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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iunpw 4401* | 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 4402 | 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 4403 | 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 4404 | 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 4405 | 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 4406 | The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.) |
Theorem | onun2i 4407 | 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 4408 | 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 4409 | An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
Theorem | onuni 4410 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
Theorem | orduni 4411 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
Theorem | bm2.5ii 4412* | Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
Theorem | sucexb 4413 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
Theorem | sucexg 4414 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
Theorem | sucex 4415 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
Theorem | ordsucim 4416 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
Theorem | suceloni 4417 | 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 4418 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
Theorem | sucelon 4419 | The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
Theorem | ordsucss 4420 | The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
Theorem | ordelsuc 4421 | 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 4422 | 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 4423* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
Theorem | onsucelsucr 4424 | Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4445. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6387. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Theorem | onsucsssucr 4425 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4442. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | sucunielr 4426 | Successor and union. The converse (where is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4446. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | unon 4427 | The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) |
Theorem | onuniss2 4428* | The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | limon 4429 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
Theorem | ordunisuc2r 4430* | 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 4431 | An ordinal number is a subset of . (Contributed by NM, 11-Aug-1994.) |
Theorem | onsuci 4432 | 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 4433* | 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 4434 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | ordtriexmidlem 4435 | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4437 or weak linearity in ordsoexmid 4477) with a proposition . Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmidlem2 4436* | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4437 or weak linearity in ordsoexmid 4477) with a proposition . Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmid 4437* |
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 4438* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
Theorem | 2ordpr 4439 | Version of 2on 6322 with the definition of expanded and expressed in terms of . (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | ontr2exmid 4440* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
Theorem | ordtri2or2exmidlem 4441* | A set which is if or if is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | onsucsssucexmid 4442* | The converse of onsucsssucr 4425 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | onsucelsucexmidlem1 4443* | Lemma for onsucelsucexmid 4445. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmidlem 4444* | Lemma for onsucelsucexmid 4445. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5765), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4435. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmid 4445* | The converse of onsucelsucr 4424 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 4424 does hold, as seen at nnsucelsuc 6387. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | ordsucunielexmid 4446* | The converse of sucunielr 4426 (where is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | regexmidlemm 4447* | Lemma for regexmid 4450. is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | regexmidlem1 4448* | Lemma for regexmid 4450. If has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmidlema 4449* | Lemma for reg2exmid 4451. If has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Theorem | regexmid 4450* |
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 4452. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmid 4451* | If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Axiom | ax-setind 4452* |
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 4453* | -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.) |
Theorem | setind 4454* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
Theorem | setind2 4455 | 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 4456 |
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 4452, we could redefine (df-iord 4288) to also require (df-frind 4254) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4457 (which under that definition would presumably not need ax-setind 4452 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 4457. To encourage ordirr 4457 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 4457 | 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 4452. If in the definition of ordinals df-iord 4288, we also required that membership be well-founded on any ordinal (see df-frind 4254), then we could prove ordirr 4457 without ax-setind 4452. (Contributed by NM, 2-Jan-1994.) |
Theorem | onirri 4458 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | nordeq 4459 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
Theorem | ordn2lp 4460 | 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 4461 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
Theorem | orddif 4462 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Theorem | elirrv 4463 | 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 4464 | 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 4465 | The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
Theorem | ruALT 4466 | Alternate proof of Russell's Paradox ru 2908, simplified using (indirectly) the Axiom of Set Induction ax-setind 4452. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | onprc 4467 | 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 4402), 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 4468 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
Theorem | en2lp 4469 | 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 4470 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Theorem | opthreg 4471 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4452 (via the preleq 4470 step). See df-op 3536 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
Theorem | suc11g 4472 | 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.) |
Theorem | suc11 4473 | The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) |
Theorem | dtruex 4474* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4115 can also be summarized as "at least two sets exist", the difference is that dtruarb 4115 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific , we can construct a set which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | dtru 4475* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4474. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | eunex 4476 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | ordsoexmid 4477 | Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.) |
Theorem | ordsuc 4478 | The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.) |
Theorem | onsucuni2 4479 | A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | 0elsucexmid 4480* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
Theorem | nlimsucg 4481 | A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | ordpwsucss 4482 |
The collection of ordinals in the power class of an ordinal is a
superset of its successor.
We can think of as another possible definition of successor, which would be equivalent to df-suc 4293 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4354) and (onuniss2 4428). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4485). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Theorem | onnmin 4483 | No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.) |
Theorem | ssnel 4484 | Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.) |
Theorem | ordpwsucexmid 4485* | The subset in ordpwsucss 4482 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
Theorem | ordtri2or2exmid 4486* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | onintexmid 4487* | If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.) |
Theorem | zfregfr 4488 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
Theorem | ordfr 4489 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
Theorem | ordwe 4490 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
Theorem | wetriext 4491* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
Theorem | wessep 4492 | A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.) |
Theorem | reg3exmidlemwe 4493* | Lemma for reg3exmid 4494. Our counterexample satisfies . (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | reg3exmid 4494* | If any inhabited set satisfying df-wetr 4256 for has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | dcextest 4495* | If it is decidable whether is a set, then is decidable (where does not occur in ). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.) |
DECID DECID | ||
Theorem | tfi 4496* |
The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring]
p. 39. This principle states that if is a class of ordinal
numbers with the property that every ordinal number included in
also belongs to , then every ordinal number is in .
(Contributed by NM, 18-Feb-2004.) |
Theorem | tfis 4497* | Transfinite Induction Schema. If all ordinal numbers less than a given number have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.) |
Theorem | tfis2f 4498* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis2 4499* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis3 4500* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
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