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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | unixp 6101 | The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.) |
⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) | ||
Theorem | unixp0 6102 | A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.) |
⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅) | ||
Theorem | unixpid 6103 | Field of a Cartesian square. (Contributed by FL, 10-Oct-2009.) |
⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴 | ||
Theorem | ressn 6104 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) | ||
Theorem | cnviin 6105* | The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.) |
⊢ (𝐴 ≠ ∅ → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵) | ||
Theorem | cnvpo 6106 | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴) | ||
Theorem | cnvso 6107 | The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.) |
⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | ||
Theorem | xpco 6108 | Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.) |
⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶)) | ||
Theorem | xpcoid 6109 | Composition of two Cartesian squares. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴) | ||
Theorem | elsnxp 6110* | Membership in a Cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = 〈𝑋, 𝑦〉)) | ||
Theorem | reu3op 6111* | There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 1-Jul-2023.) |
⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒 ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉))) | ||
Theorem | reuop 6112* | There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 23-Jun-2023.) |
⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝜓 ↔ 𝜒)) & ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝜃 → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉))) | ||
Theorem | opreu2reurex 6113* | There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.) |
⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒)) | ||
Theorem | opreu2reu 6114* | If there is a unique ordered pair fulfilling a wff, then there is a double restricted unique existential qualification fulfilling a corresponding wff. (Contributed by AV, 25-Jun-2023.) (Revised by AV, 2-Jul-2023.) |
⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒) | ||
Syntax | cpred 6115 | The predecessors symbol. |
class Pred(𝑅, 𝐴, 𝑋) | ||
Definition | df-pred 6116 | Define the predecessor class of a binary relation. This is the class of all elements 𝑦 of 𝐴 such that 𝑦𝑅𝑋 (see elpred 6129) . (Contributed by Scott Fenton, 29-Jan-2011.) |
⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | ||
Theorem | predeq123 6117 | Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌)) | ||
Theorem | predeq1 6118 | Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋)) | ||
Theorem | predeq2 6119 | Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) | ||
Theorem | predeq3 6120 | Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌)) | ||
Theorem | nfpred 6121 | Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝑋 ⇒ ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋) | ||
Theorem | predpredss 6122 | If 𝐴 is a subset of 𝐵, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋)) | ||
Theorem | predss 6123 | The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴 | ||
Theorem | sspred 6124 | Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.) |
⊢ ((𝐵 ⊆ 𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) | ||
Theorem | dfpred2 6125* | An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 8-Feb-2011.) |
⊢ 𝑋 ∈ V ⇒ ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) | ||
Theorem | dfpred3 6126* | 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 6127* | 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 | elpredim 6128 | Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) |
⊢ 𝑋 ∈ V ⇒ ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋) | ||
Theorem | elpred 6129 | Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) |
⊢ 𝑌 ∈ V ⇒ ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) | ||
Theorem | elpredg 6130 | Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) |
⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋)) | ||
Theorem | predasetex 6131 | The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) |
⊢ 𝐴 ∈ V ⇒ ⊢ Pred(𝑅, 𝐴, 𝑋) ∈ V | ||
Theorem | dffr4 6132* | 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 6133 | 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 | predpo 6134 | Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.) |
⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))) | ||
Theorem | predso 6135 | Property of the predecessor class for strict orderings. (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 | predbrg 6136 | Closed form of elpredim 6128. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋) | ||
Theorem | setlikespec 6137 | If 𝑅 is set-like in 𝐴, then all predecessors 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 6138 | 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 6139 | 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 6140 | 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 6141 | 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 6142 | 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 | preddowncl 6143* | 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 6144 | 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 6145 | 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 6146 | The predecessor class over ∅ is always ∅. (Contributed by Scott Fenton, 16-Apr-2011.) (Proof shortened by AV, 11-Jun-2021.) |
⊢ Pred(𝑅, ∅, 𝑋) = ∅ | ||
Theorem | tz6.26 6147* | 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.) |
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) | ||
Theorem | tz6.26i 6148* | All nonempty subclasses of a class having a well-founded 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 6149* | The Principle of Well-Founded 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 | wfii 6150* | The Principle of Well-Founded 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 6151* | Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (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 6152* | Well-Founded Induction Schema. If all elements less than a given set 𝑥 of the well-founded 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 6153* | Well-Founded Induction Schema, using implicit substitution. (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 6154* | Well Founded 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 6155* | Well-Founded 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 6156* | Well Founded 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 6157* | Well Founded 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 6158 | Extend the definition of a wff to include the ordinal predicate. |
wff Ord 𝐴 | ||
Syntax | con0 6159 | 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 6160 | Extend the definition of a wff to include the limit ordinal predicate. |
wff Lim 𝐴 | ||
Syntax | csuc 6161 | Extend class notation to include the successor function. |
class suc 𝐴 | ||
Definition | df-ord 6162 | 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. (Contributed by NM, 17-Sep-1993.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | ||
Definition | df-on 6163 | 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 6164 | 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 6215, dflim3 7542, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.) |
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | ||
Definition | df-suc 6165 | Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 8139). Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. 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 6234), so that the successor of any ordinal class is still an ordinal class (ordsuc 7509), 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 6166 | Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.) |
⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) | ||
Theorem | elong 6167 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴)) | ||
Theorem | elon 6168 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ On ↔ Ord 𝐴) | ||
Theorem | eloni 6169 | An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ On → Ord 𝐴) | ||
Theorem | elon2 6170 | An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.) |
⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | ||
Theorem | limeq 6171 | 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 6172 | 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 6173 | An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.) |
⊢ (Ord 𝐴 → Tr 𝐴) | ||
Theorem | ordfr 6174 | 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 6175 | 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 6176 | 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 6177 | 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. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.) |
⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | nordeq 6178 | 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 6179 | 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 6180* | 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 6181 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | ||
Theorem | tron 6182 | The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.) |
⊢ Tr On | ||
Theorem | ordelon 6183 | An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | ||
Theorem | onelon 6184 | An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | ||
Theorem | tz7.7 6185 | 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 6186 | 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 6187 | 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 6188 | 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 6189 | 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 6190 | 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 6191 | A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (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 6192 | 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 6193 | A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) | ||
Theorem | ordtri2 6194 | A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) | ||
Theorem | ordtri3 6195 | 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 6196 | 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 6197 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | ||
Theorem | onfr 6198 | The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7478 (through epweon 7477) in order to eliminate the need for the axiom of regularity. (Contributed by NM, 17-May-1994.) |
⊢ E Fr On | ||
Theorem | onelpss 6199 | 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 6200 | 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))) |
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