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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | freq2 4101 | Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.) |
Theorem | frforeq3 4102 | Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.) |
FrFor FrFor | ||
Theorem | nffrfor 4103 | Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
FrFor | ||
Theorem | nffr 4104 | Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Theorem | frirrg 4105 | A well-founded relation is irreflexive. This is the case where exists. (Contributed by Jim Kingdon, 21-Sep-2021.) |
Theorem | fr0 4106 | Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.) |
Theorem | frind 4107* | Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.) |
Theorem | efrirr 4108 | 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.) |
Theorem | tz7.2 4109 | Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent . (Contributed by NM, 4-May-1994.) |
Theorem | nfwe 4110 | Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Theorem | weeq1 4111 | Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.) |
Theorem | weeq2 4112 | Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.) |
Theorem | wefr 4113 | A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.) |
Theorem | wepo 4114 | A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.) |
Theorem | wetrep 4115* | An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.) |
Theorem | we0 4116 | Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.) |
Syntax | word 4117 | Extend the definition of a wff to include the ordinal predicate. |
Syntax | con0 4118 | 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.) |
Syntax | wlim 4119 | Extend the definition of a wff to include the limit ordinal predicate. |
Syntax | csuc 4120 | Extend class notation to include the successor function. |
Definition | df-iord 4121* | 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 4122 instead for naming consistency with set.mm. (New usage is discouraged.) |
Theorem | dford3 4122* | Alias for df-iord 4121. Use it instead of df-iord 4121 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.) |
Definition | df-on 4123 | Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.) |
Definition | df-ilim 4124 | 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 to (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 4125 instead for naming consistency with set.mm. (New usage is discouraged.) |
Theorem | dflim2 4125 | Alias for df-ilim 4124. Use it instead of df-ilim 4124 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.) |
Definition | df-suc 4126 | 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 4167). 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.) |
Theorem | ordeq 4127 | Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.) |
Theorem | elong 4128 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
Theorem | elon 4129 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
Theorem | eloni 4130 | An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.) |
Theorem | elon2 4131 | An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.) |
Theorem | limeq 4132 | Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Theorem | ordtr 4133 | An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.) |
Theorem | ordelss 4134 | An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.) |
Theorem | trssord 4135 | A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.) |
Theorem | ordelord 4136 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.) |
Theorem | tron 4137 | The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.) |
Theorem | ordelon 4138 | An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.) |
Theorem | onelon 4139 | An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.) |
Theorem | ordin 4140 | The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.) |
Theorem | onin 4141 | The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.) |
Theorem | onelss 4142 | 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.) |
Theorem | ordtr1 4143 | Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) |
Theorem | ontr1 4144 | Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.) |
Theorem | onintss 4145* | 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.) |
Theorem | ord0 4146 | The empty set is an ordinal class. (Contributed by NM, 11-May-1994.) |
Theorem | 0elon 4147 | The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.) |
Theorem | inton 4148 | The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.) |
Theorem | nlim0 4149 | The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Theorem | limord 4150 | A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.) |
Theorem | limuni 4151 | A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.) |
Theorem | limuni2 4152 | The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.) |
Theorem | 0ellim 4153 | A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.) |
Theorem | limelon 4154 | A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.) |
Theorem | onn0 4155 | The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.) |
Theorem | onm 4156 | The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.) |
Theorem | suceq 4157 | Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Theorem | elsuci 4158 | Membership in a successor. This one-way implication does not require that either or be sets. (Contributed by NM, 6-Jun-1994.) |
Theorem | elsucg 4159 | Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.) |
Theorem | elsuc2g 4160 | Variant of membership in a successor, requiring that rather than be a set. (Contributed by NM, 28-Oct-2003.) |
Theorem | elsuc 4161 | Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.) |
Theorem | elsuc2 4162 | Membership in a successor. (Contributed by NM, 15-Sep-2003.) |
Theorem | nfsuc 4163 | Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.) |
Theorem | elelsuc 4164 | Membership in a successor. (Contributed by NM, 20-Jun-1998.) |
Theorem | sucel 4165* | Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.) |
Theorem | suc0 4166 | The successor of the empty set. (Contributed by NM, 1-Feb-2005.) |
Theorem | sucprc 4167 | A proper class is its own successor. (Contributed by NM, 3-Apr-1995.) |
Theorem | unisuc 4168 | 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.) |
Theorem | unisucg 4169 | 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.) |
Theorem | sssucid 4170 | 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.) |
Theorem | sucidg 4171 | Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
Theorem | sucid 4172 | 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.) |
Theorem | nsuceq0g 4173 | No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.) |
Theorem | eqelsuc 4174 | A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.) |
Theorem | iunsuc 4175* | Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Theorem | suctr 4176 | The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) |
Theorem | trsuc 4177 | 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.) |
Theorem | trsucss 4178 | A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.) |
Theorem | sucssel 4179 | A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.) |
Theorem | orduniss 4180 | An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.) |
Theorem | onordi 4181 | An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) |
Theorem | ontrci 4182 | An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) |
Theorem | oneli 4183 | A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | onelssi 4184 | A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.) |
Theorem | onelini 4185 | An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.) |
Theorem | oneluni 4186 | An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
Theorem | onunisuci 4187 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
Axiom | ax-un 4188* |
Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory.
It states that a set exists that includes the union of a given set
i.e. the
collection of all members of the members of . The
variant axun2 4190 states that the union itself exists. A
version with the
standard abbreviation for union is uniex2 4191. A version using class
notation is uniex 4192.
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 3899), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 262). The union of a class df-uni 3602 should not be confused with the union of two classes df-un 2977. Their relationship is shown in unipr 3615. (Contributed by NM, 23-Dec-1993.) |
Theorem | zfun 4189* | Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) |
Theorem | axun2 4190* | A variant of the Axiom of Union ax-un 4188. For any set , there exists a set whose members are exactly the members of the members of i.e. the union of . Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
Theorem | uniex2 4191* | The Axiom of Union using the standard abbreviation for union. Given any set , its union exists. (Contributed by NM, 4-Jun-2006.) |
Theorem | uniex 4192 | The Axiom of Union in class notation. This says that if is a set i.e. (see isset 2605), then the union of is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.) |
Theorem | uniexg 4193 | The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent instead of to make the theorem more general and thus shorten some proofs; obviously the universal class constant is one possible substitution for class variable . (Contributed by NM, 25-Nov-1994.) |
Theorem | unex 4194 | The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.) |
Theorem | unexb 4195 | Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
Theorem | unexg 4196 | A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) |
Theorem | tpexg 4197 | An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.) |
Theorem | unisn3 4198* | Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.) |
Theorem | snnex 4199* | The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) |
Theorem | opeluu 4200 | Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.) |
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