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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsyl6eqbrr 4001 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 4002 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 4003 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 4004 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 4005 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 4006 Deduction version of bound-variable hypothesis builder nfbr 4007. (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 4007 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 4008* 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 4009 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
 |-  ( A ( R  u.  S ) B  <-> 
 ( A R B  \/  A S B ) )
 
Theorembrin 4010 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
 |-  ( A ( R  i^i  S ) B  <-> 
 ( A R B  /\  A S B ) )
 
Theorembrdif 4011 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( A ( R 
 \  S ) B  <-> 
 ( A R B  /\  -.  A S B ) )
 
Theoremsbcbrg 4012 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 4013* 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 4014* 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 4015* 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 4016 Extend class notation to include ordered-pair class abstraction (class builder).
 class  { <. x ,  y >.  |  ph }
 
Syntaxcmpt 4017 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 4018* 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 (see dfid2 4248 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 6073. For example,  R  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1 )  =  y ) }  ->  3 R 4 (ex-opab 20727). (Contributed by NM, 4-Jul-1994.)
 |- 
 { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y ( z  = 
 <. x ,  y >.  /\  ph ) }
 
Definitiondf-mpt 4019* 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. An example is the square function for complex numbers,  ( x  e.  CC  |->  ( x ^ 2 ) ). 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 4020* 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 4021 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 4022* 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 4023 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
 |-  ( ph  <->  ps )   =>    |- 
 { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }
 
Theoremnfopab 4024* 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 4025 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 4026 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 4027* 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 4028* 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 4029* 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 4030* 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 4031* 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 4032* 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 4033* 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 4034* 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 4035 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 4036 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 4037* 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 4038* 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 4039* 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 4040* 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 4041* 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 4042 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 4043 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 4044 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 4045 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 4046* 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 4047* 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 4048* 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 4049 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
 |-  F/_ x ( x  e.  A  |->  B )
 
Theoremcbvmpt 4050* 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 4051* 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 4052* 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 4053 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
 wff  Tr  A
 
Definitiondf-tr 4054 Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5008). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4055 (which is suggestive of the word "transitive"), dftr3 4057, dftr4 4058, dftr5 4056, and (when  A is a set) unisuc 4405. 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 4055* 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 4056* 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 4057* 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 4058 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 4059 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B ) )
 
Theoremtrel 4060 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 4061 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 4062 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 4063 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
 |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
 
Theoremtr0 4064 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
 |- 
 Tr  (/)
 
Theoremtrv 4065 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
 |- 
 Tr  _V
 
Theoremtriun 4066* 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 4067* 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 4068* 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 )
 
Theoremtrintss 4069 If  A is transitive and non-null, then  |^| A is a subset of  A. (Contributed by Scott Fenton, 3-Mar-2011.)
 |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  |^| A  C_  A )
 
Theoremtrint0 4070 Any non-empty transitive class includes its intersection. Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)
 |-  ( ( Tr  A  /\  A  =/=  (/) )  ->  |^| A  C_  A )
 
2.2  ZF Set Theory - add the Axiom of Replacement
 
2.2.1  Introduce the Axiom of Replacement
 
Axiomax-rep 4071* Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory. Axiom 5 of [TakeutiZaring] p. 19. It tells us that the image of any set under a function is also a set (see the variant funimaex 5233). Although  ph may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and  ph encodes the predicate "the value of the function at  w is  z." Thus  ph will ordinarily have free variables 
w and  z- think of it informally as  ph ( w ,  z ). We prefix  ph with the quantifier  A. y in order to "protect" the axiom from any  ph containing  y, thus allowing us to eliminate any restrictions on  ph. This makes the axiom usable in a formalization that omits the logically redundant axiom ax-17 1628. Another common variant is derived as axrep5 4076, where you can find some further remarks. A slightly more compact version is shown as axrep2 4073. A quite different variant is zfrep6 5647, which if used in place of ax-rep 4071 would also require that the Separation Scheme axsep 4080 be stated as a separate axiom.

There is very a strong generalization of Replacement that doesn't demand function-like behavior of  ph. Two versions of this generalization are called the Collection Principle cp 7494 and the Boundedness Axiom bnd 7495.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4080, Null Set axnul 4088, and Pairing axpr 4151, all of which we derive from Replacement. In order to make it easier to identify the uses of those redundant axioms, we restate them as axioms ax-sep 4081, ax-nul 4089, and ax-pr 4152 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

 |-  ( A. w E. y A. z ( A. y ph  ->  z  =  y )  ->  E. y A. z ( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) ) )
 
Theoremaxrep1 4072* The version of the Axiom of Replacement used in the Metamath Solitaire applet http://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 4071 
-> axrep1 4072 
-> axrep2 4073 
-> axrepnd 8149 
-> zfcndrep 8169 = ax-rep 4071. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 |- 
 E. x ( E. y A. z ( ph  ->  z  =  y ) 
 ->  A. z ( z  e.  x  <->  E. x ( x  e.  y  /\  ph )
 ) )
 
Theoremaxrep2 4073* Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on  ph. (Contributed by NM, 15-Aug-2003.)
 |- 
 E. x ( E. y A. z ( ph  ->  z  =  y ) 
 ->  A. z ( z  e.  x  <->  E. x ( x  e.  y  /\  A. y ph ) ) )
 
Theoremaxrep3 4074* Axiom of Replacement slightly strengthened from axrep2 4073; 
w may occur free in  ph. (Contributed by NM, 2-Jan-1997.)
 |- 
 E. x ( E. y A. z ( ph  ->  z  =  y ) 
 ->  A. z ( z  e.  x  <->  E. x ( x  e.  w  /\  A. y ph ) ) )
 
Theoremaxrep4 4075* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
 |- 
 F/ z ph   =>    |-  ( A. x E. z A. y ( ph  ->  y  =  z ) 
 ->  E. z A. y
 ( y  e.  z  <->  E. x ( x  e.  w  /\  ph )
 ) )
 
Theoremaxrep5 4076* Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us 
ph is analogous to a "function" from  x to  y (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set  z that corresponds to the "image" of  ph restricted to some other set  w. The hypothesis says  z must not be free in  ph. (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |- 
 F/ z ph   =>    |-  ( A. x ( x  e.  w  ->  E. z A. y (
 ph  ->  y  =  z ) )  ->  E. z A. y ( y  e.  z  <->  E. x ( x  e.  w  /\  ph )
 ) )
 
Theoremzfrepclf 4077* An inference rule based on the Axiom of Replacement. Typically,  ph defines a function from  x to  y. (Contributed by NM, 26-Nov-1995.)
 |-  F/_ x A   &    |-  A  e.  _V   &    |-  ( x  e.  A  ->  E. z A. y (
 ph  ->  y  =  z ) )   =>    |- 
 E. z A. y
 ( y  e.  z  <->  E. x ( x  e.  A  /\  ph )
 )
 
Theoremzfrep3cl 4078* An inference rule based on the Axiom of Replacement. Typically,  ph defines a function from  x to  y. (Contributed by NM, 26-Nov-1995.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  E. z A. y (
 ph  ->  y  =  z ) )   =>    |- 
 E. z A. y
 ( y  e.  z  <->  E. x ( x  e.  A  /\  ph )
 )
 
Theoremzfrep4 4079* A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)
 |- 
 { x  |  ph }  e.  _V   &    |-  ( ph  ->  E. z A. y ( ps  ->  y  =  z ) )   =>    |-  { y  | 
 E. x ( ph  /\ 
 ps ) }  e.  _V
 
2.2.2  Derive the Axiom of Separation
 
Theoremaxsep 4080* Separation Scheme, which is an axiom scheme of Zermelo's original theory. Scheme Sep of [BellMachover] p. 463. As we show here, it is redundant if we assume Replacement in the form of ax-rep 4071. Some textbooks present Separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger Replacement. 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 2934. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

The variable  x can appear free in the wff  ph, which in textbooks is often written  ph ( x ). To specify this in the Metamath language, we omit the distinct variable requirement ($d) that  x not appear in  ph.

For a version using a class variable, see zfauscl 4083, which requires the Axiom of Extensionality as well as Replacement for its derivation.

If we omit the requirement that  y not occur in  ph, we can derive a contradiction, as notzfaus 4123 shows (contradicting zfauscl 4083). However, as axsep2 4082 shows, we can eliminate the restriction that  z not occur in  ph.

Note: the distinct variable restriction that  z not occur in  ph is actually redundant in this particular proof, but we keep it since its purpose is to demonstrate the derivation of the exact ax-sep 4081 from ax-rep 4071.

This theorem should not be referenced by any proof. Instead, use ax-sep 4081 below so that the uses of the Axiom of Separation can be more easily identified. (Contributed by NM, 11-Sep-2006.) (New usage is discouraged.)

 |- 
 E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph ) )
 
Axiomax-sep 4081* The Axiom of Separation of ZF set theory. See axsep 4080 for more information. It was derived as axsep 4080 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 11-Sep-2006.)
 |- 
 E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph ) )
 
Theoremaxsep2 4082* A less restrictive version of the Separation Scheme axsep 4080, 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 4081 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 4083* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4081, we invoke the Axiom of Extensionality (indirectly via vtocl 2789), which is needed for the justification of class variable notation.

If we omit the requirement that  y not occur in  ph, we can derive a contradiction, as notzfaus 4123 shows. (Contributed by NM, 5-Aug-1993.)

 |-  A  e.  _V   =>    |-  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
 )
 
Theorembm1.3ii 4084* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4081. 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 )
 
Theoremax9vsep 4085* Derive a weakened version of ax-9 1684 ( i.e. ax-9v 1632), where  x and  y must be distinct, from Separation ax-sep 4081 and Extensionality ax-ext 2237. See ax9 1683 for the derivation of ax-9 1684 from ax-9v 1632. (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 4086* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2241 to strengthen the hypothesis in the form of axnul 4088). (Contributed by NM, 22-Dec-2007.)
 |- 
 E. x A. y  -.  y  e.  x   =>    |-  E! x A. y  -.  y  e.  x
 
TheoremaxnulALT 4087* Prove axnul 4088 directly from ax-rep 4071 without using any equality axioms (ax-9 1684 thru ax-16 1927) if we accept ax-4 1692 as an axiom. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.)
 |- 
 E. x A. y  -.  y  e.  x
 
Theoremaxnul 4088* 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 4081. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tells us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 4086).

This proof, suggested by Jeff Hoffman (3-Feb-2008), uses only ax-5 1533 and ax-gen 1536 from predicate calculus, which are valid in "free logic" i.e. logic holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] p. 277). Thus our ax-sep 4081 implies the existence of at least one set. Note that Kunen's version of ax-sep 4081 (Axiom 3 of [Kunen] p. 11) does not imply the existence of a set because his is universally closed i.e. prefixed with universal quantifiers to eliminate all free variables. His existence is provided by a separate axiom stating  E. x x  =  x (Axiom 0 of [Kunen] p. 10).

See axnulALT 4087 for a proof directly from ax-rep 4071.

This theorem should not be referenced by any proof. Instead, use ax-nul 4089 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by NM, 7-Aug-2003.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.)

 |- 
 E. x A. y  -.  y  e.  x
 
Axiomax-nul 4089* The Null Set Axiom of ZF set theory. It was derived as axnul 4088 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 7-Aug-2003.)
 |- 
 E. x A. y  -.  y  e.  x
 
Theorem0ex 4090 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 4089. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  (/)  e.  _V
 
2.2.4  Theorems requiring subset and intersection existence
 
Theoremnalset 4091* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
 |- 
 -.  E. x A. y  y  e.  x
 
Theoremvprc 4092 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 4093 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)
 |- 
 -.  _V  e.  A
 
Theoremvnex 4094 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)
 |- 
 -.  E. x  x  =  _V
 
Theoreminex1 4095 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 4096 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
 |-  A  e.  _V   =>    |-  ( B  i^i  A )  e.  _V
 
Theoreminex1g 4097 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
 |-  ( A  e.  V  ->  ( A  i^i  B )  e.  _V )
 
Theoremssex 4098 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 4081 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
 |-  B  e.  _V   =>    |-  ( A  C_  B  ->  A  e.  _V )
 
Theoremssexi 4099 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 4100 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 )
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