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
Theorem	trintssm 4201*	Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)
|- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A )
2.2  IZF Set Theory - add the Axioms of Collection and Separation
2.2.1  Introduce the Axiom of Collection
Axiom	ax-coll 4202*	Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4260 but uses a freeness hypothesis in place of one of the distinct variable conditions. (Contributed by Jim Kingdon, 23-Aug-2018.)
|- F/ b ph   =>    |- ( A. x e. a E. y ph -> E. b A. x e. a E. y e. b ph )
Theorem	repizf 4203*	Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). In our context this is not an axiom, but a theorem proved from ax-coll 4202. It is identical to zfrep6 4204 except for the choice of a freeness hypothesis rather than a disjoint variable condition between b and ph. (Contributed by Jim Kingdon, 23-Aug-2018.)
|- F/ b ph   =>    |- ( A. x e. a E! y ph -> E. b A. x e. a E. y e. b ph )
Theorem	zfrep6 4204*	A version of the Axiom of Replacement. Normally ph would have free variables x and y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4205 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version. (Contributed by NM, 10-Oct-2003.)
|- ( A. x e. z E! y ph -> E. w A. x e. z E. y e. w ph )
2.2.2  Introduce the Axiom of Separation
Axiom	ax-sep 4205*	The Axiom of Separation of IZF set theory. Axiom 6 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed, and with a F/ y ph condition replaced by a disjoint variable condition between y and ph). The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with x e. z) so that it asserts the existence of a collection only if it is smaller than some other collection z that already exists. This prevents Russell's paradox ru 3028. In some texts, this scheme is called "Aussonderung" or the Subset Axiom. (Contributed by NM, 11-Sep-2006.)
|- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
Theorem	axsep2 4206*	A less restrictive version of the Separation Scheme ax-sep 4205, where variables x and z can both appear free in the wff ph, which can therefore be thought of as ph ( x , z ). This version was derived from the more restrictive ax-sep 4205 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
Theorem	zfauscl 4207*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4205, we invoke the Axiom of Extensionality (indirectly via vtocl 2856), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)
|- A e. _V   =>    |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) )
Theorem	bm1.3ii 4208*	Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4205. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)
|- E. x A. y ( ph -> y e. x )   =>    |- E. x A. y ( y e. x <-> ph )
Theorem	a9evsep 4209*	Derive a weakened version of ax-i9 1576, where x and y must be distinct, from Separation ax-sep 4205 and Extensionality ax-ext 2211. The theorem -. A. x -. x = y also holds (ax9vsep 4210), but in intuitionistic logic E. x x = y is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- E. x x = y
Theorem	ax9vsep 4210*	Derive a weakened version of ax-9 1577, where x and y must be distinct, from Separation ax-sep 4205 and Extensionality ax-ext 2211. In intuitionistic logic a9evsep 4209 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. A. x -. x = y
2.2.3  Derive the Null Set Axiom
Theorem	zfnuleu 4211*	Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2214 to strengthen the hypothesis in the form of axnul 4212). (Contributed by NM, 22-Dec-2007.)
|- E. x A. y -. y e. x   =>    |- E! x A. y -. y e. x
Theorem	axnul 4212*	The Null Set Axiom of ZF set theory: there exists a set with no elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks, this is presented as a separate axiom; here we show it can be derived from Separation ax-sep 4205. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tell us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 4211). This theorem should not be referenced by any proof. Instead, use ax-nul 4213 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
|- E. x A. y -. y e. x
Axiom	ax-nul 4213*	The Null Set Axiom of IZF set theory. It was derived as axnul 4212 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. Axiom 4 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by NM, 7-Aug-2003.)
|- E. x A. y -. y e. x
Theorem	0ex 4214	The Null Set Axiom of ZF set theory: the empty set exists. Corollary 5.16 of [TakeutiZaring] p. 20. For the unabbreviated version, see ax-nul 4213. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- (/) e. _V
Theorem	csbexga 4215	The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- ( ( A e. V /\ A. x B e. W ) -> [_ A / x ]_ B e. _V )
Theorem	csbexa 4216	The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- A e. _V   &    |- B e. _V   =>    |- [_ A / x ]_ B e. _V
2.2.4  Theorems requiring subset and intersection existence
Theorem	nalset 4217*	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
|- -. E. x A. y y e. x
Theorem	vnex 4218	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
|- -. E. x x = _V
Theorem	vprc 4219	The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.)
|- -. _V e. _V
Theorem	nvel 4220	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
|- -. _V e. A
Theorem	inex1 4221	Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
|- A e. _V   =>    |- ( A i^i B ) e. _V
Theorem	inex2 4222	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
|- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	inex1g 4223	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
|- ( A e. V -> ( A i^i B ) e. _V )
Theorem	ssex 4224	The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4205 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
|- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	ssexi 4225	The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
|- B e. _V   &    |- A C_ B   =>    |- A e. _V
Theorem	ssexg 4226	The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
|- ( ( A C_ B /\ B e. C ) -> A e. _V )
Theorem	ssexd 4227	A subclass of a set is a set. Deduction form of ssexg 4226. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> B e. C )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> A e. _V )
Theorem	prcssprc 4228	The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
|- ( ( A C_ B /\ A e/ _V ) -> B e/ _V )
Theorem	difexg 4229	Existence of a difference. (Contributed by NM, 26-May-1998.)
|- ( A e. V -> ( A \ B ) e. _V )
Theorem	zfausab 4230*	Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
|- A e. _V   =>    |- { x | ( x e. A /\ ph ) } e. _V
Theorem	rabexg 4231*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
|- ( A e. V -> { x e. A | ph } e. _V )
Theorem	rabex 4232*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
|- A e. _V   =>    |- { x e. A | ph } e. _V
Theorem	rabexd 4233*	Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4234. (Contributed by AV, 16-Jul-2019.)
|- B = { x e. A | ps }   &    |- ( ph -> A e. V )   =>    |- ( ph -> B e. _V )
Theorem	rabex2 4234*	Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
|- B = { x e. A | ps }   &    |- A e. _V   =>    |- B e. _V
Theorem	rab2ex 4235*	A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
|- B = { y e. A | ps }   &    |- A e. _V   =>    |- { x e. B | ph } e. _V
Theorem	elssabg 4236*	Membership in a class abstraction involving a subset. Unlike elabg 2950, A does not have to be a set. (Contributed by NM, 29-Aug-2006.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( B e. V -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )
Theorem	inteximm 4237*	The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( E. x x e. A -> |^| A e. _V )
Theorem	intexr 4238	If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( |^| A e. _V -> A =/= (/) )
Theorem	intnexr 4239	If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( |^| A = _V -> -. |^| A e. _V )
Theorem	intexabim 4240	The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( E. x ph -> |^| { x | ph } e. _V )
Theorem	intexrabim 4241	The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( E. x e. A ph -> |^| { x e. A | ph } e. _V )
Theorem	iinexgm 4242*	The existence of an indexed union. x is normally a free-variable parameter in B, which should be read B ( x ). (Contributed by Jim Kingdon, 28-Aug-2018.)
|- ( ( E. x x e. A /\ A. x e. A B e. C ) -> |^|_ x e. A B e. _V )
Theorem	inuni 4243*	The intersection of a union U. A with a class B is equal to the union of the intersections of each element of A with B. (Contributed by FL, 24-Mar-2007.)
|- ( U. A i^i B ) = U. { x | E. y e. A x = ( y i^i B ) }
Theorem	elpw2g 4244	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
|- ( B e. V -> ( A e. ~P B <-> A C_ B ) )
Theorem	elpw2 4245	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
|- B e. _V   =>    |- ( A e. ~P B <-> A C_ B )
Theorem	elpwi2 4246	Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
|- B e. V   &    |- A C_ B   =>    |- A e. ~P B
Theorem	pwnss 4247	The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. V -> -. ~P A C_ A )
Theorem	pwne 4248	No set equals its power set. The sethood antecedent is necessary; compare pwv 3890. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
|- ( A e. V -> ~P A =/= A )
Theorem	repizf2lem 4249	Lemma for repizf2 4250. If we have a function-like proposition which provides at most one value of y for each x in a set w, we can change "at most one" to "exactly one" by restricting the values of x to those values for which the proposition provides a value of y. (Contributed by Jim Kingdon, 7-Sep-2018.)
|- ( A. x e. w E* y ph <-> A. x e. { x e. w | E. y ph } E! y ph )
Theorem	repizf2 4250*	Replacement. This version of replacement is stronger than repizf 4203 in the sense that ph does not need to map all values of x in w to a value of y. The resulting set contains those elements for which there is a value of y and in that sense, this theorem combines repizf 4203 with ax-sep 4205. Another variation would be A. x e. w E* y ph -> { y | E. x ( x e. w /\ ph ) } e. _V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
|- F/ z ph   =>    |- ( A. x e. w E* y ph -> E. z A. x e. { x e. w | E. y ph } E. y e. z ph )
2.2.5  Theorems requiring empty set existence
Theorem	class2seteq 4251*	Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
|- ( A e. V -> { x e. A | A e. _V } = A )
Theorem	0elpw 4252	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
|- (/) e. ~P A
Theorem	0nep0 4253	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
|- (/) =/= { (/) }
Theorem	0inp0 4254	Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)
|- ( A = (/) -> -. A = { (/) } )
Theorem	unidif0 4255	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
|- U. ( A \ { (/) } ) = U. A
Theorem	iin0imm 4256*	An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
|- ( E. y y e. A -> |^|_ x e. A (/) = (/) )
Theorem	iin0r 4257*	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 4258	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
|- |^| _V = (/)
Theorem	axpweq 4259*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4262 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 4260*	A very strong generalization of the Axiom of Replacement (compare zfrep6 4204). 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 4202. (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 4261*	A variant of the Boundedness Axiom bnd 4260 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 4262*	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 4208). The variant axpow2 4264 uses explicit subset notation. A version using class notation is pwex 4271. (Contributed by NM, 5-Aug-1993.)
|- E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y )
Theorem	zfpow 4263*	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 4264*	A variant of the Axiom of Power Sets ax-pow 4262 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 4265*	A variant of the Axiom of Power Sets ax-pow 4262. 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 4266*	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 4267	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 4268 from vpwex 4267. (Revised by BJ, 10-Aug-2022.)
|- ~P x e. _V
Theorem	pwexg 4268	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 4269	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 4270*	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 4271	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
|- A e. _V   =>    |- ~P A e. _V
Theorem	snexg 4272	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 4273	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 4274	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 4275	A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
|- -. -. { A } e. _V
Theorem	p0ex 4276	The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
|- { (/) } e. _V
Theorem	pp0ex 4277	{ (/) , { (/) } } (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
|- { (/) , { (/) } } e. _V
Theorem	ord3ex 4278	The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
|- { (/) , { (/) } , { (/) , { (/) } } } e. _V
Theorem	dtruarb 4279*	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 4655 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 4280	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 4281*	Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3573 and undifdcss 7108 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 4282	Formula for an abbreviation of excluded middle.
wff EXMID
Definition	df-exmid 4283	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 4281 with exmidundif 4294. 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 4284 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 4205, in which case EXMID means that all propositions are decidable (see exmidexmid 4284 and notice that it relies on ax-sep 4205). If we instead work with ax-bdsep 16415, 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 4284	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 848, peircedc 919, or condc 858. (Contributed by Jim Kingdon, 18-Jun-2022.)
|- (EXMID -> DECID ph )
Theorem	ss1o0el1 4285	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 4286	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 4287	A slight strengthening of pwtrufal 16534. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
|- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) )
Theorem	exmid1dc 4288*	A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4281 or ordtriexmid 4617. 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 4289*	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 4290*	Excluded middle is equivalent to the biconditionalized version of sssnr 3834 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
|- (EXMID <-> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
Theorem	exmidsssnc 4291*	Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4286 but lets you choose any set as the element of the singleton rather than just (/). It is similar to exmidsssn 4290 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 4292	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 4293*	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 4294*	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 3573 and undifdcss 7108 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 4295*	Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4294 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 4296*	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 4297*	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 4208). (Contributed by NM, 14-Nov-2006.)
|- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
Theorem	zfpair2 4298	Derive the abbreviated version of the Axiom of Pairing from ax-pr 4297. (Contributed by NM, 14-Nov-2006.)
|- { x , y } e. _V
Theorem	prexg 4299	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 3780, prprc1 3778, and prprc2 3779. (Contributed by Jim Kingdon, 16-Sep-2018.)
|- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
Theorem	snelpwg 4300	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 4213. (Revised by BJ, 17-Jan-2025.)
|- ( A e. V -> ( A e. B <-> { A } e. ~P B ) )