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Type | Label | Description |
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Statement | ||
Theorem | onsuc 4501 | The successor of an ordinal number is an ordinal number. Closed form of onsuci 4516. Forward implication of onsucb 4503. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
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Theorem | ordsucg 4502 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
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Theorem | onsucb 4503 | A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4501. (Contributed by NM, 9-Sep-2003.) |
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Theorem | ordsucss 4504 | 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 4505 | 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 4506 | 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 4507* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
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Theorem | onsucelsucr 4508 |
Membership is inherited by predecessors. The converse, for all ordinals,
implies excluded middle, as shown at onsucelsucexmid 4530. However, the
converse does hold where ![]() |
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Theorem | onsucsssucr 4509 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4527. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
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Theorem | sucunielr 4510 |
Successor and union. The converse (where ![]() |
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Theorem | unon 4511 | 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 4512* | 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 4513 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
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Theorem | ordunisuc2r 4514* | 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 4515 |
An ordinal number is a subset of ![]() |
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Theorem | onsuci 4516 | The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4501 and onsucb 4503. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
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Theorem | onintonm 4517* | 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 4518 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | ordtriexmidlem 4519 |
Lemma for decidability and ordinals. The set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ordtriexmidlem2 4520* |
Lemma for decidability and ordinals. The set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ordtriexmid 4521* |
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 7238 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 4522* | Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4521. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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Theorem | ordtri2orexmid 4523* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
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Theorem | 2ordpr 4524 |
Version of 2on 6426 with the definition of ![]() ![]() |
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Theorem | ontr2exmid 4525* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
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Theorem | ordtri2or2exmidlem 4526* |
A set which is ![]() ![]() ![]() ![]() ![]() |
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Theorem | onsucsssucexmid 4527* | The converse of onsucsssucr 4509 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
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Theorem | onsucelsucexmidlem1 4528* | Lemma for onsucelsucexmid 4530. (Contributed by Jim Kingdon, 2-Aug-2019.) |
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Theorem | onsucelsucexmidlem 4529* |
Lemma for onsucelsucexmid 4530. The set
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Theorem | onsucelsucexmid 4530* |
The converse of onsucelsucr 4508 implies excluded middle. On the other
hand, if ![]() |
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Theorem | ordsucunielexmid 4531* |
The converse of sucunielr 4510 (where ![]() |
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Theorem | regexmidlemm 4532* |
Lemma for regexmid 4535. ![]() |
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Theorem | regexmidlem1 4533* |
Lemma for regexmid 4535. If ![]() |
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Theorem | reg2exmidlema 4534* |
Lemma for reg2exmid 4536. If ![]() ![]() |
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Theorem | regexmid 4535* |
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 4537. (Contributed by Jim Kingdon, 3-Sep-2019.) |
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Theorem | reg2exmid 4536* |
If any inhabited set has a minimal element (when expressed by ![]() |
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Axiom | ax-setind 4537* |
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 4538* |
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Theorem | setind 4539* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
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Theorem | setind2 4540 | 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 4541 |
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 4537, 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 4542 | 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 4537. If in the definition of ordinals df-iord 4367, we also required that membership be well-founded on any ordinal (see df-frind 4333), then we could prove ordirr 4542 without ax-setind 4537. (Contributed by NM, 2-Jan-1994.) |
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Theorem | onirri 4543 | 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 4544 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
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Theorem | ordn2lp 4545 | 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 4546 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
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Theorem | orddif 4547 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
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Theorem | elirrv 4548 | 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 4549 | 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 4550 |
The Russell class is equal to the universe ![]() |
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Theorem | ruALT 4551 | Alternate proof of Russell's Paradox ru 2962, simplified using (indirectly) the Axiom of Set Induction ax-setind 4537. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | onprc 4552 | 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 4486), 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 4553 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
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Theorem | en2lp 4554 | 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 4555 | 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 4556 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4537 (via the preleq 4555 step). See df-op 3602 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 4557 | 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 4558 | 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 4559* |
At least two sets exist (or in terms of first-order logic, the universe
of discourse has two or more objects). Although dtruarb 4192 can also be
summarized as "at least two sets exist", the difference is
that
dtruarb 4192 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 4560* | 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 4559. (Contributed by Jim Kingdon, 29-Dec-2018.) |
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Theorem | eunex 4561 | 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 4562 | 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 4563 | 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 4564 | 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 4565* | 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 4566 | 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 4567 |
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 4568 | 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 4569 | 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 4570* | The subset in ordpwsucss 4567 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 4571* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
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Theorem | ontri2orexmidim 4572* | Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4571. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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Theorem | onintexmid 4573* | 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 4574 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
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Theorem | ordfr 4575 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
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Theorem | ordwe 4576 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
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Theorem | wetriext 4577* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
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Theorem | wessep 4578 | 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 4579* |
Lemma for reg3exmid 4580. Our counterexample ![]() ![]() |
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Theorem | reg3exmid 4580* |
If any inhabited set satisfying df-wetr 4335 for ![]() |
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Theorem | dcextest 4581* |
If it is decidable whether ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tfi 4582* |
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 4583* |
Transfinite Induction Schema. If all ordinal numbers less than a given
number ![]() |
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Theorem | tfis2f 4584* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
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Theorem | tfis2 4585* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
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Theorem | tfis3 4586* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
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Theorem | tfisi 4587* | 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.) |
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Axiom | ax-iinf 4588* | Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
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Theorem | zfinf2 4589* | A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.) |
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Syntax | com 4590 | Extend class notation to include the class of natural numbers. |
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Definition | df-iom 4591* |
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 We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7193. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4592 instead for naming consistency with set.mm. (New usage is discouraged.) |
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Theorem | dfom3 4592* | Alias for df-iom 4591. Use it instead of df-iom 4591 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
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Theorem | omex 4593 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
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Theorem | peano1 4594 | 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.) |
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Theorem | peano2 4595 | 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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Theorem | peano3 4596 | 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.) |
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Theorem | peano4 4597 | 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.) |
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Theorem | peano5 4598* | 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 4603. (Contributed by NM, 18-Feb-2004.) |
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Theorem | find 4599* |
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 ![]() ![]() ![]() ![]() ![]() |
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Theorem | finds 4600* | 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.) |
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