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Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcbvopab 3901* 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 3902* 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 3903* 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 3904* 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 3905* 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 3906* 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 3907* 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 3908* 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 3909 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 3910 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 3911* 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 3912* 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 3913* 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 3914* 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 3915* 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 3916 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 3917 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 3918 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 3919 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 3920* 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 3921* 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 3922* 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 3923 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
 |-  F/_ x ( x  e.  A  |->  B )
 
Theoremcbvmptf 3924* 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 )
 
Theoremcbvmpt 3925* 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 3926* 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 3927* 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 3928 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
 wff  Tr  A
 
Definitiondf-tr 3929 Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 3930 (which is suggestive of the word "transitive"), dftr3 3932, dftr4 3933, and dftr5 3931. 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 3930* 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 3931* 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 3932* 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 3933 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 3934 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B ) )
 
Theoremtrel 3935 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 3936 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 3937 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 3938 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
 |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
 
Theoremtr0 3939 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
 |- 
 Tr  (/)
 
Theoremtrv 3940 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
 |- 
 Tr  _V
 
Theoremtriun 3941* 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 3942* 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 3943* 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 3944* 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 3945* Obsolete version of trintssm 3944 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 3946* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3999 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 )
 
Theoremrepizf 3947* 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 3946. It is identical to zfrep6 3948 except for the choice of a freeness hypothesis rather than a distinct variable constraint 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 )
 
Theoremzfrep6 3948* 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 3949 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
 
Axiomax-sep 3949* 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 distinct variable constraint 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 2837. 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 ) )
 
Theoremaxsep2 3950* A less restrictive version of the Separation Scheme ax-sep 3949, 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 3949 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 ) )
 
Theoremzfauscl 3951* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3949, we invoke the Axiom of Extensionality (indirectly via vtocl 2673), 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 )
 )
 
Theorembm1.3ii 3952* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3949. 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 )
 
Theorema9evsep 3953* Derive a weakened version of ax-i9 1468, where  x and  y must be distinct, from Separation ax-sep 3949 and Extensionality ax-ext 2070. The theorem  -.  A. x -.  x  =  y also holds (ax9vsep 3954), 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
 
Theoremax9vsep 3954* Derive a weakened version of ax-9 1469, where  x and  y must be distinct, from Separation ax-sep 3949 and Extensionality ax-ext 2070. In intuitionistic logic a9evsep 3953 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
 
Theoremzfnuleu 3955* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2073 to strengthen the hypothesis in the form of axnul 3956). (Contributed by NM, 22-Dec-2007.)
 |- 
 E. x A. y  -.  y  e.  x   =>    |-  E! x A. y  -.  y  e.  x
 
Theoremaxnul 3956* 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 3949. 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 3955).

This theorem should not be referenced by any proof. Instead, use ax-nul 3957 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
 
Axiomax-nul 3957* The Null Set Axiom of IZF set theory. It was derived as axnul 3956 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
 
Theorem0ex 3958 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 3957. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  (/)  e.  _V
 
Theoremcsbexga 3959 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 )
 
Theoremcsbexa 3960 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
 
Theoremnalset 3961* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
 |- 
 -.  E. x A. y  y  e.  x
 
Theoremvnex 3962 The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
 |- 
 -.  E. x  x  =  _V
 
Theoremvprc 3963 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
 
Theoremnvel 3964 The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
 |- 
 -.  _V  e.  A
 
Theoreminex1 3965 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
 
Theoreminex2 3966 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
 |-  A  e.  _V   =>    |-  ( B  i^i  A )  e.  _V
 
Theoreminex1g 3967 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
 |-  ( A  e.  V  ->  ( A  i^i  B )  e.  _V )
 
Theoremssex 3968 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 3949 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
 |-  B  e.  _V   =>    |-  ( A  C_  B  ->  A  e.  _V )
 
Theoremssexi 3969 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
 |-  B  e.  _V   &    |-  A  C_  B   =>    |-  A  e.  _V
 
Theoremssexg 3970 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 )
 
Theoremssexd 3971 A subclass of a set is a set. Deduction form of ssexg 3970. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  A  e.  _V )
 
Theoremdifexg 3972 Existence of a difference. (Contributed by NM, 26-May-1998.)
 |-  ( A  e.  V  ->  ( A  \  B )  e.  _V )
 
Theoremzfausab 3973* 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
 
Theoremrabexg 3974* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
 |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
 
Theoremrabex 3975* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
 |-  A  e.  _V   =>    |-  { x  e.  A  |  ph }  e.  _V
 
Theoremelssabg 3976* Membership in a class abstraction involving a subset. Unlike elabg 2759,  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 ) ) )
 
Theoreminteximm 3977* The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
 
Theoremintexr 3978 If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( |^| A  e.  _V 
 ->  A  =/=  (/) )
 
Theoremintnexr 3979 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 )
 
Theoremintexabim 3980 The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( E. x ph  -> 
 |^| { x  |  ph }  e.  _V )
 
Theoremintexrabim 3981 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 )
 
Theoremiinexgm 3982* 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 )
 
Theoreminuni 3983* 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 ) }
 
Theoremelpw2g 3984 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 )
 )
 
Theoremelpw2 3985 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 )
 
Theorempwnss 3986 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 )
 
Theorempwne 3987 No set equals its power set. The sethood antecedent is necessary; compare pwv 3647. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  ( A  e.  V  ->  ~P A  =/=  A )
 
Theoremrepizf2lem 3988 Lemma for repizf2 3989. 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 )
 
Theoremrepizf2 3989* Replacement. This version of replacement is stronger than repizf 3947 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 3947 with ax-sep 3949. 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
 
Theoremclass2seteq 3990* 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 )
 
Theorem0elpw 3991 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
 |-  (/)  e.  ~P A
 
Theorem0nep0 3992 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
 |-  (/)  =/=  { (/) }
 
Theorem0inp0 3993 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  =  { (/) } )
 
Theoremunidif0 3994 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
 |- 
 U. ( A  \  { (/) } )  = 
 U. A
 
Theoremiin0imm 3995* 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  (/)  =  (/) )
 
Theoremiin0r 3996* 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  =/=  (/) )
 
Theoremintv 3997 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
 |- 
 |^| _V  =  (/)
 
Theoremaxpweq 3998* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4001 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
 
Theorembnd 3999* A very strong generalization of the Axiom of Replacement (compare zfrep6 3948). 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 3946. (Contributed by NM, 17-Oct-2004.)
 |-  ( A. x  e.  z  E. y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
 
Theorembnd2 4000* A variant of the Boundedness Axiom bnd 3999 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 ) )
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