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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | orddisj 6401 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | ||
Theorem | onfr 6402 | The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7771 (through epweon 7769) in order to eliminate the need for the axiom of regularity. (Contributed by NM, 17-May-1994.) |
⊢ E Fr On | ||
Theorem | onelpss 6403 | 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 6404 | 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 6405 | 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 6406 | Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) |
⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | ||
Theorem | ordtr2 6407 | 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 6408 | 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 6409 | 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 6410 | 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 6411* | 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 6412 | 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 6413 | 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 6414* | 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 6415* | 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 6416 | The empty set is an ordinal class. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 11-May-1994.) |
⊢ Ord ∅ | ||
Theorem | 0elon 6417 | 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 6418 | 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 6419 | An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.) |
⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | ||
Theorem | dflim2 6420 | An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.) |
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | ||
Theorem | inton 6421 | The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.) |
⊢ ∩ On = ∅ | ||
Theorem | nlim0 6422 | 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 6423 | A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.) |
⊢ (Lim 𝐴 → Ord 𝐴) | ||
Theorem | limuni 6424 | 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 6425 | The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.) |
⊢ (Lim 𝐴 → Lim ∪ 𝐴) | ||
Theorem | 0ellim 6426 | A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.) |
⊢ (Lim 𝐴 → ∅ ∈ 𝐴) | ||
Theorem | limelon 6427 | 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 6428 | The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.) |
⊢ On ≠ ∅ | ||
Theorem | suceq 6429 | 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 6430 | 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 6431 | 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 6432 | 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 6433 | 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 6434 | Membership in a successor. (Contributed by NM, 15-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | ||
Theorem | nfsuc 6435 | Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 suc 𝐴 | ||
Theorem | elelsuc 6436 | Membership in a successor. (Contributed by NM, 20-Jun-1998.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵) | ||
Theorem | sucel 6437* | 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 6438 | The successor of the empty set. (Contributed by NM, 1-Feb-2005.) |
⊢ suc ∅ = {∅} | ||
Theorem | sucprc 6439 | A proper class is its own successor. (Contributed by NM, 3-Apr-1995.) |
⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | ||
Theorem | unisucs 6440 | 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 6442. (Revised by BJ, 28-Dec-2024.) |
⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)) | ||
Theorem | unisucg 6441 | 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 6442. (Revised by BJ, 28-Dec-2024.) |
⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)) | ||
Theorem | unisuc 6442 | 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 6443 | 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 6444 | 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 6445 | 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 6446 | No successor is empty. (Contributed by NM, 3-Apr-1995.) |
⊢ suc 𝐴 ≠ ∅ | ||
Theorem | eqelsuc 6447 | 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 6448* | 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 6449 | 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 6450 | 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 6451 | 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 6452 | 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 6453 | 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 6454 | 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 6455 | 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 6456 | 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 6457 | 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 6458 | 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 6459 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | ||
Theorem | orduniss 6460 | An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.) |
⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | ||
Theorem | ordtri2or 6461 | 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 6462 | A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | ||
Theorem | ordtri2or3 6463 | A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6462. (Contributed by David Moews, 1-May-2017.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) | ||
Theorem | ordelinel 6464 | 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 6465 | 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 6466 | 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 6467 | 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 6468 | 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 6469 | 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 6470 | 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 6471 | The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) | ||
Theorem | ontr 6472 | 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 6473 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6483. (Revised by BJ, 28-Dec-2024.) |
⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) | ||
Theorem | onordi 6474 | An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ Ord 𝐴 | ||
Theorem | ontrciOLD 6475 | Obsolete version of ontr 6472 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 6476 | 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 6477 | 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 6478 | 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 6479 | An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) | ||
Theorem | onssnel2i 6480 | An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵) | ||
Theorem | onelini 6481 | 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 6482 | An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴) | ||
Theorem | onunisuci 6483 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ ∪ suc 𝐴 = 𝐴 | ||
Theorem | onsseli 6484 | 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 6485 | 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 6486 | 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 6487 | An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.) |
⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | ||
Theorem | snsn0non 6488 | The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7866). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6489. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ ¬ {{∅}} ∈ On | ||
Theorem | onxpdisj 6489 | 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 6488. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (On ∩ (V × V)) = ∅ | ||
Theorem | onnev 6490 | 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 6491 | Obsolete version of onnev 6490 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 6492 | Extend class notation with Russell's definition description binder (inverted iota). |
class (℩𝑥𝜑) | ||
Theorem | iotajust 6493* | Soundness justification theorem for df-iota 6494. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | ||
Definition | df-iota 6494* |
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 6513);
otherwise, it evaluates to the empty set (see iotanul 6520). 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 7387 (or iotacl 6528 for unbounded iota), as demonstrated in the proof of supub 9468. This can be easier than applying riotasbc 7389 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}} | ||
Theorem | dfiota2 6495* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} | ||
Theorem | nfiota1 6496 | Bound-variable hypothesis builder for the ℩ class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥(℩𝑥𝜑) | ||
Theorem | nfiotadw 6497* | Deduction version of nfiotaw 6498. Version of nfiotad 6499 with a disjoint variable condition, which does not require ax-13 2366. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2366. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiotaw 6498* | Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6500 with a disjoint variable condition, which does not require ax-13 2366. (Contributed by NM, 23-Aug-2011.) Avoid ax-13 2366. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) | ||
Theorem | nfiotad 6499 | Deduction version of nfiota 6500. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker nfiotadw 6497 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) | ||
Theorem | nfiota 6500 | Bound-variable hypothesis builder for the ℩ class. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker nfiotaw 6498 when possible. (Contributed by NM, 23-Aug-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥(℩𝑦𝜑) |
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