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	nbrne1 4101	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
|- ( ( A R B /\ -. A R C ) -> B =/= C )
Theorem	nbrne2 4102	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
|- ( ( A R C /\ -. B R C ) -> A =/= B )
Theorem	eqbrtri 4103	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B R C   =>    |- A R C
Theorem	eqbrtrd 4104	Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	eqbrtrri 4105	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- A R C   =>    |- B R C
Theorem	eqbrtrrd 4106	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> A R C )   =>    |- ( ph -> B R C )
Theorem	breqtri 4107	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- B = C   =>    |- A R C
Theorem	breqtrd 4108	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A R C )
Theorem	breqtrri 4109	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- C = B   =>    |- A R C
Theorem	breqtrrd 4110	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A R C )
Theorem	3brtr3i 4111	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
|- A R B   &    |- A = C   &    |- B = D   =>    |- C R D
Theorem	3brtr4i 4112	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
|- A R B   &    |- C = A   &    |- D = B   =>    |- C R D
Theorem	3brtr3d 4113	Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C R D )
Theorem	3brtr4d 4114	Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
|- ( ph -> A R B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C R D )
Theorem	3brtr3g 4115	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
|- ( ph -> A R B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C R D )
Theorem	3brtr4g 4116	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
|- ( ph -> A R B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C R D )
Theorem	eqbrtrid 4117	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- A = B   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	eqbrtrrid 4118	B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
|- B = A   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	breqtrid 4119	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- A R B   &    |- ( ph -> B = C )   =>    |- ( ph -> A R C )
Theorem	breqtrrid 4120	B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
|- A R B   &    |- ( ph -> C = B )   =>    |- ( ph -> A R C )
Theorem	eqbrtrdi 4121	A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
|- ( ph -> A = B )   &    |- B R C   =>    |- ( ph -> A R C )
Theorem	eqbrtrrdi 4122	A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> B = A )   &    |- B R C   =>    |- ( ph -> A R C )
Theorem	breqtrdi 4123	A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- ( ph -> A R B )   &    |- B = C   =>    |- ( ph -> A R C )
Theorem	breqtrrdi 4124	A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
|- ( ph -> A R B )   &    |- C = B   =>    |- ( ph -> A R C )
Theorem	ssbrd 4125	Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( C A D -> C B D ) )
Theorem	ssbr 4126	Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.)
|- ( A C_ B -> ( C A D -> C B D ) )
Theorem	ssbri 4127	Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
|- A C_ B   =>    |- ( C A D -> C B D )
Theorem	nfbrd 4128	Deduction version of bound-variable hypothesis builder nfbr 4129. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x R )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/ x A R B )
Theorem	nfbr 4129	Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x A   &    |- F/_ x R   &    |- F/_ x B   =>    |- F/ x A R B
Theorem	brab1 4130*	Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
|- ( x R A <-> x e. { z | z R A } )
Theorem	br0 4131	The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
|- -. A (/) B
Theorem	brne0 4132	If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4133. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
|- ( A R B -> R =/= (/) )
Theorem	brm 4133*	If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
|- ( A R B -> E. x x e. R )
Theorem	brun 4134	The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
|- ( A ( R u. S ) B <-> ( A R B \/ A S B ) )
Theorem	brin 4135	The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
|- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )
Theorem	brdif 4136	The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
|- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) )
Theorem	sbcbrg 4137	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
Theorem	sbcbr12g 4138*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
Theorem	sbcbr1g 4139*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R C ) )
Theorem	sbcbr2g 4140*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
|- ( A e. D -> ( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) )
Theorem	brralrspcev 4141*	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 4142*	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 4143	Extend class notation to include ordered-pair class abstraction (class builder).
class { <. x , y >. | ph }
Syntax	cmpt 4144	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 4145*	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 4146*	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 4147*	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 4148	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 4149*	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 4150	Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
|- ( ph <-> ps )   =>    |- { <. x , y >. | ph } = { <. x , y >. | ps }
Theorem	nfopab 4151*	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 4152	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 4153	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 4154*	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 4155*	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 4156*	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 4157*	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 4158*	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 4159*	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 4160*	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 4161*	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 4162	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 4163	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 4164*	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 4165*	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 4166*	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 4167*	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 4168*	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 4169	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 4170	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 4171	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 4172	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 4173*	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 4174*	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 4175*	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 4176	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
|- F/_ x ( x e. A |-> B )
Theorem	cbvmptf 4177*	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 4178*	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 4179*	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 4180*	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 4181	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr A
Definition	df-tr 4182	Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4183 (which is suggestive of the word "transitive"), dftr3 4185, dftr4 4186, and dftr5 4184. 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 4183*	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 4184*	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 4185*	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 4186	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 4187	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
|- ( A = B -> ( Tr A <-> Tr B ) )
Theorem	trel 4188	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 4189	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 4190	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 4191	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
|- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )
Theorem	tr0 4192	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
|- Tr (/)
Theorem	trv 4193	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
|- Tr _V
Theorem	triun 4194*	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 4195*	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 4196*	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 4197*	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 4198*	Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4255 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 4199*	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 4198. It is identical to zfrep6 4200 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 4200*	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 4201 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 )