Theorem oeeu 7683

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 2622	. . . . . 6 ⊢ ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} = ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)}
2	1	oeeulem 7681	. . . . 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 1073	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑎)} ∈ On)
4		elex 3212	. . . 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		fvexd 6203	. . 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)
7		fvexd 6203	. . 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)
8		eqid 2622	. . . 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))
9		eqid 2622	. . . 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)))
10		eqid 2622	. . . 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)))
11	1, 8, 9, 10	oeeui 7682	. . 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))))))
12	5, 6, 7, 11	euotd 4975	. 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)))
13		df-3an 1039	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)))
14		ancom 466	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
15	13, 14	bitri 264	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴 ↑𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴 ↑𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
16	15	anbi1i 731	. . . . . . . . 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))
17	16	anbi2i 730	. . . . . . . 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)))
18		an12 838	. . . . . . . 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)))
19		anass 681	. . . . . . . 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))))
20	17, 18, 19	3bitri 286	. . . . . . 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))))
21	20	exbii 1774	. . . . . 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))))
22		df-rex 2918	. . . . . 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))))
23		r19.42v 3092	. . . . . 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)))
24	21, 22, 23	3bitr2i 288	. . . . 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)))
25	24	2exbii 1775	. . . 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)))
26		r2ex 3061	. . . 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)))
27	25, 26	bitr4i 267	. . 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))
28	27	eubii 2492	. 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))
29	12, 28	sylib 208	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 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ⟨cop 4183  ⟨cotp 4185  ∪ cuni 4436  ∩ cint 4475  Oncon0 5723  suc csuc 5725  ℩cio 5849  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  1_𝑜c1o 7553  2_𝑜c2o 7554   +_𝑜 coa 7557   ·_𝑜 comu 7558   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by: (None)