Theorem oeeui 7682

Description: The division algorithm for ordinal exponentiation. (This version of oeeu 7683 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 7665.) (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
oeeu.1	⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}
oeeu.2	⊢ 𝑃 = (℩𝑤∃𝑦 ∈ On ∃𝑧 ∈ (𝐴 ↑𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
oeeu.3	⊢ 𝑌 = (1st ‘𝑃)
oeeu.4	⊢ 𝑍 = (2nd ‘𝑃)

Assertion

Ref	Expression
oeeui	⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜) ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝐶)) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))

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

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑦,𝑧,𝑤)   𝑃(𝑥,𝑦,𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)   𝑋(𝑥)   𝑌(𝑥,𝑦,𝑧,𝑤)   𝑍(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem oeeui

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

Step	Hyp	Ref	Expression
1		eldifi 3732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
2	1	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐴 ∈ On)
3	2	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐴 ∈ On)
4		simprl 794	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐶 ∈ On)
5		oecl 7617	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 𝐶) ∈ On)
6	3, 4, 5	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 𝐶) ∈ On)
7		om1 7622	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ↑𝑜 𝐶) ∈ On → ((𝐴 ↑𝑜 𝐶) ·𝑜 1𝑜) = (𝐴 ↑𝑜 𝐶))
8	6, 7	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 1𝑜) = (𝐴 ↑𝑜 𝐶))
9		df1o2 7572	. . . . . . . . . . . . . . . 16 ⊢ 1𝑜 = {∅}
10		dif1o 7580	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (𝐴 ∖ 1𝑜) ↔ (𝐷 ∈ 𝐴 ∧ 𝐷 ≠ ∅))
11	10	simprbi 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (𝐴 ∖ 1𝑜) → 𝐷 ≠ ∅)
12	11	ad2antll 765	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐷 ≠ ∅)
13		eldifi 3732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (𝐴 ∖ 1𝑜) → 𝐷 ∈ 𝐴)
14	13	ad2antll 765	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐷 ∈ 𝐴)
15		onelon 5748	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ 𝐴) → 𝐷 ∈ On)
16	3, 14, 15	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐷 ∈ On)
17		on0eln0 5780	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 ↔ 𝐷 ≠ ∅))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (∅ ∈ 𝐷 ↔ 𝐷 ≠ ∅))
19	12, 18	mpbird 247	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ∅ ∈ 𝐷)
20	19	snssd 4340	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → {∅} ⊆ 𝐷)
21	9, 20	syl5eqss 3649	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 1𝑜 ⊆ 𝐷)
22		1on 7567	. . . . . . . . . . . . . . . . 17 ⊢ 1𝑜 ∈ On
23	22	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 1𝑜 ∈ On)
24		omwordi 7651	. . . . . . . . . . . . . . . 16 ⊢ ((1𝑜 ∈ On ∧ 𝐷 ∈ On ∧ (𝐴 ↑𝑜 𝐶) ∈ On) → (1𝑜 ⊆ 𝐷 → ((𝐴 ↑𝑜 𝐶) ·𝑜 1𝑜) ⊆ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷)))
25	23, 16, 6, 24	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (1𝑜 ⊆ 𝐷 → ((𝐴 ↑𝑜 𝐶) ·𝑜 1𝑜) ⊆ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷)))
26	21, 25	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 1𝑜) ⊆ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷))
27	8, 26	eqsstr3d 3640	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 𝐶) ⊆ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷))
28		omcl 7616	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑𝑜 𝐶) ∈ On ∧ 𝐷 ∈ On) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ On)
29	6, 16, 28	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ On)
30		simplrl 800	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐸 ∈ (𝐴 ↑𝑜 𝐶))
31		onelon 5748	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑𝑜 𝐶) ∈ On ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝐶)) → 𝐸 ∈ On)
32	6, 30, 31	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐸 ∈ On)
33		oaword1 7632	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ⊆ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸))
34	29, 32, 33	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ⊆ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸))
35		simplrr 801	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)
36	34, 35	sseqtrd 3641	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ⊆ 𝐵)
37	27, 36	sstrd 3613	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 𝐶) ⊆ 𝐵)
38		oeeu.1	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}
39	38	oeeulem 7681	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝑋 ∈ On ∧ (𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)))
40	39	simp3d 1075	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋))
41	40	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋))
42	39	simp1d 1073	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ On)
43	42	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝑋 ∈ On)
44		suceloni 7013	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ On → suc 𝑋 ∈ On)
45	43, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → suc 𝑋 ∈ On)
46		oecl 7617	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ suc 𝑋 ∈ On) → (𝐴 ↑𝑜 suc 𝑋) ∈ On)
47	3, 45, 46	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 suc 𝑋) ∈ On)
48		ontr2 5772	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑𝑜 𝐶) ∈ On ∧ (𝐴 ↑𝑜 suc 𝑋) ∈ On) → (((𝐴 ↑𝑜 𝐶) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)) → (𝐴 ↑𝑜 𝐶) ∈ (𝐴 ↑𝑜 suc 𝑋)))
49	6, 47, 48	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴 ↑𝑜 𝐶) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)) → (𝐴 ↑𝑜 𝐶) ∈ (𝐴 ↑𝑜 suc 𝑋)))
50	37, 41, 49	mp2and 715	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 𝐶) ∈ (𝐴 ↑𝑜 suc 𝑋))
51		simplll 798	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐴 ∈ (On ∖ 2𝑜))
52		oeord 7668	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ suc 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐶 ∈ suc 𝑋 ↔ (𝐴 ↑𝑜 𝐶) ∈ (𝐴 ↑𝑜 suc 𝑋)))
53	4, 45, 51, 52	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐶 ∈ suc 𝑋 ↔ (𝐴 ↑𝑜 𝐶) ∈ (𝐴 ↑𝑜 suc 𝑋)))
54	50, 53	mpbird 247	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐶 ∈ suc 𝑋)
55		onsssuc 5813	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶 ⊆ 𝑋 ↔ 𝐶 ∈ suc 𝑋))
56	4, 43, 55	syl2anc 693	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐶 ⊆ 𝑋 ↔ 𝐶 ∈ suc 𝑋))
57	54, 56	mpbird 247	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐶 ⊆ 𝑋)
58	39	simp2d 1074	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 𝑋) ⊆ 𝐵)
59	58	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 𝑋) ⊆ 𝐵)
60		eloni 5733	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → Ord 𝐴)
61	3, 60	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → Ord 𝐴)
62		ordsucss 7018	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐴 → (𝐷 ∈ 𝐴 → suc 𝐷 ⊆ 𝐴))
63	61, 14, 62	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → suc 𝐷 ⊆ 𝐴)
64		suceloni 7013	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ On → suc 𝐷 ∈ On)
65	16, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → suc 𝐷 ∈ On)
66		dif20el 7585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
67	51, 66	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ∅ ∈ 𝐴)
68		oen0 7666	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝐶))
69	3, 4, 67, 68	syl21anc 1325	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ∅ ∈ (𝐴 ↑𝑜 𝐶))
70		omword 7650	. . . . . . . . . . . . . . . 16 ⊢ (((suc 𝐷 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝐶) ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝐶)) → (suc 𝐷 ⊆ 𝐴 ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 suc 𝐷) ⊆ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴)))
71	65, 3, 6, 69, 70	syl31anc 1329	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (suc 𝐷 ⊆ 𝐴 ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 suc 𝐷) ⊆ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴)))
72	63, 71	mpbid 222	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 suc 𝐷) ⊆ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
73		oaord 7627	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ On ∧ (𝐴 ↑𝑜 𝐶) ∈ On ∧ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ On) → (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ↔ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴 ↑𝑜 𝐶))))
74	32, 6, 29, 73	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ↔ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴 ↑𝑜 𝐶))))
75	30, 74	mpbid 222	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴 ↑𝑜 𝐶)))
76	35, 75	eqeltrrd 2702	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴 ↑𝑜 𝐶)))
77		odi 7659	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ↑𝑜 𝐶) ∈ On ∧ 𝐷 ∈ On ∧ 1𝑜 ∈ On) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐷 +𝑜 1𝑜)) = (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 ((𝐴 ↑𝑜 𝐶) ·𝑜 1𝑜)))
78	6, 16, 23, 77	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐷 +𝑜 1𝑜)) = (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 ((𝐴 ↑𝑜 𝐶) ·𝑜 1𝑜)))
79		oa1suc 7611	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ On → (𝐷 +𝑜 1𝑜) = suc 𝐷)
80	16, 79	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐷 +𝑜 1𝑜) = suc 𝐷)
81	80	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐷 +𝑜 1𝑜)) = ((𝐴 ↑𝑜 𝐶) ·𝑜 suc 𝐷))
82	8	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 ((𝐴 ↑𝑜 𝐶) ·𝑜 1𝑜)) = (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴 ↑𝑜 𝐶)))
83	78, 81, 82	3eqtr3d 2664	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → ((𝐴 ↑𝑜 𝐶) ·𝑜 suc 𝐷) = (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 (𝐴 ↑𝑜 𝐶)))
84	76, 83	eleqtrrd 2704	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 suc 𝐷))
85	72, 84	sseldd 3604	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
86		oesuc 7607	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 suc 𝐶) = ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
87	3, 4, 86	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 suc 𝐶) = ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
88	85, 87	eleqtrrd 2704	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐶))
89		oecl 7617	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑𝑜 𝑋) ∈ On)
90	3, 43, 89	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 𝑋) ∈ On)
91		suceloni 7013	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
92	91	ad2antrl 764	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → suc 𝐶 ∈ On)
93		oecl 7617	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴 ↑𝑜 suc 𝐶) ∈ On)
94	3, 92, 93	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 suc 𝐶) ∈ On)
95		ontr2 5772	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ (𝐴 ↑𝑜 suc 𝐶) ∈ On) → (((𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐶)) → (𝐴 ↑𝑜 𝑋) ∈ (𝐴 ↑𝑜 suc 𝐶)))
96	90, 94, 95	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (((𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐶)) → (𝐴 ↑𝑜 𝑋) ∈ (𝐴 ↑𝑜 suc 𝐶)))
97	59, 88, 96	mp2and 715	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐴 ↑𝑜 𝑋) ∈ (𝐴 ↑𝑜 suc 𝐶))
98		oeord 7668	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ On ∧ suc 𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑋 ∈ suc 𝐶 ↔ (𝐴 ↑𝑜 𝑋) ∈ (𝐴 ↑𝑜 suc 𝐶)))
99	43, 92, 51, 98	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝑋 ∈ suc 𝐶 ↔ (𝐴 ↑𝑜 𝑋) ∈ (𝐴 ↑𝑜 suc 𝐶)))
100	97, 99	mpbird 247	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝑋 ∈ suc 𝐶)
101		onsssuc 5813	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ On ∧ 𝐶 ∈ On) → (𝑋 ⊆ 𝐶 ↔ 𝑋 ∈ suc 𝐶))
102	43, 4, 101	syl2anc 693	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝑋 ⊆ 𝐶 ↔ 𝑋 ∈ suc 𝐶))
103	100, 102	mpbird 247	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝑋 ⊆ 𝐶)
104	57, 103	eqssd 3620	. . . . . . . 8 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → 𝐶 = 𝑋)
105	104, 16	jca 554	. . . . . . 7 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜))) → (𝐶 = 𝑋 ∧ 𝐷 ∈ On))
106		simprl 794	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐶 = 𝑋)
107	42	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝑋 ∈ On)
108	106, 107	eqeltrd 2701	. . . . . . . 8 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐶 ∈ On)
109	2	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐴 ∈ On)
110	109, 108, 5	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑𝑜 𝐶) ∈ On)
111		simprr 796	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
112	110, 111, 28	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ On)
113		simplrl 800	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐸 ∈ (𝐴 ↑𝑜 𝐶))
114	110, 113, 31	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐸 ∈ On)
115	112, 114, 33	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ⊆ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸))
116		simplrr 801	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)
117	115, 116	sseqtrd 3641	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ⊆ 𝐵)
118	40	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋))
119		suceq 5790	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = 𝑋 → suc 𝐶 = suc 𝑋)
120	119	ad2antrl 764	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → suc 𝐶 = suc 𝑋)
121	120	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑𝑜 suc 𝐶) = (𝐴 ↑𝑜 suc 𝑋))
122	109, 108, 86	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑𝑜 suc 𝐶) = ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
123	121, 122	eqtr3d 2658	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑𝑜 suc 𝑋) = ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
124	118, 123	eleqtrd 2703	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐵 ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
125		omcl 7616	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑𝑜 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ On)
126	110, 109, 125	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ On)
127		ontr2 5772	. . . . . . . . . . . 12 ⊢ ((((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ On ∧ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ On) → ((((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ⊆ 𝐵 ∧ 𝐵 ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴)))
128	112, 126, 127	syl2anc 693	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ⊆ 𝐵 ∧ 𝐵 ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴)))
129	117, 124, 128	mp2and 715	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
130	66	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∅ ∈ 𝐴)
131	130	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ∅ ∈ 𝐴)
132	109, 108, 131, 68	syl21anc 1325	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ∅ ∈ (𝐴 ↑𝑜 𝐶))
133		omord2 7647	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝐶) ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝐶)) → (𝐷 ∈ 𝐴 ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴)))
134	111, 109, 110, 132, 133	syl31anc 1329	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐷 ∈ 𝐴 ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) ∈ ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴)))
135	129, 134	mpbird 247	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐷 ∈ 𝐴)
136	106	oveq2d 6666	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑𝑜 𝐶) = (𝐴 ↑𝑜 𝑋))
137	58	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑𝑜 𝑋) ⊆ 𝐵)
138	136, 137	eqsstrd 3639	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑𝑜 𝐶) ⊆ 𝐵)
139		eldifi 3732	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ∈ On)
140	139	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ On)
141	140	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐵 ∈ On)
142		ontri1 5757	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑𝑜 𝐶) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑𝑜 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝐶)))
143	110, 141, 142	syl2anc 693	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑𝑜 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝐶)))
144	138, 143	mpbid 222	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝐶))
145		om0 7597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ↑𝑜 𝐶) ∈ On → ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) = ∅)
146	110, 145	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) = ∅)
147	146	oveq1d 6665	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸) = (∅ +𝑜 𝐸))
148		oa0r 7618	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ On → (∅ +𝑜 𝐸) = 𝐸)
149	114, 148	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (∅ +𝑜 𝐸) = 𝐸)
150	147, 149	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸) = 𝐸)
151	150, 113	eqeltrd 2701	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸) ∈ (𝐴 ↑𝑜 𝐶))
152		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = ∅ → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) = ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅))
153	152	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝐷 = ∅ → (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = (((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸))
154	153	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ (𝐷 = ∅ → ((((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (𝐴 ↑𝑜 𝐶) ↔ (((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) +𝑜 𝐸) ∈ (𝐴 ↑𝑜 𝐶)))
155	151, 154	syl5ibrcom 237	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐷 = ∅ → (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (𝐴 ↑𝑜 𝐶)))
156	116	eleq1d 2686	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) ∈ (𝐴 ↑𝑜 𝐶) ↔ 𝐵 ∈ (𝐴 ↑𝑜 𝐶)))
157	155, 156	sylibd 229	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐷 = ∅ → 𝐵 ∈ (𝐴 ↑𝑜 𝐶)))
158	157	necon3bd 2808	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (¬ 𝐵 ∈ (𝐴 ↑𝑜 𝐶) → 𝐷 ≠ ∅))
159	144, 158	mpd 15	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐷 ≠ ∅)
160	135, 159, 10	sylanbrc 698	. . . . . . . 8 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐷 ∈ (𝐴 ∖ 1𝑜))
161	108, 160	jca 554	. . . . . . 7 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)))
162	105, 161	impbida 877	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) → ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ↔ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)))
163	162	ex 450	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) → ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ↔ (𝐶 = 𝑋 ∧ 𝐷 ∈ On))))
164	163	pm5.32rd 672	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ ((𝐶 = 𝑋 ∧ 𝐷 ∈ On) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))))
165		anass 681	. . . 4 ⊢ (((𝐶 = 𝑋 ∧ 𝐷 ∈ On) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))))
166	164, 165	syl6bb 276	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))))
167		3anass 1042	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))
168		oveq2 6658	. . . . . . . 8 ⊢ (𝐶 = 𝑋 → (𝐴 ↑𝑜 𝐶) = (𝐴 ↑𝑜 𝑋))
169	168	eleq2d 2687	. . . . . . 7 ⊢ (𝐶 = 𝑋 → (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ↔ 𝐸 ∈ (𝐴 ↑𝑜 𝑋)))
170	168	oveq1d 6665	. . . . . . . . 9 ⊢ (𝐶 = 𝑋 → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) = ((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷))
171	170	oveq1d 6665	. . . . . . . 8 ⊢ (𝐶 = 𝑋 → (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸))
172	171	eqeq1d 2624	. . . . . . 7 ⊢ (𝐶 = 𝑋 → ((((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵 ↔ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))
173	169, 172	3anbi23d 1402	. . . . . 6 ⊢ (𝐶 = 𝑋 → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 ∈ On ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝑋) ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))
174	167, 173	syl5bbr 274	. . . . 5 ⊢ (𝐶 = 𝑋 → ((𝐷 ∈ On ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐷 ∈ On ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝑋) ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))
175	2, 42, 89	syl2anc 693	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 𝑋) ∈ On)
176		oen0 7666	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑋 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝑋))
177	2, 42, 130, 176	syl21anc 1325	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∅ ∈ (𝐴 ↑𝑜 𝑋))
178		ne0i 3921	. . . . . . 7 ⊢ (∅ ∈ (𝐴 ↑𝑜 𝑋) → (𝐴 ↑𝑜 𝑋) ≠ ∅)
179	177, 178	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 𝑋) ≠ ∅)
180		omeu 7665	. . . . . . 7 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ 𝐵 ∈ On ∧ (𝐴 ↑𝑜 𝑋) ≠ ∅) → ∃!𝑎∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
181		oeeu.2	. . . . . . . . 9 ⊢ 𝑃 = (℩𝑤∃𝑦 ∈ On ∃𝑧 ∈ (𝐴 ↑𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
182		opeq1 4402	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → ⟨𝑦, 𝑧⟩ = ⟨𝑑, 𝑧⟩)
183	182	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → (𝑤 = ⟨𝑦, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑧⟩))
184		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) = ((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑))
185	184	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧))
186	185	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → ((((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵 ↔ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = 𝐵))
187	183, 186	anbi12d 747	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑑 → ((𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ (𝑤 = ⟨𝑑, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = 𝐵)))
188		opeq2 4403	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑒 → ⟨𝑑, 𝑧⟩ = ⟨𝑑, 𝑒⟩)
189	188	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑒 → (𝑤 = ⟨𝑑, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑒⟩))
190		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑒 → (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒))
191	190	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑒 → ((((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = 𝐵 ↔ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
192	189, 191	anbi12d 747	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑒 → ((𝑤 = ⟨𝑑, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑧) = 𝐵) ↔ (𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵)))
193	187, 192	cbvrex2v 3180	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ On ∃𝑧 ∈ (𝐴 ↑𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑𝑜 𝑋)(𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
194		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑎 → (𝑤 = ⟨𝑑, 𝑒⟩ ↔ 𝑎 = ⟨𝑑, 𝑒⟩))
195	194	anbi1d 741	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑎 → ((𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵) ↔ (𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵)))
196	195	2rexbidv 3057	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑎 → (∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑𝑜 𝑋)(𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵)))
197	193, 196	syl5bb 272	. . . . . . . . . 10 ⊢ (𝑤 = 𝑎 → (∃𝑦 ∈ On ∃𝑧 ∈ (𝐴 ↑𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵)))
198	197	cbviotav 5857	. . . . . . . . 9 ⊢ (℩𝑤∃𝑦 ∈ On ∃𝑧 ∈ (𝐴 ↑𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) = (℩𝑎∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
199	181, 198	eqtri 2644	. . . . . . . 8 ⊢ 𝑃 = (℩𝑎∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵))
200		oeeu.3	. . . . . . . 8 ⊢ 𝑌 = (1st ‘𝑃)
201		oeeu.4	. . . . . . . 8 ⊢ 𝑍 = (2nd ‘𝑃)
202		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) = ((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷))
203	202	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝑒))
204	203	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵 ↔ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝑒) = 𝐵))
205		oveq2 6658	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝑒) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸))
206	205	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → ((((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝑒) = 𝐵 ↔ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))
207	199, 200, 201, 204, 206	opiota 7229	. . . . . . 7 ⊢ (∃!𝑎∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑𝑜 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑑) +𝑜 𝑒) = 𝐵) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝑋) ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
208	180, 207	syl 17	. . . . . 6 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ 𝐵 ∈ On ∧ (𝐴 ↑𝑜 𝑋) ≠ ∅) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝑋) ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
209	175, 140, 179, 208	syl3anc 1326	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝑋) ∧ (((𝐴 ↑𝑜 𝑋) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
210	174, 209	sylan9bbr 737	. . . 4 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ 𝐶 = 𝑋) → ((𝐷 ∈ On ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
211	210	pm5.32da 673	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵))) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍))))
212	166, 211	bitrd 268	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍))))
213		3an4anass 1291	. 2 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜) ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝐶)) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝐸 ∈ (𝐴 ↑𝑜 𝐶) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵)))
214		3anass 1042	. 2 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
215	212, 213, 214	3bitr4g 303	1 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜) ∧ 𝐸 ∈ (𝐴 ↑𝑜 𝐶)) ∧ (((𝐴 ↑𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃!weu 2470   ≠ wne 2794  ∃wrex 2913  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ cuni 4436  ∩ cint 4475  Ord word 5722  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-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:  oeeu  7683  cantnflem3  8588  cantnflem4  8589