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Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexse 4101 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( A  e.  V  ->  R Se  A )
 
Theoremsess1 4102 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  C_  S  ->  ( S Se  A  ->  R Se 
 A ) )
 
Theoremsess2 4103 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  C_  B  ->  ( R Se  B  ->  R Se 
 A ) )
 
Theoremseeq1 4104 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
 )
 
Theoremseeq2 4105 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
 )
 
Theoremnfse 4106 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R Se  A
 
Theoremepse 4107 The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)
 |- 
 _E Se  A
 
Theoremfrforeq1 4108 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( R  =  S  ->  (FrFor  R A T  <-> FrFor  S A T ) )
 
Theoremfreq1 4109 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
 
Theoremfrforeq2 4110 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( A  =  B  ->  (FrFor  R A T  <-> FrFor  R B T ) )
 
Theoremfreq2 4111 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
 
Theoremfrforeq3 4112 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( S  =  T  ->  (FrFor  R A S  <-> FrFor  R A T ) )
 
Theoremnffrfor 4113 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   &    |-  F/_ x S   =>    |- 
 F/ xFrFor  R A S
 
Theoremnffr 4114 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Fr  A
 
Theoremfrirrg 4115 A well-founded relation is irreflexive. This is the case where  A exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
 |-  ( ( R  Fr  A  /\  A  e.  V  /\  B  e.  A ) 
 ->  -.  B R B )
 
Theoremfr0 4116 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
 |-  R  Fr  (/)
 
Theoremfrind 4117* Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 ( ch  /\  x  e.  A )  ->  ( A. y  e.  A  ( y R x 
 ->  ps )  ->  ph )
 )   &    |-  ( ch  ->  R  Fr  A )   &    |-  ( ch  ->  A  e.  V )   =>    |-  ( ( ch 
 /\  x  e.  A )  ->  ph )
 
Theoremefrirr 4118 Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
 |-  (  _E  Fr  A  ->  -.  A  e.  A )
 
Theoremtz7.2 4119 Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent  _E  Fr  A. (Contributed by NM, 4-May-1994.)
 |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
 
Theoremnfwe 4120 Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  We  A
 
Theoremweeq1 4121 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  We  A  <->  S  We  A ) )
 
Theoremweeq2 4122 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  We  A  <->  R  We  B ) )
 
Theoremwefr 4123 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
 |-  ( R  We  A  ->  R  Fr  A )
 
Theoremwepo 4124 A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
 |-  ( ( R  We  A  /\  A  e.  V )  ->  R  Po  A )
 
Theoremwetrep 4125* An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
 |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  ( ( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
 
Theoremwe0 4126 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
 |-  R  We  (/)
 
2.3.9  Ordinals
 
Syntaxword 4127 Extend the definition of a wff to include the ordinal predicate.
 wff  Ord  A
 
Syntaxcon0 4128 Extend the definition of a class to include the class of all ordinal numbers. (The 0 in the name prevents creating a file called con.html, which causes problems in Windows.)
 class  On
 
Syntaxwlim 4129 Extend the definition of a wff to include the limit ordinal predicate.
 wff  Lim  A
 
Syntaxcsuc 4130 Extend class notation to include the successor function.
 class  suc  A
 
Definitiondf-iord 4131* Define the ordinal predicate, which is true for a class that is transitive and whose elements are transitive. Definition of ordinal in [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4132 instead for naming consistency with set.mm. (New usage is discouraged.)
 |-  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
 
Theoremdford3 4132* Alias for df-iord 4131. Use it instead of df-iord 4131 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)
 |-  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
 
Definitiondf-on 4133 Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)
 |- 
 On  =  { x  |  Ord  x }
 
Definitiondf-ilim 4134 Define the limit ordinal predicate, which is true for an ordinal that has the empty set as an element and is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then changes  A  =/=  (/) to  (/)  e.  A (which would be equivalent given the law of the excluded middle, but which is not for us). (Contributed by Jim Kingdon, 11-Nov-2018.) Use its alias dflim2 4135 instead for naming consistency with set.mm. (New usage is discouraged.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
 
Theoremdflim2 4135 Alias for df-ilim 4134. Use it instead of df-ilim 4134 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
 
Definitiondf-suc 4136 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1". Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 4177). Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.)
 |- 
 suc  A  =  ( A  u.  { A }
 )
 
Theoremordeq 4137 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
 
Theoremelong 4138 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
 
Theoremelon 4139 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  On 
 <-> 
 Ord  A )
 
Theoremeloni 4140 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  On  ->  Ord  A )
 
Theoremelon2 4141 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
 |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
 
Theoremlimeq 4142 Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
 
Theoremordtr 4143 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
 |-  ( Ord  A  ->  Tr  A )
 
Theoremordelss 4144 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  B  C_  A )
 
Theoremtrssord 4145 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
 |-  ( ( Tr  A  /\  A  C_  B  /\  Ord 
 B )  ->  Ord  A )
 
Theoremordelord 4146 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  Ord  B )
 
Theoremtron 4147 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
 |- 
 Tr  On
 
Theoremordelon 4148 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  B  e.  On )
 
Theoremonelon 4149 An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
 |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )
 
Theoremordin 4150 The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
 
Theoremonin 4151 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  B )  e.  On )
 
Theoremonelss 4152 An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  e.  On  ->  ( B  e.  A  ->  B  C_  A )
 )
 
Theoremordtr1 4153 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
 |-  ( Ord  C  ->  ( ( A  e.  B  /\  B  e.  C ) 
 ->  A  e.  C ) )
 
Theoremontr1 4154 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
 |-  ( C  e.  On  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
 
Theoremonintss 4155* If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  On  ->  ( ps  ->  |^|
 { x  e.  On  |  ph }  C_  A ) )
 
Theoremord0 4156 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
 |- 
 Ord  (/)
 
Theorem0elon 4157 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
 |-  (/)  e.  On
 
Theoreminton 4158 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
 |- 
 |^| On  =  (/)
 
Theoremnlim0 4159 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |- 
 -.  Lim  (/)
 
Theoremlimord 4160 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
 |-  ( Lim  A  ->  Ord 
 A )
 
Theoremlimuni 4161 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
 |-  ( Lim  A  ->  A  =  U. A )
 
Theoremlimuni2 4162 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
 |-  ( Lim  A  ->  Lim  U. A )
 
Theorem0ellim 4163 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
 |-  ( Lim  A  ->  (/)  e.  A )
 
Theoremlimelon 4164 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
 |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )
 
Theoremonn0 4165 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
 |- 
 On  =/=  (/)
 
Theoremonm 4166 The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
 |- 
 E. x  x  e. 
 On
 
Theoremsuceq 4167 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  =  B  ->  suc  A  =  suc  B )
 
Theoremelsuci 4168 Membership in a successor. This one-way implication does not require that either  A or  B be sets. (Contributed by NM, 6-Jun-1994.)
 |-  ( A  e.  suc  B 
 ->  ( A  e.  B  \/  A  =  B ) )
 
Theoremelsucg 4169 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
 |-  ( A  e.  V  ->  ( A  e.  suc  B  <-> 
 ( A  e.  B  \/  A  =  B ) ) )
 
Theoremelsuc2g 4170 Variant of membership in a successor, requiring that  B rather than  A be a set. (Contributed by NM, 28-Oct-2003.)
 |-  ( B  e.  V  ->  ( A  e.  suc  B  <-> 
 ( A  e.  B  \/  A  =  B ) ) )
 
Theoremelsuc 4171 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( A  e.  suc 
 B 
 <->  ( A  e.  B  \/  A  =  B ) )
 
Theoremelsuc2 4172 Membership in a successor. (Contributed by NM, 15-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( B  e.  suc 
 A 
 <->  ( B  e.  A  \/  B  =  A ) )
 
Theoremnfsuc 4173 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
 |-  F/_ x A   =>    |-  F/_ x  suc  A
 
Theoremelelsuc 4174 Membership in a successor. (Contributed by NM, 20-Jun-1998.)
 |-  ( A  e.  B  ->  A  e.  suc  B )
 
Theoremsucel 4175* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
 |-  ( suc  A  e.  B 
 <-> 
 E. x  e.  B  A. y ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
 
Theoremsuc0 4176 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
 |- 
 suc  (/)  =  { (/) }
 
Theoremsucprc 4177 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
 |-  ( -.  A  e.  _V 
 ->  suc  A  =  A )
 
Theoremunisuc 4178 A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( Tr  A  <->  U.
 suc  A  =  A )
 
Theoremunisucg 4179 A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by Jim Kingdon, 18-Aug-2019.)
 |-  ( A  e.  V  ->  ( Tr  A  <->  U. suc  A  =  A ) )
 
Theoremsssucid 4180 A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
 |-  A  C_  suc  A
 
Theoremsucidg 4181 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
 |-  ( A  e.  V  ->  A  e.  suc  A )
 
Theoremsucid 4182 A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
 |-  A  e.  _V   =>    |-  A  e.  suc  A
 
Theoremnsuceq0g 4183 No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
 |-  ( A  e.  V  ->  suc  A  =/=  (/) )
 
Theoremeqelsuc 4184 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
 |-  A  e.  _V   =>    |-  ( A  =  B  ->  A  e.  suc  B )
 
Theoremiunsuc 4185* Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  U_ x  e.  suc  A B  =  ( U_ x  e.  A  B  u.  C )
 
Theoremsuctr 4186 The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
 |-  ( Tr  A  ->  Tr 
 suc  A )
 
Theoremtrsuc 4187 A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
 
Theoremtrsucss 4188 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
 |-  ( Tr  A  ->  ( B  e.  suc  A  ->  B  C_  A )
 )
 
Theoremsucssel 4189 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
 |-  ( A  e.  V  ->  ( suc  A  C_  B  ->  A  e.  B ) )
 
Theoremorduniss 4190 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
 |-  ( Ord  A  ->  U. A  C_  A )
 
Theoremonordi 4191 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  Ord  A
 
Theoremontrci 4192 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  Tr  A
 
Theoremoneli 4193 A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  ( B  e.  A  ->  B  e.  On )
 
Theoremonelssi 4194 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
 |-  A  e.  On   =>    |-  ( B  e.  A  ->  B  C_  A )
 
Theoremonelini 4195 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  ( B  e.  A  ->  B  =  ( B  i^i  A ) )
 
Theoremoneluni 4196 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
 |-  A  e.  On   =>    |-  ( B  e.  A  ->  ( A  u.  B )  =  A )
 
Theoremonunisuci 4197 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
 |-  A  e.  On   =>    |-  U. suc  A  =  A
 
2.4  IZF Set Theory - add the Axiom of Union
 
2.4.1  Introduce the Axiom of Union
 
Axiomax-un 4198* Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set  y exists that includes the union of a given set  x i.e. the collection of all members of the members of  x. The variant axun2 4200 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4201. A version using class notation is uniex 4202.

This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3906), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 257).

The union of a class df-uni 3609 should not be confused with the union of two classes df-un 2950. Their relationship is shown in unipr 3622. (Contributed by NM, 23-Dec-1993.)

 |- 
 E. y A. z
 ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
 
Theoremzfun 4199* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
 |- 
 E. x A. y
 ( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x )
 
Theoremaxun2 4200* A variant of the Axiom of Union ax-un 4198. For any set  x, there exists a set  y whose members are exactly the members of the members of  x i.e. the union of  x. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
 |- 
 E. y A. z
 ( z  e.  y  <->  E. w ( z  e.  w  /\  w  e.  x ) )
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