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Type | Label | Description |
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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 | ||
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}} |
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