Theorem oeeui 8540

Description: The division algorithm for ordinal exponentiation. (This version of oeeu 8541 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 8522.) (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 4085	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2	1	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐴 ∈ On)
3	2	ad2antrr 727	. . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 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 8474	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
6	3, 4, 5	syl2anc 585	. . . . . . . . . . . . . . 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 8479	. . . . . . . . . . . . . . 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 8414	. . . . . . . . . . . . . . . 16 ⊢ 1o = {∅}
10		dif1o 8437	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (𝐴 ∖ 1o) ↔ (𝐷 ∈ 𝐴 ∧ 𝐷 ≠ ∅))
11	10	simprbi 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (𝐴 ∖ 1o) → 𝐷 ≠ ∅)
12	11	ad2antll 730	. . . . . . . . . . . . . . . . . 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 4085	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (𝐴 ∖ 1o) → 𝐷 ∈ 𝐴)
14	13	ad2antll 730	. . . . . . . . . . . . . . . . . . . 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 6350	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ 𝐴) → 𝐷 ∈ On)
16	3, 14, 15	syl2anc 585	. . . . . . . . . . . . . . . . . . 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 6382	. . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . 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 4767	. . . . . . . . . . . . . . . 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	eqsstrid 3974	. . . . . . . . . . . . . . 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 8419	. . . . . . . . . . . . . . . . 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 8508	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 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	eqsstrrd 3971	. . . . . . . . . . . . 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 8473	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ 𝐷 ∈ On) → ((𝐴 ↑o 𝐶) ·o 𝐷) ∈ On)
29	6, 16, 28	syl2anc 585	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . 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 6350	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ 𝐸 ∈ (𝐴 ↑o 𝐶)) → 𝐸 ∈ On)
32	6, 30, 31	syl2anc 585	. . . . . . . . . . . . . . 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 8489	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . 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 3946	. . . . . . . . . . . 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 8539	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝑋 ∈ On ∧ (𝐴 ↑o 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑o suc 𝑋)))
40	39	simp3d 1145	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ (𝐴 ↑o suc 𝑋))
41	40	ad2antrr 727	. . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ On)
43	42	ad2antrr 727	. . . . . . . . . . . . . . 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		onsuc 7765	. . . . . . . . . . . . . . 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 8474	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ suc 𝑋 ∈ On) → (𝐴 ↑o suc 𝑋) ∈ On)
47	3, 45, 46	syl2anc 585	. . . . . . . . . . . . 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 6373	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 700	. . . . . . . . . . 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 775	. . . . . . . . . . . 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 8526	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 257	. . . . . . . . . 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 6417	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶 ⊆ 𝑋 ↔ 𝐶 ∈ suc 𝑋))
56	4, 43, 55	syl2anc 585	. . . . . . . . . 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 257	. . . . . . . . 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 1144	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ⊆ 𝐵)
59	58	ad2antrr 727	. . . . . . . . . . . 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 6335	. . . . . . . . . . . . . . . . 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 7770	. . . . . . . . . . . . . . . 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		onsuc 7765	. . . . . . . . . . . . . . . . 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 8442	. . . . . . . . . . . . . . . . . 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 8524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐶))
69	3, 4, 67, 68	syl21anc 838	. . . . . . . . . . . . . . . 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 8507	. . . . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 8484	. . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . 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 8516	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 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 8468	. . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . 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 8464	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
87	3, 4, 86	syl2anc 585	. . . . . . . . . . . . 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 2840	. . . . . . . . . . . 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 8474	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
90	3, 43, 89	syl2anc 585	. . . . . . . . . . . . 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		onsuc 7765	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
92	91	ad2antrl 729	. . . . . . . . . . . . . 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 8474	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) ∈ On)
94	3, 92, 93	syl2anc 585	. . . . . . . . . . . . 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 6373	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 700	. . . . . . . . . . 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 8526	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 257	. . . . . . . . . 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 6417	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ On ∧ 𝐶 ∈ On) → (𝑋 ⊆ 𝐶 ↔ 𝑋 ∈ suc 𝐶))
102	43, 4, 101	syl2anc 585	. . . . . . . . . 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 257	. . . . . . . . 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 3953	. . . . . . . 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 511	. . . . . . 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 771	. . . . . . . . 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 727	. . . . . . . . 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 2837	. . . . . . . 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 727	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 778	. . . . . . . . . . . 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 3972	. . . . . . . . . . 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 727	. . . . . . . . . . . 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 6393	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = 𝑋 → suc 𝐶 = suc 𝑋)
120	119	ad2antrl 729	. . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 8473	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝐶) ·o 𝐴) ∈ On)
126	110, 109, 125	syl2anc 585	. . . . . . . . . . . 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 6373	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 700	. . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∅ ∈ 𝐴)
131	130	ad2antrr 727	. . . . . . . . . . . 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 838	. . . . . . . . . . 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 8504	. . . . . . . . . . 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 1376	. . . . . . . . . 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 257	. . . . . . . . 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 7384	. . . . . . . . . . . 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 727	. . . . . . . . . . . 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 3970	. . . . . . . . . . 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 4085	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (On ∖ 1o) → 𝐵 ∈ On)
140	139	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ On)
141	140	ad2antrr 727	. . . . . . . . . . . 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 6359	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑o 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑o 𝐶)))
143	110, 141, 142	syl2anc 585	. . . . . . . . . . 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 232	. . . . . . . . . 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 8454	. . . . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . 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 8475	. . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = ∅ → ((𝐴 ↑o 𝐶) ·o 𝐷) = ((𝐴 ↑o 𝐶) ·o ∅))
153	152	oveq1d 7383	. . . . . . . . . . . . . 14 ⊢ (𝐷 = ∅ → (((𝐴 ↑o 𝐶) ·o 𝐷) +o 𝐸) = (((𝐴 ↑o 𝐶) ·o ∅) +o 𝐸))
154	153	eleq1d 2822	. . . . . . . . . . . . 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 247	. . . . . . . . . . . 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 2822	. . . . . . . . . . . 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 239	. . . . . . . . . . 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 2947	. . . . . . . . . 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 584	. . . . . . . 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 511	. . . . . . 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 801	. . . . . 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 412	. . . . 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 578	. . . 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 468	. . . 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	bitrdi 287	. . 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 1095	. . . . . 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 7376	. . . . . . . 8 ⊢ (𝐶 = 𝑋 → (𝐴 ↑o 𝐶) = (𝐴 ↑o 𝑋))
169	168	eleq2d 2823	. . . . . . 7 ⊢ (𝐶 = 𝑋 → (𝐸 ∈ (𝐴 ↑o 𝐶) ↔ 𝐸 ∈ (𝐴 ↑o 𝑋)))
170	168	oveq1d 7383	. . . . . . . . 9 ⊢ (𝐶 = 𝑋 → ((𝐴 ↑o 𝐶) ·o 𝐷) = ((𝐴 ↑o 𝑋) ·o 𝐷))
171	170	oveq1d 7383	. . . . . . . 8 ⊢ (𝐶 = 𝑋 → (((𝐴 ↑o 𝐶) ·o 𝐷) +o 𝐸) = (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝐸))
172	171	eqeq1d 2739	. . . . . . 7 ⊢ (𝐶 = 𝑋 → ((((𝐴 ↑o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵 ↔ (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵))
173	169, 172	3anbi23d 1442	. . . . . 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	bitr3id 285	. . . . 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 585	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ∈ On)
176		oen0 8524	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑋 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝑋))
177	2, 42, 130, 176	syl21anc 838	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∅ ∈ (𝐴 ↑o 𝑋))
178	177	ne0d 4296	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ≠ ∅)
179		omeu 8522	. . . . . . 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))
180		oeeu.2	. . . . . . . . 9 ⊢ 𝑃 = (℩𝑤∃𝑦 ∈ On ∃𝑧 ∈ (𝐴 ↑o 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵))
181		opeq1 4831	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → ⟨𝑦, 𝑧⟩ = ⟨𝑑, 𝑧⟩)
182	181	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → (𝑤 = ⟨𝑦, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑧⟩))
183		oveq2 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑑 → ((𝐴 ↑o 𝑋) ·o 𝑦) = ((𝐴 ↑o 𝑋) ·o 𝑑))
184	183	oveq1d 7383	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → (((𝐴 ↑o 𝑋) ·o 𝑦) +o 𝑧) = (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑧))
185	184	eqeq1d 2739	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → ((((𝐴 ↑o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵 ↔ (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑧) = 𝐵))
186	182, 185	anbi12d 633	. . . . . . . . . . . 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)))
187		opeq2 4832	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑒 → ⟨𝑑, 𝑧⟩ = ⟨𝑑, 𝑒⟩)
188	187	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑒 → (𝑤 = ⟨𝑑, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑒⟩))
189		oveq2 7376	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑒 → (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑧) = (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑒))
190	189	eqeq1d 2739	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑒 → ((((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑧) = 𝐵 ↔ (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵))
191	188, 190	anbi12d 633	. . . . . . . . . . . 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)))
192	186, 191	cbvrex2vw 3221	. . . . . . . . . . 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))
193		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑎 → (𝑤 = ⟨𝑑, 𝑒⟩ ↔ 𝑎 = ⟨𝑑, 𝑒⟩))
194	193	anbi1d 632	. . . . . . . . . . . 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)))
195	194	2rexbidv 3203	. . . . . . . . . . 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)))
196	192, 195	bitrid 283	. . . . . . . . . 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)))
197	196	cbviotavw 6464	. . . . . . . . 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))
198	180, 197	eqtri 2760	. . . . . . . 8 ⊢ 𝑃 = (℩𝑎∃𝑑 ∈ On ∃𝑒 ∈ (𝐴 ↑o 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵))
199		oeeu.3	. . . . . . . 8 ⊢ 𝑌 = (1st ‘𝑃)
200		oeeu.4	. . . . . . . 8 ⊢ 𝑍 = (2nd ‘𝑃)
201		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝐴 ↑o 𝑋) ·o 𝑑) = ((𝐴 ↑o 𝑋) ·o 𝐷))
202	201	oveq1d 7383	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑒) = (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝑒))
203	202	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵 ↔ (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝑒) = 𝐵))
204		oveq2 7376	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝑒) = (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝐸))
205	204	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → ((((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝑒) = 𝐵 ↔ (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵))
206	198, 199, 200, 203, 205	opiota 8013	. . . . . . 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)))
207	179, 206	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)))
208	175, 140, 178, 207	syl3anc 1374	. . . . 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)))
209	174, 208	sylan9bbr 510	. . . 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)))
210	209	pm5.32da 579	. . 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))))
211	166, 210	bitrd 279	. 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))))
212		3an4anass 1105	. 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)))
213		3anass 1095	. 2 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
214	211, 212, 213	3bitr4g 314	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃!weu 2569   ≠ wne 2933  ∃wrex 3062  {crab 3401   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  {csn 4582  ⟨cop 4588  ∪ cuni 4865  ∩ cint 4904  Ord word 6324  Oncon0 6325  suc csuc 6327  ℩cio 6454  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  1_oc1o 8400  2_oc2o 8401   +_o coa 8404   ·_o comu 8405   ↑_o coe 8406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-omul 8412  df-oexp 8413

This theorem is referenced by:  oeeu  8541  cantnflem3  9612  cantnflem4  9613