P. 44 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4301-4400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iin0r 4301*	If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)
|- ( |^|_ x e. A (/) = (/) -> A =/= (/) )
Theorem	intv 4302	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
|- |^| _V = (/)
Theorem	axpweq 4303*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4306 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
|- A e. _V   =>    |- ( ~P A e. _V <-> E. x A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )
2.2.6  Collection principle
Theorem	bnd 4304*	A very strong generalization of the Axiom of Replacement (compare zfrep6 4243). Its strength lies in the rather profound fact that ph ( x , y ) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. In the context of IZF, it is just a slight variation of ax-coll 4241. (Contributed by NM, 17-Oct-2004.)
|- ( A. x e. z E. y ph -> E. w A. x e. z E. y e. w ph )
Theorem	bnd2 4305*	A variant of the Boundedness Axiom bnd 4304 that picks a subset z out of a possibly proper class B in which a property is true. (Contributed by NM, 4-Feb-2004.)
|- A e. _V   =>    |- ( A. x e. A E. y e. B ph -> E. z ( z C_ B /\ A. x e. A E. y e. z ph ) )
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
2.3.1  Introduce the Axiom of Power Sets
Axiom	ax-pow 4306*	Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set y exists that includes the power set of a given set x i.e. contains every subset of x. This is Axiom 8 of [Crosilla] p. "Axioms of CZF and IZF" except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4249). The variant axpow2 4308 uses explicit subset notation. A version using class notation is pwex 4315. (Contributed by NM, 5-Aug-1993.)
|- E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y )
Theorem	zfpow 4307*	Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
|- E. x A. y ( A. x ( x e. y -> x e. z ) -> y e. x )
Theorem	axpow2 4308*	A variant of the Axiom of Power Sets ax-pow 4306 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
|- E. y A. z ( z C_ x -> z e. y )
Theorem	axpow3 4309*	A variant of the Axiom of Power Sets ax-pow 4306. For any set x, there exists a set y whose members are exactly the subsets of x i.e. the power set of x. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
|- E. y A. z ( z C_ x <-> z e. y )
Theorem	el 4310*	Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- E. y x e. y
Theorem	vpwex 4311	Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4312 from vpwex 4311. (Revised by BJ, 10-Aug-2022.)
|- ~P x e. _V
Theorem	pwexg 4312	Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
|- ( A e. V -> ~P A e. _V )
Theorem	pwexd 4313	Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> A e. V )   =>    |- ( ph -> ~P A e. _V )
Theorem	abssexg 4314*	Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A e. V -> { x | ( x C_ A /\ ph ) } e. _V )
Theorem	pwex 4315	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
|- A e. _V   =>    |- ~P A e. _V
Theorem	snexg 4316	A singleton whose element exists is a set. The A e. _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
|- ( A e. V -> { A } e. _V )
Theorem	snex 4317	A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
|- A e. _V   =>    |- { A } e. _V
Theorem	snexprc 4318	A singleton whose element is a proper class is a set. The -. A e. _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
|- ( -. A e. _V -> { A } e. _V )
Theorem	notnotsnex 4319	A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
|- -. -. { A } e. _V
Theorem	p0ex 4320	The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
|- { (/) } e. _V
Theorem	pp0ex 4321	{ (/) , { (/) } } (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
|- { (/) , { (/) } } e. _V
Theorem	ord3ex 4322	The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
|- { (/) , { (/) } , { (/) , { (/) } } } e. _V
Theorem	dtruarb 4323*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4701 in which we are given a set y and go from there to a set x which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
|- E. x E. y -. x = y
Theorem	pwuni 4324	A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
|- A C_ ~P U. A
Theorem	undifexmid 4325*	Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3605 and undifdcss 7220 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
|- ( x C_ y <-> ( x u. ( y \ x ) ) = y )   =>    |- ( ph \/ -. ph )
2.3.2  A notation for excluded middle
Syntax	wem 4326	Formula for an abbreviation of excluded middle.
wff EXMID
Definition	df-exmid 4327	The expression EXMID will be used as a readable shorthand for any form of the law of the excluded middle; this is a useful shorthand largely because it hides statements of the form "for any proposition" in a system which can only quantify over sets, not propositions. To see how this compares with other ways of expressing excluded middle, compare undifexmid 4325 with exmidundif 4338. The former may be more recognizable as excluded middle because it is in terms of propositions, and the proof may be easier to follow for much the same reason (it just has to show ph and -. ph in the the relevant parts of the proof). The latter, however, has the key advantage of being able to prove both directions of the biconditional. To state that excluded middle implies a proposition is hard to do gracefully without EXMID, because there is no way to write a hypothesis ph \/ -. ph for an arbitrary proposition; instead the hypothesis would need to be the particular instance of excluded middle which that proof needs. Or to say it another way, EXMID implies DECID ph by exmidexmid 4328 but there is no good way to express the converse. This definition and how we use it is easiest to understand (and most appropriate to assign the name "excluded middle" to) if we assume ax-sep 4244, in which case EXMID means that all propositions are decidable (see exmidexmid 4328 and notice that it relies on ax-sep 4244). If we instead work with ax-bdsep 16824, EXMID as defined here means that all bounded propositions are decidable. (Contributed by Mario Carneiro and Jim Kingdon, 18-Jun-2022.)
|- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
Theorem	exmidexmid 4328	EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions. To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 855, peircedc 926, or condc 865. (Contributed by Jim Kingdon, 18-Jun-2022.)
|- (EXMID -> DECID ph )
Theorem	ss1o0el1 4329	A subclass of { (/) } contains the empty set if and only if it equals { (/) }. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)
|- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )
Theorem	exmid01 4330	Excluded middle is equivalent to saying any subset of { (/) } is either (/) or { (/) }. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
|- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
Theorem	pwntru 4331	A slight strengthening of pwtrufal 16941. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
|- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) )
Theorem	exmid1dc 4332*	A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4325 or ordtriexmid 4663. In this context x = { (/) } can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
|- ( ( ph /\ x C_ { (/) } ) -> DECID x = { (/) } )   =>    |- ( ph -> EXMID )
Theorem	exmidn0m 4333*	Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
|- (EXMID <-> A. x ( x = (/) \/ E. y y e. x ) )
Theorem	exmidsssn 4334*	Excluded middle is equivalent to the biconditionalized version of sssnr 3873 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
|- (EXMID <-> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
Theorem	exmidsssnc 4335*	Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4330 but lets you choose any set as the element of the singleton rather than just (/). It is similar to exmidsssn 4334 but for a particular set B rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
|- ( B e. V -> (EXMID <-> A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) ) )
Theorem	exmid0el 4336	Excluded middle is equivalent to decidability of (/) being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
|- (EXMID <-> A. xDECID (/) e. x )
Theorem	exmidel 4337*	Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)
|- (EXMID <-> A. x A. yDECID x e. y )
Theorem	exmidundif 4338*	Excluded middle is equivalent to every subset having a complement. That is, the union of a subset and its relative complement being the whole set. Although special cases such as undifss 3605 and undifdcss 7220 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)
|- (EXMID <-> A. x A. y ( x C_ y <-> ( x u. ( y \ x ) ) = y ) )
Theorem	exmidundifim 4339*	Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4338 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
|- (EXMID <-> A. x A. y ( x C_ y -> ( x u. ( y \ x ) ) = y ) )
Theorem	exmid1stab 4340*	If every proposition is stable, excluded middle follows. We are thinking of x as a proposition and x = { (/) } as " x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
|- ( ( ph /\ x C_ { (/) } ) -> STAB x = { (/) } )   =>    |- ( ph -> EXMID )
2.3.3  Axiom of Pairing
Axiom	ax-pr 4341*	The Axiom of Pairing of IZF set theory. Axiom 2 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4249). (Contributed by NM, 14-Nov-2006.)
|- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
Theorem	zfpair2 4342	Derive the abbreviated version of the Axiom of Pairing from ax-pr 4341. (Contributed by NM, 14-Nov-2006.)
|- { x , y } e. _V
Theorem	vsnex 4343	A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
|- { x } e. _V
Theorem	prexg 4344	The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3818, prprc1 3816, and prprc2 3817. (Contributed by Jim Kingdon, 16-Sep-2018.)
|- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
Theorem	snelpwg 4345	A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 4254. (Revised by BJ, 17-Jan-2025.)
|- ( A e. V -> ( A e. B <-> { A } e. ~P B ) )
Theorem	snelpwi 4346	A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
|- ( A e. B -> { A } e. ~P B )
Theorem	snelpw 4347	A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
|- A e. _V   =>    |- ( A e. B <-> { A } e. ~P B )
Theorem	prelpw 4348	An unordered pair of two sets is a member of the powerclass of a class if and only if the two sets are members of that class. (Contributed by AV, 8-Jan-2020.)
|- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } e. ~P C ) )
Theorem	prelpwi 4349	A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
|- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C )
Theorem	rext 4350*	A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
|- ( A. z ( x e. z -> y e. z ) -> x = y )
Theorem	sspwb 4351	Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
|- ( A C_ B <-> ~P A C_ ~P B )
Theorem	unipw 4352	A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
|- U. ~P A = A
Theorem	pwel 4353	Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
|- ( A e. B -> ~P A e. ~P ~P U. B )
Theorem	pwtr 4354	A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
|- ( Tr A <-> Tr ~P A )
Theorem	ssextss 4355*	An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
|- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )
Theorem	ssext 4356*	An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
|- ( A = B <-> A. x ( x C_ A <-> x C_ B ) )
Theorem	nssssr 4357*	Negation of subclass relationship. Compare nssr 3308. (Contributed by Jim Kingdon, 17-Sep-2018.)
|- ( E. x ( x C_ A /\ -. x C_ B ) -> -. A C_ B )
Theorem	pweqb 4358	Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
|- ( A = B <-> ~P A = ~P B )
Theorem	intid 4359*	The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
|- A e. _V   =>    |- |^| { x | A e. x } = { A }
Theorem	euabex 4360	The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
|- ( E! x ph -> { x | ph } e. _V )
Theorem	mss 4361*	An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
|- ( E. y y e. A -> E. x ( x C_ A /\ E. z z e. x ) )
Theorem	exss 4362*	Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
|- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) )
Theorem	opexg 4363	An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)
|- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
Theorem	opex 4364	An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)
|- A e. _V   &    |- B e. _V   =>    |- <. A , B >. e. _V
Theorem	otexg 4365	An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)
|- ( ( A e. U /\ B e. V /\ C e. W ) -> <. A , B , C >. e. _V )
Theorem	elop 4366	An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   =>    |- ( A e. <. B , C >. <-> ( A = { B } \/ A = { B , C } ) )
Theorem	opi1 4367	One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- { A } e. <. A , B >.
Theorem	opi2 4368	One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- { A , B } e. <. A , B >.
2.3.4  Ordered pair theorem
Theorem	opm 4369*	An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)
|- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
Theorem	opnzi 4370	An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4369). (Contributed by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- <. A , B >. =/= (/)
Theorem	opth1 4371	Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. = <. C , D >. -> A = C )
Theorem	opth 4372	The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that C and D are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )
Theorem	opthg 4373	Ordered pair theorem. C and D are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )
Theorem	opthg2 4374	Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( C e. V /\ D e. W ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )
Theorem	opth2 4375	Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
|- C e. _V   &    |- D e. _V   =>    |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )
Theorem	otth2 4376	Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- R e. _V   =>    |- ( <. <. A , B >. , R >. = <. <. C , D >. , S >. <-> ( A = C /\ B = D /\ R = S ) )
Theorem	otth 4377	Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- R e. _V   =>    |- ( <. A , B , R >. = <. C , D , S >. <-> ( A = C /\ B = D /\ R = S ) )
Theorem	eqvinop 4378*	A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
|- B e. _V   &    |- C e. _V   =>    |- ( A = <. B , C >. <-> E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) )
Theorem	copsexg 4379*	Substitution of class A for ordered pair <. x , y >.. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
Theorem	copsex2t 4380*	Closed theorem form of copsex2g 4381. (Contributed by NM, 17-Feb-2013.)
|- ( ( 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 ) )
Theorem	copsex2g 4381*	Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
Theorem	copsex4g 4382*	An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
|- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ( ph <-> ps ) )   =>    |- ( ( ( 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 ) )
Theorem	0nelop 4383	A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- -. (/) e. <. A , B >.
Theorem	opwo0id 4384	An ordered pair is equal to the ordered pair without the empty set. This is because no ordered pair contains the empty set. (Contributed by AV, 15-Nov-2021.)
|- <. X , Y >. = ( <. X , Y >. \ { (/) } )
Theorem	opeqex 4385	Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
|- ( <. A , B >. = <. C , D >. -> ( ( A e. _V /\ B e. _V ) <-> ( C e. _V /\ D e. _V ) ) )
Theorem	opcom 4386	An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. = <. B , A >. <-> A = B )
Theorem	moop2 4387*	"At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- B e. _V   =>    |- E* x A = <. B , x >.
Theorem	opeqsn 4388	Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   =>    |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) )
Theorem	opeqpr 4389	Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( <. A , B >. = { C , D } <-> ( ( C = { A } /\ D = { A , B } ) \/ ( C = { A , B } /\ D = { A } ) ) )
Theorem	euotd 4390*	Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
|- ( ph -> A e. _V )   &    |- ( ph -> B e. _V )   &    |- ( ph -> C e. _V )   &    |- ( ph -> ( ps <-> ( a = A /\ b = B /\ c = C ) ) )   =>    |- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) )
Theorem	uniop 4391	The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- U. <. A , B >. = { A , B }
Theorem	uniopel 4392	Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. e. C -> U. <. A , B >. e. U. C )
2.3.5  Ordered-pair class abstractions (cont.)
Theorem	opabid 4393	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
Theorem	opabidw 4394*	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4393 with a disjoint variable condition. (Contributed by NM, 14-Apr-1995.) (Revised by GG, 26-Jan-2024.)
|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
Theorem	elopab 4395*	Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.)
|- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )
Theorem	opelopabsbALT 4396*	The law of concretion in terms of substitutions. Less general than opelopabsb 4397, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )
Theorem	opelopabsb 4397*	The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
|- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
Theorem	brabsb 4398*	The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
|- R = { <. x , y >. | ph }   =>    |- ( A R B <-> [. A / x ]. [. B / y ]. ph )
Theorem	opelopabt 4399*	Closed theorem form of opelopab 4409. (Contributed by NM, 19-Feb-2013.)
|- ( ( 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 ) )
Theorem	opelopabga 4400*	The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )