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Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxwlim 4101 Extend the definition of a wff to include the limit ordinal predicate.

Syntaxcsuc 4102 Extend class notation to include the successor function.

Definitiondf-iord 4103* 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 4104 instead for naming consistency with set.mm. (New usage is discouraged.)

Theoremdford3 4104* Alias for df-iord 4103. Use it instead of df-iord 4103 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)

Definitiondf-on 4105 Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)

Definitiondf-ilim 4106 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 4107 instead for naming consistency with set.mm. (New usage is discouraged.)

Theoremdflim2 4107 Alias for df-ilim 4106. Use it instead of df-ilim 4106 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)

Definitiondf-suc 4108 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 4149). 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.)

Theoremordeq 4109 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)

Theoremelong 4110 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Theoremelon 4111 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Theoremeloni 4112 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)

Theoremelon2 4113 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)

Theoremlimeq 4114 Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordtr 4115 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)

Theoremordelss 4116 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)

Theoremtrssord 4117 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)

Theoremordelord 4118 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)

Theoremtron 4119 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)

Theoremordelon 4120 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)

Theoremonelon 4121 An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)

Theoremordin 4122 The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)

Theoremonin 4123 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)

Theoremonelss 4124 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.)

Theoremordtr1 4125 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)

Theoremontr1 4126 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)

Theoremonintss 4127* 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.)

Theoremord0 4128 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)

Theorem0elon 4129 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)

Theoreminton 4130 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)

Theoremnlim0 4131 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremlimord 4132 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)

Theoremlimuni 4133 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)

Theoremlimuni2 4134 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)

Theorem0ellim 4135 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)

Theoremlimelon 4136 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)

Theoremonn0 4137 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)

Theoremonm 4138 The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)

Theoremsuceq 4139 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremelsuci 4140 Membership in a successor. This one-way implication does not require that either or be sets. (Contributed by NM, 6-Jun-1994.)

Theoremelsucg 4141 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)

Theoremelsuc2g 4142 Variant of membership in a successor, requiring that rather than be a set. (Contributed by NM, 28-Oct-2003.)

Theoremelsuc 4143 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)

Theoremelsuc2 4144 Membership in a successor. (Contributed by NM, 15-Sep-2003.)

Theoremnfsuc 4145 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)

Theoremelelsuc 4146 Membership in a successor. (Contributed by NM, 20-Jun-1998.)

Theoremsucel 4147* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)

Theoremsuc0 4148 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)

Theoremsucprc 4149 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)

Theoremunisuc 4150 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.)

Theoremunisucg 4151 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.)

Theoremsssucid 4152 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.)

Theoremsucidg 4153 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)

Theoremsucid 4154 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.)

Theoremnsuceq0g 4155 No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)

Theoremeqelsuc 4156 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)

Theoremiunsuc 4157* Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremsuctr 4158 The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)

Theoremtrsuc 4159 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.)

Theoremtrsucss 4160 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)

Theoremsucssel 4161 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)

Theoremorduniss 4162 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)

Theoremonordi 4163 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)

Theoremontrci 4164 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)

Theoremoneli 4165 A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Theoremonelssi 4166 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)

Theoremonelini 4167 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)

Theoremoneluni 4168 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)

Theoremonunisuci 4169 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)

2.4  IZF Set Theory - add the Axiom of Union

2.4.1  Introduce the Axiom of Union

Axiomax-un 4170* 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 4172 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4173. A version using class notation is uniex 4174.

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 3878), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 253).

The union of a class df-uni 3581 should not be confused with the union of two classes df-un 2922. Their relationship is shown in unipr 3594. (Contributed by NM, 23-Dec-1993.)

Theoremzfun 4171* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Theoremaxun2 4172* A variant of the Axiom of Union ax-un 4170. 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.)

Theoremuniex2 4173* The Axiom of Union using the standard abbreviation for union. Given any set , its union exists. (Contributed by NM, 4-Jun-2006.)

Theoremuniex 4174 The Axiom of Union in class notation. This says that if is a set i.e. (see isset 2561), then the union of is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)

Theoremuniexg 4175 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.)

Theoremunex 4176 The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)

Theoremunexb 4177 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)

Theoremunexg 4178 A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)

Theoremtpexg 4179 An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)

Theoremunisn3 4180* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)

Theoremsnnex 4181* The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)

Theoremopeluu 4182 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.)

Theoremuniuni 4183* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)

Theoremeusv1 4184* Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.)

Theoremeusvnf 4185* Even if is free in , it is effectively bound when is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremeusvnfb 4186* Two ways to say that is a set expression that does not depend on . (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2i 4187* Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2nf 4188* Two ways to express single-valuedness of a class expression . (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2 4189* Two ways to express single-valuedness of a class expression . (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremreusv1 4190* Two ways to express single-valuedness of a class expression . (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremreusv3i 4191* Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)

Theoremreusv3 4192* Two ways to express single-valuedness of a class expression . See reusv1 4190 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)

Theoremalxfr 4193* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 18-Feb-2007.)

Theoremralxfrd 4194* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremrexxfrd 4195* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)

Theoremralxfr2d 4196* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.)

Theoremrexxfr2d 4197* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremralxfr 4198* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

TheoremralxfrALT 4199* Transfer universal quantification from a variable to another variable contained in expression . This proof does not use ralxfrd 4194. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremrexxfr 4200* Transfer existence from a variable to another variable contained in expression . (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

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