Theorem oaass 8560

Description: Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. Theorem 4.2 of [Schloeder] p. 11. (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 7416	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅))
2		oveq2 7416	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
3	2	oveq2d 7424	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o ∅)))
4	1, 3	eqeq12d 2748	. . . 4 ⊢ (𝑥 = ∅ → (((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)) ↔ ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅))))
5		oveq2 7416	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦))
6		oveq2 7416	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
7	6	oveq2d 7424	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝑦)))
8	5, 7	eqeq12d 2748	. . . 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 7416	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦))
10		oveq2 7416	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
11	10	oveq2d 7424	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o suc 𝑦)))
12	9, 11	eqeq12d 2748	. . . 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 7416	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶))
14		oveq2 7416	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
15	14	oveq2d 7424	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝐶)))
16	13, 15	eqeq12d 2748	. . . 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 8534	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
18		oa0 8515	. . . . . 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 8515	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
21	20	oveq2d 7424	. . . . . 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 2775	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅)))
24		suceq 6430	. . . . . 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 8523	. . . . . . . 8 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
26	17, 25	sylan 580	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
27		oasuc 8523	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
28	27	oveq2d 7424	. . . . . . . . . 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 8534	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
31		oasuc 8523	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
32	30, 31	sylan2 593	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
33	29, 32	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
34	33	anassrs 468	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
35	26, 34	eqeq12d 2748	. . . . . 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	imbitrrid 245	. . . . 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 414	. . . 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		iuneq2 5016	. . . . . . 7 ⊢ (∀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)))
39	38	adantl 482	. . . . . 6 ⊢ (((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)))
40		vex 3478	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
41		oalim 8531	. . . . . . . . . 10 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 +o 𝐵) +o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +o 𝐵) +o 𝑦))
42	40, 41	mpanr1 701	. . . . . . . . 9 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +o 𝐵) +o 𝑦))
43	17, 42	sylan 580	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +o 𝐵) +o 𝑦))
44	43	ancoms 459	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 +o 𝐵) +o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +o 𝐵) +o 𝑦))
45	44	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))
46		oalimcl 8559	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +o 𝑥))
47	40, 46	mpanr1 701	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +o 𝑥))
48	47	ancoms 459	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → Lim (𝐵 +o 𝑥))
49		ovex 7441	. . . . . . . . . . 11 ⊢ (𝐵 +o 𝑥) ∈ V
50		oalim 8531	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ ((𝐵 +o 𝑥) ∈ V ∧ Lim (𝐵 +o 𝑥))) → (𝐴 +o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
51	49, 50	mpanr1 701	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)) → (𝐴 +o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
52	48, 51	sylan2 593	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
53		limelon 6428	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
54	40, 53	mpan 688	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
55		oacl 8534	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +o 𝑥) ∈ On)
56	55	ancoms 459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑥) ∈ On)
57		onelon 6389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 +o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
58	57	ex 413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 +o 𝑥) ∈ On → (𝑧 ∈ (𝐵 +o 𝑥) → 𝑧 ∈ On))
59	56, 58	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → 𝑧 ∈ On))
60	59	adantld 491	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
61	60	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
62		0ellim 6427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
63		onelss 6406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵 ∈ On → (𝑧 ∈ 𝐵 → 𝑧 ⊆ 𝐵))
64	20	sseq2d 4014	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵 ∈ On → (𝑧 ⊆ (𝐵 +o ∅) ↔ 𝑧 ⊆ 𝐵))
65	63, 64	sylibrd 258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵 ∈ On → (𝑧 ∈ 𝐵 → 𝑧 ⊆ (𝐵 +o ∅)))
66	65	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ 𝐵) → 𝑧 ⊆ (𝐵 +o ∅))
67		oveq2 7416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → (𝐵 +o 𝑦) = (𝐵 +o ∅))
68	67	sseq2d 4014	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +o 𝑦) ↔ 𝑧 ⊆ (𝐵 +o ∅)))
69	68	rspcev 3612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∅ ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +o ∅)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
70	62, 66, 69	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Lim 𝑥 ∧ (𝐵 ∈ On ∧ 𝑧 ∈ 𝐵)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
71	70	expr 457	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
72	71	adantrl 714	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
73	72	adantrr 715	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
74		oawordex 8556	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵 ⊆ 𝑧 ↔ ∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧))
75	74	ad2ant2l 744	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵 ⊆ 𝑧 ↔ ∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧))
76		oaord 8546	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
77	76	3expb 1120	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦 ∈ 𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
78		eleq1 2821	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐵 +o 𝑦) = 𝑧 → ((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
79	77, 78	sylan9bb 510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
80	79	an32s 650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
81	80	biimpar 478	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑦 ∈ 𝑥)
82		eqimss2 4041	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐵 +o 𝑦) = 𝑧 → 𝑧 ⊆ (𝐵 +o 𝑦))
83	82	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ⊆ (𝐵 +o 𝑦))
84	81, 83	jca 512	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +o 𝑦)))
85	84	anasss 467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥))) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +o 𝑦)))
86	85	expcom 414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +o 𝑦))))
87	86	reximdv2 3164	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
88	87	adantrr 715	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
89	75, 88	sylbid 239	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵 ⊆ 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
90	89	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝐵 ⊆ 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
91		eloni 6374	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ On → Ord 𝑧)
92		eloni 6374	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐵 ∈ On → Ord 𝐵)
93		ordtri2or 6462	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Ord 𝑧 ∧ Ord 𝐵) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
94	91, 92, 93	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
95	94	ad2ant2l 744	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
96	95	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
97	73, 90, 96	mpjaod 858	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
98	97	exp45 439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Lim 𝑥 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))))
99	98	imp 407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
100	99	adantld 491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
101	100	imp32 419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
102		simplrr 776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ On)
103		onelon 6389	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
104	103, 30	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ 𝑥)) → (𝐵 +o 𝑦) ∈ On)
105	104	exp32 421	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐵 ∈ On → (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (𝐵 +o 𝑦) ∈ On)))
106	105	com12 32	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ On → (𝐵 ∈ On → (𝑦 ∈ 𝑥 → (𝐵 +o 𝑦) ∈ On)))
107	106	imp31 418	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ 𝑥) → (𝐵 +o 𝑦) ∈ On)
108	107	ad4ant24 752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → (𝐵 +o 𝑦) ∈ On)
109		simpll 765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On) → 𝐴 ∈ On)
110	109	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → 𝐴 ∈ On)
111		oaword 8548	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ On ∧ (𝐵 +o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ (𝐵 +o 𝑦) ↔ (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
112	102, 108, 110, 111	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((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))))
113	112	rexbidva 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((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))))
114	101, 113	mpbid 231	. . . . . . . . . . . . . . . . . . 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	114	exp32 421	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))))
116	61, 115	mpdd 43	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
117	116	exp32 421	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑥 → (𝑥 ∈ On → (𝐵 ∈ On → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))))
118	54, 117	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝑥 → (𝐵 ∈ On → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))))
119	118	exp4a 432	. . . . . . . . . . . . . 14 ⊢ (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))))
120	119	imp31 418	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
121	120	ralrimiv 3145	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))
122		iunss2 5052	. . . . . . . . . . . 12 ⊢ (∀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)))
123	121, 122	syl 17	. . . . . . . . . . 11 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)))
124	123	ancoms 459	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)))
125		oaordi 8545	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 → (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
126	125	anim1d 611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ 𝑥 ∧ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))) → ((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ∧ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)))))
127		oveq2 7416	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐵 +o 𝑦) → (𝐴 +o 𝑧) = (𝐴 +o (𝐵 +o 𝑦)))
128	127	eleq2d 2819	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐵 +o 𝑦) → (𝑤 ∈ (𝐴 +o 𝑧) ↔ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))))
129	128	rspcev 3612	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ∧ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))
130	126, 129	syl6 35	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ 𝑥 ∧ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧)))
131	130	expd 416	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 → (𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))))
132	131	rexlimdv 3153	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦 ∈ 𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧)))
133		eliun 5001	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) ↔ ∃𝑦 ∈ 𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)))
134		eliun 5001	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ↔ ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))
135	132, 133, 134	3imtr4g 295	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑤 ∈ ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) → 𝑤 ∈ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧)))
136	135	ssrdv 3988	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
137	54, 136	sylan 580	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
138	137	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
139	124, 138	eqssd 3999	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)))
140	52, 139	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)))
141	140	an12s 647	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)))
142	141	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)))
143	39, 45, 142	3eqtr4d 2782	. . . . 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)))
144	143	exp31 420	. . . 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)))))
145	4, 8, 12, 16, 23, 37, 144	tfinds3 7853	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))))
146	145	com12 32	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))))
147	146	3impia 1117	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  ∪ ciun 4997  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7408   +_o coa 8462

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-oadd 8469

This theorem is referenced by:  odi  8578  oaabs  8646  oaabs2  8647  oaabsb  42034  omabs2  42072  ofoaass  42100  naddwordnexlem3  42140  naddwordnexlem4  42142