Theorem omopth2 8595

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 1245	. . . . . . 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 6372	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
3	1, 2	syl 18	. . . . . 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 1247	. . . . . . 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 6372	. . . . . . 7 ⊢ (𝐷 ∈ On → Ord 𝐷)
6	4, 5	syl 18	. . . . . 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 6395	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐷) → (𝐵 ∈ 𝐷 ∨ 𝐵 = 𝐷 ∨ 𝐷 ∈ 𝐵))
8	3, 6, 7	syl2anc 596	. . . . 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 490	. . . . . . . . 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 1243	. . . . . . . . . . . 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 8547	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐷) ∈ On)
12	10, 4, 11	syl2anc 596	. . . . . . . . . . 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 1248	. . . . . . . . . . . 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 6387	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → 𝐸 ∈ On)
15	10, 13, 14	syl2anc 596	. . . . . . . . . . 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 8546	. . . . . . . . . . 11 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ·o 𝐷) +o 𝐸) ∈ On)
17	12, 15, 16	syl2anc 596	. . . . . . . . . 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 6372	. . . . . . . . . 10 ⊢ (((𝐴 ·o 𝐷) +o 𝐸) ∈ On → Ord ((𝐴 ·o 𝐷) +o 𝐸))
19		ordirr 6380	. . . . . . . . . 10 ⊢ (Ord ((𝐴 ·o 𝐷) +o 𝐸) → ¬ ((𝐴 ·o 𝐷) +o 𝐸) ∈ ((𝐴 ·o 𝐷) +o 𝐸))
20	17, 18, 19	3syl 19	. . . . . . . . 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 2880	. . . . . . . 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 881	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐷 → (𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)))
23		omeulem2 8594	. . . . . . . . . 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 486	. . . . . . . . 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 35	. . . . . . . 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 122	. . . . . 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 25	. . . . . 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 2883	. . . . . . . 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 881	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐵 → (𝐷 ∈ 𝐵 ∨ (𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶)))
31		simpl1r 1244	. . . . . . . . . 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 1246	. . . . . . . . . 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 8594	. . . . . . . . . 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 1413	. . . . . . . . 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 35	. . . . . . . 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 122	. . . . . 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 1456	. . . . 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 16	. . . 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 6387	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
41		eloni 6372	. . . . . . . 8 ⊢ (𝐶 ∈ On → Ord 𝐶)
42	40, 41	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
43	10, 32, 42	syl2anc 596	. . . . . 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 6372	. . . . . . . 8 ⊢ (𝐸 ∈ On → Ord 𝐸)
45	14, 44	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → Ord 𝐸)
46	10, 13, 45	syl2anc 596	. . . . . 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 6395	. . . . . 6 ⊢ ((Ord 𝐶 ∧ Ord 𝐸) → (𝐶 ∈ 𝐸 ∨ 𝐶 = 𝐸 ∨ 𝐸 ∈ 𝐶))
48	43, 46, 47	syl2anc 596	. . . . 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 882	. . . . . . . . . 10 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → (𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)))
50	49, 24	syl5 35	. . . . . . . . 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 708	. . . . . . . 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 122	. . . . . 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 25	. . . . . 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 2766	. . . . . . . . 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 882	. . . . . . . . . 10 ⊢ ((𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶) → (𝐷 ∈ 𝐵 ∨ (𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶)))
57	56, 34	syl5 35	. . . . . . . . 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 708	. . . . . . . 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 122	. . . . . 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 1456	. . . . 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 16	. . . 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 521	. . 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 418	. 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 7428	. . 3 ⊢ (𝐵 = 𝐷 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐷))
66		id 23	. . 3 ⊢ (𝐶 = 𝐸 → 𝐶 = 𝐸)
67	65, 66	oveqan12d 7439	. 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∅c0 4278  Ord word 6361  Oncon0 6362  (class class class)co 7420   +_o coa 8476   ·_o comu 8477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-oadd 8483  df-omul 8484

This theorem is used by:  omeu  8596  dfac12lem2  10226