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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | onuni 4301 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
Theorem | orduni 4302 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
Theorem | bm2.5ii 4303* | Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
Theorem | sucexb 4304 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
Theorem | sucexg 4305 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
Theorem | sucex 4306 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
Theorem | ordsucim 4307 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
Theorem | suceloni 4308 | 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 4309 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
Theorem | sucelon 4310 | The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
Theorem | ordsucss 4311 | The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
Theorem | ordelsuc 4312 | 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 4313 | 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 4314* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
Theorem | onsucelsucr 4315 | Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4336. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6234. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Theorem | onsucsssucr 4316 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4333. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | sucunielr 4317 | Successor and union. The converse (where is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4337. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | unon 4318 | The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) |
Theorem | onuniss2 4319* | The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | limon 4320 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
Theorem | ordunisuc2r 4321* | 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 4322 | An ordinal number is a subset of . (Contributed by NM, 11-Aug-1994.) |
Theorem | onsuci 4323 | 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 4324* | 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 4325 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | ordtriexmidlem 4326 | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4328 or weak linearity in ordsoexmid 4368) with a proposition . Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmidlem2 4327* | Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4328 or weak linearity in ordsoexmid 4368) with a proposition . Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Theorem | ordtriexmid 4328* |
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 4329* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
Theorem | 2ordpr 4330 | Version of 2on 6172 with the definition of expanded and expressed in terms of . (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | ontr2exmid 4331* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
Theorem | ordtri2or2exmidlem 4332* | A set which is if or if is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | onsucsssucexmid 4333* | The converse of onsucsssucr 4316 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Theorem | onsucelsucexmidlem1 4334* | Lemma for onsucelsucexmid 4336. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmidlem 4335* | Lemma for onsucelsucexmid 4336. The set appears as in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5625), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4326. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | onsucelsucexmid 4336* | The converse of onsucelsucr 4315 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 4315 does hold, as seen at nnsucelsuc 6234. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | ordsucunielexmid 4337* | The converse of sucunielr 4317 (where is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Theorem | regexmidlemm 4338* | Lemma for regexmid 4341. is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | regexmidlem1 4339* | Lemma for regexmid 4341. If has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmidlema 4340* | Lemma for reg2exmid 4342. If has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Theorem | regexmid 4341* |
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 4343. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Theorem | reg2exmid 4342* | If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Axiom | ax-setind 4343* |
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 4344* | -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.) |
Theorem | setind 4345* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
Theorem | setind2 4346 | 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 4347 |
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 4343, we could redefine (df-iord 4184) to also require (df-frind 4150) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4348 (which under that definition would presumably not need ax-setind 4343 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 4348. To encourage ordirr 4348 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 4348 | 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 4343. If in the definition of ordinals df-iord 4184, we also required that membership be well-founded on any ordinal (see df-frind 4150), then we could prove ordirr 4348 without ax-setind 4343. (Contributed by NM, 2-Jan-1994.) |
Theorem | onirri 4349 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | nordeq 4350 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
Theorem | ordn2lp 4351 | 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 4352 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
Theorem | orddif 4353 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Theorem | elirrv 4354 | 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 4355 | 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 4356 | The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
Theorem | ruALT 4357 | Alternate proof of Russell's Paradox ru 2837, simplified using (indirectly) the Axiom of Set Induction ax-setind 4343. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | onprc 4358 | 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 4293), 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 4359 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
Theorem | en2lp 4360 | 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 4361 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Theorem | opthreg 4362 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4343 (via the preleq 4361 step). See df-op 3450 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
Theorem | suc11g 4363 | 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 4364 | 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 4365* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4017 can also be summarized as "at least two sets exist", the difference is that dtruarb 4017 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 4366* | 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 4365. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | eunex 4367 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | ordsoexmid 4368 | 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 4369 | 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 4370 | 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 4371* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
Theorem | nlimsucg 4372 | A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | ordpwsucss 4373 |
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 4189 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4250) and (onuniss2 4319). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4376). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Theorem | onnmin 4374 | 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 4375 | 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 4376* | The subset in ordpwsucss 4373 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
Theorem | ordtri2or2exmid 4377* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | onintexmid 4378* | 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 4379 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
Theorem | ordfr 4380 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
Theorem | ordwe 4381 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
Theorem | wetriext 4382* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
Theorem | wessep 4383 | 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 4384* | Lemma for reg3exmid 4385. Our counterexample satisfies . (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | reg3exmid 4385* | If any inhabited set satisfying df-wetr 4152 for has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | dcextest 4386* | 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 4387* |
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 4388* | 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 4389* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis2 4390* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis3 4391* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
Theorem | tfisi 4392* | 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 4393* | Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
Theorem | zfinf2 4394* | 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 4395 | Extend class notation to include the class of natural numbers. |
Definition | df-iom 4396* |
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 4186. 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 4397 instead for naming consistency with set.mm. (New usage is discouraged.) |
Theorem | dfom3 4397* | Alias for df-iom 4396. Use it instead of df-iom 4396 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
Theorem | omex 4398 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
Theorem | peano1 4399 | 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 4400 | 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.) |
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