P. 43 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)

Type	Label	Description
Statement
2.2.6  Collection principle
Theorem	bnd 4201*	A very strong generalization of the Axiom of Replacement (compare zfrep6 4146). 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 4144. (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 4202*	A variant of the Boundedness Axiom bnd 4201 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 4203*	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 4150). The variant axpow2 4205 uses explicit subset notation. A version using class notation is pwex 4212. (Contributed by NM, 5-Aug-1993.)
|- E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y )
Theorem	zfpow 4204*	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 4205*	A variant of the Axiom of Power Sets ax-pow 4203 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 4206*	A variant of the Axiom of Power Sets ax-pow 4203. 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 4207*	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 4208	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 4209 from vpwex 4208. (Revised by BJ, 10-Aug-2022.)
|- ~P x e. _V
Theorem	pwexg 4209	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 4210	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 4211*	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 4212	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
|- A e. _V   =>    |- ~P A e. _V
Theorem	snexg 4213	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 4214	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 4215	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 4216	A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
|- -. -. { A } e. _V
Theorem	p0ex 4217	The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
|- { (/) } e. _V
Theorem	pp0ex 4218	{ (/) , { (/) } } (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
|- { (/) , { (/) } } e. _V
Theorem	ord3ex 4219	The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
|- { (/) , { (/) } , { (/) , { (/) } } } e. _V
Theorem	dtruarb 4220*	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 4591 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 4221	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 4222*	Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3527 and undifdcss 6979 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 4223	Formula for an abbreviation of excluded middle.
wff EXMID
Definition	df-exmid 4224	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 4222 with exmidundif 4235. 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 4225 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 4147, in which case EXMID means that all propositions are decidable (see exmidexmid 4225 and notice that it relies on ax-sep 4147). If we instead work with ax-bdsep 15376, 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 4225	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 844, peircedc 915, or condc 854. (Contributed by Jim Kingdon, 18-Jun-2022.)
|- (EXMID -> DECID ph )
Theorem	ss1o0el1 4226	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 4227	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 4228	A slight strengthening of pwtrufal 15488. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
|- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) )
Theorem	exmid1dc 4229*	A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4222 or ordtriexmid 4553. 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 4230*	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 4231*	Excluded middle is equivalent to the biconditionalized version of sssnr 3779 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
|- (EXMID <-> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
Theorem	exmidsssnc 4232*	Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4227 but lets you choose any set as the element of the singleton rather than just (/). It is similar to exmidsssn 4231 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 4233	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 4234*	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 4235*	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 3527 and undifdcss 6979 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 4236*	Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4235 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 4237*	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 4238*	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 4150). (Contributed by NM, 14-Nov-2006.)
|- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
Theorem	zfpair2 4239	Derive the abbreviated version of the Axiom of Pairing from ax-pr 4238. (Contributed by NM, 14-Nov-2006.)
|- { x , y } e. _V
Theorem	prexg 4240	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 3728, prprc1 3726, and prprc2 3727. (Contributed by Jim Kingdon, 16-Sep-2018.)
|- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
Theorem	snelpwi 4241	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 4242	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	prelpwi 4243	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 4244*	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 4245	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 4246	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 4247	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 4248	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 4249*	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 4250*	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 4251*	Negation of subclass relationship. Compare nssr 3239. (Contributed by Jim Kingdon, 17-Sep-2018.)
|- ( E. x ( x C_ A /\ -. x C_ B ) -> -. A C_ B )
Theorem	pweqb 4252	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 4253*	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 4254	The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
|- ( E! x ph -> { x | ph } e. _V )
Theorem	mss 4255*	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 4256*	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 4257	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 4258	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 4259	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 4260	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 4261	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 4262	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 4263*	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 4264	An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4263). (Contributed by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- <. A , B >. =/= (/)
Theorem	opth1 4265	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 4266	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 4267	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 4268	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 4269	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 4270	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 4271	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 4272*	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 4273*	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 4274*	Closed theorem form of copsex2g 4275. (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 4275*	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 4276*	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 4277	A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- -. (/) e. <. A , B >.
Theorem	opeqex 4278	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 4279	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 4280*	"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 4281	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 4282	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 4283*	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 4284	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 4285	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 4286	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 4287*	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4286 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 4288*	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 4289*	The law of concretion in terms of substitutions. Less general than opelopabsb 4290, 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 4290*	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 4291*	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 4292*	Closed theorem form of opelopab 4302. (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 4293*	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 ) )
Theorem	brabga 4294*	The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   &    |- R = { <. x , y >. | ph }   =>    |- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) )
Theorem	opelopab2a 4295*	Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )
Theorem	opelopaba 4296*	The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps )
Theorem	braba 4297*	The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   &    |- R = { <. x , y >. | ph }   =>    |- ( A R B <-> ps )
Theorem	opelopabg 4298*	The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )
Theorem	brabg 4299*	The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- R = { <. x , y >. | ph }   =>    |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )
Theorem	opelopabgf 4300*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4298 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
|- F/ x ps   &    |- F/ y ch   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )