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Theorem List for Metamath Proof Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremorduni 4601 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)

Theoremonint 4602 The intersection (infimum) of a non-empty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)

Theoremonint0 4603 The intersection of a class of ordinal numbers is zero iff the class contains zero. (Contributed by NM, 24-Apr-2004.)

Theoremonssmin 4604* A non-empty class of ordinal numbers has the smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.)

Theoremonminesb 4605 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 29-Sep-2003.)

Theoremonminsb 4606 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)

Theoremoninton 4607 The intersection of a non-empty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by NM, 29-Jan-1997.)

Theoremonintrab 4608 The intersection of a class of ordinal numbers exists iff it is an ordinal number. (Contributed by NM, 6-Nov-2003.)

Theoremonintrab2 4609 An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.)

Theoremonnmin 4610 No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)

Theoremonnminsb 4611* An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. is the wff resulting from the substitution of for in wff . (Contributed by NM, 9-Nov-2003.)

Theoremoneqmin 4612* A way to show that an ordinal number equals the minimum of a non-empty collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)

Theorembm2.5ii 4613* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)

Theoremonminex 4614* If a wff is true for an ordinal number, there is the smallest ordinal number for which it is true. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Mario Carneiro, 20-Nov-2016.)

Theoremsucon 4615 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)

Theoremsucexb 4616 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)

Theoremsucexg 4617 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)

Theoremsucex 4618 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)

Theoremonmindif2 4619 The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed. (Contributed by NM, 20-Nov-2003.)

Theoremsuceloni 4620 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)

Theoremordsuc 4621 The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.)

Theoremordpwsuc 4622 The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005.)

Theoremonpwsuc 4623 The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.)

Theoremsucelon 4624 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)

Theoremordsucss 4625 The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)

Theoremonpsssuc 4626 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremordelsuc 4627 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.)

Theoremonsucmin 4628* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)

Theoremordsucelsuc 4629 Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (Contributed by NM, 22-Jun-1998.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremordsucsssuc 4630 The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)

Theoremordsucuniel 4631 Given an element of the union of an ordinal , is an element of itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)

Theoremordsucun 4632 The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.)

Theoremordunpr 4633 The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)

Theoremordunel 4634 The maximum of two ordinals belongs to a third if each of them do. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremonsucuni 4635 A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.)

Theoremordsucuni 4636 An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.)

Theoremorduniorsuc 4637 An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.)

Theoremunon 4638 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)

Theoremordunisuc 4639 An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremorduniss2 4640* The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.)

Theoremonsucuni2 4641 A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theorem0elsuc 4642 The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.)

Theoremlimon 4643 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)

Theoremonssi 4644 An ordinal number is a subset of . (Contributed by NM, 11-Aug-1994.)

Theoremonsuci 4645 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)

Theoremonuniorsuci 4646 An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994.)

Theoremonuninsuci 4647* A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)

Theoremonsucssi 4648 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.)

Theoremnlimsucg 4649 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremorduninsuc 4650* An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)

Theoremordunisuc2 4651* An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)

Theoremordzsl 4652* An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)

Theoremonzsl 4653* An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdflim3 4654* An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdflim4 4655* An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)

Theoremlimsuc 4656 The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)

Theoremlimsssuc 4657 A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)

Theoremnlimon 4658* Two ways to express the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol KI for this class. (Contributed by NM, 1-Nov-2004.)

Theoremlimuni3 4659* The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)

2.4.3  Transfinite induction

Theoremtfi 4660* 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 .

See theorem tfindes 4669 or tfinds 4666 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

Theoremtfis 4661* 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.)

Theoremtfis2f 4662* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)

Theoremtfis2 4663* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)

Theoremtfis3 4664* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)

Theoremtfisi 4665* 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.)

Theoremtfinds 4666* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremtfindsg 4667* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. The basis of this version is an arbitrary ordinal instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.)

Theoremtfindsg2 4668* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. The basis of this version is an arbitrary ordinal instead of zero. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 5-Jan-2005.)

Theoremtfindes 4669* Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction hypothesis for successors, and the third is the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)

Theoremtfinds2 4670* Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)

Theoremtfinds3 4671* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)

2.4.4  The natural numbers (i.e. finite ordinals)

Syntaxcom 4672 Extend class notation to include the class of natural numbers.

Definitiondf-om 4673* Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e. all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 4674 for an alternate definition. Later, when we assume the Axiom of Infinity, we show is a set in omex 7360, and can then be defined per dfom3 7364 (the smallest inductive set) and dfom4 7366.

Note: the natural numbers are a subset of the ordinal numbers df-on 4412. They are completely different from the natural numbers (df-nn 9763) that are a subset of the complex numbers defined much later in our development, although the two sets have analogous properties and operations defined on them. (Contributed by NM, 15-May-1994.)

Theoremdfom2 4674 An alternate definition of the set of natural numbers . Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the inner class builder of non-limit ordinal numbers (see nlimon 4658). (Contributed by NM, 1-Nov-2004.)

Theoremelom 4675* Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 7365. (Contributed by NM, 15-May-1994.)

Theoremomsson 4676 Omega is a subset of . (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremlimomss 4677 The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.)

Theoremnnon 4678 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)

Theoremnnoni 4679 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)

Theoremnnord 4680 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)

Theoremordom 4681 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelnn 4682 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)

Theoremomon 4683 The class of natural numbers is either an ordinal number (if we accept the Axiom of Infinity) or the proper class of all ordinal numbers (if we deny the Axiom of Infinity). Remark in [TakeutiZaring] p. 43. (Contributed by NM, 10-May-1998.)

Theoremomelon2 4684 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)

Theoremnnlim 4685 A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)

Theoremomssnlim 4686 The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremlimom 4687 Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Our proof, however, does not require the Axiom of Infinity. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theorempeano2b 4688 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)

Theoremnnsuc 4689* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)

Theoremssnlim 4690* An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.)

2.4.5  Peano's postulates

Theorempeano1 4691 Zero is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 4691 through peano5 4695 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.)

Theorempeano2 4692 The successor of any natural number is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)

Theorempeano3 4693 The successor of any natural number is not zero. One of Peano's 5 postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)

Theorempeano4 4694 Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's 5 postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)

Theorempeano5 4695* The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's 5 postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction hypothesis, is derived from this theorem as theorem findes 4702. (Contributed by NM, 18-Feb-2004.)

Theoremnn0suc 4696* A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)

2.4.6  Finite induction (for finite ordinals)

Theoremfind 4697* 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.)

Theoremfinds 4698* 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 hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 14-Apr-1995.)

Theoremfindsg 4699* 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 hypothesis. The basis of this version is an arbitrary natural number instead of zero. (Contributed by NM, 16-Sep-1995.)

Theoremfinds2 4700* 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 hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)

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