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

Theoremsucexg 4601 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)

Theoremsucex 4602 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)

Theoremonmindif2 4603 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 4604 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)

Theoremordsuc 4605 The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.)

Theoremordpwsuc 4606 The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005.)

Theoremonpwsuc 4607 The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.)

Theoremsucelon 4608 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)

Theoremordsucss 4609 The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)

Theoremonpsssuc 4610 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremordelsuc 4611 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 4612* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)

Theoremordsucelsuc 4613 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 4614 The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)

Theoremordsucuniel 4615 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 4616 The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.)

Theoremordunpr 4617 The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)

Theoremordunel 4618 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 4619 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 4620 An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.)

Theoremorduniorsuc 4621 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 4622 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)

Theoremordunisuc 4623 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 4624* The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.)

Theoremonsucuni2 4625 A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theorem0elsuc 4626 The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.)

Theoremlimon 4627 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)

Theoremonssi 4628 An ordinal number is a subset of . (Contributed by NM, 11-Aug-1994.)

Theoremonsuci 4629 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)

Theoremonuniorsuci 4630 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 4631* A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)

Theoremonsucssi 4632 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 4633 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremorduninsuc 4634* An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)

Theoremordunisuc2 4635* An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)

Theoremordzsl 4636* An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)

Theoremonzsl 4637* 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 4638* 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 4639* An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)

Theoremlimsuc 4640 The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)

Theoremlimsssuc 4641 A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)

Theoremnlimon 4642* 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 4643* 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 4644* 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 4653 or tfinds 4650 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

Theoremtfis 4645* 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 4646* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)

Theoremtfis2 4647* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)

Theoremtfis3 4648* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)

Theoremtfisi 4649* 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 4650* 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 4651* 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 4652* 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 4653* 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 4654* 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 4655* 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 4656 Extend class notation to include the class of natural numbers.

Definitiondf-om 4657* 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 4658 for an alternate definition. Later, when we assume the Axiom of Infinity, we show is a set in omex 7344, and can then be defined per dfom3 7348 (the smallest inductive set) and dfom4 7350.

Note: the natural numbers are a subset of the ordinal numbers df-on 4396. They are completely different from the natural numbers (df-nn 9747) 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 4658 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 4642). (Contributed by NM, 1-Nov-2004.)

Theoremelom 4659* Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 7349. (Contributed by NM, 15-May-1994.)

Theoremomsson 4660 Omega is a subset of . (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremlimomss 4661 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 4662 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)

Theoremnnoni 4663 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)

Theoremnnord 4664 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)

Theoremordom 4665 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 4666 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)

Theoremomon 4667 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 4668 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)

Theoremnnlim 4669 A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)

Theoremomssnlim 4670 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 4671 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 4672 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)

Theoremnnsuc 4673* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)

Theoremssnlim 4674* 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 4675 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 4675 through peano5 4679 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 4676 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 4677 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 4678 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 4679* 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 4686. (Contributed by NM, 18-Feb-2004.)

Theoremnn0suc 4680* A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)

2.4.6  Finite induction (for finite ordinals)

Theoremfind 4681* 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 4682* 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 4683* 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 4684* 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.)

Theoremfinds1 4685* 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, 22-Mar-2006.)

Theoremfindes 4686 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 4653 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)

2.4.7  Relations

Syntaxcxp 4687 Extend the definition of a class to include the cross product.

Syntaxccnv 4688 Extend the definition of a class to include the converse of a class.

Syntaxcdm 4689 Extend the definition of a class to include the domain of a class.

Syntaxcrn 4690 Extend the definition of a class to include the range of a class.

Syntaxcres 4691 Extend the definition of a class to include the restriction of a class. (Read: The restriction of to .)

Syntaxcima 4692 Extend the definition of a class to include the image of a class. (Read: The image of under .)

Syntaxccom 4693 Extend the definition of a class to include the composition of two classes. (Read: The composition of and .)

Syntaxwrel 4694 Extend the definition of a wff to include the relation predicate. (Read: is a relation.)

Definitiondf-xp 4695* Define the cross product of two classes. Definition 9.11 of [Quine] p. 64. For example, (ex-xp 20823). Another example is that the set of rational numbers are defined in df-q 10317 using the cross-product ; the left- and right-hand sides of the cross-product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)

Definitiondf-rel 4696 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5124 and dfrel3 5131. (Contributed by NM, 1-Aug-1994.)

Definitiondf-cnv 4697* Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if and then , as proven in brcnv 4864 (see df-br 4024 and df-rel 4696 for more on relations). For example, (ex-cnv 20824). We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)

Definitiondf-co 4698* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, (ex-co 20825) because (see cos0 12430) and (see df-e 12350). Note that Definition 7 of [Suppes] p. 63 reverses and , uses instead of , and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.)

Definitiondf-dm 4699* Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, (ex-dm 20826). Another example is the domain of the complex arctangent, arctan (for proof see atandm 20172). Contrast with range (defined in df-rn 4700). For alternate definitions see dfdm2 5204, dfdm3 4867, and dfdm4 4872. The notation " " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)

Definitiondf-rn 4700 Define the range of a class. For example, (ex-rn 20827). Contrast with domain (defined in df-dm 4699). For alternate definitions, see dfrn2 4868, dfrn3 4869, and dfrn4 5134. The notation " " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)

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