Theorem oeeui 8211

Description: The division algorithm for ordinal exponentiation. (This version of oeeu 8212 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 8194.) (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 4054	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2	1	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐴 ∈ On)
3	2	ad2antrr 725	. . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . 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 8145	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
6	3, 4, 5	syl2anc 587	. . . . . . . . . . . . . . 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 8151	. . . . . . . . . . . . . . 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 8099	. . . . . . . . . . . . . . . 16 ⊢ 1o = {∅}
10		dif1o 8108	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (𝐴 ∖ 1o) ↔ (𝐷 ∈ 𝐴 ∧ 𝐷 ≠ ∅))
11	10	simprbi 500	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (𝐴 ∖ 1o) → 𝐷 ≠ ∅)
12	11	ad2antll 728	. . . . . . . . . . . . . . . . . 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 4054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (𝐴 ∖ 1o) → 𝐷 ∈ 𝐴)
14	13	ad2antll 728	. . . . . . . . . . . . . . . . . . . 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 6184	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ 𝐴) → 𝐷 ∈ On)
16	3, 14, 15	syl2anc 587	. . . . . . . . . . . . . . . . . . 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 6214	. . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . 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 4702	. . . . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . 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 8092	. . . . . . . . . . . . . . . . 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 8180	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . 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 8144	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ 𝐷 ∈ On) → ((𝐴 ↑o 𝐶) ·o 𝐷) ∈ On)
29	6, 16, 28	syl2anc 587	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . 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 6184	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ 𝐸 ∈ (𝐴 ↑o 𝐶)) → 𝐸 ∈ On)
32	6, 30, 31	syl2anc 587	. . . . . . . . . . . . . . 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 8161	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . 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 3925	. . . . . . . . . . . 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 8210	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝑋 ∈ On ∧ (𝐴 ↑o 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑o suc 𝑋)))
40	39	simp3d 1141	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ (𝐴 ↑o suc 𝑋))
41	40	ad2antrr 725	. . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ On)
43	42	ad2antrr 725	. . . . . . . . . . . . . . 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 7508	. . . . . . . . . . . . . . 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 8145	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ suc 𝑋 ∈ On) → (𝐴 ↑o suc 𝑋) ∈ On)
47	3, 45, 46	syl2anc 587	. . . . . . . . . . . . 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 6206	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 698	. . . . . . . . . . 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 774	. . . . . . . . . . . 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 8197	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 260	. . . . . . . . . 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 6246	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶 ⊆ 𝑋 ↔ 𝐶 ∈ suc 𝑋))
56	4, 43, 55	syl2anc 587	. . . . . . . . . 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 260	. . . . . . . . 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 1140	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ⊆ 𝐵)
59	58	ad2antrr 725	. . . . . . . . . . . 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 6169	. . . . . . . . . . . . . . . . 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 7513	. . . . . . . . . . . . . . . 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 7508	. . . . . . . . . . . . . . . . 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 8113	. . . . . . . . . . . . . . . . . 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 8195	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐶))
69	3, 4, 67, 68	syl21anc 836	. . . . . . . . . . . . . . . 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 8179	. . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 8156	. . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . 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 2891	. . . . . . . . . . . . . . 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 8188	. . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . 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 8139	. . . . . . . . . . . . . . . . . 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 7151	. . . . . . . . . . . . . . . 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 7151	. . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . 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 2893	. . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . 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 8135	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
87	3, 4, 86	syl2anc 587	. . . . . . . . . . . . 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 2893	. . . . . . . . . . . 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 8145	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
90	3, 43, 89	syl2anc 587	. . . . . . . . . . . . 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 7508	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
92	91	ad2antrl 727	. . . . . . . . . . . . . 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 8145	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) ∈ On)
94	3, 92, 93	syl2anc 587	. . . . . . . . . . . . 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 6206	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 698	. . . . . . . . . . 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 8197	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 260	. . . . . . . . . 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 6246	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ On ∧ 𝐶 ∈ On) → (𝑋 ⊆ 𝐶 ↔ 𝑋 ∈ suc 𝐶))
102	43, 4, 101	syl2anc 587	. . . . . . . . . 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 260	. . . . . . . . 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 3932	. . . . . . . 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 515	. . . . . . 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 770	. . . . . . . . 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 725	. . . . . . . . 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 2890	. . . . . . . 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 725	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 777	. . . . . . . . . . . 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 3955	. . . . . . . . . . 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 725	. . . . . . . . . . . 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 6224	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = 𝑋 → suc 𝐶 = suc 𝑋)
120	119	ad2antrl 727	. . . . . . . . . . . . . 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 7151	. . . . . . . . . . . . 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 587	. . . . . . . . . . . . 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 2835	. . . . . . . . . . . 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 2892	. . . . . . . . . . 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 8144	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝐶) ·o 𝐴) ∈ On)
126	110, 109, 125	syl2anc 587	. . . . . . . . . . . 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 6206	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 698	. . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∅ ∈ 𝐴)
131	130	ad2antrr 725	. . . . . . . . . . . 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 836	. . . . . . . . . . 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 8176	. . . . . . . . . . 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 1370	. . . . . . . . . 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 260	. . . . . . . . 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 7151	. . . . . . . . . . . 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 725	. . . . . . . . . . . 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 3953	. . . . . . . . . . 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 4054	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (On ∖ 1o) → 𝐵 ∈ On)
140	139	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ On)
141	140	ad2antrr 725	. . . . . . . . . . . 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 6193	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑o 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑o 𝐶)))
143	110, 141, 142	syl2anc 587	. . . . . . . . . . 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 235	. . . . . . . . . 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 8125	. . . . . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . . . 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 8146	. . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . 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 2890	. . . . . . . . . . . . 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 7143	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = ∅ → ((𝐴 ↑o 𝐶) ·o 𝐷) = ((𝐴 ↑o 𝐶) ·o ∅))
153	152	oveq1d 7150	. . . . . . . . . . . . . 14 ⊢ (𝐷 = ∅ → (((𝐴 ↑o 𝐶) ·o 𝐷) +o 𝐸) = (((𝐴 ↑o 𝐶) ·o ∅) +o 𝐸))
154	153	eleq1d 2874	. . . . . . . . . . . . 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 250	. . . . . . . . . . . 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 2874	. . . . . . . . . . . 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 242	. . . . . . . . . . 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 3001	. . . . . . . . . 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 586	. . . . . . . 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 515	. . . . . . 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 800	. . . . . 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 416	. . . . 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 581	. . . 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 472	. . . 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 290	. . 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 1092	. . . . . 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 7143	. . . . . . . 8 ⊢ (𝐶 = 𝑋 → (𝐴 ↑o 𝐶) = (𝐴 ↑o 𝑋))
169	168	eleq2d 2875	. . . . . . 7 ⊢ (𝐶 = 𝑋 → (𝐸 ∈ (𝐴 ↑o 𝐶) ↔ 𝐸 ∈ (𝐴 ↑o 𝑋)))
170	168	oveq1d 7150	. . . . . . . . 9 ⊢ (𝐶 = 𝑋 → ((𝐴 ↑o 𝐶) ·o 𝐷) = ((𝐴 ↑o 𝑋) ·o 𝐷))
171	170	oveq1d 7150	. . . . . . . 8 ⊢ (𝐶 = 𝑋 → (((𝐴 ↑o 𝐶) ·o 𝐷) +o 𝐸) = (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝐸))
172	171	eqeq1d 2800	. . . . . . 7 ⊢ (𝐶 = 𝑋 → ((((𝐴 ↑o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵 ↔ (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵))
173	169, 172	3anbi23d 1436	. . . . . 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 288	. . . . 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 587	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ∈ On)
176		oen0 8195	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑋 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝑋))
177	2, 42, 130, 176	syl21anc 836	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∅ ∈ (𝐴 ↑o 𝑋))
178	177	ne0d 4251	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ≠ ∅)
179		omeu 8194	. . . . . . 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 4763	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → ⟨𝑦, 𝑧⟩ = ⟨𝑑, 𝑧⟩)
182	181	eqeq2d 2809	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → (𝑤 = ⟨𝑦, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑧⟩))
183		oveq2 7143	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑑 → ((𝐴 ↑o 𝑋) ·o 𝑦) = ((𝐴 ↑o 𝑋) ·o 𝑑))
184	183	oveq1d 7150	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → (((𝐴 ↑o 𝑋) ·o 𝑦) +o 𝑧) = (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑧))
185	184	eqeq1d 2800	. . . . . . . . . . . . 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 4765	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑒 → ⟨𝑑, 𝑧⟩ = ⟨𝑑, 𝑒⟩)
188	187	eqeq2d 2809	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑒 → (𝑤 = ⟨𝑑, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑒⟩))
189		oveq2 7143	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑒 → (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑧) = (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑒))
190	189	eqeq1d 2800	. . . . . . . . . . . . 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 3409	. . . . . . . . . . 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 2802	. . . . . . . . . . . . 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 3259	. . . . . . . . . . 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	syl5bb 286	. . . . . . . . . 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 6291	. . . . . . . . 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 2821	. . . . . . . 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 7143	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝐴 ↑o 𝑋) ·o 𝑑) = ((𝐴 ↑o 𝑋) ·o 𝐷))
202	201	oveq1d 7150	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑒) = (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝑒))
203	202	eqeq1d 2800	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((((𝐴 ↑o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵 ↔ (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝑒) = 𝐵))
204		oveq2 7143	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝑒) = (((𝐴 ↑o 𝑋) ·o 𝐷) +o 𝐸))
205	204	eqeq1d 2800	. . . . . . . 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 7739	. . . . . . 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 1368	. . . . 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 514	. . . 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 582	. . 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 282	. 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 1102	. 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 1092	. 2 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌 ∧ 𝐸 = 𝑍)))
214	211, 212, 213	3bitr4g 317	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃!weu 2628   ≠ wne 2987  ∃wrex 3107  {crab 3110   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {csn 4525  ⟨cop 4531  ∪ cuni 4800  ∩ cint 4838  Ord word 6158  Oncon0 6159  suc csuc 6161  ℩cio 6281  ‘cfv 6324  (class class class)co 7135  1^st c1st 7669  2^nd c2nd 7670  1_oc1o 8078  2_oc2o 8079   +_o coa 8082   ·_o comu 8083   ↑_o coe 8084

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-oexp 8091

This theorem is referenced by:  oeeu  8212  cantnflem3  9138  cantnflem4  9139