P. 78 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eldifpw 7701	Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
⊢ 𝐶 ∈ V    ⇒   ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵))
Theorem	elpwun 7702	Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)
Theorem	pwuncl 7703	Power classes are closed under union. (Contributed by AV, 27-Feb-2024.)
⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋)
Theorem	iunpw 7704*	An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)
Theorem	fr3nr 7705	A well-founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 10-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))
Theorem	epne3 7706	A well-founded class contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵))
Theorem	dfwe2 7707*	Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
2.4.2  Ordinals (continued)
Theorem	epweon 7708	The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of [Mendelson] p. 244. For a shorter proof requiring ax-un 7668, see epweonALT 7709. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7668. (Revised by BTernaryTau, 30-Nov-2024.)
⊢ E We On
Theorem	epweonALT 7709	Alternate proof of epweon 7708, shorter but requiring ax-un 7668. (Contributed by NM, 1-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ E We On
Theorem	ordon 7710	The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
⊢ Ord On
Theorem	onprc 7711	No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 7710), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)
⊢ ¬ On ∈ V
Theorem	ssorduni 7712	The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
Theorem	ssonuni 7713	The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 1-Nov-2003.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
Theorem	ssonunii 7714	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)
Theorem	ordeleqon 7715	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))
Theorem	ordsson 7716	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 𝐴 → 𝐴 ⊆ On)
Theorem	dford5 7717	A class is ordinal iff it is a subclass of On and transitive. (Contributed by Scott Fenton, 21-Nov-2021.)
⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴))
Theorem	onss 7718	An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
Theorem	predon 7719	The predecessor of an ordinal under E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by BJ, 16-Oct-2024.)
⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
Theorem	ssonprc 7720	Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On))
Theorem	onuni 7721	The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
Theorem	orduni 7722	The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
⊢ (Ord 𝐴 → Ord ∪ 𝐴)
Theorem	onint 7723	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 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)
Theorem	onint0 7724	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))
Theorem	onssmin 7725*	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 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
Theorem	onminesb 7726	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)
Theorem	onminsb 7727	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.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ On 𝜑 → 𝜓)
Theorem	oninton 7728	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)
Theorem	onintrab 7729	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)
Theorem	onintrab2 7730	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)
Theorem	onnmin 7731	No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)
⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴)
Theorem	onnminsb 7732*	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
Theorem	oneqmin 7733*	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 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))
Theorem	uniordint 7734*	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 ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onminex 7735*	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓))
Theorem	sucon 7736	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
⊢ suc On = On
Theorem	sucexb 7737	A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
Theorem	sucexg 7738	The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
Theorem	sucex 7739	The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ suc 𝐴 ∈ V
Theorem	onmindif2 7740	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}))
Theorem	ordsuci 7741	The successor of an ordinal class is an ordinal class. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) Extract and adapt from a subproof of onsuc 7743. (Revised by BTernaryTau, 6-Jan-2025.) (Proof shortened by BJ, 11-Jan-2025.)
⊢ (Ord 𝐴 → Ord suc 𝐴)
Theorem	sucexeloni 7742	If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7743 does not require ax-un 7668. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)
Theorem	onsuc 7743	The successor of an ordinal number is an ordinal number. Closed form of onsuci 7769. Forward implication of onsucb 7747. Proposition 7.24 of [TakeutiZaring] p. 41. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) (Proof shortened by BTernaryTau, 30-Nov-2024.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
Theorem	ordsuc 7744	A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995.) Avoid ax-un 7668. (Revised by BTernaryTau, 6-Jan-2025.)
⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
Theorem	ordpwsuc 7745	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)
Theorem	onpwsuc 7746	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)
Theorem	onsucb 7747	A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 7743. (Contributed by NM, 9-Sep-2003.)
⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
Theorem	ordsucss 7748	The successor of an element of an ordinal class is a subset of it. Lemma 1.14 of [Schloeder] p. 2. (Contributed by NM, 21-Jun-1998.)
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
Theorem	onpsssuc 7749	An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)
Theorem	ordelsuc 7750	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.)
⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵))
Theorem	onsucmin 7751*	The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥})
Theorem	ordsucelsuc 7752	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 → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
Theorem	ordsucsssuc 7753	The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
Theorem	ordsucuniel 7754	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 𝐴 ∈ 𝐵))
Theorem	ordsucun 7755	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 (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))
Theorem	ordunpr 7756	The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶})
Theorem	ordunel 7757	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 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 ∪ 𝐶) ∈ 𝐴)
Theorem	onsucuni 7758	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)
Theorem	ordsucuni 7759	An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.)
⊢ (Ord 𝐴 → 𝐴 ⊆ suc ∪ 𝐴)
Theorem	orduniorsuc 7760	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 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
Theorem	unon 7761	The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
⊢ ∪ On = On
Theorem	ordunisuc 7762	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)
Theorem	orduniss2 7763*	The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.)
⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)
Theorem	onsucuni2 7764	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)
Theorem	0elsuc 7765	The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.)
⊢ (Ord 𝐴 → ∅ ∈ suc 𝐴)
Theorem	limon 7766	The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
⊢ Lim On
Theorem	onuniorsuc 7767	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.) Put in closed form. (Revised by BJ, 11-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
Theorem	onssi 7768	An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ 𝐴 ⊆ On
Theorem	onsuci 7769	The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7743 and onsucb 7747. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ suc 𝐴 ∈ On
Theorem	onuninsuci 7770*	An ordinal is equal to its union if and only if it is not the successor of an ordinal. A closed-form generalization of this result is orduninsuc 7773. (Contributed by NM, 18-Feb-2004.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Theorem	onsucssi 7771	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    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)
Theorem	nlimsucg 7772	A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)
Theorem	orduninsuc 7773*	An ordinal class is equal to its union if and only if it is not the successor of an ordinal. Closed-form generalization of onuninsuci 7770. (Contributed by NM, 18-Feb-2004.)
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
Theorem	ordunisuc2 7774*	An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
Theorem	ordzsl 7775*	An ordinal is zero, a successor ordinal, or a limit ordinal. Remark 1.12 of [Schloeder] p. 2. (Contributed by NM, 1-Oct-2003.)
⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
Theorem	onzsl 7776*	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)))
Theorem	dflim3 7777*	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)))
Theorem	dflim4 7778*	An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
Theorem	limsuc 7779	The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
Theorem	limsssuc 7780	A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
⊢ (Lim 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))
Theorem	nlimon 7781*	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}
Theorem	limuni3 7782*	The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Lim 𝑥) → Lim ∪ 𝐴)
2.4.3  Transfinite induction
Theorem	tfi 7783*	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 7793 or tfinds 7790 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)
Theorem	tfisg 7784*	A closed form of tfis 7785. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥 ∈ On 𝜑)
Theorem	tfis 7785*	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 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis2f 7786*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis2 7787*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis3 7788*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ On → 𝜒)
Theorem	tfisi 7789*	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ On)    &   ⊢ ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅 ⊆ 𝑇) ∧ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)    &   ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	tfinds 7790*	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. Theorem 1.19 of [Schloeder] p. 3. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ On → (𝜒 → 𝜃))    &   ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))    ⇒   ⊢ (𝐴 ∈ On → 𝜏)
Theorem	tfindsg 7791*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝐵 ∈ On → 𝜓)    &   ⊢ (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))    &   ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))    ⇒   ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → 𝜏)
Theorem	tfindsg2 7792*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝐵 ∈ On → 𝜓)    &   ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ 𝑦) → (𝜒 → 𝜃))    &   ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝜏)
Theorem	tfindes 7793*	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 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfinds2 7794*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝜏 → 𝜓)    &   ⊢ (𝑦 ∈ On → (𝜏 → (𝜒 → 𝜃)))    &   ⊢ (Lim 𝑥 → (𝜏 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))    ⇒   ⊢ (𝑥 ∈ On → (𝜏 → 𝜑))
Theorem	tfinds3 7795*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝜂 → 𝜓)    &   ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃)))    &   ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))    ⇒   ⊢ (𝐴 ∈ On → (𝜂 → 𝜏))
2.4.4  The natural numbers (i.e., finite ordinals)
Syntax	com 7796	Extend class notation to include the class of natural numbers.
class ω
Definition	df-om 7797*	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 7798 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 9533, and ω can then be defined per dfom3 9537 (the smallest inductive set) and dfom4 9539. Note: the natural numbers ω are a subset of the ordinal numbers df-on 6310. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-nn 12123) with analogous properties and operations, but they will be different sets. (Contributed by NM, 15-May-1994.)
⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
Theorem	dfom2 7798	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 7781). (Contributed by NM, 1-Nov-2004.)
⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
Theorem	elom 7799*	Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9538. (Contributed by NM, 15-May-1994.)
⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))
Theorem	omsson 7800	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ω ⊆ On