Theorem omopth2 8193

Description: An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
omopth2	⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸) ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))

Proof of Theorem omopth2

Step	Hyp	Ref	Expression
1		simpl2l 1223	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐵 ∈ On)
2		eloni 6169	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
3	1, 2	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → Ord 𝐵)
4		simpl3l 1225	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐷 ∈ On)
5		eloni 6169	. . . . . . 7 ⊢ (𝐷 ∈ On → Ord 𝐷)
6	4, 5	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → Ord 𝐷)
7		ordtri3or 6191	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐷) → (𝐵 ∈ 𝐷 ∨ 𝐵 = 𝐷 ∨ 𝐷 ∈ 𝐵))
8	3, 6, 7	syl2anc 587	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐵 ∈ 𝐷 ∨ 𝐵 = 𝐷 ∨ 𝐷 ∈ 𝐵))
9		simpr 488	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸))
10		simpl1l 1221	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐴 ∈ On)
11		omcl 8144	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐷) ∈ On)
12	10, 4, 11	syl2anc 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐴 ·o 𝐷) ∈ On)
13		simpl3r 1226	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐸 ∈ 𝐴)
14		onelon 6184	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → 𝐸 ∈ On)
15	10, 13, 14	syl2anc 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐸 ∈ On)
16		oacl 8143	. . . . . . . . . . 11 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ·o 𝐷) +o 𝐸) ∈ On)
17	12, 15, 16	syl2anc 587	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ((𝐴 ·o 𝐷) +o 𝐸) ∈ On)
18		eloni 6169	. . . . . . . . . 10 ⊢ (((𝐴 ·o 𝐷) +o 𝐸) ∈ On → Ord ((𝐴 ·o 𝐷) +o 𝐸))
19		ordirr 6177	. . . . . . . . . 10 ⊢ (Ord ((𝐴 ·o 𝐷) +o 𝐸) → ¬ ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐷) +o 𝐸))
20	17, 18, 19	3syl 18	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ¬ ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐷) +o 𝐸))
21	9, 20	eqneltrd 2909	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ¬ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸))
22		orc 864	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐷 → (𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)))
23		omeulem2 8192	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
24	23	adantr 484	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ((𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
25	22, 24	syl5 34	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐵 ∈ 𝐷 → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
26	21, 25	mtod 201	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ¬ 𝐵 ∈ 𝐷)
27	26	pm2.21d 121	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐵 ∈ 𝐷 → 𝐵 = 𝐷))
28		idd 24	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐵 = 𝐷 → 𝐵 = 𝐷))
29	20, 9	neleqtrrd 2912	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ¬ ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐵) +o 𝐶))
30		orc 864	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐵 → (𝐷 ∈ 𝐵 ∨ (𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶)))
31		simpl1r 1222	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐴 ≠ ∅)
32		simpl2r 1224	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐶 ∈ 𝐴)
33		omeulem2 8192	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴)) → ((𝐷 ∈ 𝐵 ∨ (𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶)) → ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐵) +o 𝐶)))
34	10, 31, 4, 13, 1, 32, 33	syl222anc 1383	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ((𝐷 ∈ 𝐵 ∨ (𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶)) → ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐵) +o 𝐶)))
35	30, 34	syl5 34	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐷 ∈ 𝐵 → ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐵) +o 𝐶)))
36	29, 35	mtod 201	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ¬ 𝐷 ∈ 𝐵)
37	36	pm2.21d 121	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐷 ∈ 𝐵 → 𝐵 = 𝐷))
38	27, 28, 37	3jaod 1425	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ((𝐵 ∈ 𝐷 ∨ 𝐵 = 𝐷 ∨ 𝐷 ∈ 𝐵) → 𝐵 = 𝐷))
39	8, 38	mpd 15	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐵 = 𝐷)
40		onelon 6184	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
41		eloni 6169	. . . . . . . 8 ⊢ (𝐶 ∈ On → Ord 𝐶)
42	40, 41	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
43	10, 32, 42	syl2anc 587	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → Ord 𝐶)
44		eloni 6169	. . . . . . . 8 ⊢ (𝐸 ∈ On → Ord 𝐸)
45	14, 44	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → Ord 𝐸)
46	10, 13, 45	syl2anc 587	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → Ord 𝐸)
47		ordtri3or 6191	. . . . . 6 ⊢ ((Ord 𝐶 ∧ Ord 𝐸) → (𝐶 ∈ 𝐸 ∨ 𝐶 = 𝐸 ∨ 𝐸 ∈ 𝐶))
48	43, 46, 47	syl2anc 587	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐶 ∈ 𝐸 ∨ 𝐶 = 𝐸 ∨ 𝐸 ∈ 𝐶))
49		olc 865	. . . . . . . . . 10 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → (𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)))
50	49, 24	syl5 34	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
51	39, 50	mpand 694	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐶 ∈ 𝐸 → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
52	21, 51	mtod 201	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ¬ 𝐶 ∈ 𝐸)
53	52	pm2.21d 121	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐶 ∈ 𝐸 → 𝐶 = 𝐸))
54		idd 24	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐶 = 𝐸 → 𝐶 = 𝐸))
55	39	eqcomd 2804	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐷 = 𝐵)
56		olc 865	. . . . . . . . . 10 ⊢ ((𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶) → (𝐷 ∈ 𝐵 ∨ (𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶)))
57	56, 34	syl5 34	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ((𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶) → ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐵) +o 𝐶)))
58	55, 57	mpand 694	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐸 ∈ 𝐶 → ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐵) +o 𝐶)))
59	29, 58	mtod 201	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ¬ 𝐸 ∈ 𝐶)
60	59	pm2.21d 121	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐸 ∈ 𝐶 → 𝐶 = 𝐸))
61	53, 54, 60	3jaod 1425	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → ((𝐶 ∈ 𝐸 ∨ 𝐶 = 𝐸 ∨ 𝐸 ∈ 𝐶) → 𝐶 = 𝐸))
62	48, 61	mpd 15	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → 𝐶 = 𝐸)
63	39, 62	jca 515	. . 3 ⊢ ((((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐵 = 𝐷 ∧ 𝐶 = 𝐸))
64	63	ex 416	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸) → (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))
65		oveq2 7143	. . 3 ⊢ (𝐵 = 𝐷 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐷))
66		id 22	. . 3 ⊢ (𝐶 = 𝐸 → 𝐶 = 𝐸)
67	65, 66	oveqan12d 7154	. 2 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 = 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸))
68	64, 67	impbid1 228	1 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸) ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∅c0 4243  Ord word 6158  Oncon0 6159  (class class class)co 7135   +_o coa 8082   ·_o comu 8083

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-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-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-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-omul 8090

This theorem is referenced by:  omeu  8194  dfac12lem2  9555