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Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssonunii 7501 The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 ∈ On)

Theoremordeleqon 7502 A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse. (Contributed by NM, 1-Jun-2003.)
(Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))

Theoremordsson 7503 Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
(Ord 𝐴𝐴 ⊆ On)

Theoremonss 7504 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → 𝐴 ⊆ On)

Theorempredon 7505 The predecessor of an ordinal under E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011.)
(𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Theoremssonprc 7506 Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ⊆ On → (𝐴 ∉ V ↔ 𝐴 = On))

Theoremonuni 7507 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
(𝐴 ∈ On → 𝐴 ∈ On)

Theoremorduni 7508 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴 → Ord 𝐴)

Theoremonint 7509 The intersection (infimum) of a nonempty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)
((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)

Theoremonint0 7510 The intersection of a class of ordinal numbers is zero iff the class contains zero. (Contributed by NM, 24-Apr-2004.)
(𝐴 ⊆ On → ( 𝐴 = ∅ ↔ ∅ ∈ 𝐴))

Theoremonssmin 7511* A nonempty class of ordinal numbers has the smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.)
((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremonminesb 7512 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 7513 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 7514 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 7515 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 7516 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 7517 No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)
((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)

Theoremonnminsb 7518* 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 7519* 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 7520* 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 7521* 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 7522 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
suc On = On

Theoremsucexb 7523 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
(𝐴 ∈ V ↔ suc 𝐴 ∈ V)

Theoremsucexg 7524 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → suc 𝐴 ∈ V)

Theoremsucex 7525 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       suc 𝐴 ∈ V

Theoremonmindif2 7526 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 7527 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 7528 The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.)
(Ord 𝐴 ↔ Ord suc 𝐴)

Theoremordpwsuc 7529 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 7530 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 7531 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
(𝐴 ∈ On ↔ suc 𝐴 ∈ On)

Theoremordsucss 7532 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 7533 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)

Theoremordelsuc 7534 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 7535* 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 7536 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 7537 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 7538 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 7539 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 7540 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 7541 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 7542 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 7543 An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴𝐴 ⊆ suc 𝐴)

Theoremorduniorsuc 7544 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 7545 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 7546 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 7547* 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 7548 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 7549 The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.)
(Ord 𝐴 → ∅ ∈ suc 𝐴)

Theoremlimon 7550 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
Lim On

Theoremonssi 7551 An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       𝐴 ⊆ On

Theoremonsuci 7552 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 7553 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 7554* A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)
𝐴 ∈ On       (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Theoremonsucssi 7555 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 7556 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 7557* 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 7558* 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 7559* 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 7560* 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 7561* 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 7562* An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))

Theoremlimsuc 7563 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 7564 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 7565* 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 7566* 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 7567* 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 7576 or tfinds 7573 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)

Theoremtfis 7568* 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 7569* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)

Theoremtfis2 7570* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)

Theoremtfis3 7571* 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 7572* 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 7573* 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 7574* 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 7575* 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 7576* 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 7577* 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 7578* 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 7579 Extend class notation to include the class of natural numbers.
class ω

Definitiondf-om 7580* 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 7581 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 9139, and ω can then be defined per dfom3 9143 (the smallest inductive set) and dfom4 9145.

Note: the natural numbers ω are a subset of the ordinal numbers df-on 6173. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-nn 11675) with analogous properties and operations, but they will be different sets. (Contributed by NM, 15-May-1994.)

ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}

Theoremdfom2 7581 An alternate definition of the set of natural numbers ω. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the restricted class abstraction of non-limit ordinal numbers (see nlimon 7565). (Contributed by NM, 1-Nov-2004.)
ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}

Theoremelom 7582* Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9144. (Contributed by NM, 15-May-1994.)
(𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥𝐴𝑥)))

Theoremomsson 7583 Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
ω ⊆ On

Theoremlimomss 7584 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 7585 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
(𝐴 ∈ ω → 𝐴 ∈ On)

Theoremnnoni 7586 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
𝐴 ∈ ω       𝐴 ∈ On

Theoremnnord 7587 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
(𝐴 ∈ ω → Ord 𝐴)

Theoremordom 7588 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 7589 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)

Theoremomon 7590 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 7591 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
(ω ∈ V → ω ∈ On)

Theoremnnlim 7592 A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
(𝐴 ∈ ω → ¬ Lim 𝐴)

Theoremomssnlim 7593 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 7594 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 7595 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
(𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)

Theoremnnsuc 7596* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)

Theoremomsucne 7597 A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.)
(𝐴 ∈ ω → 𝐴 ≠ suc 𝐴)

Theoremssnlim 7598* 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 7599* 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 7600 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 7600 through peano5 7604 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

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