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Type | Label | Description |
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Statement | ||
Theorem | ifelpwun 4501 | Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.) |
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Theorem | ifexd 4502 | Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.) |
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Theorem | ordon 4503 | 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.) |
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Theorem | ssorduni 4504 | 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.) |
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Theorem | ssonuni 4505 | The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.) |
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Theorem | ssonunii 4506 | 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.) |
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Theorem | onun2 4507 | The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.) |
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Theorem | onun2i 4508 | The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.) |
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Theorem | ordsson 4509 | 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.) |
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Theorem | onss 4510 | An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
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Theorem | onuni 4511 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
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Theorem | orduni 4512 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
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Theorem | bm2.5ii 4513* | Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
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Theorem | sucexb 4514 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
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Theorem | sucexg 4515 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
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Theorem | sucex 4516 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
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Theorem | ordsucim 4517 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
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Theorem | onsuc 4518 | The successor of an ordinal number is an ordinal number. Closed form of onsuci 4533. Forward implication of onsucb 4520. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
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Theorem | ordsucg 4519 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
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Theorem | onsucb 4520 | A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4518. (Contributed by NM, 9-Sep-2003.) |
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Theorem | ordsucss 4521 | The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
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Theorem | ordelsuc 4522 | 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.) |
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Theorem | onsucssi 4523 | 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.) |
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Theorem | onsucmin 4524* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
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Theorem | onsucelsucr 4525 |
Membership is inherited by predecessors. The converse, for all ordinals,
implies excluded middle, as shown at onsucelsucexmid 4547. However, the
converse does hold where ![]() |
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Theorem | onsucsssucr 4526 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4544. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
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Theorem | sucunielr 4527 |
Successor and union. The converse (where ![]() |
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Theorem | unon 4528 | The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) |
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Theorem | onuniss2 4529* | The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.) |
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Theorem | limon 4530 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
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Theorem | ordunisuc2r 4531* | An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.) |
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Theorem | onssi 4532 |
An ordinal number is a subset of ![]() |
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Theorem | onsuci 4533 | The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4518 and onsucb 4520. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
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Theorem | onintonm 4534* | 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.) |
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Theorem | onintrab2im 4535 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | ordtriexmidlem 4536 |
Lemma for decidability and ordinals. The set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ordtriexmidlem2 4537* |
Lemma for decidability and ordinals. The set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ordtriexmid 4538* |
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 7269 which is much the same theorem but biconditionalized and using the EXMID notation. (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.) |
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Theorem | ontriexmidim 4539* | Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4538. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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Theorem | ordtri2orexmid 4540* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
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Theorem | 2ordpr 4541 |
Version of 2on 6451 with the definition of ![]() ![]() |
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Theorem | ontr2exmid 4542* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
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Theorem | ordtri2or2exmidlem 4543* |
A set which is ![]() ![]() ![]() ![]() ![]() |
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Theorem | onsucsssucexmid 4544* | The converse of onsucsssucr 4526 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
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Theorem | onsucelsucexmidlem1 4545* | Lemma for onsucelsucexmid 4547. (Contributed by Jim Kingdon, 2-Aug-2019.) |
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Theorem | onsucelsucexmidlem 4546* |
Lemma for onsucelsucexmid 4547. The set
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Theorem | onsucelsucexmid 4547* |
The converse of onsucelsucr 4525 implies excluded middle. On the other
hand, if ![]() |
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Theorem | ordsucunielexmid 4548* |
The converse of sucunielr 4527 (where ![]() |
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Theorem | regexmidlemm 4549* |
Lemma for regexmid 4552. ![]() |
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Theorem | regexmidlem1 4550* |
Lemma for regexmid 4552. If ![]() |
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Theorem | reg2exmidlema 4551* |
Lemma for reg2exmid 4553. If ![]() ![]() |
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Theorem | regexmid 4552* |
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
For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4554. (Contributed by Jim Kingdon, 3-Sep-2019.) |
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Theorem | reg2exmid 4553* |
If any inhabited set has a minimal element (when expressed by ![]() |
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Axiom | ax-setind 4554* |
Axiom of ![]() 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.) |
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Theorem | setindel 4555* |
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Theorem | setind 4556* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
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Theorem | setind2 4557 | Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.) |
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Theorem | elirr 4558 |
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 4554, we could redefine
(Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.) |
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Theorem | ordirr 4559 | 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 4554. If in the definition of ordinals df-iord 4384, we also required that membership be well-founded on any ordinal (see df-frind 4350), then we could prove ordirr 4559 without ax-setind 4554. (Contributed by NM, 2-Jan-1994.) |
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Theorem | onirri 4560 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
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Theorem | nordeq 4561 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
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Theorem | ordn2lp 4562 | 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.) |
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Theorem | orddisj 4563 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
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Theorem | orddif 4564 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
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Theorem | elirrv 4565 | The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.) |
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Theorem | sucprcreg 4566 | 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.) |
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Theorem | ruv 4567 |
The Russell class is equal to the universe ![]() |
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Theorem | ruALT 4568 | Alternate proof of Russell's Paradox ru 2976, simplified using (indirectly) the Axiom of Set Induction ax-setind 4554. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | onprc 4569 | 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 4503), 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.) |
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Theorem | sucon 4570 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
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Theorem | en2lp 4571 | 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.) |
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Theorem | preleq 4572 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
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Theorem | opthreg 4573 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4554 (via the preleq 4572 step). See df-op 3616 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
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Theorem | suc11g 4574 | 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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Theorem | suc11 4575 | The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) |
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Theorem | dtruex 4576* |
At least two sets exist (or in terms of first-order logic, the universe
of discourse has two or more objects). Although dtruarb 4209 can also be
summarized as "at least two sets exist", the difference is
that
dtruarb 4209 shows the existence of two sets which are not
equal to each
other, but this theorem says that given a specific ![]() ![]() |
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Theorem | dtru 4577* | 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 4576. (Contributed by Jim Kingdon, 29-Dec-2018.) |
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Theorem | eunex 4578 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
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Theorem | ordsoexmid 4579 | 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.) |
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Theorem | ordsuc 4580 | 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.) |
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Theorem | onsucuni2 4581 | A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | 0elsucexmid 4582* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
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Theorem | nlimsucg 4583 | A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | ordpwsucss 4584 |
The collection of ordinals in the power class of an ordinal is a
superset of its successor.
We can think of
Constructively |
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Theorem | onnmin 4585 | 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.) |
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Theorem | ssnel 4586 | 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.) |
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Theorem | ordpwsucexmid 4587* | The subset in ordpwsucss 4584 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
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Theorem | ordtri2or2exmid 4588* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
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Theorem | ontri2orexmidim 4589* | Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4588. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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Theorem | onintexmid 4590* | 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.) |
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Theorem | zfregfr 4591 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
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Theorem | ordfr 4592 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
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Theorem | ordwe 4593 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
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Theorem | wetriext 4594* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
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Theorem | wessep 4595 | A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.) |
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Theorem | reg3exmidlemwe 4596* |
Lemma for reg3exmid 4597. Our counterexample ![]() ![]() |
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Theorem | reg3exmid 4597* |
If any inhabited set satisfying df-wetr 4352 for ![]() |
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Theorem | dcextest 4598* |
If it is decidable whether ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tfi 4599* |
The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring]
p. 39. This principle states that if ![]() ![]() ![]() ![]() (Contributed by NM, 18-Feb-2004.) |
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Theorem | tfis 4600* |
Transfinite Induction Schema. If all ordinal numbers less than a given
number ![]() |
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