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Theorem oeeui 7569

Description: The division algorithm for ordinal exponentiation. (This version of oeeu 7570 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 7552.) (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 = (1^st ‘𝑃)
oeeu.4	⊢ 𝑍 = (2^nd ‘𝑃)

**Assertion**
Ref	Expression
oeeui	⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜) ∧ 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 3694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2_𝑜) → 𝐴 ∈ On)
2	1	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → 𝐴 ∈ On)
3	2	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐴 ∈ On)
4		simprl 790	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐶 ∈ On)
5		oecl 7504	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑_𝑜 𝐶) ∈ On)
6	3, 4, 5	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 𝐶) ∈ On)
7		om1 7509	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ↑_𝑜 𝐶) ∈ On → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 1_𝑜) = (𝐴 ↑_𝑜 𝐶))
8	6, 7	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 1_𝑜) = (𝐴 ↑_𝑜 𝐶))
9		df1o2 7459	. . . . . . . . . . . . . . . 16 ⊢ 1_𝑜 = {∅}
10		dif1o 7467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (𝐴 ∖ 1_𝑜) ↔ (𝐷 ∈ 𝐴 ∧ 𝐷 ≠ ∅))
11	10	simprbi 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (𝐴 ∖ 1_𝑜) → 𝐷 ≠ ∅)
12	11	ad2antll 761	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐷 ≠ ∅)
13		eldifi 3694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (𝐴 ∖ 1_𝑜) → 𝐷 ∈ 𝐴)
14	13	ad2antll 761	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐷 ∈ 𝐴)
15		onelon 5665	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ 𝐴) → 𝐷 ∈ On)
16	3, 14, 15	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐷 ∈ On)
17		on0eln0 5697	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ On → (∅ ∈ 𝐷 ↔ 𝐷 ≠ ∅))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (∅ ∈ 𝐷 ↔ 𝐷 ≠ ∅))
19	12, 18	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ∅ ∈ 𝐷)
20	19	snssd 4281	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → {∅} ⊆ 𝐷)
21	9, 20	syl5eqss 3612	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 1_𝑜 ⊆ 𝐷)
22		1on 7454	. . . . . . . . . . . . . . . . 17 ⊢ 1_𝑜 ∈ On
23	22	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 1_𝑜 ∈ On)
24		omwordi 7538	. . . . . . . . . . . . . . . 16 ⊢ ((1_𝑜 ∈ On ∧ 𝐷 ∈ On ∧ (𝐴 ↑_𝑜 𝐶) ∈ On) → (1_𝑜 ⊆ 𝐷 → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 1_𝑜) ⊆ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷)))
25	23, 16, 6, 24	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (1_𝑜 ⊆ 𝐷 → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 1_𝑜) ⊆ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷)))
26	21, 25	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 1_𝑜) ⊆ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷))
27	8, 26	eqsstr3d 3603	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 𝐶) ⊆ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷))
28		omcl 7503	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑_𝑜 𝐶) ∈ On ∧ 𝐷 ∈ On) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) ∈ On)
29	6, 16, 28	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) ∈ On)
30		simplrl 796	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐸 ∈ (𝐴 ↑_𝑜 𝐶))
31		onelon 5665	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑_𝑜 𝐶) ∈ On ∧ 𝐸 ∈ (𝐴 ↑_𝑜 𝐶)) → 𝐸 ∈ On)
32	6, 30, 31	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐸 ∈ On)
33		oaword1 7519	. . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) ⊆ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸))
35		simplrr 797	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)
36	34, 35	sseqtrd 3604	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) ⊆ 𝐵)
37	27, 36	sstrd 3578	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 𝐶) ⊆ 𝐵)
38		oeeu.1	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑_𝑜 𝑥)}
39	38	oeeulem 7568	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (𝑋 ∈ On ∧ (𝐴 ↑_𝑜 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑_𝑜 suc 𝑋)))
40	39	simp3d 1068	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → 𝐵 ∈ (𝐴 ↑_𝑜 suc 𝑋))
41	40	ad2antrr 758	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐵 ∈ (𝐴 ↑_𝑜 suc 𝑋))
42	39	simp1d 1066	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → 𝑋 ∈ On)
43	42	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝑋 ∈ On)
44		suceloni 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ On → suc 𝑋 ∈ On)
45	43, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → suc 𝑋 ∈ On)
46		oecl 7504	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ suc 𝑋 ∈ On) → (𝐴 ↑_𝑜 suc 𝑋) ∈ On)
47	3, 45, 46	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 suc 𝑋) ∈ On)
48		ontr2 5689	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (((𝐴 ↑_𝑜 𝐶) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑_𝑜 suc 𝑋)) → (𝐴 ↑_𝑜 𝐶) ∈ (𝐴 ↑_𝑜 suc 𝑋)))
50	37, 41, 49	mp2and 711	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 𝐶) ∈ (𝐴 ↑_𝑜 suc 𝑋))
51		simplll 794	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐴 ∈ (On ∖ 2_𝑜))
52		oeord 7555	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ suc 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2_𝑜)) → (𝐶 ∈ suc 𝑋 ↔ (𝐴 ↑_𝑜 𝐶) ∈ (𝐴 ↑_𝑜 suc 𝑋)))
53	4, 45, 51, 52	syl3anc 1318	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐶 ∈ suc 𝑋 ↔ (𝐴 ↑_𝑜 𝐶) ∈ (𝐴 ↑_𝑜 suc 𝑋)))
54	50, 53	mpbird 246	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐶 ∈ suc 𝑋)
55		onsssuc 5730	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶 ⊆ 𝑋 ↔ 𝐶 ∈ suc 𝑋))
56	4, 43, 55	syl2anc 691	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐶 ⊆ 𝑋 ↔ 𝐶 ∈ suc 𝑋))
57	54, 56	mpbird 246	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐶 ⊆ 𝑋)
58	39	simp2d 1067	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (𝐴 ↑_𝑜 𝑋) ⊆ 𝐵)
59	58	ad2antrr 758	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 𝑋) ⊆ 𝐵)
60		eloni 5650	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → Ord 𝐴)
61	3, 60	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → Ord 𝐴)
62		ordsucss 6910	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐴 → (𝐷 ∈ 𝐴 → suc 𝐷 ⊆ 𝐴))
63	61, 14, 62	sylc 63	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → suc 𝐷 ⊆ 𝐴)
64		suceloni 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ On → suc 𝐷 ∈ On)
65	16, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → suc 𝐷 ∈ On)
66		dif20el 7472	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2_𝑜) → ∅ ∈ 𝐴)
67	51, 66	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ∅ ∈ 𝐴)
68		oen0 7553	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑_𝑜 𝐶))
69	3, 4, 67, 68	syl21anc 1317	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ∅ ∈ (𝐴 ↑_𝑜 𝐶))
70		omword 7537	. . . . . . . . . . . . . . . 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 1321	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (suc 𝐷 ⊆ 𝐴 ↔ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 suc 𝐷) ⊆ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐴)))
72	63, 71	mpbid 221	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 suc 𝐷) ⊆ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐴))
73		oaord 7514	. . . . . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ↔ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) ∈ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 (𝐴 ↑_𝑜 𝐶))))
75	30, 74	mpbid 221	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) ∈ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 (𝐴 ↑_𝑜 𝐶)))
76	35, 75	eqeltrrd 2689	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐵 ∈ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 (𝐴 ↑_𝑜 𝐶)))
77		odi 7546	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ↑_𝑜 𝐶) ∈ On ∧ 𝐷 ∈ On ∧ 1_𝑜 ∈ On) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 (𝐷 +_𝑜 1_𝑜)) = (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 ((𝐴 ↑_𝑜 𝐶) ·_𝑜 1_𝑜)))
78	6, 16, 23, 77	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 (𝐷 +_𝑜 1_𝑜)) = (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 ((𝐴 ↑_𝑜 𝐶) ·_𝑜 1_𝑜)))
79		oa1suc 7498	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ On → (𝐷 +_𝑜 1_𝑜) = suc 𝐷)
80	16, 79	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐷 +_𝑜 1_𝑜) = suc 𝐷)
81	80	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 (𝐷 +_𝑜 1_𝑜)) = ((𝐴 ↑_𝑜 𝐶) ·_𝑜 suc 𝐷))
82	8	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 ((𝐴 ↑_𝑜 𝐶) ·_𝑜 1_𝑜)) = (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 (𝐴 ↑_𝑜 𝐶)))
83	78, 81, 82	3eqtr3d 2652	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 suc 𝐷) = (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 (𝐴 ↑_𝑜 𝐶)))
84	76, 83	eleqtrrd 2691	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐵 ∈ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 suc 𝐷))
85	72, 84	sseldd 3569	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐵 ∈ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐴))
86		oesuc 7494	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑_𝑜 suc 𝐶) = ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐴))
87	3, 4, 86	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 suc 𝐶) = ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐴))
88	85, 87	eleqtrrd 2691	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐵 ∈ (𝐴 ↑_𝑜 suc 𝐶))
89		oecl 7504	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑_𝑜 𝑋) ∈ On)
90	3, 43, 89	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 𝑋) ∈ On)
91		suceloni 6905	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
92	91	ad2antrl 760	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → suc 𝐶 ∈ On)
93		oecl 7504	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴 ↑_𝑜 suc 𝐶) ∈ On)
94	3, 92, 93	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 suc 𝐶) ∈ On)
95		ontr2 5689	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (((𝐴 ↑_𝑜 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑_𝑜 suc 𝐶)) → (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 suc 𝐶)))
97	59, 88, 96	mp2and 711	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 suc 𝐶))
98		oeord 7555	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ On ∧ suc 𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2_𝑜)) → (𝑋 ∈ suc 𝐶 ↔ (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 suc 𝐶)))
99	43, 92, 51, 98	syl3anc 1318	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝑋 ∈ suc 𝐶 ↔ (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 suc 𝐶)))
100	97, 99	mpbird 246	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝑋 ∈ suc 𝐶)
101		onsssuc 5730	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ On ∧ 𝐶 ∈ On) → (𝑋 ⊆ 𝐶 ↔ 𝑋 ∈ suc 𝐶))
102	43, 4, 101	syl2anc 691	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝑋 ⊆ 𝐶 ↔ 𝑋 ∈ suc 𝐶))
103	100, 102	mpbird 246	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝑋 ⊆ 𝐶)
104	57, 103	eqssd 3585	. . . . . . . 8 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → 𝐶 = 𝑋)
105	104, 16	jca 553	. . . . . . 7 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜))) → (𝐶 = 𝑋 ∧ 𝐷 ∈ On))
106		simprl 790	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐶 = 𝑋)
107	42	ad2antrr 758	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝑋 ∈ On)
108	106, 107	eqeltrd 2688	. . . . . . . 8 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐶 ∈ On)
109	2	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐴 ∈ On)
110	109, 108, 5	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑_𝑜 𝐶) ∈ On)
111		simprr 792	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
112	110, 111, 28	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) ∈ On)
113		simplrl 796	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 691	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐸 ∈ On)
115	112, 114, 33	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 797	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 3604	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 758	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐵 ∈ (𝐴 ↑_𝑜 suc 𝑋))
119		suceq 5707	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = 𝑋 → suc 𝐶 = suc 𝑋)
120	119	ad2antrl 760	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → suc 𝐶 = suc 𝑋)
121	120	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 691	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 2646	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 2690	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐵 ∈ ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐴))
125		omcl 7503	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑_𝑜 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐴) ∈ On)
126	110, 109, 125	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐴) ∈ On)
127		ontr2 5689	. . . . . . . . . . . 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 691	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 711	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → ∅ ∈ 𝐴)
131	130	ad2antrr 758	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ∅ ∈ 𝐴)
132	109, 108, 131, 68	syl21anc 1317	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ∅ ∈ (𝐴 ↑_𝑜 𝐶))
133		omord2 7534	. . . . . . . . . . 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 1321	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 246	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐷 ∈ 𝐴)
136	106	oveq2d 6565	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 758	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑_𝑜 𝑋) ⊆ 𝐵)
138	136, 137	eqsstrd 3602	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐴 ↑_𝑜 𝐶) ⊆ 𝐵)
139		eldifi 3694	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (On ∖ 1_𝑜) → 𝐵 ∈ On)
140	139	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → 𝐵 ∈ On)
141	140	ad2antrr 758	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐵 ∈ On)
142		ontri1 5674	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑_𝑜 𝐶) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑_𝑜 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑_𝑜 𝐶)))
143	110, 141, 142	syl2anc 691	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 221	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ¬ 𝐵 ∈ (𝐴 ↑_𝑜 𝐶))
145		om0 7484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ↑_𝑜 𝐶) ∈ On → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 ∅) = ∅)
146	110, 145	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 ∅) = ∅)
147	146	oveq1d 6564	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 7505	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ On → (∅ +_𝑜 𝐸) = 𝐸)
149	114, 148	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (∅ +_𝑜 𝐸) = 𝐸)
150	147, 149	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 2688	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 6557	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = ∅ → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) = ((𝐴 ↑_𝑜 𝐶) ·_𝑜 ∅))
153	152	oveq1d 6564	. . . . . . . . . . . . . 14 ⊢ (𝐷 = ∅ → (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = (((𝐴 ↑_𝑜 𝐶) ·_𝑜 ∅) +_𝑜 𝐸))
154	153	eleq1d 2672	. . . . . . . . . . . . 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 236	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 2672	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 228	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐷 = ∅ → 𝐵 ∈ (𝐴 ↑_𝑜 𝐶)))
158	157	necon3bd 2796	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (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 ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐷 ≠ ∅)
160	135, 159, 10	sylanbrc 695	. . . . . . . 8 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → 𝐷 ∈ (𝐴 ∖ 1_𝑜))
161	108, 160	jca 553	. . . . . . 7 ⊢ ((((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜)))
162	105, 161	impbida 873	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) → ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜)) ↔ (𝐶 = 𝑋 ∧ 𝐷 ∈ On)))
163	162	ex 449	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → ((𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵) → ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜)) ↔ (𝐶 = 𝑋 ∧ 𝐷 ∈ On))))
164	163	pm5.32rd 670	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜)) ∧ (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 679	. . . 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 275	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜)) ∧ (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 1035	. . . . . 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 6557	. . . . . . . 8 ⊢ (𝐶 = 𝑋 → (𝐴 ↑_𝑜 𝐶) = (𝐴 ↑_𝑜 𝑋))
169	168	eleq2d 2673	. . . . . . 7 ⊢ (𝐶 = 𝑋 → (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ↔ 𝐸 ∈ (𝐴 ↑_𝑜 𝑋)))
170	168	oveq1d 6564	. . . . . . . . 9 ⊢ (𝐶 = 𝑋 → ((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) = ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷))
171	170	oveq1d 6564	. . . . . . . 8 ⊢ (𝐶 = 𝑋 → (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷) +_𝑜 𝐸))
172	171	eqeq1d 2612	. . . . . . 7 ⊢ (𝐶 = 𝑋 → ((((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵 ↔ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵))
173	169, 172	3anbi23d 1394	. . . . . 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 273	. . . . 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 691	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (𝐴 ↑_𝑜 𝑋) ∈ On)
176		oen0 7553	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑋 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑_𝑜 𝑋))
177	2, 42, 130, 176	syl21anc 1317	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → ∅ ∈ (𝐴 ↑_𝑜 𝑋))
178		ne0i 3880	. . . . . . 7 ⊢ (∅ ∈ (𝐴 ↑_𝑜 𝑋) → (𝐴 ↑_𝑜 𝑋) ≠ ∅)
179	177, 178	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (𝐴 ↑_𝑜 𝑋) ≠ ∅)
180		omeu 7552	. . . . . . 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 4340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → ⟨𝑦, 𝑧⟩ = ⟨𝑑, 𝑧⟩)
183	182	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → (𝑤 = ⟨𝑦, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑧⟩))
184		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑦) = ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑))
185	184	oveq1d 6564	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑦) +_𝑜 𝑧) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) +_𝑜 𝑧))
186	185	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → ((((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑦) +_𝑜 𝑧) = 𝐵 ↔ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) +_𝑜 𝑧) = 𝐵))
187	183, 186	anbi12d 743	. . . . . . . . . . . 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 4341	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑒 → ⟨𝑑, 𝑧⟩ = ⟨𝑑, 𝑒⟩)
189	188	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑒 → (𝑤 = ⟨𝑑, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑒⟩))
190		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑒 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) +_𝑜 𝑧) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) +_𝑜 𝑒))
191	190	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑒 → ((((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) +_𝑜 𝑧) = 𝐵 ↔ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) +_𝑜 𝑒) = 𝐵))
192	189, 191	anbi12d 743	. . . . . . . . . . . 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 3156	. . . . . . . . . . 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 2614	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑎 → (𝑤 = ⟨𝑑, 𝑒⟩ ↔ 𝑎 = ⟨𝑑, 𝑒⟩))
195	194	anbi1d 737	. . . . . . . . . . . 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 3039	. . . . . . . . . . 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 271	. . . . . . . . . 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 5774	. . . . . . . . 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 2632	. . . . . . . 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 = (1^st ‘𝑃)
201		oeeu.4	. . . . . . . 8 ⊢ 𝑍 = (2^nd ‘𝑃)
202		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) = ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷))
203	202	oveq1d 6564	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) +_𝑜 𝑒) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷) +_𝑜 𝑒))
204	203	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑑) +_𝑜 𝑒) = 𝐵 ↔ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷) +_𝑜 𝑒) = 𝐵))
205		oveq2 6557	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷) +_𝑜 𝑒) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷) +_𝑜 𝐸))
206	205	eqeq1d 2612	. . . . . . . 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 7118	. . . . . . 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 1318	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴 ↑_𝑜 𝑋) ∧ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵) ↔ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
210	174, 209	sylan9bbr 733	. . . 4 ⊢ (((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) ∧ 𝐶 = 𝑋) → ((𝐷 ∈ On ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ↔ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
211	210	pm5.32da 671	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → ((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 267	. 2 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍))))
213		3an4anass 1283	. 2 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝐸 ∈ (𝐴 ↑_𝑜 𝐶)) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵) ↔ ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜)) ∧ (𝐸 ∈ (𝐴 ↑_𝑜 𝐶) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵)))
214		3anass 1035	. 2 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
215	212, 213, 214	3bitr4g 302	1 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐵 ∈ (On ∖ 1_𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝐸 ∈ (𝐴 ↑_𝑜 𝐶)) ∧ (((𝐴 ↑_𝑜 𝐶) ·_𝑜 𝐷) +_𝑜 𝐸) = 𝐵) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃!weu 2458 ≠ wne 2780 ∃wrex 2897 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 {csn 4125 ⟨cop 4131 ∪ cuni 4372 ∩ cint 4410 Ord word 5639 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-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: oeeu 7570 cantnflem3 8471 cantnflem4 8472

W3C validator