P. 46 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eusv2nf 4501*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   =>    |- ( E! y E. x y = A <-> F/_ x A )
Theorem	eusv2 4502*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   =>    |- ( E! y E. x y = A <-> E! y A. x y = A )
Theorem	reusv1 4503*	Two ways to express single-valuedness of a class expression C ( y ). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- ( E. y e. B ph -> ( E! x e. A A. y e. B ( ph -> x = C ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )
Theorem	reusv3i 4504*	Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
|- ( y = z -> ( ph <-> ps ) )   &    |- ( y = z -> C = D )   =>    |- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
Theorem	reusv3 4505*	Two ways to express single-valuedness of a class expression C ( y ). See reusv1 4503 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 4506*	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 4507*	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 4508*	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 4509*	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 4510*	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 4511*	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 4512*	Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4507. (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 4513*	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 4514*	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 4515*	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 4516*	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 4517*	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 4518	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 4519	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 4520	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 4521	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 4522	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 4523	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 4524	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 4525*	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 4526	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 4527	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 4528	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 4529	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 4530	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 4531	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 4532	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 4533	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 4534	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 4535	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 4536	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 4537	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 4538	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 4539	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 4540	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 4541	The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
|- ( Ord A -> Ord U. A )
Theorem	bm2.5ii 4542*	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 4543	A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
|- ( A e. _V <-> suc A e. _V )
Theorem	sucexg 4544	The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
|- ( A e. V -> suc A e. _V )
Theorem	sucex 4545	The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- suc A e. _V
Theorem	ordsucim 4546	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
|- ( Ord A -> Ord suc A )
Theorem	onsuc 4547	The successor of an ordinal number is an ordinal number. Closed form of onsuci 4562. Forward implication of onsucb 4549. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
|- ( A e. On -> suc A e. On )
Theorem	ordsucg 4548	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 4549	A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4547. (Contributed by NM, 9-Sep-2003.)
|- ( A e. On <-> suc A e. On )
Theorem	ordsucss 4550	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 4551	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 4552	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 4553*	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 4554	Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4576. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6567. (Contributed by Jim Kingdon, 17-Jul-2019.)
|- ( B e. On -> ( suc A e. suc B -> A e. B ) )
Theorem	onsucsssucr 4555	The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4573. (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 4556	Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4577. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- ( suc A e. B -> A e. U. B )
Theorem	unon 4557	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 4558*	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 4559	The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
|- Lim On
Theorem	ordunisuc2r 4560*	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 4561	An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
|- A e. On   =>    |- A C_ On
Theorem	onsuci 4562	The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4547 and onsucb 4549. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
|- A e. On   =>    |- suc A e. On
Theorem	onintonm 4563*	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 4564	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 4565	Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4567 or weak linearity in ordsoexmid 4608) 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 4566*	Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4567 or weak linearity in ordsoexmid 4608) 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 4567*	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 7333 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 4568*	Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4567. (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 4569*	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 4570	Version of 2on 6501 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- Ord { (/) , { (/) } }
Theorem	ontr2exmid 4571*	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 4572*	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 4573*	The converse of onsucsssucr 4555 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 4574*	Lemma for onsucelsucexmid 4576. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
Theorem	onsucelsucexmidlem 4575*	Lemma for onsucelsucexmid 4576. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5925), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4565. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On
Theorem	onsucelsucexmid 4576*	The converse of onsucelsucr 4554 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 4554 does hold, as seen at nnsucelsuc 6567. (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 4577*	The converse of sucunielr 4556 (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 4578*	Lemma for regexmid 4581. A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- E. y y e. A
Theorem	regexmidlem1 4579*	Lemma for regexmid 4581. 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 4580*	Lemma for reg2exmid 4582. 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 4581*	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 4583. (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 4582*	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 4583*	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 4584*	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 4585*	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 4586	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 4587	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 4583, we could redefine Ord A (df-iord 4411) to also require _E Fr A (df-frind 4377) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4588 (which under that definition would presumably not need ax-setind 4583 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 4588. To encourage ordirr 4588 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 4588	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 4583. If in the definition of ordinals df-iord 4411, we also required that membership be well-founded on any ordinal (see df-frind 4377), then we could prove ordirr 4588 without ax-setind 4583. (Contributed by NM, 2-Jan-1994.)
|- ( Ord A -> -. A e. A )
Theorem	onirri 4589	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 4590	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 4591	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 4592	An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
|- ( Ord A -> ( A i^i { A } ) = (/) )
Theorem	orddif 4593	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
|- ( Ord A -> A = ( suc A \ { A } ) )
Theorem	elirrv 4594	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 4595	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 4596	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 4597	Alternate proof of Russell's Paradox ru 2996, simplified using (indirectly) the Axiom of Set Induction ax-setind 4583. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- { x | x e/ x } e/ _V
Theorem	onprc 4598	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 4532), 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 4599	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
|- suc On = On
Theorem	en2lp 4600	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 )