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

Theorem0ellim 4101 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)

Theoremlimelon 4102 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)

Theoremonn0 4103 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)

Theoremonm 4104 The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)

Theoremsuceq 4105 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremelsuci 4106 Membership in a successor. This one-way implication does not require that either or be sets. (Contributed by NM, 6-Jun-1994.)

Theoremelsucg 4107 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)

Theoremelsuc2g 4108 Variant of membership in a successor, requiring that rather than be a set. (Contributed by NM, 28-Oct-2003.)

Theoremelsuc 4109 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)

Theoremelsuc2 4110 Membership in a successor. (Contributed by NM, 15-Sep-2003.)

Theoremnfsuc 4111 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)

Theoremelelsuc 4112 Membership in a successor. (Contributed by NM, 20-Jun-1998.)

Theoremsucel 4113* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)

Theoremsuc0 4114 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)

Theoremsucprc 4115 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)

Theoremunisuc 4116 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 4117 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 4118 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 4119 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 4120 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 4121 No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)

Theoremeqelsuc 4122 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)

Theoremiunsuc 4123* 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 4124 The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)

Theoremtrsuc 4125 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 4126 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)

Theoremsucssel 4127 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)

Theoremorduniss 4128 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)

Theoremonordi 4129 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)

Theoremontrci 4130 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)

Theoremoneli 4131 A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Theoremonelssi 4132 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)

Theoremonelini 4133 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)

Theoremoneluni 4134 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)

Theoremonunisuci 4135 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 4136* 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 4138 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4139. A version using class notation is uniex 4140.

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 3869), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 253).

The union of a class df-uni 3572 should not be confused with the union of two classes df-un 2916. Their relationship is shown in unipr 3585. (Contributed by NM, 23-Dec-1993.)

Theoremzfun 4137* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Theoremaxun2 4138* A variant of the Axiom of Union ax-un 4136. 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 4139* The Axiom of Union using the standard abbreviation for union. Given any set , its union exists. (Contributed by NM, 4-Jun-2006.)

Theoremuniex 4140 The Axiom of Union in class notation. This says that if is a set i.e. (see isset 2555), then the union of is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)

Theoremuniexg 4141 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 4142 The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)

Theoremunexb 4143 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)

Theoremunexg 4144 A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)

Theoremtpexg 4145 An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)

Theoremunisn3 4146* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)

Theoremsnnex 4147* The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)

Theoremopeluu 4148 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 4149* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)

Theoremeusv1 4150* Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.)

Theoremeusvnf 4151* 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 4152* Two ways to say that is a set expression that does not depend on . (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2i 4153* Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2nf 4154* Two ways to express single-valuedness of a class expression . (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2 4155* 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 4156* 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 4157* Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)

Theoremreusv3 4158* Two ways to express single-valuedness of a class expression . See reusv1 4156 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)

Theoremalxfr 4159* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 18-Feb-2007.)

Theoremralxfrd 4160* 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 4161* 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 4162* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.)

Theoremrexxfr2d 4163* 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 4164* 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 4165* Transfer universal quantification from a variable to another variable contained in expression . This proof does not use ralxfrd 4160. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremrexxfr 4166* Transfer existence from a variable to another variable contained in expression . (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

Theoremrabxfrd 4167* Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 16-Jan-2012.)

Theoremrabxfr 4168* Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 10-Jun-2005.)

Theoremreuhypd 4169* A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)

Theoremreuhyp 4170* A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)

Theoremuniexb 4171 The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)

Theorempwexb 4172 The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)

Theoremuniv 4173 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Theoremeldifpw 4174 Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)

Theoremop1stb 4175 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.)

Theoremop1stbg 4176 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)

Theoremiunpw 4177* An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)

2.4.2  Ordinals (continued)

Theoremordon 4178 The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)

Theoremssorduni 4179 The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremssonuni 4180 The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)

Theoremssonunii 4181 The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)

Theoremonun2 4182 The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)

Theoremonun2i 4183 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.)

Theoremordsson 4184 Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.)

Theoremonss 4185 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)

Theoremonuni 4186 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)

Theoremorduni 4187 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)

Theorembm2.5ii 4188* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)

Theoremsucexb 4189 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)

Theoremsucexg 4190 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)

Theoremsucex 4191 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)

Theoremordsucim 4192 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)

Theoremsuceloni 4193 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)

Theoremordsucg 4194 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)

Theoremsucelon 4195 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)

Theoremordsucss 4196 The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)

Theoremordelsuc 4197 A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)

Theoremonsucmin 4198* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)

Theoremonsucelsucr 4199 Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4215. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6009. (Contributed by Jim Kingdon, 17-Jul-2019.)

Theoremonsucsssucr 4200 The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4212. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)

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