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Theorem List for Intuitionistic Logic Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembreq12i 3801 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A R C  <->  B R D )
 
Theorembreq1d 3802 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A R C  <->  B R C ) )
 
Theorembreqd 3803 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D  <->  C B D ) )
 
Theorembreq2d 3804 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C R A  <->  C R B ) )
 
Theorembreq12d 3805 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A R C  <->  B R D ) )
 
Theorembreq123d 3806 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  R  =  S )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A R C  <->  B S D ) )
 
Theorembreqan12d 3807 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 ) )
 
Theorembreqan12rd 3808 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 ) )
 
Theoremnbrne1 3809 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 )
 
Theoremnbrne2 3810 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 )
 
Theoremeqbrtri 3811 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B R C   =>    |-  A R C
 
Theoremeqbrtrd 3812 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 )
 
Theoremeqbrtrri 3813 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  A R C   =>    |-  B R C
 
Theoremeqbrtrrd 3814 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 )
 
Theorembreqtri 3815 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  B  =  C   =>    |-  A R C
 
Theorembreqtrd 3816 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 )
 
Theorembreqtrri 3817 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  C  =  B   =>    |-  A R C
 
Theorembreqtrrd 3818 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 )
 
Theorem3brtr3i 3819 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
 
Theorem3brtr4i 3820 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
 
Theorem3brtr3d 3821 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 )
 
Theorem3brtr4d 3822 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 )
 
Theorem3brtr3g 3823 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 )
 
Theorem3brtr4g 3824 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 )
 
Theoremsyl5eqbr 3825 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  A  =  B   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5eqbrr 3826 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5breq 3827 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  A R B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5breqr 3828 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
 |-  A R B   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6eqbr 3829 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  B R C   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6eqbrr 3830 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  B  =  A )   &    |-  B R C   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6breq 3831 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  B  =  C   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6breqr 3832 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
 |-  ( ph  ->  A R B )   &    |-  C  =  B   =>    |-  ( ph  ->  A R C )
 
Theoremssbrd 3833 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 ) )
 
Theoremssbri 3834 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 )
 
Theoremnfbrd 3835 Deduction version of bound-variable hypothesis builder nfbr 3836. (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 )
 
Theoremnfbr 3836 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
 
Theorembrab1 3837* 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 }
 )
 
Theorembrun 3838 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
 |-  ( A ( R  u.  S ) B  <-> 
 ( A R B  \/  A S B ) )
 
Theorembrin 3839 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
 |-  ( A ( R  i^i  S ) B  <-> 
 ( A R B  /\  A S B ) )
 
Theorembrdif 3840 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( A ( R 
 \  S ) B  <-> 
 ( A R B  /\  -.  A S B ) )
 
Theoremsbcbrg 3841 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 ) )
 
Theoremsbcbr12g 3842* 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 ) )
 
Theoremsbcbr1g 3843* 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 ) )
 
Theoremsbcbr2g 3844* 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 ) )
 
2.1.23  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 3845 Extend class notation to include ordered-pair class abstraction (class builder).
 class  { <. x ,  y >.  |  ph }
 
Syntaxcmpt 3846 Extend the definition of a class to include maps-to notation for defining a function via a rule.
 class  ( x  e.  A  |->  B )
 
Definitiondf-opab 3847* 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 ) }
 
Definitiondf-mpt 3848* 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 ) }
 
Theoremopabss 3849* 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
 
Theoremopabbid 3850 Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (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 } )
 
Theoremopabbidv 3851* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ps }  =  { <. x ,  y >.  |  ch } )
 
Theoremopabbii 3852 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
 |-  ( ph  <->  ps )   =>    |- 
 { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }
 
Theoremnfopab 3853* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
 |- 
 F/ z ph   =>    |-  F/_ z { <. x ,  y >.  |  ph }
 
Theoremnfopab1 3854 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 }
 
Theoremnfopab2 3855 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 }
 
Theoremcbvopab 3856* 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 }
 
Theoremcbvopabv 3857* 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 }
 
Theoremcbvopab1 3858* 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 }
 
Theoremcbvopab2 3859* 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 }
 
Theoremcbvopab1s 3860* 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 }
 
Theoremcbvopab1v 3861* 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 }
 
Theoremcbvopab2v 3862* 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 }
 
Theoremcsbopabg 3863* 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 } )
 
Theoremunopab 3864 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
 |-  ( { <. x ,  y >.  |  ph }  u.  {
 <. x ,  y >.  |  ps } )  =  { <. x ,  y >.  |  ( ph  \/  ps ) }
 
Theoremmpteq12f 3865 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 ) )
 
Theoremmpteq12dva 3866* 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 ) )
 
Theoremmpteq12dv 3867* 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 ) )
 
Theoremmpteq12 3868* 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 ) )
 
Theoremmpteq1 3869* 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 ) )
 
Theoremmpteq1d 3870* 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 ) )
 
Theoremmpteq2ia 3871 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 )
 
Theoremmpteq2i 3872 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 )
 
Theoremmpteq12i 3873 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 )
 
Theoremmpteq2da 3874 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 ) )
 
Theoremmpteq2dva 3875* 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 ) )
 
Theoremmpteq2dv 3876* 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 ) )
 
Theoremnfmpt 3877* 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 )
 
Theoremnfmpt1 3878 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
 |-  F/_ x ( x  e.  A  |->  B )
 
Theoremcbvmpt 3879* 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 )
 
Theoremcbvmptv 3880* 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 )
 
Theoremmptv 3881* 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
 
Syntaxwtr 3882 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
 wff  Tr  A
 
Definitiondf-tr 3883 Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 3884 (which is suggestive of the word "transitive"), dftr3 3886, dftr4 3887, and dftr5 3885. 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 )
 
Theoremdftr2 3884* 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 ) )
 
Theoremdftr5 3885* 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 )
 
Theoremdftr3 3886* 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 )
 
Theoremdftr4 3887 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 )
 
Theoremtreq 3888 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B ) )
 
Theoremtrel 3889 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 ) )
 
Theoremtrel3 3890 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 ) )
 
Theoremtrss 3891 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 ) )
 
Theoremtrin 3892 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
 |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
 
Theoremtr0 3893 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
 |- 
 Tr  (/)
 
Theoremtrv 3894 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
 |- 
 Tr  _V
 
Theoremtriun 3895* 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 )
 
Theoremtruni 3896* 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 )
 
Theoremtrint 3897* 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 )
 
Theoremtrintssm 3898* 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 )
 
TheoremtrintssmOLD 3899* Obsolete version of trintssm 3898 as of 30-Oct-2021. (Contributed by Jim Kingdon, 22-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( E. x  x  e.  A  /\  Tr  A )  ->  |^| A  C_  A )
 
2.2  IZF Set Theory - add the Axioms of Collection and Separation
 
2.2.1  Introduce the Axiom of Collection
 
Axiomax-coll 3900* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3953 but uses a freeness hypothesis in place of one of the distinct variable constraints. (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 )
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