTH=6 HEIGHT=19 TITLE='l' ALIGN=TOP>";
|
|
althtmldef "l" as '<I><FONT COLOR="#FF0000">l</FONT></I>';
|
|
latexdef "l" as "l";
|
|
|
|
htmldef "Fix" as
|
|
"<IMG SRC='_fix.gif' WIDTH=21 HEIGHT=19 TITLE='Fix' ALIGN=TOP>";
|
|
althtmldef "Fix" as '<FONT FACE=sans-serif> Fix </FONT>';
|
|
latexdef "Fix" as "{\rm Fix}";
|
|
htmldef "<<" as
|
|
"<IMG SRC='llangle.gif' WIDTH=6 HEIGHT=19 TITLE='<<' ALIGN=TOP>";
|
|
althtmldef "<<" as "⟪";
|
|
latexdef "<<" as "\langle\langle";
|
|
htmldef ">>" as
|
|
"<IMG SRC='rrangle.gif' WIDTH=6 HEIGHT=19 TITLE='>>' ALIGN=TOP>";
|
|
althtmldef ">>" as "⟫";
|
|
latexdef ">>" as "\rangle\rangle";
|
|
htmldef "(x)" as
|
|
" <IMG SRC='otimes.gif' WIDTH=13 HEIGHT=19 TITLE='(x)' ALIGN=TOP> ";
|
|
althtmldef "(x)" as " ⊗ ";
|
|
latexdef "(x)" as "\otimes";
|
|
htmldef "Image" as "Image";
|
|
althtmldef "Image" as "Image";
|
|
latexdef "Image" as "{\rm Image}";
|
|
|
|
htmldef "Image_k" as "Image<SUB><I>k</I></SUB>";
|
|
althtmldef "Image_k" as "Image<SUB><I>k</I></SUB>";
|
|
latexdef "Image_k" as "{\rm Image}_k";
|
|
|
|
htmldef "~P1" as
|
|
"<IMG SRC='scrp.gif' WIDTH=16 HEIGHT=19 TITLE='~P' ALIGN=TOP><SUB>1</SUB> ";
|
|
althtmldef "~P1" as '<FONT FACE=sans-serif>℘</FONT><SUB>1</SUB>';
|
|
latexdef "~P1" as "{\cal P}_1";
|
|
|
|
htmldef "X._k" as
|
|
" <IMG SRC='times.gif' WIDTH=9 HEIGHT=19 TITLE='X.'" +
|
|
"ALIGN=TOP><SUB><I>k</I></SUB> ";
|
|
althtmldef "X._k" as ' ×<SUB><I>k</I></SUB> ';
|
|
latexdef "X._k" as "\times_k";
|
|
htmldef "`'_k" as "<IMG SRC='_cnv.gif' WIDTH=10 HEIGHT=19 TITLE=" + '"' +
|
|
"`'" + '"' + " ALIGN=TOP><SUB><I>k</I></SUB>";
|
|
/*htmldef "`'" as
|
|
"<IMG SRC='smallsmile.gif' WIDTH=12 HEIGHT=19 TITLE=" +
|
|
'"' + "`'" + '"' + " ALIGN=TOP>";*/
|
|
althtmldef "`'_k" as
|
|
'<FONT SIZE="-1"><SUP>◡</SUP></FONT><SUB><I>k</I></SUB>';
|
|
latexdef "`'_k" as "{}^{\smallsmile}_k";
|
|
htmldef '"_k' as "<IMG SRC='backquote.gif' WIDTH=7 HEIGHT=19 TITLE='" + '"' +
|
|
"' ALIGN=TOP><SUB><I>k</I></SUB>";
|
|
althtmldef '"_k' as ' “<SUB><I>k</I></SUB> ';
|
|
latexdef '"_k' as "``_k";
|
|
htmldef "o._k" as
|
|
" <IMG SRC='circ.gif' WIDTH=8 HEIGHT=19 TITLE='o.' ALIGN=TOP>" +
|
|
"<SUB><I>k</I></SUB> ";
|
|
althtmldef "o._k" as
|
|
' <FONT FACE=sans-serif>∘</FONT><SUB><I>k</I></SUB> ';
|
|
latexdef "o._k" as "\circ_k";
|
|
htmldef "SI" as
|
|
"<FONT FACE=sans-serif> SI </FONT>";
|
|
althtmldef "SI" as '<FONT FACE=sans-serif> SI </FONT>';
|
|
latexdef "SI" as "{\rm SI}";
|
|
htmldef "Clos1" as
|
|
"<FONT FACE=sans-serif> Clos1 </FONT>";
|
|
althtmldef "Clos1" as '<FONT FACE=sans-serif> Clos1 </FONT>';
|
|
latexdef "Clos1" as "{\rm Clos1}";
|
|
htmldef "Phi" as
|
|
"<FONT FACE=sans-serif> Phi </FONT>";
|
|
althtmldef "Phi" as '<FONT FACE=sans-serif> Phi </FONT>';
|
|
latexdef "Phi" as "{\rm Phi}";
|
|
htmldef "Proj1" as
|
|
"<FONT FACE=sans-serif> Proj1 </FONT>";
|
|
althtmldef "Proj1" as '<FONT FACE=sans-serif> Proj1 </FONT>';
|
|
latexdef "Proj1" as "{\rm Proj1}";
|
|
htmldef "Proj2" as
|
|
"<FONT FACE=sans-serif> Proj2 </FONT>";
|
|
althtmldef "Proj2" as '<FONT FACE=sans-serif> Proj2 </FONT>';
|
|
latexdef "Proj2" as "{\rm Proj2}";
|
|
htmldef "_S" as
|
|
"<FONT FACE=sans-serif> S </FONT>";
|
|
althtmldef "_S" as '<FONT FACE=sans-serif> S </FONT>';
|
|
latexdef "_S" as "{\rm S}";
|
|
htmldef "U.1" as '⋃<SUB>1</SUB>';
|
|
althtmldef "U.1" as '⋃<SUB>1</SUB>';
|
|
latexdef "U.1" as "\bigcup_1";
|
|
htmldef "_I_k" as
|
|
" <IMG SRC='rmci.gif' WIDTH=4 HEIGHT=19 TITLE='_I_k' ALIGN=TOP>" +
|
|
"<SUB><I>k</I></SUB> ";
|
|
althtmldef "_I_k" as ' I<SUB><I>k</I></SUB> ';
|
|
latexdef "_I_k" as "{\rm I}_k";
|
|
htmldef "_S_k" as
|
|
" <FONT FACE=sans-serif>S</FONT><SUB><I>k</I></SUB> ";
|
|
althtmldef "_S_k" as
|
|
' <FONT FACE=sans-serif>S</FONT><SUB><I>k</I></SUB> ';
|
|
latexdef "_S_k" as "{\rm S}_k";
|
|
htmldef "Ins2_k" as
|
|
" <FONT FACE=sans-serif>Ins2</FONT><SUB><I>k</I></SUB> ";
|
|
althtmldef "Ins2_k" as
|
|
' <FONT FACE=sans-serif>Ins2</FONT><SUB><I>k</I></SUB> ';
|
|
latexdef "Ins2_k" as "{\rm Ins2}_k";
|
|
htmldef "Ins3_k" as
|
|
" <FONT FACE=sans-serif>Ins3</FONT><SUB><I>k</I></SUB> ";
|
|
althtmldef "Ins3_k" as
|
|
' <FONT FACE=sans-serif>Ins3</FONT><SUB><I>k</I></SUB> ';
|
|
latexdef "Ins3_k" as "{\rm Ins3}_k";
|
|
htmldef "SI_k" as
|
|
" <FONT FACE=sans-serif>SI</FONT><SUB><I>k</I></SUB> ";
|
|
althtmldef "SI_k" as ' <FONT FACE=sans-serif>SI</FONT><SUB><I>k</I></SUB> ';
|
|
latexdef "SI_k" as "{\rm SI}_k";
|
|
|
|
|
|
htmldef "Ins2" as
|
|
" <FONT FACE=sans-serif>Ins2</FONT> ";
|
|
althtmldef "Ins2" as ' <FONT FACE=sans-serif>Ins2</FONT> ';
|
|
latexdef "Ins2" as "{\rm Ins2}";
|
|
htmldef "Ins3" as
|
|
" <FONT FACE=sans-serif>Ins3</FONT> ";
|
|
althtmldef "Ins3" as ' <FONT FACE=sans-serif>Ins3</FONT> ';
|
|
latexdef "Ins3" as "{\rm Ins3}";
|
|
htmldef "Ins4" as
|
|
" <FONT FACE=sans-serif>Ins4</FONT> ";
|
|
althtmldef "Ins4" as ' <FONT FACE=sans-serif>Ins4</FONT> ';
|
|
latexdef "Ins4" as "{\rm Ins4}";
|
|
htmldef "Cup" as
|
|
" <FONT FACE=sans-serif>Cup</FONT> ";
|
|
althtmldef "Cup" as ' <FONT FACE=sans-serif>Cup</FONT> ';
|
|
latexdef "Cup" as "{\rm Cup}";
|
|
htmldef "Compose" as
|
|
" <FONT FACE=sans-serif>Compose</FONT> ";
|
|
althtmldef "Compose" as ' <FONT FACE=sans-serif>Compose</FONT> ';
|
|
latexdef "Compose" as "{\rm Compose}";
|
|
htmldef "Disj" as
|
|
" <FONT FACE=sans-serif>Disj</FONT> ";
|
|
althtmldef "Disj" as ' <FONT FACE=sans-serif>Disj</FONT> ';
|
|
latexdef "Disj" as "{\rm Disj}";
|
|
htmldef "AddC" as
|
|
" <FONT FACE=sans-serif>AddC</FONT> ";
|
|
althtmldef "AddC" as ' <FONT FACE=sans-serif>AddC</FONT> ';
|
|
latexdef "AddC" as "{\rm AddC}";
|
|
|
|
htmldef "SI_3" as
|
|
" <FONT FACE=sans-serif>SI</FONT><SUB><I>3</I></SUB> ";
|
|
althtmldef "SI_3" as ' <FONT FACE=sans-serif>SI</FONT><SUB><I>3</I></SUB> ';
|
|
latexdef "SI_3" as "{\rm SI}_3";
|
|
|
|
|
|
htmldef "P6" as
|
|
" <FONT FACE=sans-serif>P6</FONT> ";
|
|
althtmldef "P6" as ' <FONT FACE=sans-serif>P6</FONT> ';
|
|
latexdef "P6" as "{\rm P6}";
|
|
|
|
|
|
htmldef "Nn" as
|
|
" <FONT FACE=sans-serif>Nn</FONT> ";
|
|
althtmldef "Nn" as ' <FONT FACE=sans-serif>Nn</FONT> ';
|
|
latexdef "Nn" as "{\rm Nn}";
|
|
htmldef "Fin" as
|
|
" <FONT FACE=sans-serif>Fin</FONT> ";
|
|
althtmldef "Fin" as ' <FONT FACE=sans-serif>Fin</FONT> ';
|
|
latexdef "Fin" as "{\rm Fin}";
|
|
|
|
htmldef "<_[fin]" as
|
|
" <IMG SRC='le.gif' WIDTH=11 HEIGHT=19 ALT='<_' ALIGN=TOP>" +
|
|
"<SUB>fin</SUB> ";
|
|
althtmldef "<_[fin]" as ' ≤<SUB>fin</SUB> ';
|
|
latexdef "<_[fin]" as "{\le}_{\rm fin}";
|
|
htmldef "<[fin]" as
|
|
" <IMG SRC='lt.gif' WIDTH=11 HEIGHT=19 ALT='<' ALIGN=TOP>" +
|
|
"<SUB>fin</SUB> ";
|
|
althtmldef "<[fin]" as ' <<SUB>fin</SUB> ';
|
|
latexdef "<[fin]" as "<_{\rm fin}";
|
|
htmldef "Nc[fin]" as
|
|
" <FONT FACE=sans-serif>Nc</FONT><SUB>fin</SUB> ";
|
|
althtmldef "Nc[fin]" as ' <FONT FACE=sans-serif>Nc</FONT><SUB>fin</SUB> ';
|
|
latexdef "Nc[fin]" as "{\rm Nc}_{\rm fin}";
|
|
htmldef "_T[fin]" as
|
|
" <FONT FACE=sans-serif>T</FONT><SUB>fin</SUB> ";
|
|
althtmldef "_T[fin]" as ' <FONT FACE=sans-serif>T</FONT><SUB>fin</SUB> ';
|
|
latexdef "_T[fin]" as "{\rm T}_{\rm fin}";
|
|
htmldef "Even[fin]" as
|
|
" <FONT FACE=sans-serif>Even</FONT><SUB>fin</SUB> ";
|
|
althtmldef "Even[fin]" as
|
|
' <FONT FACE=sans-serif>Even</FONT><SUB>fin</SUB> ';
|
|
latexdef "Even[fin]" as "{\rm Even}_{\rm fin}";
|
|
htmldef "Odd[fin]" as
|
|
" <FONT FACE=sans-serif>Odd</FONT><SUB>fin</SUB> ";
|
|
althtmldef "Odd[fin]" as ' <FONT FACE=sans-serif>Odd</FONT><SUB>fin</SUB> ';
|
|
latexdef "Odd[fin]" as "{\rm Odd}_{\rm fin}";
|
|
htmldef "_S[fin]" as
|
|
" <FONT FACE=sans-serif>S</FONT><SUB>fin</SUB> ";
|
|
althtmldef "_S[fin]" as ' <FONT FACE=sans-serif>S</FONT><SUB>fin</SUB> ';
|
|
latexdef "_S[fin]" as "{\rm S}_{\rm fin}";
|
|
htmldef "Sp[fin]" as
|
|
" <FONT FACE=sans-serif>Sp</FONT><SUB>fin</SUB> ";
|
|
althtmldef "Sp[fin]" as ' <FONT FACE=sans-serif>Sp</FONT><SUB>fin</SUB> ';
|
|
latexdef "Sp[fin]" as "{\rm Sp}_{\rm fin}";
|
|
|
|
htmldef "Funs" as
|
|
" <FONT FACE=sans-serif>Funs</FONT> ";
|
|
althtmldef "Funs" as ' <FONT FACE=sans-serif>Funs</FONT> ';
|
|
latexdef "Funs" as "{\rm Funs}";
|
|
htmldef "Fns" as
|
|
" <FONT FACE=sans-serif>Fns</FONT> ";
|
|
althtmldef "Fns" as ' <FONT FACE=sans-serif>Fns</FONT> ';
|
|
latexdef "Fns" as "{\rm Fns}";
|
|
htmldef "PProd" as
|
|
" <FONT FACE=sans-serif>PProd</FONT> ";
|
|
althtmldef "PProd" as ' <FONT FACE=sans-serif>PProd</FONT> ';
|
|
latexdef "PProd" as "{\rm PProd}";
|
|
htmldef "Cross" as
|
|
" <FONT FACE=sans-serif>Cross</FONT> ";
|
|
althtmldef "Cross" as ' <FONT FACE=sans-serif>Cross</FONT> ';
|
|
latexdef "Cross" as "{\rm Cross}";
|
|
htmldef "Pw1Fn" as
|
|
" <FONT FACE=sans-serif>Pw1Fn</FONT> ";
|
|
althtmldef "Pw1Fn" as ' <FONT FACE=sans-serif>Pw1Fn</FONT> ';
|
|
latexdef "Pw1Fn" as "{\rm Pw1Fn}";
|
|
htmldef "FullFun" as
|
|
" <FONT FACE=sans-serif>FullFun</FONT> ";
|
|
althtmldef "FullFun" as ' <FONT FACE=sans-serif>FullFun</FONT> ';
|
|
latexdef "FullFun" as "{\rm FullFun}";
|
|
|
|
htmldef "Trans" as
|
|
" <FONT FACE=sans-serif>Trans</FONT> ";
|
|
althtmldef "Trans" as ' <FONT FACE=sans-serif>Trans</FONT> ';
|
|
latexdef "Trans" as "{\rm Trans}";
|
|
htmldef "Ref" as
|
|
" <FONT FACE=sans-serif>Ref</FONT> ";
|
|
althtmldef "Ref" as ' <FONT FACE=sans-serif>Ref</FONT> ';
|
|
latexdef "Ref" as "{\rm Ref}";
|
|
htmldef "Antisym" as
|
|
" <FONT FACE=sans-serif>Antisym</FONT> ";
|
|
althtmldef "Antisym" as ' <FONT FACE=sans-serif>Antisym</FONT> ';
|
|
latexdef "Antisym" as "{\rm Antisym}";
|
|
htmldef "Po" as
|
|
" <FONT FACE=sans-serif>Po</FONT> ";
|
|
althtmldef "Po" as ' <FONT FACE=sans-serif>Po</FONT> ';
|
|
latexdef "Po" as "{\rm Po}";
|
|
htmldef "Connex" as
|
|
" <FONT FACE=sans-serif>Connex</FONT> ";
|
|
althtmldef "Connex" as ' <FONT FACE=sans-serif>Connex</FONT> ';
|
|
latexdef "Connex" as "{\rm Connex}";
|
|
htmldef "Or" as
|
|
" <FONT FACE=sans-serif>Or</FONT> ";
|
|
althtmldef "Or" as ' <FONT FACE=sans-serif>Or</FONT> ';
|
|
latexdef "Or" as "{\rm Or}";
|
|
htmldef "Fr" as
|
|
" <FONT FACE=sans-serif>Fr</FONT> ";
|
|
althtmldef "Fr" as ' <FONT FACE=sans-serif>Fr</FONT> ';
|
|
latexdef "Fr" as "{\rm Fr}";
|
|
htmldef "We" as
|
|
" <FONT FACE=sans-serif>We</FONT> ";
|
|
althtmldef "We" as ' <FONT FACE=sans-serif>We</FONT> ';
|
|
latexdef "We" as "{\rm We}";
|
|
htmldef "Ext" as
|
|
" <FONT FACE=sans-serif>Ext</FONT> ";
|
|
althtmldef "Ext" as ' <FONT FACE=sans-serif>Ext</FONT> ';
|
|
latexdef "Ext" as "{\rm Ext}";
|
|
htmldef "Sym" as
|
|
" <FONT FACE=sans-serif>Sym</FONT> ";
|
|
althtmldef "Sym" as ' <FONT FACE=sans-serif>Sym</FONT> ';
|
|
latexdef "Sym" as "{\rm Sym}";
|
|
htmldef "Er" as
|
|
" <FONT FACE=sans-serif>Er</FONT> ";
|
|
althtmldef "Er" as ' <FONT FACE=sans-serif>Er</FONT> ';
|
|
latexdef "Er" as "{\rm Er}";
|
|
|
|
htmldef "/." as
|
|
"<IMG SRC='diagup.gif' WIDTH=14 HEIGHT=19 TITLE='/.' ALIGN=TOP>";
|
|
althtmldef "/." as ' <B>/</B> ';
|
|
latexdef "/." as "\diagup";
|
|
|
|
htmldef "~~" as
|
|
" <IMG SRC='approx.gif' WIDTH=13 HEIGHT=19 TITLE='~~' ALIGN=TOP> ";
|
|
althtmldef "~~" as ' ≈ '; /* ≈ */
|
|
latexdef "~~" as "\approx";
|
|
|
|
htmldef "^m" as
|
|
" <IMG SRC='_hatm.gif' WIDTH=15 HEIGHT=19 TITLE='^m' ALIGN=TOP> ";
|
|
althtmldef "^m" as ' ↑<SUB><I>m</I></SUB> ';
|
|
latexdef "^m" as "\uparrow_m";
|
|
htmldef "^pm" as
|
|
" <IMG SRC='_hatpm.gif' WIDTH=21 HEIGHT=19 TITLE='^pm' ALIGN=TOP> ";
|
|
althtmldef "^pm" as ' ↑<SUB><I>pm</I></SUB> ';
|
|
latexdef "^pm" as "\uparrow_{pm}";
|
|
|
|
htmldef "NC" as
|
|
" <FONT FACE=sans-serif>NC</FONT> ";
|
|
althtmldef "NC" as ' <FONT FACE=sans-serif>NC</FONT> ';
|
|
latexdef "NC" as "{\rm NC}";
|
|
htmldef "<_c" as
|
|
" <IMG SRC='le.gif' WIDTH=11 HEIGHT=19 ALT='<_' ALIGN=TOP>" +
|
|
"<SUB>c</SUB> ";
|
|
althtmldef "<_c" as ' ≤<SUB>c</SUB> ';
|
|
latexdef "<_c" as "{\le}_c";
|
|
htmldef "<c" as
|
|
" <IMG SRC='lt.gif' WIDTH=11 HEIGHT=19 ALT='<' ALIGN=TOP><SUB>c</SUB> ";
|
|
althtmldef "<c" as ' <<SUB>c</SUB> ';
|
|
latexdef "<c" as "<_c ";
|
|
htmldef "Nc" as
|
|
" <FONT FACE=sans-serif>Nc</FONT> ";
|
|
althtmldef "Nc" as ' <FONT FACE=sans-serif>Nc</FONT> ';
|
|
latexdef "Nc" as "{\rm Nc}";
|
|
htmldef ".c" as ' ·<SUB><I>c</I></SUB> ';
|
|
althtmldef ".c" as ' ·<SUB><I>c</I></SUB> ';
|
|
latexdef ".c" as "\cdot_c";
|
|
htmldef "T_c" as
|
|
" <FONT FACE=sans-serif>T</FONT><SUB>c</SUB> ";
|
|
althtmldef "T_c" as ' <FONT FACE=sans-serif>T</FONT><SUB>c</SUB> ';
|
|
latexdef "T_c" as "{\rm T}_c ";
|
|
htmldef "2c" as '2<SUB><I>c</I></SUB>';
|
|
althtmldef "2c" as '2<SUB><I>c</I></SUB>';
|
|
latexdef "2c" as "2_c";
|
|
htmldef "3c" as '3<SUB><I>c</I></SUB>';
|
|
althtmldef "3c" as '3<SUB><I>c</I></SUB>';
|
|
latexdef "3c" as "3_c";
|
|
htmldef "^c" as ' ↑<SUB><I>c</I></SUB> ';
|
|
althtmldef "^c" as ' ↑<SUB><I>c</I></SUB> ';
|
|
latexdef "^c" as "\uparrow_c";
|
|
|
|
htmldef "Sp[ac]" as
|
|
" <FONT FACE=sans-serif>Sp</FONT><SUB>ac</SUB> ";
|
|
althtmldef "Sp[ac]" as ' <FONT FACE=sans-serif>Sp</FONT><SUB>ac</SUB> ';
|
|
latexdef "Sp[ac]" as "{\rm Sp}_{\rm ac}";
|
|
|
|
htmldef "TcFn" as "TcFn";
|
|
althtmldef "TcFn" as "TcFn";
|
|
latexdef "TcFn" as "{\rm TcFn}";
|
|
|
|
htmldef "FRec" as " <FONT FACE=sans-serif>FRec</FONT> ";
|
|
althtmldef "FRec" as " <FONT FACE=sans-serif>FRec</FONT> ";
|
|
latexdef "FRec" as "{\rm FRec}";
|
|
|
|
htmldef "Dom" as " <FONT FACE=sans-serif>Dom</FONT> ";
|
|
althtmldef "Dom" as " <FONT FACE=sans-serif>Dom</FONT> ";
|
|
latexdef "Dom" as "{\rm Dom}";
|
|
|
|
htmldef "Ran" as " <FONT FACE=sans-serif>Ran</FONT> ";
|
|
althtmldef "Ran" as " <FONT FACE=sans-serif>Ran</FONT> ";
|
|
latexdef "Ran" as "{\rm Ran}";
|
|
|
|
|
|
htmldef "Can" as
|
|
" <FONT FACE=sans-serif>Can</FONT> ";
|
|
althtmldef "Can" as ' <FONT FACE=sans-serif>Can</FONT> ';
|
|
latexdef "Can" as "{\rm Can}";
|
|
|
|
htmldef "SCan" as
|
|
" <FONT FACE=sans-serif>SCan</FONT> ";
|
|
althtmldef "SCan" as ' <FONT FACE=sans-serif>SCan</FONT> ';
|
|
latexdef "SCan" as "{\rm SCan}";
|
|
$)
|