P. 434 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43301-43400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	omnord1 43301*	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 43302	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 8558. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ↑o 𝐵))
Theorem	oege2 43303	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 8560. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))
Theorem	rp-oelim2 43304	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 8567. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ↑o 𝐵))
Theorem	oeord2lim 43305	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 8554. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
Theorem	oeord2i 43306	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 43307	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 43308	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 43309	Ordinal 3 is the unordered triple containing ordinals 0, 1, and 2. (Contributed by RP, 8-Jul-2021.)
⊢ 3o = {∅, 1o, 2o}
Theorem	df3o3 43310	Ordinal 3, fully expanded. (Contributed by RP, 8-Jul-2021.)
⊢ 3o = {∅, {∅}, {∅, {∅}}}
Theorem	oenord1ex 43311	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 43312*	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 43313*	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 43314	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 43315*	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 43316	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 43317*	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 43318*	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 43319*	Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27704. (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 43320*	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 43321*	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 43322*	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 8524 or oawordeu 8522. (Contributed by RP, 7-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = 𝐶)
Theorem	nnawordexg 43323*	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 8604. (Contributed by RP, 7-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +o ω)) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
Theorem	succlg 43324	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 43325*	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 43326	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 43327	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 43328	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 43329	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 43330	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 43331	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 43398, limsssuc 7829. (Contributed by RP, 22-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
Theorem	tfsconcatlem 43332*	Lemma for tfsconcatun 43333. (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 43333*	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 43334*	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 43335*	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 43336*	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 43337*	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 43338*	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 43339*	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 43340*	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 43341*	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 43342*	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 43343*	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 43344*	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 43345*	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 43346*	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 43347*	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 43348	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 43349*	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 43350*	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 43351	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 43352	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 43353	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 43354	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 43355	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 43356	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 43357	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 43358	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 43359	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 43360	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 43361	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 43362	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 43363	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 43364	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 43365	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 43366*	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 43367	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 43368	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 43369	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 43370*	The class of all ordinal sums of elements from two ordinals is ordinal. Lemma for oaun3 43378. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)})
Theorem	oaun3lem2 43371*	The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 43378. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
Theorem	oaun3lem3 43372*	The class of all ordinal sums of elements from two ordinals is an ordinal. Lemma for oaun3 43378. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ∈ On)
Theorem	oaun3lem4 43373*	The class of all ordinal sums of elements from two ordinals is less than the successor to the sum. Lemma for oaun3 43378. (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 43374*	Two ways to express a class. (Contributed by RP, 13-Feb-2025.)
⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
Theorem	oadif1lem 43375*	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 43376*	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 43377*	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 43378*	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 43379*	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 43380*	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 43381*	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 43382*	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 43383*	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 43384*	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 43385*	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 43386*	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 43387*	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 43388	Natural addition with 1 is same as successor. (Contributed by RP, 31-Dec-2024.)
⊢ (𝐴 ∈ On → (𝐴 +no 1o) = suc 𝐴)
Theorem	naddass1 43389	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)))
Theorem	naddgeoa 43390	Natural addition results in a value greater than or equal than that of ordinal addition. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵))
Theorem	naddonnn 43391	Natural addition with a natural number on the right results in a value equal to that of ordinal addition. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐴 +no 𝐵))
Theorem	naddwordnexlem0 43392	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, (ω ·o suc 𝐶) lies between 𝐴 and 𝐵. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))
Theorem	naddwordnexlem1 43393	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, 𝐵 is equal to or larger than 𝐴. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	naddwordnexlem2 43394	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, 𝐵 is larger than 𝐴. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐵)
Theorem	naddwordnexlem3 43395*	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, every natural sum of 𝐴 with a natural number is less that 𝐵. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐴 +no 𝑥) ∈ 𝐵)
Theorem	oawordex3 43396*	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, some ordinal sum of 𝐴 is equal to 𝐵. This is a specialization of oawordex 8524. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Theorem	naddwordnexlem4 43397*	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, there exists a product with omega such that the ordinal sum with 𝐴 is less than or equal to 𝐵 while the natural sum is larger than 𝐵. (Contributed by RP, 15-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))    &   ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑀)    &   ⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +o 𝑦)}    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (On ∖ 1o)((𝐶 +o 𝑥) = 𝐷 ∧ (𝐴 +o (ω ·o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·o 𝑥))))
Theorem	ordsssucim 43398	If an ordinal is less than or equal to the successor of another, then the first is either less than or equal to the second or the first is equal to the successor of the second. Theorem 1 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 See also ordsssucb 43331 for a biimplication when 𝐴 is a set. (Contributed by RP, 3-Jan-2025.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
Theorem	insucid 43399	The intersection of a class and its successor is itself. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∩ suc 𝐴) = 𝐴
Theorem	om2 43400	Two ways to double an ordinal. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 +o 𝐴) = (𝐴 ·o 2o))