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Type | Label | Description |
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Statement | ||
Theorem | onssi 4401 | An ordinal number is a subset of . (Contributed by NM, 11-Aug-1994.) |
Theorem | onsuci 4402 | 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 4403* | 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 4404 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | ordtriexmidlem 4405 | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4407 or weak linearity in ordsoexmid 4447) with a proposition . Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmidlem2 4406* | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4407 or weak linearity in ordsoexmid 4447) with a proposition . Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmid 4407* |
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 4408* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
Theorem | 2ordpr 4409 | Version of 2on 6290 with the definition of expanded and expressed in terms of . (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | ontr2exmid 4410* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
Theorem | ordtri2or2exmidlem 4411* | A set which is if or if is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | onsucsssucexmid 4412* | The converse of onsucsssucr 4395 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | onsucelsucexmidlem1 4413* | Lemma for onsucelsucexmid 4415. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmidlem 4414* | Lemma for onsucelsucexmid 4415. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5733), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4405. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmid 4415* | The converse of onsucelsucr 4394 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 4394 does hold, as seen at nnsucelsuc 6355. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | ordsucunielexmid 4416* | The converse of sucunielr 4396 (where is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | regexmidlemm 4417* | Lemma for regexmid 4420. is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | regexmidlem1 4418* | Lemma for regexmid 4420. If has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmidlema 4419* | Lemma for reg2exmid 4421. If has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Theorem | regexmid 4420* |
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 4422. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmid 4421* | If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Axiom | ax-setind 4422* |
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 4423* | -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.) |
Theorem | setind 4424* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
Theorem | setind2 4425 | 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 4426 |
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 4422, we could redefine (df-iord 4258) to also require (df-frind 4224) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4427 (which under that definition would presumably not need ax-setind 4422 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 4427. To encourage ordirr 4427 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 4427 | 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 4422. If in the definition of ordinals df-iord 4258, we also required that membership be well-founded on any ordinal (see df-frind 4224), then we could prove ordirr 4427 without ax-setind 4422. (Contributed by NM, 2-Jan-1994.) |
Theorem | onirri 4428 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | nordeq 4429 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
Theorem | ordn2lp 4430 | 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 4431 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
Theorem | orddif 4432 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Theorem | elirrv 4433 | 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 4434 | 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 4435 | The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
Theorem | ruALT 4436 | Alternate proof of Russell's Paradox ru 2881, simplified using (indirectly) the Axiom of Set Induction ax-setind 4422. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | onprc 4437 | 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 4372), 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 4438 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
Theorem | en2lp 4439 | 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 4440 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Theorem | opthreg 4441 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4422 (via the preleq 4440 step). See df-op 3506 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
Theorem | suc11g 4442 | 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 4443 | 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 4444* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4085 can also be summarized as "at least two sets exist", the difference is that dtruarb 4085 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 4445* | 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 4444. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | eunex 4446 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | ordsoexmid 4447 | 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 4448 | 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 4449 | 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 4450* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
Theorem | nlimsucg 4451 | A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | ordpwsucss 4452 |
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 4263 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4324) and (onuniss2 4398). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4455). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Theorem | onnmin 4453 | 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 4454 | 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 4455* | The subset in ordpwsucss 4452 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
Theorem | ordtri2or2exmid 4456* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | onintexmid 4457* | 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 4458 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
Theorem | ordfr 4459 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
Theorem | ordwe 4460 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
Theorem | wetriext 4461* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
Theorem | wessep 4462 | 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 4463* | Lemma for reg3exmid 4464. Our counterexample satisfies . (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | reg3exmid 4464* | If any inhabited set satisfying df-wetr 4226 for has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | dcextest 4465* | 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 4466* |
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 4467* | 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 4468* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis2 4469* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis3 4470* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
Theorem | tfisi 4471* | A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
Axiom | ax-iinf 4472* | Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
Theorem | zfinf2 4473* | A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.) |
Syntax | com 4474 | Extend class notation to include the class of natural numbers. |
Definition | df-iom 4475* |
Define the class of natural numbers as the smallest inductive set, which
is valid provided we assume the Axiom of Infinity. Definition 6.3 of
[Eisenberg] p. 82.
Note: the natural numbers are a subset of the ordinal numbers df-on 4260. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers with analogous properties and operations, but they will be different sets. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4476 instead for naming consistency with set.mm. (New usage is discouraged.) |
Theorem | dfom3 4476* | Alias for df-iom 4475. Use it instead of df-iom 4475 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
Theorem | omex 4477 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
Theorem | peano1 4478 | Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.) |
Theorem | peano2 4479 | The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
Theorem | peano3 4480 | The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
Theorem | peano4 4481 | Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.) |
Theorem | peano5 4482* | The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as theorem findes 4487. (Contributed by NM, 18-Feb-2004.) |
Theorem | find 4483* | The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that is a set of natural numbers, zero belongs to , and given any member of the member's successor also belongs to . The conclusion is that every natural number is in . (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | finds 4484* | Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.) |
Theorem | finds2 4485* | Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.) |
Theorem | finds1 4486* | Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.) |
Theorem | findes 4487 | Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.) |
Theorem | nn0suc 4488* | A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.) |
Theorem | elnn 4489 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
Theorem | ordom 4490 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
Theorem | omelon2 4491 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
Theorem | omelon 4492 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
Theorem | nnon 4493 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Theorem | nnoni 4494 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Theorem | nnord 4495 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
Theorem | omsson 4496 | Omega is a subset of . (Contributed by NM, 13-Jun-1994.) |
Theorem | limom 4497 | Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.) |
Theorem | peano2b 4498 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
Theorem | nnsuc 4499* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
Theorem | nnsucpred 4500 | The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.) |
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