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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | onm 4201 | The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.) |
Theorem | suceq 4202 | Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Theorem | elsuci 4203 | Membership in a successor. This one-way implication does not require that either or be sets. (Contributed by NM, 6-Jun-1994.) |
Theorem | elsucg 4204 | Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.) |
Theorem | elsuc2g 4205 | Variant of membership in a successor, requiring that rather than be a set. (Contributed by NM, 28-Oct-2003.) |
Theorem | elsuc 4206 | Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.) |
Theorem | elsuc2 4207 | Membership in a successor. (Contributed by NM, 15-Sep-2003.) |
Theorem | nfsuc 4208 | Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.) |
Theorem | elelsuc 4209 | Membership in a successor. (Contributed by NM, 20-Jun-1998.) |
Theorem | sucel 4210* | Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.) |
Theorem | suc0 4211 | The successor of the empty set. (Contributed by NM, 1-Feb-2005.) |
Theorem | sucprc 4212 | A proper class is its own successor. (Contributed by NM, 3-Apr-1995.) |
Theorem | unisuc 4213 | 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 4214 | 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 4215 | 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 4216 | 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 4217 | 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 4218 | No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.) |
Theorem | eqelsuc 4219 | A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.) |
Theorem | iunsuc 4220* | 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 4221 | The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) |
Theorem | trsuc 4222 | 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 4223 | A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.) |
Theorem | sucssel 4224 | A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.) |
Theorem | orduniss 4225 | An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.) |
Theorem | onordi 4226 | An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) |
Theorem | ontrci 4227 | An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) |
Theorem | oneli 4228 | 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 4229 | A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.) |
Theorem | onelini 4230 | An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.) |
Theorem | oneluni 4231 | An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
Theorem | onunisuci 4232 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
Axiom | ax-un 4233* |
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 4235 states that the union itself exists. A
version with the
standard abbreviation for union is uniex2 4236. A version using class
notation is uniex 4237.
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 3934), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 262). The union of a class df-uni 3637 should not be confused with the union of two classes df-un 2992. Their relationship is shown in unipr 3650. (Contributed by NM, 23-Dec-1993.) |
Theorem | zfun 4234* | Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) |
Theorem | axun2 4235* | A variant of the Axiom of Union ax-un 4233. 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 4236* | The Axiom of Union using the standard abbreviation for union. Given any set , its union exists. (Contributed by NM, 4-Jun-2006.) |
Theorem | uniex 4237 | The Axiom of Union in class notation. This says that if is a set i.e. (see isset 2619), then the union of is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.) |
Theorem | uniexg 4238 | 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 4239 | The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.) |
Theorem | unexb 4240 | Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
Theorem | unexg 4241 | A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) |
Theorem | tpexg 4242 | An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.) |
Theorem | unisn3 4243* | Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.) |
Theorem | snnex 4244* | 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 4245 | 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.) |
Theorem | uniuni 4246* | Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.) |
Theorem | eusv1 4247* | Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.) |
Theorem | eusvnf 4248* | 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.) |
Theorem | eusvnfb 4249* | Two ways to say that is a set expression that does not depend on . (Contributed by Mario Carneiro, 18-Nov-2016.) |
Theorem | eusv2i 4250* | Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.) |
Theorem | eusv2nf 4251* | Two ways to express single-valuedness of a class expression . (Contributed by Mario Carneiro, 18-Nov-2016.) |
Theorem | eusv2 4252* | Two ways to express single-valuedness of a class expression . (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Theorem | reusv1 4253* | Two ways to express single-valuedness of a class expression . (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Theorem | reusv3i 4254* | Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.) |
Theorem | reusv3 4255* | Two ways to express single-valuedness of a class expression . See reusv1 4253 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.) |
Theorem | alxfr 4256* | Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 18-Feb-2007.) |
Theorem | ralxfrd 4257* | 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.) |
Theorem | rexxfrd 4258* | 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.) |
Theorem | ralxfr2d 4259* | Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.) |
Theorem | rexxfr2d 4260* | 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.) |
Theorem | ralxfr 4261* | 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.) |
Theorem | ralxfrALT 4262* | Transfer universal quantification from a variable to another variable contained in expression . This proof does not use ralxfrd 4257. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rexxfr 4263* | Transfer existence from a variable to another variable contained in expression . (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
Theorem | rabxfrd 4264* | Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 16-Jan-2012.) |
Theorem | rabxfr 4265* | Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 10-Jun-2005.) |
Theorem | reuhypd 4266* | A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.) |
Theorem | reuhyp 4267* | A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.) |
Theorem | uniexb 4268 | The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.) |
Theorem | pwexb 4269 | 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.) |
Theorem | univ 4270 | The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
Theorem | eldifpw 4271 | Membership in a power class difference. (Contributed by NM, 25-Mar-2007.) |
Theorem | op1stb 4272 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.) |
Theorem | op1stbg 4273 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.) |
Theorem | iunpw 4274* | 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.) |
Theorem | ordon 4275 | 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.) |
Theorem | ssorduni 4276 | 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.) |
Theorem | ssonuni 4277 | The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.) |
Theorem | ssonunii 4278 | 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.) |
Theorem | onun2 4279 | The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.) |
Theorem | onun2i 4280 | The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.) |
Theorem | ordsson 4281 | 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.) |
Theorem | onss 4282 | An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
Theorem | onuni 4283 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
Theorem | orduni 4284 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
Theorem | bm2.5ii 4285* | Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
Theorem | sucexb 4286 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
Theorem | sucexg 4287 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
Theorem | sucex 4288 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
Theorem | ordsucim 4289 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
Theorem | suceloni 4290 | The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
Theorem | ordsucg 4291 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
Theorem | sucelon 4292 | The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
Theorem | ordsucss 4293 | The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
Theorem | ordelsuc 4294 | 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.) |
Theorem | onsucssi 4295 | A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.) |
Theorem | onsucmin 4296* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
Theorem | onsucelsucr 4297 | Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4318. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6200. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Theorem | onsucsssucr 4298 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4315. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | sucunielr 4299 | Successor and union. The converse (where is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4319. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | unon 4300 | The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) |
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