P. 435 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43401-43500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	oaabsb 43401	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 43402	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 43403*	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 43404	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 8497. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵))
Theorem	omge2 43405	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 8498. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵))
Theorem	omlim2 43406	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 43407	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 43408	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 43409	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 43410	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 43411	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 43412*	When the same non-zero 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 43413	Any non-zero ordinal power is greater-than-or-equal to the term on the left. Lemma 3.19 of [Schloeder] p. 10. See oewordi 8515. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ↑o 𝐵))
Theorem	oege2 43414	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 8517. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))
Theorem	rp-oelim2 43415	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 8524. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ↑o 𝐵))
Theorem	oeord2lim 43416	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 8511. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
Theorem	oeord2i 43417	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 43418	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 43419	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 43420	Ordinal 3 is the unordered triple containing ordinals 0, 1, and 2. (Contributed by RP, 8-Jul-2021.)
⊢ 3o = {∅, 1o, 2o}
Theorem	df3o3 43421	Ordinal 3, fully expanded. (Contributed by RP, 8-Jul-2021.)
⊢ 3o = {∅, {∅}, {∅, {∅}}}
Theorem	oenord1ex 43422	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 43423*	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 43424*	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 43425	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 43426*	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 43427	For terms of the form of a power of omega times a non-zero 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 43428*	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 43429*	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 43430*	Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27735. (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 43431*	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 43432*	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)))
21.36.2  Natural addition of Cantor normal forms
Theorem	oawordex2 43433*	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 8481 or oawordeu 8479. (Contributed by RP, 7-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = 𝐶)
Theorem	nnawordexg 43434*	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 8561. (Contributed by RP, 7-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +o ω)) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
Theorem	succlg 43435	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 43436*	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 43437	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 43438	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 43439	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 43440	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 43441	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 43442	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 43509, limsssuc 7789. (Contributed by RP, 22-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
Theorem	tfsconcatlem 43443*	Lemma for tfsconcatun 43444. (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 43444*	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 43445*	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 43446*	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 43447*	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 43448*	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 43449*	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 43450*	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))
Theorem	tfsconcatb0 43451*	The concatentation with the empty series leaves the series unchanged. (Contributed by RP, 25-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)) → (𝐵 = ∅ ↔ (𝐴 + 𝐵) = 𝐴))
Theorem	tfsconcat0i 43452*	The concatentation with the empty series leaves the series unchanged. (Contributed by RP, 28-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 = ∅ → (𝐴 + 𝐵) = 𝐵))
Theorem	tfsconcat0b 43453*	The concatentation with the empty series leaves the finite series unchanged. (Contributed by RP, 1-Mar-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 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵))
Theorem	tfsconcat00 43454*	The concatentation of two empty series results in an empty series. (Contributed by RP, 25-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 + 𝐵) = ∅))
Theorem	tfsconcatrev 43455*	If the domain of a transfinite sequence is an ordinal sum, the sequence can be decomposed into two sequences with domains corresponding to the addends. Theorem 2 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-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)))}))    ⇒   ⊢ ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∃𝑢 ∈ (ran 𝐹 ↑m 𝐶)∃𝑣 ∈ (ran 𝐹 ↑m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))
Theorem	tfsconcatrnss12 43456*	The range of the concatenation of transfinite sequences is a superset of the ranges of both sequences. Theorem 3 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-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 ⊆ ran (𝐴 + 𝐵) ∧ ran 𝐵 ⊆ ran (𝐴 + 𝐵)))
Theorem	tfsconcatrnss 43457*	The concatenation of transfinite sequences yields elements from a class iff both sequences yield elements from that class. (Contributed by RP, 2-Mar-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) ⊆ 𝑋 ↔ (ran 𝐴 ⊆ 𝑋 ∧ ran 𝐵 ⊆ 𝑋)))
Theorem	tfsconcatrnsson 43458*	The concatenation of transfinite sequences yields ordinals iff both sequences yield ordinals. Theorem 4 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-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) ⊆ On ↔ (ran 𝐴 ⊆ On ∧ ran 𝐵 ⊆ On)))
Theorem	tfsnfin 43459	A transfinite sequence is infinite iff its domain is greater than or equal to omega. Theorem 5 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 1-Mar-2025.)
⊢ ((𝐴 Fn 𝐵 ∧ 𝐵 ∈ On) → (¬ 𝐴 ∈ Fin ↔ ω ⊆ 𝐵))
Theorem	rp-tfslim 43460*	The limit of a sequence of ordinals is the union of its range. (Contributed by RP, 1-Mar-2025.)
⊢ (𝐴 Fn 𝐵 → ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran 𝐴)
Theorem	ofoafg 43461*	Addition operator for functions from sets into ordinals results in a function from the intersection of sets into an ordinal. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐹 ↑m 𝐶))
Theorem	ofoaf 43462	Addition operator for functions from sets into power of omega results in a function from the intersection of sets to that power of omega. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷))) → ( ∘f +o ↾ ((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐸 ↑m 𝐶))
Theorem	ofoafo 43463	Addition operator for functions from a set into a power of omega is an onto binary operator. (Contributed by RP, 5-Jan-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))–onto→(𝐶 ↑m 𝐴))
Theorem	ofoacl 43464	Closure law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) ∈ (𝐶 ↑m 𝐴))
Theorem	ofoaid1 43465	Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐹 ∘f +o (𝐴 × {∅})) = 𝐹)
Theorem	ofoaid2 43466	Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)
Theorem	ofoaass 43467	Component-wise addition of ordinal-yielding functions is associative. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
Theorem	ofoacom 43468	Component-wise addition of natural numnber-yielding functions commutes. (Contributed by RP, 5-Jan-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
Theorem	naddcnff 43469	Addition operator for Cantor normal forms is a function into Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
Theorem	naddcnffn 43470	Addition operator for Cantor normal forms is a function. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
Theorem	naddcnffo 43471	Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)
Theorem	naddcnfcl 43472	Closure law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) ∈ 𝑆)
Theorem	naddcnfcom 43473	Component-wise ordinal addition of Cantor normal forms commutes. (Contributed by RP, 2-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
Theorem	naddcnfid1 43474	Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
Theorem	naddcnfid2 43475	Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)
Theorem	naddcnfass 43476	Component-wise addition of Cantor normal forms is associative. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
Theorem	onsucunifi 43477*	The successor to the union of any non-empty, finite subset of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥)
Theorem	sucunisn 43478	The successor to the union of any singleton of a set is the successor of the set. (Contributed by RP, 11-Feb-2025.)
⊢ (𝐴 ∈ 𝑉 → suc ∪ {𝐴} = suc 𝐴)
Theorem	onsucunipr 43479	The successor to the union of any pair of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})
Theorem	onsucunitp 43480	The successor to the union of any triple of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})
Theorem	oaun3lem1 43481*	The class of all ordinal sums of elements from two ordinals is ordinal. Lemma for oaun3 43489. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)})
Theorem	oaun3lem2 43482*	The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 43489. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
Theorem	oaun3lem3 43483*	The class of all ordinal sums of elements from two ordinals is an ordinal. Lemma for oaun3 43489. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ∈ On)
Theorem	oaun3lem4 43484*	The class of all ordinal sums of elements from two ordinals is less than the successor to the sum. Lemma for oaun3 43489. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ∈ suc (𝐴 +o 𝐵))
Theorem	rp-abid 43485*	Two ways to express a class. (Contributed by RP, 13-Feb-2025.)
⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
Theorem	oadif1lem 43486*	Express the set difference of a continuous sum and its left addend as a class of sums. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊕ 𝐵) ∈ On)    &   ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 ⊕ 𝑏) ∈ On)    &   ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵))) → ∃𝑏 ∈ 𝐵 (𝐴 ⊕ 𝑏) = 𝑦)    &   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 ⊕ 𝑏) ∈ (𝐴 ⊕ 𝐵)))    &   ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 ⊕ 𝑏))    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ⊕ 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 ⊕ 𝑏)})
Theorem	oadif1 43487*	Express the set difference of an ordinal sum and its left addend as a class of sums. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +o 𝑏)})
Theorem	oaun2 43488*	Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}})
Theorem	oaun3 43489*	Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
Theorem	naddov4 43490*	Alternate expression for natural addition. (Contributed by RP, 19-Dec-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥}))
Theorem	nadd2rabtr 43491*	The set of ordinals which have a natural sum less than some ordinal is transitive. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
Theorem	nadd2rabord 43492*	The set of ordinals which have a natural sum less than some ordinal is an ordinal. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Ord {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})
Theorem	nadd2rabex 43493*	The class of ordinals which have a natural sum less than some ordinal is a set. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ∈ V)
Theorem	nadd2rabon 43494*	The set of ordinals which have a natural sum less than some ordinal is an ordinal number. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ∈ On)
Theorem	nadd1rabtr 43495*	The set of ordinals which have a natural sum less than some ordinal is transitive. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶})
Theorem	nadd1rabord 43496*	The set of ordinals which have a natural sum less than some ordinal is an ordinal. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Ord {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶})
Theorem	nadd1rabex 43497*	The class of ordinals which have a natural sum less than some ordinal is a set. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶} ∈ V)
Theorem	nadd1rabon 43498*	The set of ordinals which have a natural sum less than some ordinal is an ordinal number. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶} ∈ On)
Theorem	nadd1suc 43499	Natural addition with 1 is same as successor. (Contributed by RP, 31-Dec-2024.)
⊢ (𝐴 ∈ On → (𝐴 +no 1o) = suc 𝐴)
Theorem	naddass1 43500	Natural addition of ordinal numbers is associative when the third element is 1. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) +no 1o) = (𝐴 +no (𝐵 +no 1o)))