P. 443 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44201-44300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onexoegt 44201*	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 44202*	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 44203*	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 44204*	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 44205*	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 44206*	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 44207	Sufficient condition when the supremum of a set of ordinals is the maximum element of that set. See ordunifi 9265. (Contributed by RP, 27-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴))
Theorem	onelord 44208	Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6380 and eloni 6365. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
Theorem	onepsuc 44209	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 44210	The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7778 and weso 5642. (Contributed by RP, 15-Jan-2025.)
⊢ E Or On
Theorem	epirron 44211	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 44212	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 44213	The elementhood relation on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. See ontr1 6403. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	oneptri 44214	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 44215	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 44216	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 44217	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 44218	Ordinal zero is less than every nonzero ordinal. Theorem 1.10 of [Schloeder] p. 2. Closely related to ord0eln0 6412. (Contributed by RP, 16-Jan-2025.)
⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
Theorem	ondif1i 44219	Ordinal zero is less than every nonzero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8493. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
Theorem	onsucelab 44220*	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 44221*	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 44222*	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 44223	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 7818. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
Theorem	ordnexbtwnsuc 44224*	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 44225	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 44226*	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 44227*	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 44228*	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 44229*	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 44230*	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 7848. (Contributed by RP, 17-Jan-2025.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅))
Theorem	onov0suclim 44231	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 44232*	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 8508, oasuc 8516, and oalim 8524. (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 44233*	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 8509, omsuc 8518, and omlim 8525. (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 44234*	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 8514, oesuc 8519, oe0m1 8513, and oelim 8526. (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 44235	The operations of addition, multiplication, and exponentiation are closed. Remark 2.8 of [Schloeder] p. 5. See oacl 8527, omcl 8528, oecl 8529. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ On ∧ (𝐴 ·o 𝐵) ∈ On ∧ (𝐴 ↑o 𝐵) ∈ On))
Theorem	onsupsucismax 44236*	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 44237*	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 44238	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 44239*	A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6418. (Contributed by RP, 27-Jan-2025.)
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)
Theorem	limexissupab 44240*	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 44241	Ordinal one is both a left and right identity of ordinal multiplication. Lemma 2.15 of [Schloeder] p. 5. See om1 8534 and om1r 8535 for individual statements. (Contributed by RP, 29-Jan-2025.)
⊢ (𝐴 ∈ On → ((1o ·o 𝐴) = (𝐴 ·o 1o) ∧ (𝐴 ·o 1o) = 𝐴))
Theorem	oe0rif 44242	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 44243*	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 44244	Natural numbers are closed under ordinal addition, multiplication, and exponentiation. Theorem 2.20 of [Schloeder] p. 6. See nnacl 8604, nnmcl 8605, nnecl 8606. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ↑o 𝐵) ∈ ω))
Theorem	onsucwordi 44245	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 44246	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 44247	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 44248	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 44249	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 44250	Ordinal one plus omega is equal to omega. See oaabs 8641 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 8642 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 44251	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 44252	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 44253*	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 44254	Any nonzero ordinal product is greater-than-or-equal to the term on the left. Lemma 3.11 of [Schloeder] p. 8. See omword1 8565. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵))
Theorem	omge2 44255	Any nonzero ordinal product is greater-than-or-equal to the term on the right. Lemma 3.12 of [Schloeder] p. 9. See omword2 8566. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵))
Theorem	omlim2 44256	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 44257	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 44258	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 44259	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 44260	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 44261	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 44262*	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))
Theorem	oege1 44263	Any nonzero ordinal power is greater-than-or-equal to the term on the left. Lemma 3.19 of [Schloeder] p. 10. See oewordi 8584. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ↑o 𝐵))
Theorem	oege2 44264	Any power of an ordinal at least as large as two is greater-than-or-equal to the term on the right. Lemma 3.20 of [Schloeder] p. 10. See oeworde 8586. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))
Theorem	rp-oelim2 44265	The power of an ordinal at least as large as two with a limit ordinal on thr right is a limit ordinal. Lemma 3.21 of [Schloeder] p. 10. See oelimcl 8593. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ↑o 𝐵))
Theorem	oeord2lim 44266	Given a limit ordinal, the power of any base at least as large as two raised to an ordinal less than that limit ordinal is less than the power of that base raised to the limit ordinal . Lemma 3.22 of [Schloeder] p. 10. See oeordi 8580. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
Theorem	oeord2i 44267	Ordinal exponentiation of the same base at least as large as two preserves the ordering of the exponents. Lemma 3.23 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
Theorem	oeord2com 44268	When the same base at least as large as two is raised to ordinal powers, , ordering of the power is equivalent to the ordering of the exponents. Theorem 3.24 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
Theorem	nnoeomeqom 44269	Any natural number at least as large as two raised to the power of omega is omega. Lemma 3.25 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ ((𝐴 ∈ ω ∧ 1o ∈ 𝐴) → (𝐴 ↑o ω) = ω)
Theorem	df3o2 44270	Ordinal 3 is the unordered triple containing ordinals 0, 1, and 2. (Contributed by RP, 8-Jul-2021.)
⊢ 3o = {∅, 1o, 2o}
Theorem	df3o3 44271	Ordinal 3, fully expanded. (Contributed by RP, 8-Jul-2021.)
⊢ 3o = {∅, {∅}, {∅, {∅}}}
Theorem	oenord1ex 44272	When ordinals two and three are both raised to the power of omega, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))
Theorem	oenord1 44273*	When two ordinals (both at least as large as two) are raised to the same power, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 4-Feb-2025.)
⊢ ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))
Theorem	oaomoencom 44274*	Ordinal addition, multiplication, and exponentiation do not generally commute. Theorem 4.1 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))
Theorem	oenassex 44275	Ordinal two raised to two to the zeroth power is not the same as two squared then raised to the zeroth power. (Contributed by RP, 30-Jan-2025.)
⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)
Theorem	oenass 44276*	Ordinal exponentiation is not associative. Remark 4.6 of [Schloeder] p. 14. (Contributed by RP, 30-Jan-2025.)
⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐)
Theorem	cantnftermord 44277	For terms of the form of a power of omega times a nonzero natural number, ordering of the exponents implies ordering of the terms. Lemma 5.1 of [Schloeder] p. 15. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ (ω ∖ 1o) ∧ 𝐷 ∈ (ω ∖ 1o))) → (𝐴 ∈ 𝐵 → ((ω ↑o 𝐴) ·o 𝐶) ∈ ((ω ↑o 𝐵) ·o 𝐷)))
Theorem	cantnfub 44278*	Given a finite number of terms of the form ((ω ↑o (𝐴‘𝑛)) ·o (𝑀‘𝑛)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (ω ↑o 𝑋) when (𝐴‘𝑛) is less than 𝑋 and (𝑀‘𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 31-Jan-2025.)
⊢ (𝜑 → 𝑋 ∈ On)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝐴:𝑁–1-1→𝑋)    &   ⊢ (𝜑 → 𝑀:𝑁⟶ω)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅))    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ∧ ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋)))
Theorem	cantnfub2 44279*	Given a finite number of terms of the form ((ω ↑o (𝐴‘𝑛)) ·o (𝑀‘𝑛)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (ω ↑o suc ∪ ran 𝐴) when (𝑀‘𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 9-Feb-2025.)
⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝐴:𝑁–1-1→On)    &   ⊢ (𝜑 → 𝑀:𝑁⟶ω)    &   ⊢ 𝐹 = (𝑥 ∈ suc ∪ ran 𝐴 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅))    ⇒   ⊢ (𝜑 → (suc ∪ ran 𝐴 ∈ On ∧ 𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))
Theorem	bropabg 44280*	Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brslts 28127. (Contributed by RP, 26-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))
Theorem	cantnfresb 44281*	A Cantor normal form which sums to less than a certain power has only zeros for larger components. (Contributed by RP, 3-Feb-2025.)
⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶) ↔ ∀𝑥 ∈ (𝐵 ∖ 𝐶)(𝐹‘𝑥) = ∅))
Theorem	cantnf2 44282*	For every ordinal, 𝐴, there is a an ordinal exponent 𝑏 such that 𝐴 is less than (ω ↑o 𝑏) and for every ordinal at least as large as 𝑏 there is a unique Cantor normal form, 𝑓, with zeros for all the unnecessary higher terms, that sums to 𝐴. Theorem 5.3 of [Schloeder] p. 16. (Contributed by RP, 3-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑏 ∈ On ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))
21.38.2  Natural addition of Cantor normal forms
Theorem	oawordex2 44283*	If 𝐶 is between 𝐴 (inclusive) and (𝐴 +o 𝐵) (exclusive), there is an ordinal which equals 𝐶 when summed to 𝐴. This is a slightly different statement than oawordex 8549 or oawordeu 8547. (Contributed by RP, 7-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = 𝐶)
Theorem	nnawordexg 44284*	If an ordinal, 𝐵, is in a half-open interval between some 𝐴 and the next limit ordinal, 𝐵 is the sum of the 𝐴 and some natural number. This weakens the antecedent of nnawordex 8630. (Contributed by RP, 7-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +o ω)) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
Theorem	succlg 44285	Closure law for ordinal successor. (Contributed by RP, 8-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)))) → suc 𝐴 ∈ 𝐵)
Theorem	dflim5 44286*	A limit ordinal is either the proper class of ordinals or some nonzero product with omega. (Contributed by RP, 8-Jan-2025.)
⊢ (Lim 𝐴 ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥)))
Theorem	oacl2g 44287	Closure law for ordinal addition. Here we show that ordinal addition is closed within the empty set or any ordinal power of omega. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On))) → (𝐴 +o 𝐵) ∈ 𝐶)
Theorem	onmcl 44288	If an ordinal is less than a power of omega, the product with a natural number is also less than that power of omega. (Contributed by RP, 19-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 ∈ (ω ↑o 𝐵) → (𝐴 ·o 𝑁) ∈ (ω ↑o 𝐵)))
Theorem	omabs2 44289	Ordinal multiplication by a larger ordinal is absorbed when the larger ordinal is either 2 or ω raised to some power of ω. (Contributed by RP, 12-Jan-2025.)
⊢ (((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ 𝐵 = 2o ∨ (𝐵 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On))) → (𝐴 ·o 𝐵) = 𝐵)
Theorem	omcl2 44290	Closure law for ordinal multiplication. (Contributed by RP, 12-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
Theorem	omcl3g 44291	Closure law for ordinal multiplication. (Contributed by RP, 14-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
Theorem	ordsssucb 44292	An ordinal number is less than or equal to the successor of an ordinal class iff the ordinal number is either less than or equal to the ordinal class or the ordinal number is equal to the successor of the ordinal class. See also ordsssucim 44359, limsssuc 7850. (Contributed by RP, 22-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
Theorem	tfsconcatlem 44293*	Lemma for tfsconcatun 44294. (Contributed by RP, 23-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +o 𝐵) ∖ 𝐴)) → ∃!𝑥∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))
Theorem	tfsconcatun 44294*	The concatenation of two transfinite series is a union of functions. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
Theorem	tfsconcatfn 44295*	The concatenation of two transfinite series is a transfinite series. (Contributed by RP, 22-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷))
Theorem	tfsconcatfv1 44296*	An early value of the concatenation of two transfinite series. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴‘𝑋))
Theorem	tfsconcatfv2 44297*	A latter value of the concatenation of two transfinite series. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = (𝐵‘𝑋))
Theorem	tfsconcatfv 44298*	The value of the concatenation of two transfinite series. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +o 𝐷)) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑋))))
Theorem	tfsconcatrn 44299*	The range of the concatenation of two transfinite series. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))
Theorem	tfsconcatfo 44300*	The concatenation of two transfinite series is onto the union of the ranges. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))    ⇒   ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵):(𝐶 +o 𝐷)–onto→(ran 𝐴 ∪ ran 𝐵))