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	brralrspcev 4101*	Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
|- ( ( B e. X /\ A. y e. Y A R B ) -> E. x e. X A. y e. Y A R x )
Theorem	brimralrspcev 4102*	Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
|- ( ( B e. X /\ A. y e. Y ( ( ph /\ A R B ) -> ps ) ) -> E. x e. X A. y e. Y ( ( ph /\ A R x ) -> ps ) )
2.1.23  Ordered-pair class abstractions (class builders)
Syntax	copab 4103	Extend class notation to include ordered-pair class abstraction (class builder).
class { <. x , y >. | ph }
Syntax	cmpt 4104	Extend the definition of a class to include maps-to notation for defining a function via a rule.
class ( x e. A |-> B )
Definition	df-opab 4105*	Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually x and y are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)
|- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
Definition	df-mpt 4106*	Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from x (in A) to B ( x )". The class expression B is the value of the function at x and normally contains the variable x. Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
|- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
Theorem	opabss 4107*	The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { <. x , y >. | x R y } C_ R
Theorem	opabbid 4108	Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )
Theorem	opabbidv 4109*	Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )
Theorem	opabbii 4110	Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
|- ( ph <-> ps )   =>    |- { <. x , y >. | ph } = { <. x , y >. | ps }
Theorem	nfopab 4111*	Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)
|- F/ z ph   =>    |- F/_ z { <. x , y >. | ph }
Theorem	nfopab1 4112	The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x { <. x , y >. | ph }
Theorem	nfopab2 4113	The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ y { <. x , y >. | ph }
Theorem	cbvopab 4114*	Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
|- F/ z ph   &    |- F/ w ph   &    |- F/ x ps   &    |- F/ y ps   &    |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- { <. x , y >. | ph } = { <. z , w >. | ps }
Theorem	cbvopabv 4115*	Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- { <. x , y >. | ph } = { <. z , w >. | ps }
Theorem	cbvopab1 4116*	Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/ z ph   &    |- F/ x ps   &    |- ( x = z -> ( ph <-> ps ) )   =>    |- { <. x , y >. | ph } = { <. z , y >. | ps }
Theorem	cbvopab2 4117*	Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
|- F/ z ph   &    |- F/ y ps   &    |- ( y = z -> ( ph <-> ps ) )   =>    |- { <. x , y >. | ph } = { <. x , z >. | ps }
Theorem	cbvopab1s 4118*	Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
|- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }
Theorem	cbvopab1v 4119*	Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
|- ( x = z -> ( ph <-> ps ) )   =>    |- { <. x , y >. | ph } = { <. z , y >. | ps }
Theorem	cbvopab2v 4120*	Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
|- ( y = z -> ( ph <-> ps ) )   =>    |- { <. x , y >. | ph } = { <. x , z >. | ps }
Theorem	csbopabg 4121*	Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|- ( A e. V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
Theorem	unopab 4122	Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
|- ( { <. x , y >. | ph } u. { <. x , y >. | ps } ) = { <. x , y >. | ( ph \/ ps ) }
Theorem	mpteq12f 4123	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
|- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
Theorem	mpteq12dva 4124*	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
|- ( ph -> A = C )   &    |- ( ( ph /\ x e. A ) -> B = D )   =>    |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )
Theorem	mpteq12dv 4125*	An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
|- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )
Theorem	mpteq12 4126*	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 4127*	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 4128*	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 4129	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 4130	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 4131	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 4132	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 4133*	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 4134*	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 4135*	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 4136	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
|- F/_ x ( x e. A |-> B )
Theorem	cbvmptf 4137*	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 4138*	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 4139*	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 4140*	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 4141	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr A
Definition	df-tr 4142	Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4143 (which is suggestive of the word "transitive"), dftr3 4145, dftr4 4146, and dftr5 4144. 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 4143*	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 4144*	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 4145*	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 4146	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 4147	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
|- ( A = B -> ( Tr A <-> Tr B ) )
Theorem	trel 4148	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 4149	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 4150	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 4151	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
|- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )
Theorem	tr0 4152	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
|- Tr (/)
Theorem	trv 4153	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
|- Tr _V
Theorem	triun 4154*	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 4155*	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 4156*	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 4157*	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 4158*	Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4215 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 4159*	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 4158. It is identical to zfrep6 4160 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 4160*	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 4161 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 4161*	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 2996. 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 4162*	A less restrictive version of the Separation Scheme ax-sep 4161, 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 4161 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 4163*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4161, we invoke the Axiom of Extensionality (indirectly via vtocl 2826), 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 4164*	Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4161. 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 4165*	Derive a weakened version of ax-i9 1552, where x and y must be distinct, from Separation ax-sep 4161 and Extensionality ax-ext 2186. The theorem -. A. x -. x = y also holds (ax9vsep 4166), 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 4166*	Derive a weakened version of ax-9 1553, where x and y must be distinct, from Separation ax-sep 4161 and Extensionality ax-ext 2186. In intuitionistic logic a9evsep 4165 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 4167*	Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2189 to strengthen the hypothesis in the form of axnul 4168). (Contributed by NM, 22-Dec-2007.)
|- E. x A. y -. y e. x   =>    |- E! x A. y -. y e. x
Theorem	axnul 4168*	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 4161. 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 4167). This theorem should not be referenced by any proof. Instead, use ax-nul 4169 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 4169*	The Null Set Axiom of IZF set theory. It was derived as axnul 4168 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 4170	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 4169. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- (/) e. _V
Theorem	csbexga 4171	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 4172	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 4173*	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 4174	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
|- -. E. x x = _V
Theorem	vprc 4175	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 4176	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
|- -. _V e. A
Theorem	inex1 4177	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 4178	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
|- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	inex1g 4179	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
|- ( A e. V -> ( A i^i B ) e. _V )
Theorem	ssex 4180	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 4161 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
|- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	ssexi 4181	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 4182	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 4183	A subclass of a set is a set. Deduction form of ssexg 4182. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> B e. C )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> A e. _V )
Theorem	difexg 4184	Existence of a difference. (Contributed by NM, 26-May-1998.)
|- ( A e. V -> ( A \ B ) e. _V )
Theorem	zfausab 4185*	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 4186*	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 4187*	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 4188*	Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4189. (Contributed by AV, 16-Jul-2019.)
|- B = { x e. A | ps }   &    |- ( ph -> A e. V )   =>    |- ( ph -> B e. _V )
Theorem	rabex2 4189*	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 4190*	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 4191*	Membership in a class abstraction involving a subset. Unlike elabg 2918, 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 4192*	The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( E. x x e. A -> |^| A e. _V )
Theorem	intexr 4193	If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( |^| A e. _V -> A =/= (/) )
Theorem	intnexr 4194	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 4195	The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
|- ( E. x ph -> |^| { x | ph } e. _V )
Theorem	intexrabim 4196	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 4197*	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 4198*	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 4199	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 4200	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 )