P. 444 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44301-44400   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	ctopsep 44301	The class of separable topologies.
class TopSep
Syntax	ctoplnd 44302	The class of Lindelöf topologies.
class TopLnd
Definition	df-topsep 44303*	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 44304*	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))}
21.38  Mathbox for Noam Pasman
Theorem	r1sssucd 44305	Deductive form of r1sssuc 9793. (Contributed by Noam Pasman, 19-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴))
21.39  Mathbox for Jon Pennant
Theorem	iocunico 44306	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 44307	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 44308	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 44309*	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 26166 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 44310	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 44311*	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))
21.40  Mathbox for Richard Penner
21.40.1  Set Theory and Ordinal Numbers
Theorem	uniel 44312*	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 44313*	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 44314*	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 44315*	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 44316*	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 44317*	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 44318*	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 44319	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 44320	The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On)
Theorem	onuniintrab 44321*	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 7815. (Contributed by RP, 28-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onintunirab 44322*	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 44323	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 44324	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 44325	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 44326*	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 44327*	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 44328*	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 44329*	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 44330*	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 44331	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 44332*	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 44333*	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 44334	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 44335*	For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)
Theorem	onexomgt 44336*	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 44337	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 44338*	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 44339*	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 44340*	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 44341*	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 44342*	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 44343*	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 44344*	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 44345	Sufficient condition when the supremum of a set of ordinals is the maximum element of that set. See ordunifi 9284. (Contributed by RP, 27-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴))
Theorem	onelord 44346	Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6387 and eloni 6372. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
Theorem	onepsuc 44347	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 44348	The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7789 and weso 5641. (Contributed by RP, 15-Jan-2025.)
⊢ E Or On
Theorem	epirron 44349	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 44350	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 44351	The elementhood relation on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. See ontr1 6410. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	oneptri 44352	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 44353	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 44354	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 44355	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 44356	Ordinal zero is less than every nonzero ordinal. Theorem 1.10 of [Schloeder] p. 2. Closely related to ord0eln0 6419. (Contributed by RP, 16-Jan-2025.)
⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
Theorem	ondif1i 44357	Ordinal zero is less than every nonzero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8512. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
Theorem	onsucelab 44358*	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 44359*	A limit ordinal is a nonzero 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 44360*	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 44361	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 7829. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
Theorem	ordnexbtwnsuc 44362*	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 44363	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 44364*	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 44365*	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 44366*	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 44367*	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 44368*	A limit ordinal is a nonzero ordinal that contains all the successors of its elements. Lemma 1.18 of [Schloeder] p. 2. Closely related to dflim4 7859. (Contributed by RP, 17-Jan-2025.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅))
Theorem	onov0suclim 44369	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 44370*	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 8527, oasuc 8535, and oalim 8543. (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 44371*	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 8528, omsuc 8537, and omlim 8544. (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 44372*	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 8533, oesuc 8538, oe0m1 8532, and oelim 8545. (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 44373	The operations of addition, multiplication, and exponentiation are closed. Remark 2.8 of [Schloeder] p. 5. See oacl 8546, omcl 8547, oecl 8548. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ On ∧ (𝐴 ·o 𝐵) ∈ On ∧ (𝐴 ↑o 𝐵) ∈ On))
Theorem	onsupsucismax 44374*	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 44375*	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 44376	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 44377*	A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6425. (Contributed by RP, 27-Jan-2025.)
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)
Theorem	limexissupab 44378*	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 44379	Ordinal one is both a left and right identity of ordinal multiplication. Lemma 2.15 of [Schloeder] p. 5. See om1 8553 and om1r 8554 for individual statements. (Contributed by RP, 29-Jan-2025.)
⊢ (𝐴 ∈ On → ((1o ·o 𝐴) = (𝐴 ·o 1o) ∧ (𝐴 ·o 1o) = 𝐴))
Theorem	oe0rif 44380	Ordinal zero raised to any nonzero 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 44381*	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 44382	Natural numbers are closed under ordinal addition, multiplication, and exponentiation. Theorem 2.20 of [Schloeder] p. 6. See nnacl 8623, nnmcl 8624, nnecl 8625. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ↑o 𝐵) ∈ ω))
Theorem	onsucwordi 44383	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 44384	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 44385	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 44386	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 44387	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 44388	Ordinal one plus omega is equal to omega. See oaabs 8660 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 8661 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 44389	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 44390	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 44391*	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 44392	Any nonzero ordinal product is greater-than-or-equal to the term on the left. Lemma 3.11 of [Schloeder] p. 8. See omword1 8584. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵))
Theorem	omge2 44393	Any nonzero ordinal product is greater-than-or-equal to the term on the right. Lemma 3.12 of [Schloeder] p. 9. See omword2 8585. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵))
Theorem	omlim2 44394	The nonzero 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 44395	Given a limit ordinal, the product of any nonzero ordinal with an ordinal less than that limit ordinal is less than the product of the nonzero 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 44396	Ordinal multiplication of the same nonzero 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 44397	When the same nonzero 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 44398	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 44399	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))
Theorem	omnord1 44400*	When the same nonzero ordinal is multiplied on the right, 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, 4-Feb-2025.)
⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐))