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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | poeq1 4301 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴)) | ||
Theorem | poeq2 4302 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) | ||
Theorem | nfpo 4303 | Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Po 𝐴 | ||
Theorem | nfso 4304 | Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Or 𝐴 | ||
Theorem | pocl 4305 | Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))) | ||
Theorem | ispod 4306* | Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ⇒ ⊢ (𝜑 → 𝑅 Po 𝐴) | ||
Theorem | swopolem 4307* | Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌))) | ||
Theorem | swopo 4308* | A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → 𝑅 Po 𝐴) | ||
Theorem | poirr 4309 | A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | potr 4310 | A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) | ||
Theorem | po2nr 4311 | A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | ||
Theorem | po3nr 4312 | A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵)) | ||
Theorem | po0 4313 | Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ 𝑅 Po ∅ | ||
Theorem | pofun 4314* | A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.) |
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} & ⊢ (𝑥 = 𝑦 → 𝑋 = 𝑌) ⇒ ⊢ ((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) → 𝑆 Po 𝐴) | ||
Theorem | sopo 4315 | A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.) |
⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | ||
Theorem | soss 4316 | Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴)) | ||
Theorem | soeq1 4317 | Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴)) | ||
Theorem | soeq2 4318 | Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) | ||
Theorem | sonr 4319 | A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | sotr 4320 | A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) | ||
Theorem | issod 4321* | An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4299). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Po 𝐴) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ⇒ ⊢ (𝜑 → 𝑅 Or 𝐴) | ||
Theorem | sowlin 4322 | A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶))) | ||
Theorem | so2nr 4323 | A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | ||
Theorem | so3nr 4324 | A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵)) | ||
Theorem | sotricim 4325 | One direction of sotritric 4326 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) | ||
Theorem | sotritric 4326 | A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 28-Sep-2019.) |
⊢ 𝑅 Or 𝐴 & ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) ⇒ ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) | ||
Theorem | sotritrieq 4327 | A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.) |
⊢ 𝑅 Or 𝐴 & ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) ⇒ ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) | ||
Theorem | so0 4328 | Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ 𝑅 Or ∅ | ||
Syntax | wfrfor 4329 | Extend wff notation to include the well-founded predicate. |
wff FrFor 𝑅𝐴𝑆 | ||
Syntax | wfr 4330 | Extend wff notation to include the well-founded predicate. Read: ' 𝑅 is a well-founded relation on 𝐴.' |
wff 𝑅 Fr 𝐴 | ||
Syntax | wse 4331 | Extend wff notation to include the set-like predicate. Read: ' 𝑅 is set-like on 𝐴.' |
wff 𝑅 Se 𝐴 | ||
Syntax | wwe 4332 | Extend wff notation to include the well-ordering predicate. Read: ' 𝑅 well-orders 𝐴.' |
wff 𝑅 We 𝐴 | ||
Definition | df-frfor 4333* | Define the well-founded relation predicate where 𝐴 might be a proper class. By passing in 𝑆 we allow it potentially to be a proper class rather than a set. (Contributed by Jim Kingdon and Mario Carneiro, 22-Sep-2021.) |
⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆)) | ||
Definition | df-frind 4334* | Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because 𝑠 is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.) |
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠) | ||
Definition | df-se 4335* | Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.) |
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | ||
Definition | df-wetr 4336* | Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals do not have that as seen at ordtriexmid 4522). Given excluded middle, well-ordering is usually defined to require trichotomy (and the definition of Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.) |
⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | ||
Theorem | seex 4337* | The 𝑅-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.) |
⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) | ||
Theorem | exse 4338 | Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) | ||
Theorem | sess1 4339 | Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) | ||
Theorem | sess2 4340 | Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴)) | ||
Theorem | seeq1 4341 | Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) | ||
Theorem | seeq2 4342 | Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐴 ↔ 𝑅 Se 𝐵)) | ||
Theorem | nfse 4343 | Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Se 𝐴 | ||
Theorem | epse 4344 | The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.) |
⊢ E Se 𝐴 | ||
Theorem | frforeq1 4345 | Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.) |
⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇)) | ||
Theorem | freq1 4346 | Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴)) | ||
Theorem | frforeq2 4347 | Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇)) | ||
Theorem | freq2 4348 | Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) | ||
Theorem | frforeq3 4349 | Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.) |
⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇)) | ||
Theorem | nffrfor 4350 | Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝑆 ⇒ ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆 | ||
Theorem | nffr 4351 | Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 | ||
Theorem | frirrg 4352 | A well-founded relation is irreflexive. This is the case where 𝐴 exists. (Contributed by Jim Kingdon, 21-Sep-2021.) |
⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | fr0 4353 | Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.) |
⊢ 𝑅 Fr ∅ | ||
Theorem | frind 4354* | Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑)) & ⊢ (𝜒 → 𝑅 Fr 𝐴) & ⊢ (𝜒 → 𝐴 ∈ 𝑉) ⇒ ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → 𝜑) | ||
Theorem | efrirr 4355 | Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | tz7.2 4356 | Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.) |
⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) | ||
Theorem | nfwe 4357 | Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 We 𝐴 | ||
Theorem | weeq1 4358 | Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴)) | ||
Theorem | weeq2 4359 | Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵)) | ||
Theorem | wefr 4360 | A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.) |
⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | ||
Theorem | wepo 4361 | A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.) |
⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 Po 𝐴) | ||
Theorem | wetrep 4362* | An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.) |
⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | ||
Theorem | we0 4363 | Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.) |
⊢ 𝑅 We ∅ | ||
Syntax | word 4364 | Extend the definition of a wff to include the ordinal predicate. |
wff Ord 𝐴 | ||
Syntax | con0 4365 | 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 4366 | Extend the definition of a wff to include the limit ordinal predicate. |
wff Lim 𝐴 | ||
Syntax | csuc 4367 | Extend class notation to include the successor function. |
class suc 𝐴 | ||
Definition | df-iord 4368* |
Define the ordinal predicate, which is true for a class that is
transitive and whose elements are transitive. Definition of ordinal in
[Crosilla], p. "Set-theoretic
principles incompatible with
intuitionistic logic".
Some sources will define a notation for ordinal order corresponding to < and ≤ but we just use ∈ and ⊆ respectively. (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4369 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | ||
Theorem | dford3 4369* | Alias for df-iord 4368. Use it instead of df-iord 4368 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | ||
Definition | df-on 4370 | 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-ilim 4371 | Define the limit ordinal predicate, which is true for an ordinal that has the empty set as an element and 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, and then changes 𝐴 ≠ ∅ to ∅ ∈ 𝐴 (which would be equivalent given the law of the excluded middle, but which is not for us). (Contributed by Jim Kingdon, 11-Nov-2018.) Use its alias dflim2 4372 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | ||
Theorem | dflim2 4372 | Alias for df-ilim 4371. Use it instead of df-ilim 4371 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.) |
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | ||
Definition | df-suc 4373 | Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1". Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. 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 4414). 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 4374 | Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.) |
⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) | ||
Theorem | elong 4375 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴)) | ||
Theorem | elon 4376 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ On ↔ Ord 𝐴) | ||
Theorem | eloni 4377 | An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ On → Ord 𝐴) | ||
Theorem | elon2 4378 | An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.) |
⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | ||
Theorem | limeq 4379 | 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 | ordtr 4380 | An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.) |
⊢ (Ord 𝐴 → Tr 𝐴) | ||
Theorem | ordelss 4381 | 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 4382 | A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.) |
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) | ||
Theorem | ordelord 4383 | 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 4384 | The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.) |
⊢ Tr On | ||
Theorem | ordelon 4385 | An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | ||
Theorem | onelon 4386 | 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 | ordin 4387 | 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 4388 | The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On) | ||
Theorem | onelss 4389 | 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 4390 | Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) |
⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | ||
Theorem | ontr1 4391 | Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.) |
⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | ||
Theorem | onintss 4392* | 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 | ord0 4393 | The empty set is an ordinal class. (Contributed by NM, 11-May-1994.) |
⊢ Ord ∅ | ||
Theorem | 0elon 4394 | The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.) |
⊢ ∅ ∈ On | ||
Theorem | inton 4395 | The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.) |
⊢ ∩ On = ∅ | ||
Theorem | nlim0 4396 | 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 4397 | A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.) |
⊢ (Lim 𝐴 → Ord 𝐴) | ||
Theorem | limuni 4398 | A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.) |
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | ||
Theorem | limuni2 4399 | The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.) |
⊢ (Lim 𝐴 → Lim ∪ 𝐴) | ||
Theorem | 0ellim 4400 | A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.) |
⊢ (Lim 𝐴 → ∅ ∈ 𝐴) |
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