P. 42 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mpteq12 4101*	An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
|- ( ( A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
Theorem	mpteq1 4102*	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
|- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) )
Theorem	mpteq1d 4103*	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
|- ( ph -> A = B )   =>    |- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) )
Theorem	mpteq2ia 4104	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
|- ( x e. A -> B = C )   =>    |- ( x e. A |-> B ) = ( x e. A |-> C )
Theorem	mpteq2i 4105	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
|- B = C   =>    |- ( x e. A |-> B ) = ( x e. A |-> C )
Theorem	mpteq12i 4106	An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
|- A = C   &    |- B = D   =>    |- ( x e. A |-> B ) = ( x e. C |-> D )
Theorem	mpteq2da 4107	Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )
Theorem	mpteq2dva 4108*	Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
|- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )
Theorem	mpteq2dv 4109*	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
|- ( ph -> B = C )   =>    |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )
Theorem	nfmpt 4110*	Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( y e. A |-> B )
Theorem	nfmpt1 4111	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
|- F/_ x ( x e. A |-> B )
Theorem	cbvmptf 4112*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- ( x e. A |-> B ) = ( y e. A |-> C )
Theorem	cbvmpt 4113*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- ( x e. A |-> B ) = ( y e. A |-> C )
Theorem	cbvmptv 4114*	Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
|- ( x = y -> B = C )   =>    |- ( x e. A |-> B ) = ( y e. A |-> C )
Theorem	mptv 4115*	Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
|- ( x e. _V |-> B ) = { <. x , y >. | y = B }
2.1.24  Transitive classes
Syntax	wtr 4116	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr A
Definition	df-tr 4117	Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4118 (which is suggestive of the word "transitive"), dftr3 4120, dftr4 4121, and dftr5 4119. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
|- ( Tr A <-> U. A C_ A )
Theorem	dftr2 4118*	An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
|- ( Tr A <-> A. x A. y ( ( x e. y /\ y e. A ) -> x e. A ) )
Theorem	dftr5 4119*	An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
|- ( Tr A <-> A. x e. A A. y e. x y e. A )
Theorem	dftr3 4120*	An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
|- ( Tr A <-> A. x e. A x C_ A )
Theorem	dftr4 4121	An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
|- ( Tr A <-> A C_ ~P A )
Theorem	treq 4122	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
|- ( A = B -> ( Tr A <-> Tr B ) )
Theorem	trel 4123	In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )
Theorem	trel3 4124	In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
|- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> B e. A ) )
Theorem	trss 4125	An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
|- ( Tr A -> ( B e. A -> B C_ A ) )
Theorem	trin 4126	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
|- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )
Theorem	tr0 4127	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
|- Tr (/)
Theorem	trv 4128	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
|- Tr _V
Theorem	triun 4129*	The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
|- ( A. x e. A Tr B -> Tr U_ x e. A B )
Theorem	truni 4130*	The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
|- ( A. x e. A Tr x -> Tr U. A )
Theorem	trint 4131*	The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)
|- ( A. x e. A Tr x -> Tr |^| A )
Theorem	trintssm 4132*	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 4133*	Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4190 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 4134*	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 4133. It is identical to zfrep6 4135 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 4135*	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 4136 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 4136*	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 2976. 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 4137*	A less restrictive version of the Separation Scheme ax-sep 4136, 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 4136 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 4138*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4136, we invoke the Axiom of Extensionality (indirectly via vtocl 2806), 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 4139*	Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4136. 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 4140*	Derive a weakened version of ax-i9 1541, where x and y must be distinct, from Separation ax-sep 4136 and Extensionality ax-ext 2171. The theorem -. A. x -. x = y also holds (ax9vsep 4141), 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 4141*	Derive a weakened version of ax-9 1542, where x and y must be distinct, from Separation ax-sep 4136 and Extensionality ax-ext 2171. In intuitionistic logic a9evsep 4140 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 4142*	Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2174 to strengthen the hypothesis in the form of axnul 4143). (Contributed by NM, 22-Dec-2007.)
|- E. x A. y -. y e. x   =>    |- E! x A. y -. y e. x
Theorem	axnul 4143*	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 4136. 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 4142). This theorem should not be referenced by any proof. Instead, use ax-nul 4144 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 4144*	The Null Set Axiom of IZF set theory. It was derived as axnul 4143 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 4145	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 4144. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- (/) e. _V
Theorem	csbexga 4146	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 4147	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 4148*	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 4149	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
|- -. E. x x = _V
Theorem	vprc 4150	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 4151	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
|- -. _V e. A
Theorem	inex1 4152	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 4153	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
|- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	inex1g 4154	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
|- ( A e. V -> ( A i^i B ) e. _V )
Theorem	ssex 4155	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 4136 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
|- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	ssexi 4156	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 4157	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 4158	A subclass of a set is a set. Deduction form of ssexg 4157. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> B e. C )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> A e. _V )
Theorem	difexg 4159	Existence of a difference. (Contributed by NM, 26-May-1998.)
|- ( A e. V -> ( A \ B ) e. _V )
Theorem	zfausab 4160*	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 4161*	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 4162*	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 4163*	Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4164. (Contributed by AV, 16-Jul-2019.)
|- B = { x e. A | ps }   &    |- ( ph -> A e. V )   =>    |- ( ph -> B e. _V )
Theorem	rabex2 4164*	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 4165*	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 4166*	Membership in a class abstraction involving a subset. Unlike elabg 2898, 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 4167*	The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( E. x x e. A -> |^| A e. _V )
Theorem	intexr 4168	If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( |^| A e. _V -> A =/= (/) )
Theorem	intnexr 4169	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 4170	The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( E. x ph -> |^| { x | ph } e. _V )
Theorem	intexrabim 4171	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 4172*	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 4173*	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 4174	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 4175	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 4176	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 4177	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 4178	No set equals its power set. The sethood antecedent is necessary; compare pwv 3823. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
|- ( A e. V -> ~P A =/= A )
Theorem	repizf2lem 4179	Lemma for repizf2 4180. 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 4180*	Replacement. This version of replacement is stronger than repizf 4134 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 4134 with ax-sep 4136. 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 4181*	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 4182	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
|- (/) e. ~P A
Theorem	0nep0 4183	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
|- (/) =/= { (/) }
Theorem	0inp0 4184	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 4185	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 4186*	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 4187*	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 4188	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
|- |^| _V = (/)
Theorem	axpweq 4189*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4192 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 4190*	A very strong generalization of the Axiom of Replacement (compare zfrep6 4135). 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 4133. (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 4191*	A variant of the Boundedness Axiom bnd 4190 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 4192*	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 4139). The variant axpow2 4194 uses explicit subset notation. A version using class notation is pwex 4201. (Contributed by NM, 5-Aug-1993.)
|- E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y )
Theorem	zfpow 4193*	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 4194*	A variant of the Axiom of Power Sets ax-pow 4192 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 4195*	A variant of the Axiom of Power Sets ax-pow 4192. 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 4196*	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 4197	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 4198 from vpwex 4197. (Revised by BJ, 10-Aug-2022.)
|- ~P x e. _V
Theorem	pwexg 4198	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 4199	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 4200*	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 )