Theorem oaass 7641

Description: Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.)

Assertion

Ref	Expression
oaass	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))

Proof of Theorem oaass

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 ∅))
2		oveq2 6658	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
3	2	oveq2d 6666	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
4	1, 3	eqeq12d 2637	. . . 4 ⊢ (𝑥 = ∅ → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅))))
5		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
6		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
7	6	oveq2d 6666	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
8	5, 7	eqeq12d 2637	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
9		oveq2 6658	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦))
10		oveq2 6658	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
11	10	oveq2d 6666	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)))
12	9, 11	eqeq12d 2637	. . . 4 ⊢ (𝑥 = suc 𝑦 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦))))
13		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝐶))
14		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
15	14	oveq2d 6666	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))
16	13, 15	eqeq12d 2637	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
17		oacl 7615	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
18		oa0 7596	. . . . . 6 ⊢ ((𝐴 +𝑜 𝐵) ∈ On → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 𝐵))
19	17, 18	syl 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 𝐵))
20		oa0 7596	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
21	20	oveq2d 6666	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐴 +𝑜 (𝐵 +𝑜 ∅)) = (𝐴 +𝑜 𝐵))
22	21	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 (𝐵 +𝑜 ∅)) = (𝐴 +𝑜 𝐵))
23	19, 22	eqtr4d 2659	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
24		suceq 5790	. . . . . 6 ⊢ (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
25		oasuc 7604	. . . . . . . 8 ⊢ (((𝐴 +𝑜 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
26	17, 25	sylan 488	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
27		oasuc 7604	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
28	27	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)))
29	28	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)))
30		oacl 7615	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
31		oasuc 7604	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On) → (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
32	30, 31	sylan2 491	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
33	29, 32	eqtrd 2656	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
34	33	anassrs 680	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
35	26, 34	eqeq12d 2637	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) ↔ suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
36	24, 35	syl5ibr 236	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦))))
37	36	expcom 451	. . . 4 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)))))
38		vex 3203	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
39		oalim 7612	. . . . . . . . . 10 ⊢ (((𝐴 +𝑜 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
40	38, 39	mpanr1 719	. . . . . . . . 9 ⊢ (((𝐴 +𝑜 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
41	17, 40	sylan 488	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
42	41	ancoms 469	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
43	42	adantr 481	. . . . . 6 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
44		oalimcl 7640	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +𝑜 𝑥))
45	38, 44	mpanr1 719	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +𝑜 𝑥))
46	45	ancoms 469	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → Lim (𝐵 +𝑜 𝑥))
47		ovex 6678	. . . . . . . . . . . 12 ⊢ (𝐵 +𝑜 𝑥) ∈ V
48		oalim 7612	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ ((𝐵 +𝑜 𝑥) ∈ V ∧ Lim (𝐵 +𝑜 𝑥))) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
49	47, 48	mpanr1 719	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ Lim (𝐵 +𝑜 𝑥)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
50	46, 49	sylan2 491	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
51		limelon 5788	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
52	38, 51	mpan 706	. . . . . . . . . . . . . . . . 17 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
53		oacl 7615	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
54	53	ancoms 469	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
55		onelon 5748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐵 +𝑜 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ∈ On)
56	55	ex 450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 +𝑜 𝑥) ∈ On → (𝑧 ∈ (𝐵 +𝑜 𝑥) → 𝑧 ∈ On))
57	54, 56	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → 𝑧 ∈ On))
58	57	adantld 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ∈ On))
59	58	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ∈ On))
60		0ellim 5787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
61		onelss 5766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐵 ∈ On → (𝑧 ∈ 𝐵 → 𝑧 ⊆ 𝐵))
62	20	sseq2d 3633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐵 ∈ On → (𝑧 ⊆ (𝐵 +𝑜 ∅) ↔ 𝑧 ⊆ 𝐵))
63	61, 62	sylibrd 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵 ∈ On → (𝑧 ∈ 𝐵 → 𝑧 ⊆ (𝐵 +𝑜 ∅)))
64	63	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ 𝐵) → 𝑧 ⊆ (𝐵 +𝑜 ∅))
65		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → (𝐵 +𝑜 𝑦) = (𝐵 +𝑜 ∅))
66	65	sseq2d 3633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ 𝑧 ⊆ (𝐵 +𝑜 ∅)))
67	66	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((∅ ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +𝑜 ∅)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))
68	60, 64, 67	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Lim 𝑥 ∧ (𝐵 ∈ On ∧ 𝑧 ∈ 𝐵)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))
69	68	expr 643	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
70	69	adantrl 752	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
71	70	adantrr 753	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
72		oawordex 7637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵 ⊆ 𝑧 ↔ ∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧))
73	72	ad2ant2l 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (𝐵 ⊆ 𝑧 ↔ ∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧))
74		oaord 7627	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 ↔ (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
75	74	3expb 1266	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦 ∈ 𝑥 ↔ (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
76		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐵 +𝑜 𝑦) = 𝑧 → ((𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥) ↔ 𝑧 ∈ (𝐵 +𝑜 𝑥)))
77	75, 76	sylan9bb 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +𝑜 𝑦) = 𝑧) → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ (𝐵 +𝑜 𝑥)))
78	77	an32s 846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ (𝐵 +𝑜 𝑥)))
79	78	biimpar 502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑦 ∈ 𝑥)
80		eqimss2 3658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐵 +𝑜 𝑦) = 𝑧 → 𝑧 ⊆ (𝐵 +𝑜 𝑦))
81	80	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ⊆ (𝐵 +𝑜 𝑦))
82	79, 81	jca 554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
83	82	anasss 679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥))) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
84	83	expcom 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +𝑜 𝑦))))
85	84	reximdv2 3014	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
86	85	adantrr 753	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
87	73, 86	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (𝐵 ⊆ 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
88	87	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → (𝐵 ⊆ 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
89		eloni 5733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ On → Ord 𝑧)
90		eloni 5733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐵 ∈ On → Ord 𝐵)
91		ordtri2or 5822	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Ord 𝑧 ∧ Ord 𝐵) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
92	89, 90, 91	syl2anr 495	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
93	92	ad2ant2l 782	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
94	93	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
95	71, 88, 94	mpjaod 396	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))
96	95	exp45 642	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Lim 𝑥 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))))
97	96	imp 445	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))))
98	97	adantld 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))))
99	98	imp32 449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))
100		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ On)
101		onelon 5748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
102	101, 30	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ 𝑥)) → (𝐵 +𝑜 𝑦) ∈ On)
103	102	exp32 631	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐵 ∈ On → (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (𝐵 +𝑜 𝑦) ∈ On)))
104	103	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ On → (𝐵 ∈ On → (𝑦 ∈ 𝑥 → (𝐵 +𝑜 𝑦) ∈ On)))
105	104	imp31 448	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ 𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
106	105	adantll 750	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑦 ∈ 𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
107	106	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
108		simpll 790	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On) → 𝐴 ∈ On)
109	108	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → 𝐴 ∈ On)
110		oaword 7629	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
111	100, 107, 109, 110	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → (𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
112	111	rexbidva 3049	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) → (∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ ∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
113	99, 112	mpbid 222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
114	113	exp32 631	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))))
115	59, 114	mpdd 43	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
116	115	exp32 631	. . . . . . . . . . . . . . . . 17 ⊢ (Lim 𝑥 → (𝑥 ∈ On → (𝐵 ∈ On → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))))
117	52, 116	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑥 → (𝐵 ∈ On → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))))
118	117	exp4a 633	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 +𝑜 𝑥) → ∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))))
119	118	imp31 448	. . . . . . . . . . . . . 14 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → ∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
120	119	ralrimiv 2965	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 +𝑜 𝑥)∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
121		iunss2 4565	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ (𝐵 +𝑜 𝑥)∃𝑦 ∈ 𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
122	120, 121	syl 17	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
123	122	ancoms 469	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
124		oaordi 7626	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 → (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
125	124	anim1d 588	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ 𝑥 ∧ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ((𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥) ∧ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))))
126		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
127	126	eleq2d 2687	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐵 +𝑜 𝑦) → (𝑤 ∈ (𝐴 +𝑜 𝑧) ↔ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
128	127	rspcev 3309	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥) ∧ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧))
129	125, 128	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ 𝑥 ∧ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧)))
130	129	expd 452	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 → (𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧))))
131	130	rexlimdv 3030	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦 ∈ 𝑥 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧)))
132		eliun 4524	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ↔ ∃𝑦 ∈ 𝑥 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
133		eliun 4524	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ↔ ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧))
134	131, 132, 133	3imtr4g 285	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑤 ∈ ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → 𝑤 ∈ ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧)))
135	134	ssrdv 3609	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
136	52, 135	sylan 488	. . . . . . . . . . . 12 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
137	136	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
138	123, 137	eqssd 3620	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → ∪ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) = ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
139	50, 138	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
140	139	an12s 843	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
141	140	adantr 481	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
142		iuneq2 4537	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ∪ 𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
143	142	adantl 482	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ∪ 𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
144	141, 143	eqtr4d 2659	. . . . . 6 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
145	43, 144	eqtr4d 2659	. . . . 5 ⊢ (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)))
146	145	exp31 630	. . . 4 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)))))
147	4, 8, 12, 16, 23, 37, 146	tfinds3 7064	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
148	147	com12 32	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
149	148	3impia 1261	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  odi  7659  oaabs  7724  oaabs2  7725