P. 65 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ordtri3or 6401	A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
Theorem	ordtri1 6402	A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
Theorem	ontri1 6403	A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
Theorem	ordtri2 6404	A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
Theorem	ordtri3 6405	A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴)))
Theorem	ordtri4 6406	A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵)))
Theorem	orddisj 6407	An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
Theorem	onfr 6408	The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7779 (through epweon 7777) in order to eliminate the need for the axiom of regularity. (Contributed by NM, 17-May-1994.)
⊢ E Fr On
Theorem	onelpss 6409	Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)))
Theorem	onsseleq 6410	Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	onelss 6411	An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	ordtr1 6412	Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	ordtr2 6413	Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	ordtr3 6414	Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
Theorem	ontr1 6415	Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by NM, 11-Aug-1994.)
⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	ontr2 6416	Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)
⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	onelssex 6417*	Ordinal less than is equivalent to having an ordinal between them. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏))
Theorem	ordunidif 6418	The union of an ordinal stays the same if a subset equal to one of its elements is removed. (Contributed by NM, 10-Dec-2004.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)
Theorem	ordintdif 6419	If 𝐵 is smaller than 𝐴, then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → 𝐵 = ∩ (𝐴 ∖ 𝐵))
Theorem	onintss 6420*	If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ On → (𝜓 → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
Theorem	oneqmini 6421*	A way to show that an ordinal number equals the minimum of a 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 → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))
Theorem	ord0 6422	The empty set is an ordinal class. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 11-May-1994.)
⊢ Ord ∅
Theorem	0elon 6423	The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 17-Sep-1993.)
⊢ ∅ ∈ On
Theorem	ord0eln0 6424	A nonempty ordinal contains the empty set. Lemma 1.10 of [Schloeder] p. 2. (Contributed by NM, 25-Nov-1995.)
⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	on0eln0 6425	An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)
⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	dflim2 6426	An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
Theorem	inton 6427	The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
⊢ ∩ On = ∅
Theorem	nlim0 6428	The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ¬ Lim ∅
Theorem	limord 6429	A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
⊢ (Lim 𝐴 → Ord 𝐴)
Theorem	limuni 6430	A limit ordinal is its own supremum (union). Lemma 2.13 of [Schloeder] p. 5. (Contributed by NM, 4-May-1995.)
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
Theorem	limuni2 6431	The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
⊢ (Lim 𝐴 → Lim ∪ 𝐴)
Theorem	0ellim 6432	A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
Theorem	limelon 6433	A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
Theorem	onn0 6434	The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
⊢ On ≠ ∅
Theorem	suceq 6435	Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
Theorem	elsuci 6436	Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. Lemma 1.13 of [Schloeder] p. 2. (Contributed by NM, 6-Jun-1994.)
⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	elsucg 6437	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	elsuc2g 6438	Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	elsuc 6439	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	elsuc2 6440	Membership in a successor. (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
Theorem	nfsuc 6441	Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 suc 𝐴
Theorem	elelsuc 6442	Membership in a successor. (Contributed by NM, 20-Jun-1998.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)
Theorem	sucel 6443*	Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
Theorem	suc0 6444	The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
⊢ suc ∅ = {∅}
Theorem	sucprc 6445	A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
Theorem	unisucs 6446	The union of the successor of a set is equal to the binary union of that set with its union. (Contributed by NM, 30-Aug-1993.) Extract from unisuc 6448. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))
Theorem	unisucg 6447	A transitive class is equal to the union of its successor, closed form. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.) Generalize from unisuc 6448. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
Theorem	unisuc 6448	A transitive class is equal to the union of its successor, inference form. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)
Theorem	sssucid 6449	A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
⊢ 𝐴 ⊆ suc 𝐴
Theorem	sucidg 6450	Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). Lemma 1.7 of [Schloeder] p. 1. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
Theorem	sucid 6451	A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ suc 𝐴
Theorem	nsuceq0 6452	No successor is empty. (Contributed by NM, 3-Apr-1995.)
⊢ suc 𝐴 ≠ ∅
Theorem	eqelsuc 6453	A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵)
Theorem	iunsuc 6454*	Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)
Theorem	suctr 6455	The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ (Tr 𝐴 → Tr suc 𝐴)
Theorem	trsuc 6456	A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)
Theorem	trsucss 6457	A member of the successor of a transitive class is a subclass of it. Lemma 1.13 of [Schloeder] p. 2. (Contributed by NM, 4-Oct-2003.)
⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	ordsssuc 6458	An ordinal is a subset of another ordinal if and only if it belongs to its successor. (Contributed by NM, 28-Nov-2003.)
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
Theorem	onsssuc 6459	A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
Theorem	ordsssuc2 6460	An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
Theorem	onmindif 6461	When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)
⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵))
Theorem	ordnbtwn 6462	There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. Lemma 1.15 of [Schloeder] p. 2. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
Theorem	onnbtwn 6463	There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. Lemma 1.15 of [Schloeder] p. 2. (Contributed by NM, 9-Jun-1994.)
⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
Theorem	sucssel 6464	A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))
Theorem	orddif 6465	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
Theorem	orduniss 6466	An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴)
Theorem	ordtri2or 6467	A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	ordtri2or2 6468	A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	ordtri2or3 6469	A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6468. (Contributed by David Moews, 1-May-2017.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
Theorem	ordelinel 6470	The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))
Theorem	ordssun 6471	Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)
⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
Theorem	ordequn 6472	The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵 ∪ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
Theorem	ordun 6473	The maximum (i.e., union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵))
Theorem	onunel 6474	The union of two ordinals is in a third iff both of the first two are. (Contributed by Scott Fenton, 10-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
Theorem	ordunisssuc 6475	A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)
⊢ ((𝐴 ⊆ On ∧ Ord 𝐵) → (∪ 𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))
Theorem	suc11 6476	The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
Theorem	onun2 6477	The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
Theorem	ontr 6478	An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) Put in closed form. (Resised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ On → Tr 𝐴)
Theorem	onunisuc 6479	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6489. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
Theorem	onordi 6480	An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ Ord 𝐴
Theorem	ontrciOLD 6481	Obsolete version of ontr 6478 as of 28-Dec-2024. (Contributed by NM, 11-Jun-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ On    ⇒   ⊢ Tr 𝐴
Theorem	onirri 6482	An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ ¬ 𝐴 ∈ 𝐴
Theorem	oneli 6483	A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)
Theorem	onelssi 6484	A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)
Theorem	onssneli 6485	An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)
Theorem	onssnel2i 6486	An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵)
Theorem	onelini 6487	An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
Theorem	oneluni 6488	An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
Theorem	onunisuci 6489	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ ∪ suc 𝐴 = 𝐴
Theorem	onsseli 6490	Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)
⊢ 𝐴 ∈ On    &   ⊢ 𝐵 ∈ On    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	onun2i 6491	The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    &   ⊢ 𝐵 ∈ On    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ On
Theorem	unizlim 6492	An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
Theorem	on0eqel 6493	An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Theorem	snsn0non 6494	The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7874). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6495. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ¬ {{∅}} ∈ On
Theorem	onxpdisj 6495	Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 6494. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (On ∩ (V × V)) = ∅
Theorem	onnev 6496	The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof shortened by Wolf Lammen, 27-May-2024.)
⊢ On ≠ V
Theorem	onnevOLD 6497	Obsolete version of onnev 6496 as of 27-May-2024. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ On ≠ V
2.3.15  Definite description binder (inverted iota)
Syntax	cio 6498	Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)
Theorem	iotajust 6499*	Soundness justification theorem for df-iota 6500. (Contributed by Andrew Salmon, 29-Jun-2011.)
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
Definition	df-iota 6500*	Define Russell's definition description binder, which can be read as "the unique 𝑥 such that 𝜑", where 𝜑 ordinarily contains 𝑥 as a free variable. Our definition is meaningful only when there is exactly one 𝑥 such that 𝜑 is true (see iotaval 6519); otherwise, it evaluates to the empty set (see iotanul 6526). Russell used the inverted iota symbol ℩ to represent the binder. Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 7393 (or iotacl 6534 for unbounded iota), as demonstrated in the proof of supub 9483. This can be easier than applying riotasbc 7395 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}