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Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonminesb 7501 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.)
(∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
 
Theoremonminsb 7502 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.)
𝑥𝜓    &   (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))       (∃𝑥 ∈ On 𝜑𝜓)
 
Theoremoninton 7503 The intersection of a nonempty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by NM, 29-Jan-1997.)
((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
 
Theoremonintrab 7504 The intersection of a class of ordinal numbers exists iff it is an ordinal number. (Contributed by NM, 6-Nov-2003.)
( {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ {𝑥 ∈ On ∣ 𝜑} ∈ On)
 
Theoremonintrab2 7505 An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.)
(∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
 
Theoremonnmin 7506 No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)
((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)
 
Theoremonnminsb 7507* An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. 𝜓 is the wff resulting from the substitution of 𝐴 for 𝑥 in wff 𝜑. (Contributed by NM, 9-Nov-2003.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ On → (𝐴 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
 
Theoremoneqmin 7508* A way to show that an ordinal number equals the minimum of a nonempty 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.)
((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
 
Theoremuniordint 7509* The union of a set of ordinals is equal to the intersection of its upper bounds. Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
 
Theoremonminex 7510* If a wff is true for an ordinal number, then 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.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓))
 
Theoremsucon 7511 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
suc On = On
 
Theoremsucexb 7512 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
(𝐴 ∈ V ↔ suc 𝐴 ∈ V)
 
Theoremsucexg 7513 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → suc 𝐴 ∈ V)
 
Theoremsucex 7514 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       suc 𝐴 ∈ V
 
Theoremonmindif2 7515 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.)
((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 (𝐴 ∖ { 𝐴}))
 
Theoremsuceloni 7516 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
(𝐴 ∈ On → suc 𝐴 ∈ On)
 
Theoremordsuc 7517 The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.)
(Ord 𝐴 ↔ Ord suc 𝐴)
 
Theoremordpwsuc 7518 The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005.)
(Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
 
Theoremonpwsuc 7519 The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.)
(𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴)
 
Theoremsucelon 7520 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
(𝐴 ∈ On ↔ suc 𝐴 ∈ On)
 
Theoremordsucss 7521 The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
(Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
 
Theoremonpsssuc 7522 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)
 
Theoremordelsuc 7523 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.)
((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
 
Theoremonsucmin 7524* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
(𝐴 ∈ On → suc 𝐴 = {𝑥 ∈ On ∣ 𝐴𝑥})
 
Theoremordsucelsuc 7525 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.)
(Ord 𝐵 → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
 
Theoremordsucsssuc 7526 The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
 
Theoremordsucuniel 7527 Given an element 𝐴 of the union of an ordinal 𝐵, suc 𝐴 is an element of 𝐵 itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)
(Ord 𝐵 → (𝐴 𝐵 ↔ suc 𝐴𝐵))
 
Theoremordsucun 7528 The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.)
((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))
 
Theoremordunpr 7529 The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ {𝐵, 𝐶})
 
Theoremordunel 7530 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.)
((Ord 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ∈ 𝐴)
 
Theoremonsucuni 7531 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.)
(𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
 
Theoremordsucuni 7532 An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴𝐴 ⊆ suc 𝐴)
 
Theoremorduniorsuc 7533 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.)
(Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
 
Theoremunon 7534 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
On = On
 
Theoremordunisuc 7535 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.)
(Ord 𝐴 suc 𝐴 = 𝐴)
 
Theoremorduniss2 7536* The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.)
(Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴)
 
Theoremonsucuni2 7537 A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
 
Theorem0elsuc 7538 The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.)
(Ord 𝐴 → ∅ ∈ suc 𝐴)
 
Theoremlimon 7539 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
Lim On
 
Theoremonssi 7540 An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       𝐴 ⊆ On
 
Theoremonsuci 7541 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
𝐴 ∈ On       suc 𝐴 ∈ On
 
Theoremonuniorsuci 7542 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.)
𝐴 ∈ On       (𝐴 = 𝐴𝐴 = suc 𝐴)
 
Theoremonuninsuci 7543* A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)
𝐴 ∈ On       (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
 
Theoremonsucssi 7544 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.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵 ↔ suc 𝐴𝐵)
 
Theoremnlimsucg 7545 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉 → ¬ Lim suc 𝐴)
 
Theoremorduninsuc 7546* An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)
(Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
 
Theoremordunisuc2 7547* An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
(Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
 
Theoremordzsl 7548* An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
(Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
 
Theoremonzsl 7549* 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.)
(𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
 
Theoremdflim3 7550* 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.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
 
Theoremdflim4 7551* An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
 
Theoremlimsuc 7552 The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
(Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
 
Theoremlimsssuc 7553 A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
(Lim 𝐴 → (𝐴𝐵𝐴 ⊆ suc 𝐵))
 
Theoremnlimon 7554* 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.)
{𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥}
 
Theoremlimuni3 7555* The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 Lim 𝑥) → Lim 𝐴)
 
2.4.3  Transfinite induction
 
Theoremtfi 7556* The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if 𝐴 is a class of ordinal numbers with the property that every ordinal number included in 𝐴 also belongs to 𝐴, then every ordinal number is in 𝐴.

See theorem tfindes 7565 or tfinds 7562 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)
 
Theoremtfis 7557* Transfinite Induction Schema. If all ordinal numbers less than a given number 𝑥 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.)
(𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))       (𝑥 ∈ On → 𝜑)
 
Theoremtfis2f 7558* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)
 
Theoremtfis2 7559* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)
 
Theoremtfis3 7560* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝐴 ∈ On → 𝜒)
 
Theoremtfisi 7561* 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.)
(𝜑𝐴𝑉)    &   (𝜑𝑇 ∈ On)    &   ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅𝑇) ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜓𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝑥 = 𝑦𝑅 = 𝑆)    &   (𝑥 = 𝐴𝑅 = 𝑇)       (𝜑𝜃)
 
Theoremtfinds 7562* 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 step for successors, and the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ On → (𝜒𝜃))    &   (Lim 𝑥 → (∀𝑦𝑥 𝜒𝜑))       (𝐴 ∈ On → 𝜏)
 
Theoremtfindsg 7563* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal 𝐵 instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.)
(𝑥 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝐵 ∈ On → 𝜓)    &   (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝑦) → (𝜒𝜃))    &   (((Lim 𝑥𝐵 ∈ On) ∧ 𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))       (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵𝐴) → 𝜏)
 
Theoremtfindsg2 7564* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal suc 𝐵 instead of zero. (Contributed by NM, 5-Jan-2005.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.)
(𝑥 = suc 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝐵 ∈ On → 𝜓)    &   ((𝑦 ∈ On ∧ 𝐵𝑦) → (𝜒𝜃))    &   ((Lim 𝑥𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))       ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝜏)
 
Theoremtfindes 7565* Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction step for successors, and the third is the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)
[∅ / 𝑥]𝜑    &   (𝑥 ∈ On → (𝜑[suc 𝑥 / 𝑥]𝜑))    &   (Lim 𝑦 → (∀𝑥𝑦 𝜑[𝑦 / 𝑥]𝜑))       (𝑥 ∈ On → 𝜑)
 
Theoremtfinds2 7566* 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 𝜏 is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝜏𝜓)    &   (𝑦 ∈ On → (𝜏 → (𝜒𝜃)))    &   (Lim 𝑥 → (𝜏 → (∀𝑦𝑥 𝜒𝜑)))       (𝑥 ∈ On → (𝜏𝜑))
 
Theoremtfinds3 7567* 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 step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝜂𝜓)    &   (𝑦 ∈ On → (𝜂 → (𝜒𝜃)))    &   (Lim 𝑥 → (𝜂 → (∀𝑦𝑥 𝜒𝜑)))       (𝐴 ∈ On → (𝜂𝜏))
 
2.4.4  The natural numbers (i.e., finite ordinals)
 
Syntaxcom 7568 Extend class notation to include the class of natural numbers.
class ω
 
Definitiondf-om 7569* 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 7570 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 9095, and ω can then be defined per dfom3 9099 (the smallest inductive set) and dfom4 9101.

Note: the natural numbers ω are a subset of the ordinal numbers df-on 6189. They are completely different from the natural numbers (df-nn 11628) 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.)

ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
 
Theoremdfom2 7570 An alternate definition of the set of natural numbers ω. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the inner class builder of non-limit ordinal numbers (see nlimon 7554). (Contributed by NM, 1-Nov-2004.)
ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
 
Theoremelom 7571* Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9100. (Contributed by NM, 15-May-1994.)
(𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))
 
Theoremomsson 7572 Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
ω ⊆ On
 
Theoremlimomss 7573 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.)
(Lim 𝐴 → ω ⊆ 𝐴)
 
Theoremnnon 7574 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
(𝐴 ∈ ω → 𝐴 ∈ On)
 
Theoremnnoni 7575 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
𝐴 ∈ ω       𝐴 ∈ On
 
Theoremnnord 7576 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
(𝐴 ∈ ω → Ord 𝐴)
 
Theoremordom 7577 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Ord ω
 
Theoremelnn 7578 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
 
Theoremomon 7579 The class of natural numbers ω 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.)
(ω ∈ On ∨ ω = On)
 
Theoremomelon2 7580 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
(ω ∈ V → ω ∈ On)
 
Theoremnnlim 7581 A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
(𝐴 ∈ ω → ¬ Lim 𝐴)
 
Theoremomssnlim 7582 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.)
ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}
 
Theoremlimom 7583 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.)
Lim ω
 
Theorempeano2b 7584 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
(𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
 
Theoremnnsuc 7585* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
 
Theoremomsucne 7586 A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.)
(𝐴 ∈ ω → 𝐴 ≠ suc 𝐴)
 
Theoremssnlim 7587* 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.)
((Ord 𝐴𝐴 ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}) → 𝐴 ⊆ ω)
 
Theoremomsinds 7588* Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))       (𝐴 ∈ ω → 𝜒)
 
2.4.5  Peano's postulates
 
Theorempeano1 7589 Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 7589 through peano5 7593 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.)
∅ ∈ ω
 
Theorempeano2 7590 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.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)
 
Theorempeano3 7591 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.)
(𝐴 ∈ ω → suc 𝐴 ≠ ∅)
 
Theorempeano4 7592 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.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
 
Theorempeano5 7593* 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, 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 step, is derived from this theorem as theorem findes 7600. (Contributed by NM, 18-Feb-2004.)
((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
 
Theoremnn0suc 7594* A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
 
2.4.6  Finite induction (for finite ordinals)
 
Theoremfind 7595* 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 𝐴 is a set of natural numbers, zero belongs to 𝐴, and given any member of 𝐴 the member's successor also belongs to 𝐴. The conclusion is that every natural number is in 𝐴. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)       𝐴 = ω
 
Theoremfinds 7596* 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.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ω → (𝜒𝜃))       (𝐴 ∈ ω → 𝜏)
 
Theoremfindsg 7597* 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. The basis of this version is an arbitrary natural number 𝐵 instead of zero. (Contributed by NM, 16-Sep-1995.)
(𝑥 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝐵 ∈ ω → 𝜓)    &   (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑦) → (𝜒𝜃))       (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → 𝜏)
 
Theoremfinds2 7598* 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 step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝜏𝜓)    &   (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))       (𝑥 ∈ ω → (𝜏𝜑))
 
Theoremfinds1 7599* 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 step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   𝜓    &   (𝑦 ∈ ω → (𝜒𝜃))       (𝑥 ∈ ω → 𝜑)
 
Theoremfindes 7600 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. See tfindes 7565 for the transfinite version. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
[∅ / 𝑥]𝜑    &   (𝑥 ∈ ω → (𝜑[suc 𝑥 / 𝑥]𝜑))       (𝑥 ∈ ω → 𝜑)
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