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	breqdi 4101	Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
|- ( ph -> A = B )   &    |- ( ph -> C A D )   =>    |- ( ph -> C B D )
Theorem	breqan12d 4102	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )
Theorem	breqan12rd 4103	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ps /\ ph ) -> ( A R C <-> B R D ) )
Theorem	eqnbrtrd 4104	Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
|- ( ph -> A = B )   &    |- ( ph -> -. B R C )   =>    |- ( ph -> -. A R C )
Theorem	nbrne1 4105	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 4106	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 4107	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B R C   =>    |- A R C
Theorem	eqbrtrd 4108	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 4109	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- A R C   =>    |- B R C
Theorem	eqbrtrrd 4110	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 4111	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- B = C   =>    |- A R C
Theorem	breqtrd 4112	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 4113	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- C = B   =>    |- A R C
Theorem	breqtrrd 4114	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 4115	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 4116	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 4117	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 4118	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 4119	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 4120	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 4121	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 4122	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 4123	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 4124	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 4125	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 4126	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 4127	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 4128	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 4129	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 4130	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 4131	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 4132	Deduction version of bound-variable hypothesis builder nfbr 4133. (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 4133	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 4134*	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 4135	The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
|- -. A (/) B
Theorem	brne0 4136	If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4137. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
|- ( A R B -> R =/= (/) )
Theorem	brm 4137*	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 4138	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 4139	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 4140	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 4141	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 4142*	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 4143*	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 4144*	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 4145*	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 4146*	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 4147	Extend class notation to include ordered-pair class abstraction (class builder).
class { <. x , y >. | ph }
Syntax	cmpt 4148	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 4149*	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 4150*	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 4151*	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 4152	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 4153*	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 4154	Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
|- ( ph <-> ps )   =>    |- { <. x , y >. | ph } = { <. x , y >. | ps }
Theorem	nfopab 4155*	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 4156	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 4157	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 4158*	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 4159*	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 4160*	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 4161*	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 4162*	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 4163*	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 4164*	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 4165*	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 4166	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 4167	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 4168*	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 4169*	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 4170*	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 4171*	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 4172*	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 4173	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 4174	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 4175	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 4176	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 4177*	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 4178*	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 4179*	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 4180	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
|- F/_ x ( x e. A |-> B )
Theorem	cbvmptf 4181*	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 4182*	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 4183*	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 4184*	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 4185	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr A
Definition	df-tr 4186	Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4187 (which is suggestive of the word "transitive"), dftr3 4189, dftr4 4190, and dftr5 4188. 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 4187*	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 4188*	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 4189*	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 4190	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 4191	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
|- ( A = B -> ( Tr A <-> Tr B ) )
Theorem	trel 4192	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 4193	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 4194	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 4195	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
|- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) )
Theorem	tr0 4196	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
|- Tr (/)
Theorem	trv 4197	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
|- Tr _V
Theorem	triun 4198*	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 4199*	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 4200*	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 )