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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfopab1 4101 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 4102 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 4103* 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 4104* 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 4105* 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 4106* 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 4107* 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 4108* 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 4109* 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 4110* 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 4111 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 4112 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 4113* 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 4114* 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 4115* 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 4116* 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 4117* 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 4118 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 4119 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 4120 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 4121 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 4122* 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 4123* 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 4124* 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 4125 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
 |-  F/_ x ( x  e.  A  |->  B )
 
Theoremcbvmpt 4126* 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 4127* 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 4128* 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 4129 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
 wff  Tr  A
 
Definitiondf-tr 4130 Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5071). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4131 (which is suggestive of the word "transitive"), dftr3 4133, dftr4 4134, dftr5 4132, and (when  A is a set) unisuc 4484. 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 4131* 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 4132* 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 4133* 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 4134 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 4135 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B ) )
 
Theoremtrel 4136 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 4137 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 4138 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 4139 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
 |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
 
Theoremtr0 4140 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
 |- 
 Tr  (/)
 
Theoremtrv 4141 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
 |- 
 Tr  _V
 
Theoremtriun 4142* 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 4143* 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 4144* 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 4145 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 4146 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 4147* 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 5346). 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 1606. Another common variant is derived as axrep5 4152, where you can find some further remarks. A slightly more compact version is shown as axrep2 4149. A quite different variant is zfrep6 5764, which if used in place of ax-rep 4147 would also require that the Separation Scheme axsep 4156 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 7577 and the Boundedness Axiom bnd 7578.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4156, Null Set axnul 4164, and Pairing axpr 4229, 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 4157, ax-nul 4165, and ax-pr 4230 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 4148* 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 4147 
-> axrep1 4148 
-> axrep2 4149 
-> axrepnd 8232 
-> zfcndrep 8252 = ax-rep 4147. (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 4149* 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 4150* Axiom of Replacement slightly strengthened from axrep2 4149; 
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 4151* 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 4152* 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 4153* 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 4154* 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 4155* 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 4156* 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 4147. 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 3003. 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 4159, which requires the Axiom of Extensionality as well as Separation for its derivation.

If we omit the requirement that  y not occur in  ph, we can derive a contradiction, as notzfaus 4201 shows (contradicting zfauscl 4159). However, as axsep2 4158 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 4157 from ax-rep 4147.

This theorem should not be referenced by any proof. Instead, use ax-sep 4157 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 4157* The Axiom of Separation of ZF set theory. See axsep 4156 for more information. It was derived as axsep 4156 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 4158* A less restrictive version of the Separation Scheme axsep 4156, 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 4157 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 4159* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4157, we invoke the Axiom of Extensionality (indirectly via vtocl 2851), 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 4201 shows. (Contributed by NM, 5-Aug-1993.)

 |-  A  e.  _V   =>    |-  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
 )
 
Theorembm1.3ii 4160* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4157. 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 4161* Derive a weakened version of ax9 1902 ( i.e. ax9v 1645), where  x and  y must be distinct, from Separation ax-sep 4157 and Extensionality ax-ext 2277. See ax9 1902 for the derivation of ax9 1902 from ax9v 1645. (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 4162* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2281 to strengthen the hypothesis in the form of axnul 4164). (Contributed by NM, 22-Dec-2007.)
 |- 
 E. x A. y  -.  y  e.  x   =>    |-  E! x A. y  -.  y  e.  x
 
TheoremaxnulALT 4163* Prove axnul 4164 directly from ax-rep 4147 using none of the equality axioms ax-8 1661 through ax-15 2095 provided we accept sp 1728 as an axiom. Replace sp 1728 with the obsolete ax-4 2087 to see this in 'show traceback'. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 E. x A. y  -.  y  e.  x
 
Theoremaxnul 4164* 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 4157. 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 4162).

This proof, suggested by Jeff Hoffman, uses only ax-5 1547 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 4157 implies the existence of at least one set. Note that Kunen's version of ax-sep 4157 (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 4163 for a proof directly from ax-rep 4147.

This theorem should not be referenced by any proof. Instead, use ax-nul 4165 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 4165* The Null Set Axiom of ZF set theory. It was derived as axnul 4164 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 4166 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 4165. (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 4167* 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 4168 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 4169 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)
 |- 
 -.  _V  e.  A
 
Theoremvnex 4170 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)
 |- 
 -.  E. x  x  =  _V
 
Theoreminex1 4171 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 4172 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
 |-  A  e.  _V   =>    |-  ( B  i^i  A )  e.  _V
 
Theoreminex1g 4173 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
 |-  ( A  e.  V  ->  ( A  i^i  B )  e.  _V )
 
Theoremssex 4174 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 4157 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
 |-  B  e.  _V   =>    |-  ( A  C_  B  ->  A  e.  _V )
 
Theoremssexi 4175 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 4176 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 4177 A subclass of a set is a set. Deduction form of ssexg 4176. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  A  e.  _V )
 
Theoremdifexg 4178 Existence of a difference. (Contributed by NM, 26-May-1998.)
 |-  ( A  e.  V  ->  ( A  \  B )  e.  _V )
 
Theoremzfausab 4179* 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 4180* 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 4181* 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 4182* Membership in a class abstraction involving a subset. Unlike elabg 2928,  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 ) ) )
 
Theoremintex 4183 The intersection of a non-empty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.)
 |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
 
Theoremintnex 4184 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
 |-  ( -.  |^| A  e.  _V  <->  |^| A  =  _V )
 
Theoremintexab 4185 The intersection of a non-empty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
 |-  ( E. x ph  <->  |^| { x  |  ph }  e.  _V )
 
Theoremintexrab 4186 The intersection of a non-empty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
 |-  ( E. x  e.  A  ph  <->  |^| { x  e.  A  |  ph }  e.  _V )
 
Theoremiinexg 4187* The existence of an indexed union. 
x is normally a free-variable parameter in  B, which should be read  B ( x ). (Contributed by FL, 19-Sep-2011.)
 |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  C )  -> 
 |^|_ x  e.  A  B  e.  _V )
 
Theoremintabs 4188* Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  |^| { y  |  ps }  ->  ( ph 
 <->  ch ) )   &    |-  ( |^| { y  |  ps } 
 C_  A  /\  ch )   =>    |- 
 |^| { x  |  ( x  C_  A  /\  ph ) }  =  |^| { x  |  ph }
 
Theoreminuni 4189* 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 4190 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 4191 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 4192 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 4193 No set equals its power set. The sethood antecedent is necessary; compare pwv 3842. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  ( A  e.  V  ->  ~P A  =/=  A )
 
2.2.5  Theorems requiring empty set existence
 
Theoremclass2set 4194* Construct, from any class  A, a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (Contributed by NM, 16-Oct-2003.)
 |- 
 { x  e.  A  |  A  e.  _V }  e.  _V
 
Theoremclass2seteq 4195* Equality theorem based on class2set 4194. (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 4196 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
 |-  (/)  e.  ~P A
 
Theorem0nep0 4197 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
 |-  (/)  =/=  { (/) }
 
Theorem0inp0 4198 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 4199 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
 |- 
 U. ( A  \  { (/) } )  = 
 U. A
 
Theoremiin0 4200* An indexed intersection of the empty set, with a non-empty index set, is empty. (Contributed by NM, 20-Oct-2005.)
 |-  ( A  =/=  (/)  <->  |^|_ x  e.  A  (/) 
 =  (/) )
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