HomeTheorem List (p. 433 of 498)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hausgraph 43201 | The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
| ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×t 𝐾))) | ||
| Syntax | ctopsep 43202 | The class of separable topologies. |
| class TopSep | ||
| Syntax | ctoplnd 43203 | The class of Lindelöf topologies. |
| class TopLnd | ||
| Definition | df-topsep 43204* | A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.) |
| ⊢ TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)} | ||
| Definition | df-toplnd 43205* | A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.) |
| ⊢ TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ ∪ 𝑥 = ∪ 𝑧))} | ||
| Theorem | r1sssucd 43206 | Deductive form of r1sssuc 9743. (Contributed by Noam Pasman, 19-Jan-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ On) ⇒ ⊢ (𝜑 → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴)) | ||
| Theorem | iocunico 43207 | Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)) | ||
| Theorem | iocinico 43208 | The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝐵}) | ||
| Theorem | iocmbl 43209 | An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ∈ dom vol) | ||
| Theorem | cnioobibld 43210* | A bounded, continuous function on an open bounded interval is integrable. The function must be bounded. For a counterexample, consider 𝐹 = (𝑥 ∈ (0(,)1) ↦ (1 / 𝑥)). See cniccibl 25749 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝐿1) | ||
| Theorem | arearect 43211 | The area of a rectangle whose sides are parallel to the coordinate axes in (ℝ × ℝ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 𝐷 ∈ ℝ & ⊢ 𝐴 ≤ 𝐵 & ⊢ 𝐶 ≤ 𝐷 & ⊢ 𝑆 = ((𝐴[,]𝐵) × (𝐶[,]𝐷)) ⇒ ⊢ (area‘𝑆) = ((𝐵 − 𝐴) · (𝐷 − 𝐶)) | ||
| Theorem | areaquad 43212* | The area of a quadrilateral with two sides which are parallel to the y-axis in (ℝ × ℝ) is its width multiplied by the average height of its higher edge minus the average height of its lower edge. Co-author TA. (Contributed by Jon Pennant, 31-May-2019.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 𝐷 ∈ ℝ & ⊢ 𝐸 ∈ ℝ & ⊢ 𝐹 ∈ ℝ & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 ≤ 𝐸 & ⊢ 𝐷 ≤ 𝐹 & ⊢ 𝑈 = (𝐶 + (((𝑥 − 𝐴) / (𝐵 − 𝐴)) · (𝐷 − 𝐶))) & ⊢ 𝑉 = (𝐸 + (((𝑥 − 𝐴) / (𝐵 − 𝐴)) · (𝐹 − 𝐸))) & ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝑈[,]𝑉))} ⇒ ⊢ (area‘𝑆) = ((((𝐹 + 𝐸) / 2) − ((𝐷 + 𝐶) / 2)) · (𝐵 − 𝐴)) | ||
| Theorem | uniel 43213* | Two ways to say a union is an element of a class. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (∪ 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) | ||
| Theorem | unielss 43214* | Two ways to say the union of a class is an element of a subclass. (Contributed by RP, 29-Jan-2025.) |
| ⊢ (𝐴 ⊆ 𝐵 → (∪ 𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)) | ||
| Theorem | unielid 43215* | Two ways to say the union of a class is an element of that class. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) | ||
| Theorem | ssunib 43216* | Two ways to say a class is a subclass of a union. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) | ||
| Theorem | rp-intrabeq 43217* | Equality theorem for supremum of sets of ordinals. (Contributed by RP, 23-Jan-2025.) |
| ⊢ (𝐴 = 𝐵 → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥}) | ||
| Theorem | rp-unirabeq 43218* | Equality theorem for infimum of non-empty classes of ordinals. (Contributed by RP, 23-Jan-2025.) |
| ⊢ (𝐴 = 𝐵 → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦}) | ||
| Theorem | onmaxnelsup 43219* | Two ways to say the maximum element of a class of ordinals is also the supremum of that class. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (𝐴 ⊆ On → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)) | ||
| Theorem | onsupneqmaxlim0 43220 | If the supremum of a class of ordinals is not in that class, then the supremum is a limit ordinal or empty. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (𝐴 ⊆ On → (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)) | ||
| Theorem | onsupcl2 43221 | The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.) |
| ⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On) | ||
| Theorem | onuniintrab 43222* | The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Closed form of uniordint 7780. (Contributed by RP, 28-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) | ||
| Theorem | onintunirab 43223* | The intersection of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}) | ||
| Theorem | onsupnmax 43224 | If the union of a class of ordinals is not the maximum element of that class, then the union is a limit ordinal or empty. But this isn't a biconditional since 𝐴 could be a non-empty set where a limit ordinal or the empty set happens to be the largest element. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)) | ||
| Theorem | onsupuni 43225 | The supremum of a set of ordinals is the union of that set. Lemma 2.10 of [Schloeder] p. 5. (Contributed by RP, 19-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴) | ||
| Theorem | onsupuni2 43226 | The supremum of a set of ordinals is the union of that set. (Contributed by RP, 22-Jan-2025.) |
| ⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴) | ||
| Theorem | onsupintrab 43227* | The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Definition 2.9 of [Schloeder] p. 5. (Contributed by RP, 23-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) | ||
| Theorem | onsupintrab2 43228* | The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.) |
| ⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) | ||
| Theorem | onsupcl3 43229* | The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ On) | ||
| Theorem | onsupex3 43230* | The supremum of a set of ordinals exists. (Contributed by RP, 23-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ V) | ||
| Theorem | onuniintrab2 43231* | The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.) |
| ⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) | ||
| Theorem | oninfint 43232 | The infimum of a non-empty class of ordinals is the intersection of that class. (Contributed by RP, 23-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = ∩ 𝐴) | ||
| Theorem | oninfunirab 43233* | The infimum of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 23-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}) | ||
| Theorem | oninfcl2 43234* | The infimum of a non-empty class of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ On) | ||
| Theorem | onsupmaxb 43235 | The union of a class of ordinals is an element is an element of that class if and only if there is a maximum element of that class under the epsilon relation, which is to say that the domain of the restricted epsilon relation is not the whole class. (Contributed by RP, 25-Jan-2025.) |
| ⊢ (𝐴 ⊆ On → (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝐴)) | ||
| Theorem | onexgt 43236* | For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.) |
| ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥) | ||
| Theorem | onexomgt 43237* | For any ordinal, there is always a larger product of omega. (Contributed by RP, 1-Feb-2025.) |
| ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)) | ||
| Theorem | omlimcl2 43238 | The product of a limit ordinal with any nonzero ordinal is a limit ordinal. (Contributed by RP, 8-Jan-2025.) |
| ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴)) | ||
| Theorem | onexlimgt 43239* | For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.) |
| ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥)) | ||
| Theorem | onexoegt 43240* | For any ordinal, there is always a larger power of omega. (Contributed by RP, 1-Feb-2025.) |
| ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ↑o 𝑥)) | ||
| Theorem | oninfex2 43241* | The infimum of a non-empty class of ordinals exists. (Contributed by RP, 23-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ V) | ||
| Theorem | onsupeqmax 43242* | Condition when the supremum of a set of ordinals is the maximum element of that set. (Contributed by RP, 24-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴)) | ||
| Theorem | onsupeqnmax 43243* | Condition when the supremum of a class of ordinals is not the maximum element of that class. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (∪ 𝐴 = ∪ ∪ 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴))) | ||
| Theorem | onsuplub 43244* | The supremum of a set of ordinals is the least upper bound. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ On) → (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧)) | ||
| Theorem | onsupnub 43245* | An upper bound of a set of ordinals is not less than the supremum. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∪ 𝐴 ⊆ 𝐵) | ||
| Theorem | onfisupcl 43246 | Sufficient condition when the supremum of a set of ordinals is the maximum element of that set. See ordunifi 9244. (Contributed by RP, 27-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)) | ||
| Theorem | onelord 43247 | Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6360 and eloni 6345. (Contributed by RP, 15-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | ||
| Theorem | onepsuc 43248 | Every ordinal is less than its successor, relationship version. Lemma 1.7 of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.) |
| ⊢ (𝐴 ∈ On → 𝐴 E suc 𝐴) | ||
| Theorem | epsoon 43249 | The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7754 and weso 5632. (Contributed by RP, 15-Jan-2025.) |
| ⊢ E Or On | ||
| Theorem | epirron 43250 | The strict order on the ordinals is irreflexive. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.) |
| ⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴) | ||
| Theorem | oneptr 43251 | The strict order on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶)) | ||
| Theorem | oneltr 43252 | The elementhood relation on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. See ontr1 6382. (Contributed by RP, 15-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | ||
| Theorem | oneptri 43253 | The strict, complete (linear) order on the ordinals is complete. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵)) | ||
| Theorem | ordeldif 43254 | Membership in the difference of ordinals. (Contributed by RP, 15-Jan-2025.) |
| ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐶 ∈ (𝐴 ∖ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) | ||
| Theorem | ordeldifsucon 43255 | Membership in the difference of ordinal and successor ordinal. (Contributed by RP, 16-Jan-2025.) |
| ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))) | ||
| Theorem | ordeldif1o 43256 | Membership in the difference of ordinal and ordinal one. (Contributed by RP, 16-Jan-2025.) |
| ⊢ (Ord 𝐴 → (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅))) | ||
| Theorem | ordne0gt0 43257 | Ordinal zero is less than every non-zero ordinal. Theorem 1.10 of [Schloeder] p. 2. Closely related to ord0eln0 6391. (Contributed by RP, 16-Jan-2025.) |
| ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴) | ||
| Theorem | ondif1i 43258 | Ordinal zero is less than every non-zero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8468. (Contributed by RP, 16-Jan-2025.) |
| ⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴) | ||
| Theorem | onsucelab 43259* | The successor of every ordinal is an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.) |
| ⊢ (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}) | ||
| Theorem | dflim6 43260* | A limit ordinal is a non-zero ordinal which is not a successor ordinal. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.) |
| ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)) | ||
| Theorem | limnsuc 43261* | A limit ordinal is not an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.) |
| ⊢ (Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}) | ||
| Theorem | onsucss 43262 | If one ordinal is less than another, then the successor of the first is less than or equal to the second. Lemma 1.13 of [Schloeder] p. 2. See ordsucss 7796. (Contributed by RP, 16-Jan-2025.) |
| ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) | ||
| Theorem | ordnexbtwnsuc 43263* | For any distinct pair of ordinals, if there is no ordinal between the lesser and the greater, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.) |
| ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴)) | ||
| Theorem | orddif0suc 43264 | For any distinct pair of ordinals, if the set difference between the greater and the successor of the lesser is empty, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 17-Jan-2025.) |
| ⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴)) | ||
| Theorem | onsucf1lem 43265* | For ordinals, the successor operation is injective, so there is at most one ordinal that a given ordinal could be the successor of. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.) |
| ⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏) | ||
| Theorem | onsucf1olem 43266* | The successor operation is bijective between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏) | ||
| Theorem | onsucrn 43267* | The successor operation is surjective onto its range, the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.) |
| ⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥) ⇒ ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} | ||
| Theorem | onsucf1o 43268* | The successor operation is a bijective function between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.) |
| ⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥) ⇒ ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} | ||
| Theorem | dflim7 43269* | A limit ordinal is a non-zero ordinal that contains all the successors of its elements. Lemma 1.18 of [Schloeder] p. 2. Closely related to dflim4 7827. (Contributed by RP, 17-Jan-2025.) |
| ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅)) | ||
| Theorem | onov0suclim 43270 | Compactly express rules for binary operations on ordinals. (Contributed by RP, 18-Jan-2025.) |
| ⊢ (𝐴 ∈ On → (𝐴 ⊗ ∅) = 𝐷) & ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊗ suc 𝐶) = 𝐸) & ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 ⊗ 𝐵) = 𝐹) ⇒ ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹))) | ||
| Theorem | oa0suclim 43271* | Closed form expression of the value of ordinal addition for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.3 of [Schloeder] p. 4. See oa0 8483, oasuc 8491, and oalim 8499. (Contributed by RP, 18-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 +o 𝐵) = 𝐴) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 +o 𝐵) = suc (𝐴 +o 𝐶)) ∧ (Lim 𝐵 → (𝐴 +o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +o 𝑐)))) | ||
| Theorem | om0suclim 43272* | Closed form expression of the value of ordinal multiplication for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.5 of [Schloeder] p. 4. See om0 8484, omsuc 8493, and omlim 8500. (Contributed by RP, 18-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ·o 𝐵) = ((𝐴 ·o 𝐶) +o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ·o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 ·o 𝑐)))) | ||
| Theorem | oe0suclim 43273* | Closed form expression of the value of ordinal exponentiation for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.6 of [Schloeder] p. 4. See oe0 8489, oesuc 8494, oe0m1 8488, and oelim 8501. (Contributed by RP, 18-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ↑o 𝐵) = 1o) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐵) = ((𝐴 ↑o 𝐶) ·o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅)))) | ||
| Theorem | oaomoecl 43274 | The operations of addition, multiplication, and exponentiation are closed. Remark 2.8 of [Schloeder] p. 5. See oacl 8502, omcl 8503, oecl 8504. (Contributed by RP, 18-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ On ∧ (𝐴 ·o 𝐵) ∈ On ∧ (𝐴 ↑o 𝐵) ∈ On)) | ||
| Theorem | onsupsucismax 43275* | If the union of a set of ordinals is a successor ordinal, then that union is the maximum element of the set. This is not a bijection because sets where the maximum element is zero or a limit ordinal exist. Lemma 2.11 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴)) | ||
| Theorem | onsssupeqcond 43276* | If for every element of a set of ordinals there is an element of a subset which is at least as large, then the union of the set and the subset is the same. Lemma 2.12 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.) |
| ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐵 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏) → ∪ 𝐴 = ∪ 𝐵)) | ||
| Theorem | limexissup 43277 | An ordinal which is a limit ordinal is equal to its supremum. Lemma 2.13 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.) |
| ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup(𝐴, On, E )) | ||
| Theorem | limiun 43278* | A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6397. (Contributed by RP, 27-Jan-2025.) |
| ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥) | ||
| Theorem | limexissupab 43279* | An ordinal which is a limit ordinal is equal to the supremum of the class of all its elements. Lemma 2.13 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.) |
| ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E )) | ||
| Theorem | om1om1r 43280 | Ordinal one is both a left and right identity of ordinal multiplication. Lemma 2.15 of [Schloeder] p. 5. See om1 8509 and om1r 8510 for individual statements. (Contributed by RP, 29-Jan-2025.) |
| ⊢ (𝐴 ∈ On → ((1o ·o 𝐴) = (𝐴 ·o 1o) ∧ (𝐴 ·o 1o) = 𝐴)) | ||
| Theorem | oe0rif 43281 | Ordinal zero raised to any non-zero ordinal power is zero and zero to the zeroth power is one. Lemma 2.18 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.) |
| ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ ∈ 𝐴, ∅, 1o)) | ||
| Theorem | oasubex 43282* | While subtraction can't be a binary operation on ordinals, for any pair of ordinals there exists an ordinal that can be added to the lessor (or equal) one which will sum to the greater. Theorem 2.19 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴)) | ||
| Theorem | nnamecl 43283 | Natural numbers are closed under ordinal addition, multiplication, and exponentiation. Theorem 2.20 of [Schloeder] p. 6. See nnacl 8578, nnmcl 8579, nnecl 8580. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ↑o 𝐵) ∈ ω)) | ||
| Theorem | onsucwordi 43284 | The successor operation preserves the less-than-or-equal relationship between ordinals. Lemma 3.1 of [Schloeder] p. 7. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵)) | ||
| Theorem | oalim2cl 43285 | The ordinal sum of any ordinal with a limit ordinal on the right is a limit ordinal. (Contributed by RP, 6-Feb-2025.) |
| ⊢ ((𝐴 ∈ On ∧ Lim 𝐵 ∧ 𝐵 ∈ 𝑉) → Lim (𝐴 +o 𝐵)) | ||
| Theorem | oaltublim 43286 | Given 𝐶 is a limit ordinal, the sum of any ordinal with an ordinal less than 𝐶 is less than the sum of the first ordinal with 𝐶. Lemma 3.5 of [Schloeder] p. 7. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶)) | ||
| Theorem | oaordi3 43287 | Ordinal addition of the same number on the left preserves the ordering of the numbers on the right. Lemma 3.6 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶))) | ||
| Theorem | oaord3 43288 | When the same ordinal is added on the left, ordering of the sums is equivalent to the ordering of the ordinals on the right. Theorem 3.7 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶))) | ||
| Theorem | 1oaomeqom 43289 | Ordinal one plus omega is equal to omega. See oaabs 8615 for the sum of any natural number on the left and ordinal at least as large as omega on the right. Lemma 3.8 of [Schloeder] p. 8. See oaabs2 8616 where a power of omega is the upper bound of the left and a lower bound on the right. (Contributed by RP, 29-Jan-2025.) |
| ⊢ (1o +o ω) = ω | ||
| Theorem | oaabsb 43290 | The right addend absorbs the sum with an ordinal iff that ordinal times omega is less than or equal to the right addend. (Contributed by RP, 19-Feb-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ (𝐴 +o 𝐵) = 𝐵)) | ||
| Theorem | oaordnrex 43291 | When omega is added on the right to ordinals zero and one, ordering of the sums is not equivalent to the ordering of the ordinals on the left. Remark 3.9 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω)) | ||
| Theorem | oaordnr 43292* | When the same ordinal is added on the right, ordering of the sums is not equivalent to the ordering of the ordinals on the left. Remark 3.9 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +o 𝑐) ∈ (𝑏 +o 𝑐)) | ||
| Theorem | omge1 43293 | Any non-zero ordinal product is greater-than-or-equal to the term on the left. Lemma 3.11 of [Schloeder] p. 8. See omword1 8540. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵)) | ||
| Theorem | omge2 43294 | Any non-zero ordinal product is greater-than-or-equal to the term on the right. Lemma 3.12 of [Schloeder] p. 9. See omword2 8541. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵)) | ||
| Theorem | omlim2 43295 | The non-zero product with an limit ordinal on the right is a limit ordinal. Lemma 3.13 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.) |
| ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ·o 𝐵)) | ||
| Theorem | omord2lim 43296 | Given a limit ordinal, the product of any non-zero ordinal with an ordinal less than that limit ordinal is less than the product of the non-zero ordinal with the limit ordinal . Lemma 3.14 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.) |
| ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))) | ||
| Theorem | omord2i 43297 | Ordinal multiplication of the same non-zero number on the left preserves the ordering of the numbers on the right. Lemma 3.15 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.) |
| ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))) | ||
| Theorem | omord2com 43298 | When the same non-zero ordinal is multiplied on the left, ordering of the products is equivalent to the ordering of the ordinals on the right. Theorem 3.16 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∈ 𝐶 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))) | ||
| Theorem | 2omomeqom 43299 | Ordinal two times omega is omega. Lemma 3.17 of [Schloeder] p. 10. (Contributed by RP, 30-Jan-2025.) |
| ⊢ (2o ·o ω) = ω | ||
| Theorem | omnord1ex 43300 | When omega is multiplied on the right to ordinals one and two, ordering of the products is not equivalent to the ordering of the ordinals on the left. Remark 3.18 of [Schloeder] p. 10. (Contributed by RP, 29-Jan-2025.) |
| ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) | ||
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