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Type | Label | Description |
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Statement | ||
Theorem | onintonm 4501* | 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 4502 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | ordtriexmidlem 4503 | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4505 or weak linearity in ordsoexmid 4546) with a proposition . Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmidlem2 4504* | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4505 or weak linearity in ordsoexmid 4546) with a proposition . Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmid 4505* |
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". Also see exmidontri 7216 which is much the same theorem but biconditionalized and using the EXMID notation. (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.) |
Theorem | ontriexmidim 4506* | Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4505. (Contributed by Jim Kingdon, 26-Aug-2024.) |
DECID | ||
Theorem | ordtri2orexmid 4507* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
Theorem | 2ordpr 4508 | Version of 2on 6404 with the definition of expanded and expressed in terms of . (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | ontr2exmid 4509* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
Theorem | ordtri2or2exmidlem 4510* | A set which is if or if is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | onsucsssucexmid 4511* | The converse of onsucsssucr 4493 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | onsucelsucexmidlem1 4512* | Lemma for onsucelsucexmid 4514. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmidlem 4513* | Lemma for onsucelsucexmid 4514. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5844), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4503. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmid 4514* | The converse of onsucelsucr 4492 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 4492 does hold, as seen at nnsucelsuc 6470. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | ordsucunielexmid 4515* | The converse of sucunielr 4494 (where is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | regexmidlemm 4516* | Lemma for regexmid 4519. is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | regexmidlem1 4517* | Lemma for regexmid 4519. If has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmidlema 4518* | Lemma for reg2exmid 4520. If has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Theorem | regexmid 4519* |
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 4521. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmid 4520* | If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Axiom | ax-setind 4521* |
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 4522* | -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.) |
Theorem | setind 4523* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
Theorem | setind2 4524 | 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 4525 |
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 4521, we could redefine (df-iord 4351) to also require (df-frind 4317) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4526 (which under that definition would presumably not need ax-setind 4521 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 4526. To encourage ordirr 4526 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 4526 | 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 4521. If in the definition of ordinals df-iord 4351, we also required that membership be well-founded on any ordinal (see df-frind 4317), then we could prove ordirr 4526 without ax-setind 4521. (Contributed by NM, 2-Jan-1994.) |
Theorem | onirri 4527 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | nordeq 4528 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
Theorem | ordn2lp 4529 | 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 4530 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
Theorem | orddif 4531 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Theorem | elirrv 4532 | 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 4533 | 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 4534 | The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
Theorem | ruALT 4535 | Alternate proof of Russell's Paradox ru 2954, simplified using (indirectly) the Axiom of Set Induction ax-setind 4521. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | onprc 4536 | 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 4470), 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 4537 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
Theorem | en2lp 4538 | 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 4539 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Theorem | opthreg 4540 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4521 (via the preleq 4539 step). See df-op 3592 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
Theorem | suc11g 4541 | 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 4542 | 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 4543* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4177 can also be summarized as "at least two sets exist", the difference is that dtruarb 4177 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 4544* | 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 4543. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | eunex 4545 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | ordsoexmid 4546 | 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 4547 | 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 4548 | 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 4549* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
Theorem | nlimsucg 4550 | A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | ordpwsucss 4551 |
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 4356 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4417) and (onuniss2 4496). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4554). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Theorem | onnmin 4552 | 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 4553 | 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 4554* | The subset in ordpwsucss 4551 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
Theorem | ordtri2or2exmid 4555* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | ontri2orexmidim 4556* | Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4555. (Contributed by Jim Kingdon, 26-Aug-2024.) |
DECID | ||
Theorem | onintexmid 4557* | 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 4558 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
Theorem | ordfr 4559 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
Theorem | ordwe 4560 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
Theorem | wetriext 4561* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
Theorem | wessep 4562 | 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 4563* | Lemma for reg3exmid 4564. Our counterexample satisfies . (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | reg3exmid 4564* | If any inhabited set satisfying df-wetr 4319 for has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | dcextest 4565* | 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 4566* |
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 4567* | 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 4568* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis2 4569* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis3 4570* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
Theorem | tfisi 4571* | 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 4572* | Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
Theorem | zfinf2 4573* | 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 4574 | Extend class notation to include the class of natural numbers. |
Definition | df-iom 4575* |
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 4353. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8879) with analogous properties and operations, but they will be different sets. We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7171. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4576 instead for naming consistency with set.mm. (New usage is discouraged.) |
Theorem | dfom3 4576* | Alias for df-iom 4575. Use it instead of df-iom 4575 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
Theorem | omex 4577 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
Theorem | peano1 4578 | 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 4579 | 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 4580 | 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 4581 | 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 4582* | 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 4587. (Contributed by NM, 18-Feb-2004.) |
Theorem | find 4583* | 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 4584* | 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 4585* | 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 4586* | 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 4587 | 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 4588* | 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 | elomssom 4589 | A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4590. (Revised by BJ, 7-Aug-2024.) |
Theorem | elnn 4590 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
Theorem | ordom 4591 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
Theorem | omelon2 4592 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
Theorem | omelon 4593 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
Theorem | nnon 4594 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Theorem | nnoni 4595 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Theorem | nnord 4596 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
Theorem | omsson 4597 | Omega is a subset of . (Contributed by NM, 13-Jun-1994.) |
Theorem | limom 4598 | 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 4599 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
Theorem | nnsuc 4600* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
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