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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | uniuni 4201* | Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.) |
Theorem | eusv1 4202* | Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.) |
Theorem | eusvnf 4203* | 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 4204* | Two ways to say that is a set expression that does not depend on . (Contributed by Mario Carneiro, 18-Nov-2016.) |
Theorem | eusv2i 4205* | 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 4206* | Two ways to express single-valuedness of a class expression . (Contributed by Mario Carneiro, 18-Nov-2016.) |
Theorem | eusv2 4207* | 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 4208* | 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 4209* | Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.) |
Theorem | reusv3 4210* | Two ways to express single-valuedness of a class expression . See reusv1 4208 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.) |
Theorem | alxfr 4211* | Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 18-Feb-2007.) |
Theorem | ralxfrd 4212* | 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 4213* | 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 4214* | Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.) |
Theorem | rexxfr2d 4215* | 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 4216* | 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 4217* | Transfer universal quantification from a variable to another variable contained in expression . This proof does not use ralxfrd 4212. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rexxfr 4218* | 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 4219* | Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 16-Jan-2012.) |
Theorem | rabxfr 4220* | Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 10-Jun-2005.) |
Theorem | reuhypd 4221* | A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.) |
Theorem | reuhyp 4222* | A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.) |
Theorem | uniexb 4223 | 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 4224 | 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 4225 | The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
Theorem | eldifpw 4226 | Membership in a power class difference. (Contributed by NM, 25-Mar-2007.) |
Theorem | op1stb 4227 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.) |
Theorem | op1stbg 4228 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.) |
Theorem | iunpw 4229* | 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 4230 | 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 4231 | 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 4232 | 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 4233 | 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 4234 | The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.) |
Theorem | onun2i 4235 | 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 4236 | 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 4237 | An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
Theorem | onuni 4238 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
Theorem | orduni 4239 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
Theorem | bm2.5ii 4240* | Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
Theorem | sucexb 4241 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
Theorem | sucexg 4242 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
Theorem | sucex 4243 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
Theorem | ordsucim 4244 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
Theorem | suceloni 4245 | 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 4246 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
Theorem | sucelon 4247 | The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
Theorem | ordsucss 4248 | The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
Theorem | ordelsuc 4249 | 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 4250 | 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 4251* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
Theorem | onsucelsucr 4252 | Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4273. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6093. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Theorem | onsucsssucr 4253 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4270. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | sucunielr 4254 | Successor and union. The converse (where is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4274. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | unon 4255 | The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) |
Theorem | onuniss2 4256* | The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | limon 4257 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
Theorem | ordunisuc2r 4258* | An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.) |
Theorem | onssi 4259 | An ordinal number is a subset of . (Contributed by NM, 11-Aug-1994.) |
Theorem | onsuci 4260 | The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
Theorem | onintonm 4261* | The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.) |
Theorem | onintrab2im 4262 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | ordtriexmidlem 4263 | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4265 or weak linearity in ordsoexmid 4305) with a proposition . Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmidlem2 4264* | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4265 or weak linearity in ordsoexmid 4305) with a proposition . Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmid 4265* |
Ordinal trichotomy implies the law of the excluded middle (that is,
decidability of an arbitrary proposition).
This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.) |
Theorem | ordtri2orexmid 4266* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
Theorem | 2ordpr 4267 | Version of 2on 6032 with the definition of expanded and expressed in terms of . (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | ontr2exmid 4268* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
Theorem | ordtri2or2exmidlem 4269* | A set which is if or if is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | onsucsssucexmid 4270* | The converse of onsucsssucr 4253 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | onsucelsucexmidlem1 4271* | Lemma for onsucelsucexmid 4273. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmidlem 4272* | Lemma for onsucelsucexmid 4273. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5523), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4263. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmid 4273* | The converse of onsucelsucr 4252 implies excluded middle. On the other hand, if is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4252 does hold, as seen at nnsucelsuc 6093. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | ordsucunielexmid 4274* | The converse of sucunielr 4254 (where is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | regexmidlemm 4275* | Lemma for regexmid 4278. is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | regexmidlem1 4276* | Lemma for regexmid 4278. If has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmidlema 4277* | Lemma for reg2exmid 4279. If has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Theorem | regexmid 4278* |
The axiom of foundation implies excluded middle.
By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by ). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4280. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmid 4279* | If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Axiom | ax-setind 4280* |
Axiom of -Induction
(also known as set induction). An axiom of
Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p.
"Axioms of CZF and IZF". This replaces the Axiom of
Foundation (also
called Regularity) from Zermelo-Fraenkel set theory.
For more on axioms which might be adopted which are incompatible with this axiom (that is, Non-wellfounded Set Theory but in the absence of excluded middle), see Chapter 20 of [AczelRathjen], p. 183. (Contributed by Jim Kingdon, 19-Oct-2018.) |
Theorem | setindel 4281* | -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.) |
Theorem | setind 4282* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
Theorem | setind2 4283 | Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.) |
Theorem | elirr 4284 |
No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22.
The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4280, we could redefine (df-iord 4121) to also require (df-frind 4087) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4285 (which under that definition would presumably not need ax-setind 4280 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4285. To encourage ordirr 4285 when possible, we mark this theorem as discouraged. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.) |
Theorem | ordirr 4285 | Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4280. If in the definition of ordinals df-iord 4121, we also required that membership be well-founded on any ordinal (see df-frind 4087), then we could prove ordirr 4285 without ax-setind 4280. (Contributed by NM, 2-Jan-1994.) |
Theorem | onirri 4286 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | nordeq 4287 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
Theorem | ordn2lp 4288 | An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.) |
Theorem | orddisj 4289 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
Theorem | orddif 4290 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Theorem | elirrv 4291 | The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.) |
Theorem | sucprcreg 4292 | A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.) |
Theorem | ruv 4293 | The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
Theorem | ruALT 4294 | Alternate proof of Russell's Paradox ru 2814, simplified using (indirectly) the Axiom of Set Induction ax-setind 4280. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | onprc 4295 | No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4230), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.) |
Theorem | sucon 4296 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
Theorem | en2lp 4297 | No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.) |
Theorem | preleq 4298 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Theorem | opthreg 4299 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4280 (via the preleq 4298 step). See df-op 3407 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
Theorem | suc11g 4300 | The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.) |
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