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Type | Label | Description |
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Statement | ||
Theorem | dfpred3 6301* | An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ 𝑋 ∈ V ⇒ ⊢ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | ||
Theorem | dfpred3g 6302* | An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) | ||
Theorem | elpredgg 6303 | Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) | ||
Theorem | elpredg 6304 | Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) (Proof shortened by BJ, 16-Oct-2024.) |
⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋)) | ||
Theorem | elpredimg 6305 | Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋) | ||
Theorem | elpredim 6306 | Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) (Proof shortened by BJ, 16-Oct-2024.) |
⊢ 𝑋 ∈ V ⇒ ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋) | ||
Theorem | elpred 6307 | Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) (Proof shortened by BJ, 16-Oct-2024.) |
⊢ 𝑌 ∈ V ⇒ ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) | ||
Theorem | predexg 6308 | The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.) |
⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V) | ||
Theorem | predasetexOLD 6309 | Obsolete form of predexg 6308 as of 27-Oct-2024. (Contributed by Scott Fenton, 8-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ Pred(𝑅, 𝐴, 𝑋) ∈ V | ||
Theorem | dffr4 6310* | Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅)) | ||
Theorem | predel 6311 | Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.) |
⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴) | ||
Theorem | predbrg 6312 | Closed form of elpredim 6306. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋) | ||
Theorem | predtrss 6313 | If 𝑅 is transitive over 𝐴 and 𝑌𝑅𝑋, then Pred(𝑅, 𝐴, 𝑌) is a subclass of Pred(𝑅, 𝐴, 𝑋). (Contributed by Scott Fenton, 28-Oct-2024.) |
⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋)) | ||
Theorem | predpo 6314 | Property of the predecessor class for partial orders. (Contributed by Scott Fenton, 28-Apr-2012.) (Proof shortened by Scott Fenton, 28-Oct-2024.) |
⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))) | ||
Theorem | predso 6315 | Property of the predecessor class for strict total orders. (Contributed by Scott Fenton, 11-Feb-2011.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))) | ||
Theorem | setlikespec 6316 | If 𝑅 is set-like in 𝐴, then all predecessor classes of elements of 𝐴 exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V) | ||
Theorem | predidm 6317 | Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.) |
⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋) | ||
Theorem | predin 6318 | Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.) |
⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) | ||
Theorem | predun 6319 | Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.) |
⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋)) | ||
Theorem | preddif 6320 | Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.) |
⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋)) | ||
Theorem | predep 6321 | The predecessor under the membership relation is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋)) | ||
Theorem | trpred 6322 | The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.) |
⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋) | ||
Theorem | preddowncl 6323* | A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.) |
⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))) | ||
Theorem | predpoirr 6324 | Given a partial ordering, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) |
⊢ (𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)) | ||
Theorem | predfrirr 6325 | Given a well-founded relation, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.) |
⊢ (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)) | ||
Theorem | pred0 6326 | The predecessor class over ∅ is always ∅. (Contributed by Scott Fenton, 16-Apr-2011.) (Proof shortened by AV, 11-Jun-2021.) |
⊢ Pred(𝑅, ∅, 𝑋) = ∅ | ||
Theorem | dfse3 6327* | Alternate definition of set-like relationships. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V) | ||
Theorem | predrelss 6328 | Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024.) |
⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋)) | ||
Theorem | predprc 6329 | The predecessor of a proper class is empty. (Contributed by Scott Fenton, 25-Nov-2024.) |
⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅) | ||
Theorem | predres 6330 | Predecessor class is unaffected by restriction to the base class. (Contributed by Scott Fenton, 25-Nov-2024.) |
⊢ Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) | ||
Theorem | frpomin 6331* | Every nonempty (possibly proper) subclass of a class 𝐴 with a well-founded set-like partial order 𝑅 has a minimal element. The additional condition of partial order over frmin 9740 enables avoiding the axiom of infinity. (Contributed by Scott Fenton, 11-Feb-2022.) |
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | frpomin2 6332* | Every nonempty (possibly proper) subclass of a class 𝐴 with a well-founded set-like partial order 𝑅 has a minimal element. The additional condition of partial order over frmin 9740 enables avoiding the axiom of infinity. (Contributed by Scott Fenton, 11-Feb-2022.) |
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅) | ||
Theorem | frpoind 6333* | The principle of well-founded induction over a partial order. This theorem is a version of frind 9741 that does not require the axiom of infinity and can be used to prove wfi 6341 and tfi 7835. (Contributed by Scott Fenton, 11-Feb-2022.) |
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵) | ||
Theorem | frpoinsg 6334* | Well-Founded Induction Schema (variant). If a property passes from all elements less than 𝑦 of a well-founded set-like partial order class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2022.) |
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑦 ∈ 𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | frpoins2fg 6335* | Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022.) |
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | frpoins2g 6336* | Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022.) |
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | frpoins3g 6337* | Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 19-Aug-2024.) |
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑥)𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝐵 ∈ 𝐴) → 𝜒) | ||
Theorem | tz6.26 6338* | All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.) |
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) | ||
Theorem | tz6.26OLD 6339* | Obsolete proof of tz6.26 6338 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) | ||
Theorem | tz6.26i 6340* | All nonempty subclasses of a class having a well-ordered set-like relation 𝑅 have 𝑅-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝑅 We 𝐴 & ⊢ 𝑅 Se 𝐴 ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) | ||
Theorem | wfi 6341* | The Principle of Well-Ordered Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.) |
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵) | ||
Theorem | wfiOLD 6342* | Obsolete proof of wfi 6341 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵) | ||
Theorem | wfii 6343* | The Principle of Well-Ordered Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝑅 We 𝐴 & ⊢ 𝑅 Se 𝐴 ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)) → 𝐴 = 𝐵) | ||
Theorem | wfisg 6344* | Well-Ordered Induction Schema. If a property passes from all elements less than 𝑦 of a well-ordered class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.) |
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | wfisgOLD 6345* | Obsolete version of wfisg 6344 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 11-Feb-2011.) |
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | wfis 6346* | Well-Ordered Induction Schema. If all elements less than a given set 𝑥 of the well-ordered class 𝐴 have a property (induction hypothesis), then all elements of 𝐴 have that property. (Contributed by Scott Fenton, 29-Jan-2011.) |
⊢ 𝑅 We 𝐴 & ⊢ 𝑅 Se 𝐴 & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ⇒ ⊢ (𝑦 ∈ 𝐴 → 𝜑) | ||
Theorem | wfis2fg 6347* | Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.) |
⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | wfis2fgOLD 6348* | Obsolete version of wfis2fg 6347 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 11-Feb-2011.) |
⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | wfis2f 6349* | Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.) |
⊢ 𝑅 We 𝐴 & ⊢ 𝑅 Se 𝐴 & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) ⇒ ⊢ (𝑦 ∈ 𝐴 → 𝜑) | ||
Theorem | wfis2g 6350* | Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | ||
Theorem | wfis2 6351* | Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.) |
⊢ 𝑅 We 𝐴 & ⊢ 𝑅 Se 𝐴 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) ⇒ ⊢ (𝑦 ∈ 𝐴 → 𝜑) | ||
Theorem | wfis3 6352* | Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.) |
⊢ 𝑅 We 𝐴 & ⊢ 𝑅 Se 𝐴 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) ⇒ ⊢ (𝐵 ∈ 𝐴 → 𝜒) | ||
Syntax | word 6353 | Extend the definition of a wff to include the ordinal predicate. |
wff Ord 𝐴 | ||
Syntax | con0 6354 | Extend the definition of a class to include the class of all ordinal numbers. (The 0 in the name prevents creating a file called con.html, which causes problems in Windows.) |
class On | ||
Syntax | wlim 6355 | Extend the definition of a wff to include the limit ordinal predicate. |
wff Lim 𝐴 | ||
Syntax | csuc 6356 | Extend class notation to include the successor function. |
class suc 𝐴 | ||
Definition | df-ord 6357 |
Define the ordinal predicate, which is true for a class that is transitive
and is well-ordered by the membership relation. Variant of definition of
[BellMachover] p. 468.
Some sources will define a notation for ordinal order corresponding to < and ≤ but we just use ∈ and ⊆ respectively. (Contributed by NM, 17-Sep-1993.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | ||
Definition | df-on 6358 | Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.) |
⊢ On = {𝑥 ∣ Ord 𝑥} | ||
Definition | df-lim 6359 | Define the limit ordinal predicate, which is true for a nonempty ordinal that is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42. See dflim2 6411, dflim3 7829, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.) |
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | ||
Definition | df-suc 6360 | Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 8526). Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Definition 1.4 of [Schloeder] p. 1, similarly. Ordinal natural numbers defined using this successor function and 0 as the empty set are also called von Neumann ordinals; 0 is the empty set {}, 1 is {0, {0}}, 2 is {1, {1}}, and so on. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 6430), so that the successor of any ordinal class is still an ordinal class (ordsuc 7794), simplifying certain proofs. Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.) |
⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | ||
Theorem | ordeq 6361 | Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.) |
⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) | ||
Theorem | elong 6362 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴)) | ||
Theorem | elon 6363 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ On ↔ Ord 𝐴) | ||
Theorem | eloni 6364 | An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ On → Ord 𝐴) | ||
Theorem | elon2 6365 | An ordinal number is an ordinal set. Part of Definition 1.2 of [Schloeder] p. 1. (Contributed by NM, 8-Feb-2004.) |
⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | ||
Theorem | limeq 6366 | Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵)) | ||
Theorem | ordwe 6367 | Membership well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
⊢ (Ord 𝐴 → E We 𝐴) | ||
Theorem | ordtr 6368 | An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.) |
⊢ (Ord 𝐴 → Tr 𝐴) | ||
Theorem | ordfr 6369 | Membership is well-founded on an ordinal class. In other words, an ordinal class is well-founded. (Contributed by NM, 22-Apr-1994.) |
⊢ (Ord 𝐴 → E Fr 𝐴) | ||
Theorem | ordelss 6370 | An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) | ||
Theorem | trssord 6371 | A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.) |
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) | ||
Theorem | ordirr 6372 | No ordinal class is a member of itself. In other words, the membership relation is irreflexive on ordinal classes. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. Theorem 1.9(i) of [Schloeder] p. 1. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.) |
⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | nordeq 6373 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) | ||
Theorem | ordn2lp 6374 | An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.) |
⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | ||
Theorem | tz7.5 6375* | A nonempty subclass of an ordinal class has a minimal element. Proposition 7.5 of [TakeutiZaring] p. 36. (Contributed by NM, 18-Feb-2004.) (Revised by David Abernethy, 16-Mar-2011.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) | ||
Theorem | ordelord 6376 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. Lemma 1.3 of [Schloeder] p. 1. (Contributed by NM, 23-Apr-1994.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | ||
Theorem | tron 6377 | The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.) |
⊢ Tr On | ||
Theorem | ordelon 6378 | An element of an ordinal class is an ordinal number. Lemma 1.3 of [Schloeder] p. 1. (Contributed by NM, 26-Oct-2003.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | ||
Theorem | onelon 6379 | An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. Lemma 1.3 of [Schloeder] p. 1. (Contributed by NM, 26-Oct-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | ||
Theorem | tz7.7 6380 | A transitive class belongs to an ordinal class iff it is strictly included in it. Proposition 7.7 of [TakeutiZaring] p. 37. (Contributed by NM, 5-May-1994.) |
⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ∈ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))) | ||
Theorem | ordelssne 6381 | For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 25-Nov-1995.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))) | ||
Theorem | ordelpss 6382 | For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵)) | ||
Theorem | ordsseleq 6383 | For ordinal classes, inclusion is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | ordin 6384 | The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | ||
Theorem | onin 6385 | The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On) | ||
Theorem | ordtri3or 6386 | 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 6387 | 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 6388 | A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | ||
Theorem | ordtri2 6389 | A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) | ||
Theorem | ordtri3 6390 | 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 6391 | 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 6392 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | ||
Theorem | onfr 6393 | The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7757 (through epweon 7755) in order to eliminate the need for the axiom of regularity. (Contributed by NM, 17-May-1994.) |
⊢ E Fr On | ||
Theorem | onelpss 6394 | 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 6395 | 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 6396 | 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 6397 | Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) |
⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | ||
Theorem | ordtr2 6398 | 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 6399 | 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 6400 | 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)) |
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