Theorem oeeu 7570

Description: The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)

Assertion

Ref	Expression
oeeu	⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧

Proof of Theorem oeeu

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . . . 6 ⊢ ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} = ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}
2	1	oeeulem 7568	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} ∈ On ∧ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})))
3	2	simp1d 1066	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} ∈ On)
4		elex 3185	. . . 4 ⊢ (∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} ∈ On → ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} ∈ V)
5	3, 4	syl 17	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} ∈ V)
6		fvex 6113	. . . 4 ⊢ (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V)
8		fvex 6113	. . . 4 ⊢ (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V
9	8	a1i 11	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V)
10		eqid 2610	. . . 4 ⊢ (℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)) = (℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))
11		eqid 2610	. . . 4 ⊢ (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) = (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)))
12		eqid 2610	. . . 4 ⊢ (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) = (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)))
13	1, 10, 11, 12	oeeui 7569	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ (𝑥 = ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} ∧ 𝑦 = (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∧ 𝑧 = (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑𝑜 ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))))))
14	5, 7, 9, 13	euotd 4900	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤∃𝑥∃𝑦∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
15		df-3an 1033	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)))
16		ancom 465	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
17	15, 16	bitri 263	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
18	17	anbi1i 727	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ((𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
19	18	anbi2i 726	. . . . . . . 8 ⊢ ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
20		an12 834	. . . . . . . 8 ⊢ ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
21		anass 679	. . . . . . . 8 ⊢ (((𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
22	19, 20, 21	3bitri 285	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
23	22	exbii 1764	. . . . . 6 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
24		df-rex 2902	. . . . . 6 ⊢ (∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
25		r19.42v 3073	. . . . . 6 ⊢ (∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
26	23, 24, 25	3bitr2i 287	. . . . 5 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
27	26	2exbii 1765	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑥∃𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
28		r2ex 3043	. . . 4 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ∃𝑥∃𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
29	27, 28	bitr4i 266	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
30	29	eubii 2480	. 2 ⊢ (∃!𝑤∃𝑥∃𝑦∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃!𝑤∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
31	14, 30	sylib 207	1 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴 ↑𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ⟨cop 4131  ⟨cotp 4133  ∪ cuni 4372  ∩ cint 4410  Oncon0 5640  suc csuc 5642  ℩cio 5766  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  1_𝑜c1o 7440  2_𝑜c2o 7441   +_𝑜 coa 7444   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by: (None)