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Type | Label | Description |
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Statement | ||
Theorem | reuhyp 4401* | A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.) |
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Theorem | uniexb 4402 | The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.) |
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Theorem | pwexb 4403 | The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.) |
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Theorem | elpwpwel 4404 | A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.) |
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Theorem | univ 4405 | The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
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Theorem | eldifpw 4406 | Membership in a power class difference. (Contributed by NM, 25-Mar-2007.) |
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Theorem | op1stb 4407 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.) |
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Theorem | op1stbg 4408 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.) |
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Theorem | iunpw 4409* | 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.) |
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Theorem | ordon 4410 | 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 4411 | 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 4412 | 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 4413 | 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 4414 | The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.) |
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Theorem | onun2i 4415 | 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 4416 | 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 4417 | An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
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Theorem | onuni 4418 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
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Theorem | orduni 4419 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
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Theorem | bm2.5ii 4420* | Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
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Theorem | sucexb 4421 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
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Theorem | sucexg 4422 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
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Theorem | sucex 4423 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
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Theorem | ordsucim 4424 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
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Theorem | suceloni 4425 | The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
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Theorem | ordsucg 4426 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
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Theorem | sucelon 4427 | The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
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Theorem | ordsucss 4428 | 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 4429 | 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 4430 | 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 4431* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
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Theorem | onsucelsucr 4432 |
Membership is inherited by predecessors. The converse, for all ordinals,
implies excluded middle, as shown at onsucelsucexmid 4453. However, the
converse does hold where ![]() |
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Theorem | onsucsssucr 4433 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4450. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
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Theorem | sucunielr 4434 |
Successor and union. The converse (where ![]() |
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Theorem | unon 4435 | 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 4436* | 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 4437 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
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Theorem | ordunisuc2r 4438* | 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 4439 |
An ordinal number is a subset of ![]() |
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Theorem | onsuci 4440 | The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
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Theorem | onintonm 4441* | 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 4442 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | ordtriexmidlem 4443 |
Lemma for decidability and ordinals. The set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ordtriexmidlem2 4444* |
Lemma for decidability and ordinals. The set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ordtriexmid 4445* |
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.) |
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Theorem | ordtri2orexmid 4446* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
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Theorem | 2ordpr 4447 |
Version of 2on 6330 with the definition of ![]() ![]() |
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Theorem | ontr2exmid 4448* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
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Theorem | ordtri2or2exmidlem 4449* |
A set which is ![]() ![]() ![]() ![]() ![]() |
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Theorem | onsucsssucexmid 4450* | The converse of onsucsssucr 4433 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
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Theorem | onsucelsucexmidlem1 4451* | Lemma for onsucelsucexmid 4453. (Contributed by Jim Kingdon, 2-Aug-2019.) |
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Theorem | onsucelsucexmidlem 4452* |
Lemma for onsucelsucexmid 4453. The set
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Theorem | onsucelsucexmid 4453* |
The converse of onsucelsucr 4432 implies excluded middle. On the other
hand, if ![]() |
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Theorem | ordsucunielexmid 4454* |
The converse of sucunielr 4434 (where ![]() |
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Theorem | regexmidlemm 4455* |
Lemma for regexmid 4458. ![]() |
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Theorem | regexmidlem1 4456* |
Lemma for regexmid 4458. If ![]() |
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Theorem | reg2exmidlema 4457* |
Lemma for reg2exmid 4459. If ![]() ![]() |
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Theorem | regexmid 4458* |
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 4460. (Contributed by Jim Kingdon, 3-Sep-2019.) |
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Theorem | reg2exmid 4459* |
If any inhabited set has a minimal element (when expressed by ![]() |
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Axiom | ax-setind 4460* |
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 4461* |
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Theorem | setind 4462* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
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Theorem | setind2 4463 | 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 4464 |
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 4460, 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 4465 | 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 4460. If in the definition of ordinals df-iord 4296, we also required that membership be well-founded on any ordinal (see df-frind 4262), then we could prove ordirr 4465 without ax-setind 4460. (Contributed by NM, 2-Jan-1994.) |
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Theorem | onirri 4466 | 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 4467 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
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Theorem | ordn2lp 4468 | 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 4469 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
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Theorem | orddif 4470 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
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Theorem | elirrv 4471 | 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 4472 | 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 4473 |
The Russell class is equal to the universe ![]() |
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Theorem | ruALT 4474 | Alternate proof of Russell's Paradox ru 2912, simplified using (indirectly) the Axiom of Set Induction ax-setind 4460. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | onprc 4475 | 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 4410), 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 4476 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
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Theorem | en2lp 4477 | 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 4478 | 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 4479 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4460 (via the preleq 4478 step). See df-op 3541 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 4480 | 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 4481 | 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 4482* |
At least two sets exist (or in terms of first-order logic, the universe
of discourse has two or more objects). Although dtruarb 4123 can also be
summarized as "at least two sets exist", the difference is
that
dtruarb 4123 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 4483* | 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 4482. (Contributed by Jim Kingdon, 29-Dec-2018.) |
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Theorem | eunex 4484 | 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 4485 | 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 4486 | 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 4487 | 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 4488* | 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 4489 | 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 4490 |
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 4491 | 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 4492 | 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 4493* | The subset in ordpwsucss 4490 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 4494* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
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Theorem | onintexmid 4495* | 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 4496 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
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Theorem | ordfr 4497 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
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Theorem | ordwe 4498 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
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Theorem | wetriext 4499* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
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Theorem | wessep 4500 | 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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