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Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremseeq1 4101 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
 
Theoremseeq2 4102 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
 
Theoremnfse 4103 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Se 𝐴
 
Theoremepse 4104 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 4105 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))
 
Theoremfreq1 4106 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
 
Theoremfrforeq2 4107 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))
 
Theoremfreq2 4108 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
(𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
 
Theoremfrforeq3 4109 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))
 
Theoremnffrfor 4110 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴    &   𝑥𝑆       𝑥 FrFor 𝑅𝐴𝑆
 
Theoremnffr 4111 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Fr 𝐴
 
Theoremfrirrg 4112 A well-founded relation is irreflexive. This is the case where 𝐴 exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
((𝑅 Fr 𝐴𝐴𝑉𝐵𝐴) → ¬ 𝐵𝑅𝐵)
 
Theoremfr0 4113 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
𝑅 Fr ∅
 
Theoremfrind 4114* Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   ((𝜒𝑥𝐴) → (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑))    &   (𝜒𝑅 Fr 𝐴)    &   (𝜒𝐴𝑉)       ((𝜒𝑥𝐴) → 𝜑)
 
Theoremefrirr 4115 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 → ¬ 𝐴𝐴)
 
Theoremtz7.2 4116 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 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
 
Theoremnfwe 4117 Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 We 𝐴
 
Theoremweeq1 4118 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 We 𝐴𝑆 We 𝐴))
 
Theoremweeq2 4119 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
(𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))
 
Theoremwefr 4120 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
(𝑅 We 𝐴𝑅 Fr 𝐴)
 
Theoremwepo 4121 A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
((𝑅 We 𝐴𝐴𝑉) → 𝑅 Po 𝐴)
 
Theoremwetrep 4122* An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
(( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
 
Theoremwe0 4123 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
𝑅 We ∅
 
2.3.9  Ordinals
 
Syntaxword 4124 Extend the definition of a wff to include the ordinal predicate.
wff Ord 𝐴
 
Syntaxcon0 4125 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 4126 Extend the definition of a wff to include the limit ordinal predicate.
wff Lim 𝐴
 
Syntaxcsuc 4127 Extend class notation to include the successor function.
class suc 𝐴
 
Definitiondf-iord 4128* 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 4129 instead for naming consistency with set.mm. (New usage is discouraged.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
 
Theoremdford3 4129* Alias for df-iord 4128. Use it instead of df-iord 4128 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
 
Definitiondf-on 4130 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 4131 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 ∅ ∈ 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 4132 instead for naming consistency with set.mm. (New usage is discouraged.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))
 
Theoremdflim2 4132 Alias for df-ilim 4131. Use it instead of df-ilim 4131 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))
 
Definitiondf-suc 4133 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 4174). 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 ∪ {𝐴})
 
Theoremordeq 4134 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
 
Theoremelong 4135 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
 
Theoremelon 4136 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
𝐴 ∈ V       (𝐴 ∈ On ↔ Ord 𝐴)
 
Theoremeloni 4137 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → Ord 𝐴)
 
Theoremelon2 4138 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
(𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
 
Theoremlimeq 4139 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 4140 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → Tr 𝐴)
 
Theoremordelss 4141 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
 
Theoremtrssord 4142 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
 
Theoremordelord 4143 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
((Ord 𝐴𝐵𝐴) → Ord 𝐵)
 
Theoremtron 4144 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
Tr On
 
Theoremordelon 4145 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
 
Theoremonelon 4146 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 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
 
Theoremordin 4147 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 (𝐴𝐵))
 
Theoremonin 4148 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
 
Theoremonelss 4149 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 ∈ On → (𝐵𝐴𝐵𝐴))
 
Theoremordtr1 4150 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
(Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremontr1 4151 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
(𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremonintss 4152* 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 → (𝜑𝜓))       (𝐴 ∈ On → (𝜓 {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
 
Theoremord0 4153 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
Ord ∅
 
Theorem0elon 4154 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
∅ ∈ On
 
Theoreminton 4155 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
On = ∅
 
Theoremnlim0 4156 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
¬ Lim ∅
 
Theoremlimord 4157 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
(Lim 𝐴 → Ord 𝐴)
 
Theoremlimuni 4158 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
(Lim 𝐴𝐴 = 𝐴)
 
Theoremlimuni2 4159 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
(Lim 𝐴 → Lim 𝐴)
 
Theorem0ellim 4160 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
(Lim 𝐴 → ∅ ∈ 𝐴)
 
Theoremlimelon 4161 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
 
Theoremonn0 4162 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
On ≠ ∅
 
Theoremonm 4163 The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
𝑥 𝑥 ∈ On
 
Theoremsuceq 4164 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
 
Theoremelsuci 4165 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 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
 
Theoremelsucg 4166 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremelsuc2g 4167 Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
(𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremelsuc 4168 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
 
Theoremelsuc2 4169 Membership in a successor. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
 
Theoremnfsuc 4170 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
𝑥𝐴       𝑥 suc 𝐴
 
Theoremelelsuc 4171 Membership in a successor. (Contributed by NM, 20-Jun-1998.)
(𝐴𝐵𝐴 ∈ suc 𝐵)
 
Theoremsucel 4172* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
(suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
 
Theoremsuc0 4173 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
suc ∅ = {∅}
 
Theoremsucprc 4174 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
𝐴 ∈ V → suc 𝐴 = 𝐴)
 
Theoremunisuc 4175 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 ∈ V       (Tr 𝐴 suc 𝐴 = 𝐴)
 
Theoremunisucg 4176 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.)
(𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
 
Theoremsssucid 4177 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 ⊆ suc 𝐴
 
Theoremsucidg 4178 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
(𝐴𝑉𝐴 ∈ suc 𝐴)
 
Theoremsucid 4179 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 ∈ V       𝐴 ∈ suc 𝐴
 
Theoremnsuceq0g 4180 No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
(𝐴𝑉 → suc 𝐴 ≠ ∅)
 
Theoremeqelsuc 4181 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
𝐴 ∈ V       (𝐴 = 𝐵𝐴 ∈ suc 𝐵)
 
Theoremiunsuc 4182* 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 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)
 
Theoremsuctr 4183 The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
(Tr 𝐴 → Tr suc 𝐴)
 
Theoremtrsuc 4184 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 𝐵𝐴) → 𝐵𝐴)
 
Theoremtrsucss 4185 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
(Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))
 
Theoremsucssel 4186 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
(𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
 
Theoremorduniss 4187 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
(Ord 𝐴 𝐴𝐴)
 
Theoremonordi 4188 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Ord 𝐴
 
Theoremontrci 4189 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Tr 𝐴
 
Theoremoneli 4190 A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵 ∈ On)
 
Theoremonelssi 4191 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵𝐴)
 
Theoremonelini 4192 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵 = (𝐵𝐴))
 
Theoremoneluni 4193 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴 → (𝐴𝐵) = 𝐴)
 
Theoremonunisuci 4194 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
𝐴 ∈ On        suc 𝐴 = 𝐴
 
2.4  IZF Set Theory - add the Axiom of Union
 
2.4.1  Introduce the Axiom of Union
 
Axiomax-un 4195* 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 4197 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4198. A version using class notation is uniex 4199.

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 3903), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 257).

The union of a class df-uni 3606 should not be confused with the union of two classes df-un 2947. Their relationship is shown in unipr 3619. (Contributed by NM, 23-Dec-1993.)

𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
 
Theoremzfun 4196* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
 
Theoremaxun2 4197* A variant of the Axiom of Union ax-un 4195. 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.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 
Theoremuniex2 4198* The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.)
𝑦 𝑦 = 𝑥
 
Theoremuniex 4199 The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2576), then the union of 𝐴 is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)
𝐴 ∈ V        𝐴 ∈ V
 
Theoremuniexg 4200 The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent 𝐴𝑉 instead of 𝐴 ∈ V to make the theorem more general and thus shorten some proofs; obviously the universal class constant V is one possible substitution for class variable 𝑉. (Contributed by NM, 25-Nov-1994.)
(𝐴𝑉 𝐴 ∈ V)
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