P. 47 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reusv3 4601*	Two ways to express single-valuedness of a class expression C ( y ). See reusv1 4599 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
|- ( y = z -> ( ph <-> ps ) )   &    |- ( y = z -> C = D )   =>    |- ( E. y e. B ( ph /\ C e. A ) -> ( A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )
Theorem	alxfr 4602*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 18-Feb-2007.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A. y A e. B /\ A. x E. y x = A ) -> ( A. x ph <-> A. y ps ) )
Theorem	ralxfrd 4603*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- ( ( ph /\ y e. C ) -> A e. B )   &    |- ( ( ph /\ x e. B ) -> E. y e. C x = A )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )
Theorem	rexxfrd 4604*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( ( ph /\ y e. C ) -> A e. B )   &    |- ( ( ph /\ x e. B ) -> E. y e. C x = A )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )
Theorem	ralxfr2d 4605*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by Mario Carneiro, 20-Aug-2014.)
|- ( ( ph /\ y e. C ) -> A e. V )   &    |- ( ph -> ( x e. B <-> E. y e. C x = A ) )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )
Theorem	rexxfr2d 4606*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- ( ( ph /\ y e. C ) -> A e. V )   &    |- ( ph -> ( x e. B <-> E. y e. C x = A ) )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )
Theorem	ralxfr 4607*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. B ph <-> A. y e. C ps )
Theorem	ralxfrALT 4608*	Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4603. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. B ph <-> A. y e. C ps )
Theorem	rexxfr 4609*	Transfer existence from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x e. B ph <-> E. y e. C ps )
Theorem	rabxfrd 4610*	Class builder membership after substituting an expression A (containing y) for x in the class expression ch. (Contributed by NM, 16-Jan-2012.)
|- F/_ y B   &    |- F/_ y C   &    |- ( ( ph /\ y e. D ) -> A e. D )   &    |- ( x = A -> ( ps <-> ch ) )   &    |- ( y = B -> A = C )   =>    |- ( ( ph /\ B e. D ) -> ( C e. { x e. D | ps } <-> B e. { y e. D | ch } ) )
Theorem	rabxfr 4611*	Class builder membership after substituting an expression A (containing y) for x in the class expression ph. (Contributed by NM, 10-Jun-2005.)
|- F/_ y B   &    |- F/_ y C   &    |- ( y e. D -> A e. D )   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> A = C )   =>    |- ( B e. D -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )
Theorem	reuhypd 4612*	A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)
|- ( ( ph /\ x e. C ) -> B e. C )   &    |- ( ( ph /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )   =>    |- ( ( ph /\ x e. C ) -> E! y e. C x = A )
Theorem	reuhyp 4613*	A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)
|- ( x e. C -> B e. C )   &    |- ( ( x e. C /\ y e. C ) -> ( x = A <-> y = B ) )   =>    |- ( x e. C -> E! y e. C x = A )
Theorem	uniexb 4614	The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)
|- ( A e. _V <-> U. A e. _V )
Theorem	pwexb 4615	The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
|- ( A e. _V <-> ~P A e. _V )
Theorem	elpwpwel 4616	A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
|- ( A e. ~P ~P B <-> U. A e. ~P B )
Theorem	univ 4617	The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
|- U. _V = _V
Theorem	eldifpw 4618	Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
|- C e. _V   =>    |- ( ( A e. ~P B /\ -. C C_ B ) -> ( A u. C ) e. ( ~P ( B u. C ) \ ~P B ) )
Theorem	op1stb 4619	Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.)
|- A e. _V   &    |- B e. _V   =>    |- |^| |^| <. A , B >. = A
Theorem	op1stbg 4620	Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
|- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = A )
Theorem	iunpw 4621*	An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
|- A e. _V   =>    |- ( E. x e. A x = U. A <-> ~P U. A = U_ x e. A ~P x )
Theorem	ifelpwung 4622	Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
|- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. ~P ( A u. B ) )
Theorem	ifelpwund 4623	Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> if ( ps , A , B ) e. ~P ( A u. B ) )
Theorem	ifelpwun 4624	Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)
|- A e. _V   &    |- B e. _V   =>    |- if ( ph , A , B ) e. ~P ( A u. B )
Theorem	ifexd 4625	Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> if ( ps , A , B ) e. _V )
Theorem	ifexg 4626	Existence of the conditional operator (closed form). (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.)
|- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. _V )
Theorem	ifex 4627	Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.)
|- A e. _V   &    |- B e. _V   =>    |- if ( ph , A , B ) e. _V
2.4.2  Ordinals (continued)
Theorem	ordon 4628	The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
|- Ord On
Theorem	ssorduni 4629	The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- ( A C_ On -> Ord U. A )
Theorem	ssonuni 4630	The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
|- ( A e. V -> ( A C_ On -> U. A e. On ) )
Theorem	ssonunii 4631	The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)
|- A e. _V   =>    |- ( A C_ On -> U. A e. On )
Theorem	onun2 4632	The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On )
Theorem	onun2i 4633	The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.)
|- A e. On   &    |- B e. On   =>    |- ( A u. B ) e. On
Theorem	ordsson 4634	Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.)
|- ( Ord A -> A C_ On )
Theorem	onss 4635	An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
|- ( A e. On -> A C_ On )
Theorem	onuni 4636	The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
|- ( A e. On -> U. A e. On )
Theorem	orduni 4637	The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
|- ( Ord A -> Ord U. A )
Theorem	bm2.5ii 4638*	Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
|- A e. _V   =>    |- ( A C_ On -> U. A = |^| { x e. On | A. y e. A y C_ x } )
Theorem	sucexb 4639	A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
|- ( A e. _V <-> suc A e. _V )
Theorem	sucexg 4640	The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
|- ( A e. V -> suc A e. _V )
Theorem	sucex 4641	The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- suc A e. _V
Theorem	ordsucim 4642	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
|- ( Ord A -> Ord suc A )
Theorem	onsuc 4643	The successor of an ordinal number is an ordinal number. Closed form of onsuci 4658. Forward implication of onsucb 4645. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
|- ( A e. On -> suc A e. On )
Theorem	ordsucg 4644	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
|- ( A e. _V -> ( Ord A <-> Ord suc A ) )
Theorem	onsucb 4645	A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4643. (Contributed by NM, 9-Sep-2003.)
|- ( A e. On <-> suc A e. On )
Theorem	ordsucss 4646	The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
|- ( Ord B -> ( A e. B -> suc A C_ B ) )
Theorem	ordelsuc 4647	A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)
|- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )
Theorem	onsucssi 4648	A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
|- A e. On   &    |- B e. On   =>    |- ( A e. B <-> suc A C_ B )
Theorem	onsucmin 4649*	The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
|- ( A e. On -> suc A = |^| { x e. On | A e. x } )
Theorem	onsucelsucr 4650	Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4672. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6754. (Contributed by Jim Kingdon, 17-Jul-2019.)
|- ( B e. On -> ( suc A e. suc B -> A e. B ) )
Theorem	onsucsssucr 4651	The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4669. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
|- ( ( A e. On /\ Ord B ) -> ( suc A C_ suc B -> A C_ B ) )
Theorem	sucunielr 4652	Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4673. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- ( suc A e. B -> A e. U. B )
Theorem	unon 4653	The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
|- U. On = On
Theorem	onuniss2 4654*	The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- ( A e. On -> U. { x e. On | x C_ A } = A )
Theorem	limon 4655	The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
|- Lim On
Theorem	ordunisuc2r 4656*	An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)
|- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) )
Theorem	onssi 4657	An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
|- A e. On   =>    |- A C_ On
Theorem	onsuci 4658	The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4643 and onsucb 4645. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
|- A e. On   =>    |- suc A e. On
Theorem	onintonm 4659*	The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
|- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On )
Theorem	onintrab2im 4660	An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. x e. On ph -> |^| { x e. On | ph } e. On )
Theorem	ordtriexmidlem 4661	Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4663 or weak linearity in ordsoexmid 4704) with a proposition ph. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
|- { x e. { (/) } | ph } e. On
Theorem	ordtriexmidlem2 4662*	Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4663 or weak linearity in ordsoexmid 4704) with a proposition ph. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
|- ( { x e. { (/) } | ph } = (/) -> -. ph )
Theorem	ordtriexmid 4663*	Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition). This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Also see exmidontri 7588 which is much the same theorem but biconditionalized and using the EXMID notation. (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)
|- A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x )   =>    |- ( ph \/ -. ph )
Theorem	ontriexmidim 4664*	Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4663. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> DECID ph )
Theorem	ordtri2orexmid 4665*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
|- A. x e. On A. y e. On ( x e. y \/ y C_ x )   =>    |- ( ph \/ -. ph )
Theorem	2ordpr 4666	Version of 2on 6686 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- Ord { (/) , { (/) } }
Theorem	ontr2exmid 4667*	An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
|- A. x e. On A. y A. z e. On ( ( x C_ y /\ y e. z ) -> x e. z )   =>    |- ( ph \/ -. ph )
Theorem	ordtri2or2exmidlem 4668*	A set which is 2o if ph or (/) if -. ph is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- { x e. { (/) , { (/) } } | ph } e. On
Theorem	onsucsssucexmid 4669*	The converse of onsucsssucr 4651 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
|- A. x e. On A. y e. On ( x C_ y -> suc x C_ suc y )   =>    |- ( ph \/ -. ph )
Theorem	onsucelsucexmidlem1 4670*	Lemma for onsucelsucexmid 4672. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
Theorem	onsucelsucexmidlem 4671*	Lemma for onsucelsucexmid 4672. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 6066), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4661. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On
Theorem	onsucelsucexmid 4672*	The converse of onsucelsucr 4650 implies excluded middle. On the other hand, if y is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4650 does hold, as seen at nnsucelsuc 6754. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- A. x e. On A. y e. On ( x e. y -> suc x e. suc y )   =>    |- ( ph \/ -. ph )
Theorem	ordsucunielexmid 4673*	The converse of sucunielr 4652 (where B is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- A. x e. On A. y e. On ( x e. U. y -> suc x e. y )   =>    |- ( ph \/ -. ph )
2.5  IZF Set Theory - add the Axiom of Set Induction
2.5.1  The ZF Axiom of Foundation would imply Excluded Middle
Theorem	regexmidlemm 4674*	Lemma for regexmid 4677. A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- E. y y e. A
Theorem	regexmidlem1 4675*	Lemma for regexmid 4677. If A has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- ( E. y ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( ph \/ -. ph ) )
Theorem	reg2exmidlema 4676*	Lemma for reg2exmid 4678. If A has a minimal element (expressed by C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- ( E. u e. A A. v e. A u C_ v -> ( ph \/ -. ph ) )
Theorem	regexmid 4677*	The axiom of foundation implies excluded middle. By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by e.). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4679. (Contributed by Jim Kingdon, 3-Sep-2019.)
|- ( E. y y e. x -> E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) )   =>    |- ( ph \/ -. ph )
Theorem	reg2exmid 4678*	If any inhabited set has a minimal element (when expressed by C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- A. z ( E. w w e. z -> E. x e. z A. y e. z x C_ y )   =>    |- ( ph \/ -. ph )
2.5.2  Introduce the Axiom of Set Induction
Axiom	ax-setind 4679*	Axiom of e.-Induction (also known as set induction). An axiom of Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p. "Axioms of CZF and IZF". This replaces the Axiom of Foundation (also called Regularity) from Zermelo-Fraenkel set theory. For more on axioms which might be adopted which are incompatible with this axiom (that is, Non-wellfounded Set Theory but in the absence of excluded middle), see Chapter 20 of [AczelRathjen], p. 183. (Contributed by Jim Kingdon, 19-Oct-2018.)
|- ( A. a ( A. y e. a [ y / a ] ph -> ph ) -> A. a ph )
Theorem	setindel 4680*	e.-Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
|- ( A. x ( A. y ( y e. x -> y e. S ) -> x e. S ) -> S = _V )
Theorem	setind 4681*	Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
|- ( A. x ( x C_ A -> x e. A ) -> A = _V )
Theorem	setind2 4682	Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
|- ( ~P A C_ A -> A = _V )
Theorem	elirr 4683	No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4679, we could redefine Ord A (df-iord 4506) to also require _E Fr A (df-frind 4472) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4684 (which under that definition would presumably not need ax-setind 4679 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4684. To encourage ordirr 4684 when possible, we mark this theorem as discouraged. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.)
|- -. A e. A
Theorem	ordirr 4684	Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4679. If in the definition of ordinals df-iord 4506, we also required that membership be well-founded on any ordinal (see df-frind 4472), then we could prove ordirr 4684 without ax-setind 4679. (Contributed by NM, 2-Jan-1994.)
|- ( Ord A -> -. A e. A )
Theorem	onirri 4685	An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- -. A e. A
Theorem	nordeq 4686	A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
|- ( ( Ord A /\ B e. A ) -> A =/= B )
Theorem	ordn2lp 4687	An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> -. ( A e. B /\ B e. A ) )
Theorem	orddisj 4688	An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
|- ( Ord A -> ( A i^i { A } ) = (/) )
Theorem	orddif 4689	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
|- ( Ord A -> A = ( suc A \ { A } ) )
Theorem	elirrv 4690	The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.)
|- -. x e. x
Theorem	sucprcreg 4691	A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)
|- ( -. A e. _V <-> suc A = A )
Theorem	ruv 4692	The Russell class is equal to the universe _V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
|- { x | x e/ x } = _V
Theorem	ruALT 4693	Alternate proof of Russell's Paradox ru 3050, simplified using (indirectly) the Axiom of Set Induction ax-setind 4679. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- { x | x e/ x } e/ _V
Theorem	onprc 4694	No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4628), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)
|- -. On e. _V
Theorem	sucon 4695	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
|- suc On = On
Theorem	en2lp 4696	No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.)
|- -. ( A e. B /\ B e. A )
Theorem	preleq 4697	Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )
Theorem	opthreg 4698	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4679 (via the preleq 4697 step). See df-op 3714 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )
Theorem	suc11g 4699	The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
|- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )
Theorem	suc11 4700	The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
|- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )