Theorem omopth2 8586

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 1224	. . . . . . 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 6373	. . . . . . 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 1226	. . . . . . 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 6373	. . . . . . 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 6395	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐷) → (𝐵 ∈ 𝐷 ∨ 𝐵 = 𝐷 ∨ 𝐷 ∈ 𝐵))
8	3, 6, 7	syl2anc 582	. . . . 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 483	. . . . . . . . 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 1222	. . . . . . . . . . . 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 8538	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐷) ∈ On)
12	10, 4, 11	syl2anc 582	. . . . . . . . . . 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 1227	. . . . . . . . . . . 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 6388	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → 𝐸 ∈ On)
15	10, 13, 14	syl2anc 582	. . . . . . . . . . 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 8537	. . . . . . . . . . 11 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ·o 𝐷) +o 𝐸) ∈ On)
17	12, 15, 16	syl2anc 582	. . . . . . . . . 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 6373	. . . . . . . . . 10 ⊢ (((𝐴 ·o 𝐷) +o 𝐸) ∈ On → Ord ((𝐴 ·o 𝐷) +o 𝐸))
19		ordirr 6381	. . . . . . . . . 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 2851	. . . . . . . 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 863	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐷 → (𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)))
23		omeulem2 8585	. . . . . . . . . 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 479	. . . . . . . . 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 197	. . . . . . 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 2854	. . . . . . . 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 863	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐵 → (𝐷 ∈ 𝐵 ∨ (𝐷 = 𝐵 ∧ 𝐸 ∈ 𝐶)))
31		simpl1r 1223	. . . . . . . . . 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 1225	. . . . . . . . . 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 8585	. . . . . . . . . 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 1384	. . . . . . . . 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 197	. . . . . . 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 1426	. . . . 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 6388	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
41		eloni 6373	. . . . . . . 8 ⊢ (𝐶 ∈ On → Ord 𝐶)
42	40, 41	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
43	10, 32, 42	syl2anc 582	. . . . . 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 6373	. . . . . . . 8 ⊢ (𝐸 ∈ On → Ord 𝐸)
45	14, 44	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → Ord 𝐸)
46	10, 13, 45	syl2anc 582	. . . . . 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 582	. . . . 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 864	. . . . . . . . . 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 691	. . . . . . . 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 197	. . . . . . 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 2736	. . . . . . . . 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 864	. . . . . . . . . 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 691	. . . . . . . 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 197	. . . . . . 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 1426	. . . . 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 510	. . 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 411	. 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 7419	. . 3 ⊢ (𝐵 = 𝐷 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐷))
66		id 22	. . 3 ⊢ (𝐶 = 𝐸 → 𝐶 = 𝐸)
67	65, 66	oveqan12d 7430	. 2 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 = 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸))
68	64, 67	impbid1 224	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 205   ∧ wa 394   ∨ wo 843   ∨ w3o 1084   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∅c0 4321  Ord word 6362  Oncon0 6363  (class class class)co 7411   +_o coa 8465   ·_o comu 8466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-oadd 8472  df-omul 8473

This theorem is referenced by:  omeu  8587  dfac12lem2  10141