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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremordunel 4101 The maximum of two ordinals belongs to a third if each of them do.

Theoremonsucuni 4102 A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41.

Theoremordsucuni 4103 An ordinal class is a subclass of the successor of its union.

Theoremorduniorsuc 4104 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.

Theoremunon 4105 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40.

Theoremordunisuc 4106 An ordinal class is equal to the union of its successor. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremorduniss2 4107* The union of the ordinal subsets of an ordinal number is that number.

Theoremonsucuni2 4108 A successor ordinal is the successor of its union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theorem0elsuc 4109 The successor of an ordinal class contains the empty set.

Theoremlimon 4110 The class of ordinal numbers is a limit ordinal.

Theoremonssi 4111 An ordinal number is a subset of .

Theoremonsuci 4112 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193.

Theoremonuniorsuci 4113 An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union.

Theoremonuninsuci 4114* A limit ordinal is not a successor ordinal.

Theoremonsucssi 4115 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.

Theoremnlimsucg 4116 A successor is not a limit ordinal. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremorduninsuc 4117* An ordinal equal to its union is not a successor.

Theoremordunisuc2 4118* An ordinal equal to its union contains the successor of each of its members.

Theoremordzsl 4119* An ordinal is zero, a successor ordinal, or a limit ordinal.

Theoremonzsl 4120* An ordinal number is zero, a successor ordinal, or a limit ordinal number. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdflim3 4121* An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdflim4 4122* An alternate definition of a limit ordinal.

Theoremlimsuc 4123 The successor of a member of a limit ordinal is also a member.

Theoremlimsssuc 4124 A class includes a limit ordinal iff the successor of the class includes it.

Theoremnlimon 4125* 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.

Theoremlimuni3 4126* The union of a nonempty class of limit ordinals is a limit ordinal.

2.4.3  Transfinite induction

Theoremtfi 4127* 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 4135 or tfinds 4132 for the version involving basis and induction hypotheses.

Theoremtfis 4128* 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.

Theoremtfis2f 4129* Transfinite Induction Schema, using implicit substitition.

Theoremtfis2 4130* Transfinite Induction Schema, using implicit substitition.

Theoremtfis3 4131* Transfinite Induction Schema, using implicit substitition.

Theoremtfinds 4132* 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. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremtfindsg 4133* Transfinite Induction (inference schema), using implicit substititions. 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.

Theoremtfindsg2 4134* Transfinite Induction (inference schema), using implicit substititions. 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.)

Theoremtfindes 4135* 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.

Theoremtfinds2 4136* Transfinite Induction (inference schema), using implicit substititions. 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.

Theoremtfinds3 4137* 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. (Unnecessary distinct variable restrictions were removed by David Abernethy, 21-Jun-2011.)

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

Syntaxcom 4138 Extend class notation to include the class of natural numbers.

Definitiondf-om 4139* 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 4140 for an alternate definition. Later, when we assume the Axiom of Infinity, we show is a set in omex 6609, and can then be defined per dfom3 6613 (the smallest inductive set) and dfom4 6615.

Note: the natural numbers are a subset of the ordinal numbers df-on 3859. They are completely different from the natural numbers (df-n 8625) 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.

Theoremdfom2 4140 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 4125).

Theoremelom 4141* Membership in omega. The hypothesis can be eliminated if we assume the Axiom of Infinity; see elom3 6614.

Theoremelomg 4142* Membership in omega. The antecedent can be eliminated if we assume the Axiom of Infinity; see elom3 6614.

Theoremomsson 4143 Omega is a subset of . (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremlimomss 4144 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.

Theoremnnon 4145 A natural number is an ordinal number.

Theoremnnoni 4146 A natural number is an ordinal number.

Theoremnnord 4147 A natural number is ordinal.

Theoremordom 4148 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelnn 4149 A member of a natural number is a natural number.

Theoremomon 4150 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.

Theoremomelon2 4151 Omega is an ordinal number. (Contributed by Mario Carneiro, 9-Feb-2013.)

Theoremnnlim 4152 A natural number is not a limit ordinal.

Theoremomssnlim 4153 The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremlimom 4154 Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Our proof, however, does not require the Axiom of Infinity.

Theorempeano2b 4155 A class belongs to omega iff its successor does.

Theoremnnsuc 4156* A nonzero natural number is a successor.

Theoremssnlim 4157* An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42.

2.4.5  Peano's postulates

Theorempeano1 4158 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 4158 through peano5 4162 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity.

Theorempeano2 4159 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.

Theorempeano3 4160 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.

Theorempeano4 4161 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.

Theorempeano5 4162* 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 4169.

Theoremnn0suc 4163* A natural number is either 0 or a successor.

2.4.6  Finite induction (for finite ordinals)

Theoremfind 4164* 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 . (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Theoremfinds 4165* 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.

Theoremfindsg 4166* 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.

Theoremfinds2 4167* 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.

Theoremfinds1 4168* 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.

Theoremfindes 4169 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 4135 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)

2.4.7  Functions and relations

Syntaxcxp 4170 Extend the definition of a class to include the cross product.

Syntaxccnv 4171 Extend the definition of a class to include the converse of a class.

Syntaxcdm 4172 Extend the definition of a class to include the domain of a class.

Syntaxcrn 4173 Extend the definition of a class to include the range of a class.

Syntaxcres 4174 Extend the definition of a class to include the restriction of a class. (Read: The restriction of to .)

Syntaxcima 4175 Extend the definition of a class to include the image of a class. (Read: The image of under .)

Syntaxccom 4176 Extend the definition of a class to include the composition of two classes. (Read: The composition of and .)

Syntaxwrel 4177 Extend the definition of a wff to include the relation predicate. (Read: is a relation.)

Syntaxwfun 4178 Extend the definition of a wff to include the function predicate. (Read: is a function.)

Syntaxwfn 4179 Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .)

Syntaxwf 4180 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .)

Syntaxwf1 4181 Extend the definition of a wff to include one-to-one functions. (Read: maps one-to-one into .) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.

Syntaxwfo 4182 Extend the definition of a wff to include onto functions. (Read: maps onto .) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.

Syntaxwf1o 4183 Extend the definition of a wff to include one-to-one onto functions. (Read: maps one-to-one onto .) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.

Syntaxcfv 4184 Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .")

Syntaxwiso 4185 Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .)

Definitiondf-xp 4186* Define the cross product of two classes. Definition 9.11 of [Quine] p. 64. For example, (ex-xp 15853). Another example is that the set of rational numbers are defined in df-q 9148 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.

Definitiondf-rel 4187 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4585 and dfrel3 4613.

Definitiondf-cnv 4188* 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 4359 (see df-br 3532 and df-rel 4187 for more on relations). For example, (ex-cnv 15854). 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.

Definitiondf-co 4189* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, (ex-co 15855) because (see cos0 10576) and (see df-e 10504). Note that Definition 7 of [Suppes] p. 63 reverses and , uses instead of , and calls the operation "relative product."

Definitiondf-dm 4190* Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, (ex-dm 15856). Another example is the domain of the complex arctangent, arctan (for proof see atandm 15520). Contrast with range (defined in df-rn 4191). For alternate definitions see dfdm2 4655, dfdm3 4362, and dfdm4 4365. The notation " " is used by Enderton; other authors sometimes use script D.

Definitiondf-rn 4191 Define the range of a class. For example, (ex-rn 15857). Contrast with domain (defined in df-dm 4190). For alternate definitions, see dfrn2 4363, dfrn3 4364, and dfrn4 4616. The notation " " is used by Enderton; other authors sometimes use script R or script W.

Definitiondf-res 4192 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression (used in reeff1 10552) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 10503 defines the exponential function, which normally allows the exponent to be a complex number). Another example is that (ex-res 15858).

Definitiondf-ima 4193 Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example, (ex-ima 15859). Contrast with restriction (df-res 4192) and range (df-rn 4191). For an alternate definition, see dfima2 4487.

Definitiondf-fun 4194 Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true once we define cosine (df-cos 10506). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5369 with the maps-to notation (see df-mpt 5371 and df-mpt2 5372). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4195), a function with a given domain and codomain (df-f 4196), a one-to-one function (df-f1 4197), an onto function (df-fo 4198), or a one-to-one onto function (df-f1o 4199). For alternate definitions, see dffun2 4667, dffun3 4668, dffun4 4669, dffun5 4670, dffun6 4672, dffun7 4682, dffun8 4683, and dffun9 4684.

Definitiondf-fn 4195 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 4786, dffn3 4792, dffn4 4851, and dffn5 4949.

Definitiondf-f 4196 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 5058, dff3 5059, and dff4 5060.

Definitiondf-f1 4197 Define a one-to-one function. For equivalent definitions see dff12 4830 and dff13 5149. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow).

Definitiondf-fo 4198 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). For alternate definitions, see dffo2 4849, dffo3 5061, dffo4 5062, and dffo5 5063.

Definitiondf-f1o 4199 Define a one-to-one onto function. For equivalent definitions see dff1o2 4868, dff1o3 4869, dff1o4 4871, and dff1o5 4872. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow).

Definitiondf-fv 4200* Define the value of a function, , also known as function application. For example, (we prove this in cos0 10576 after we define cosine in df-cos 10506). Typically function is defined using maps-to notation (see df-mpt 5371 and df-mpt2 5372), but this is not required. For example, (ex-fv 15860). Note that df-ov 5208 will define two-argument functions using ordered pairs as . Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 4513), our definition apparently does not appear in the literature. However, it is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful (as shown by ndmfv 4935 and fvprc 4907). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar notation for a function's value at , i.e. " of ," but without context-dependent notational ambiguity. Alternate definitions are dffv2 4968 and dffv3 5678. For other alternate definitions (that fail to evaluate to the empty set for proper classes), see fv2 4906, fv3 4926, and fv4 5679. Restricted equivalents that require to be a function are shown in funfv 4962 and funfv2 4963. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 4947.

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