Theorem oaass 8500

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 7378	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅))
2		oveq2 7378	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
3	2	oveq2d 7386	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o ∅)))
4	1, 3	eqeq12d 2753	. . . 4 ⊢ (𝑥 = ∅ → (((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)) ↔ ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅))))
5		oveq2 7378	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦))
6		oveq2 7378	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
7	6	oveq2d 7386	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝑦)))
8	5, 7	eqeq12d 2753	. . . 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 7378	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦))
10		oveq2 7378	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
11	10	oveq2d 7386	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o suc 𝑦)))
12	9, 11	eqeq12d 2753	. . . 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 7378	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶))
14		oveq2 7378	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
15	14	oveq2d 7386	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝐶)))
16	13, 15	eqeq12d 2753	. . . 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 8474	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
18		oa0 8455	. . . . . 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 8455	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
21	20	oveq2d 7386	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
22	21	adantl 481	. . . . 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 6395	. . . . . 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 8463	. . . . . . . 8 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
26	17, 25	sylan 581	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
27		oasuc 8463	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
28	27	oveq2d 7386	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
29	28	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
30		oacl 8474	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
31		oasuc 8463	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
32	30, 31	sylan2 594	. . . . . . . . 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 467	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
35	26, 34	eqeq12d 2753	. . . . . 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 246	. . . . 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 413	. . . 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 4968	. . . . . . 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 481	. . . . . 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 3446	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
41		oalim 8471	. . . . . . . . . 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 704	. . . . . . . . 9 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +o 𝐵) +o 𝑦))
43	17, 42	sylan 581	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +o 𝐵) +o 𝑦))
44	43	ancoms 458	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 +o 𝐵) +o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 +o 𝐵) +o 𝑦))
45	44	adantr 480	. . . . . 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 8499	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +o 𝑥))
47	40, 46	mpanr1 704	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +o 𝑥))
48	47	ancoms 458	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → Lim (𝐵 +o 𝑥))
49		ovex 7403	. . . . . . . . . . 11 ⊢ (𝐵 +o 𝑥) ∈ V
50		oalim 8471	. . . . . . . . . . 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 704	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)) → (𝐴 +o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
52	48, 51	sylan2 594	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
53		limelon 6392	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
54	40, 53	mpan 691	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
55		oacl 8474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +o 𝑥) ∈ On)
56	55	ancoms 458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑥) ∈ On)
57		onelon 6352	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 +o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
58	57	ex 412	. . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
61	60	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
62		0ellim 6391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
63		onelss 6369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵 ∈ On → (𝑧 ∈ 𝐵 → 𝑧 ⊆ 𝐵))
64	20	sseq2d 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵 ∈ On → (𝑧 ⊆ (𝐵 +o ∅) ↔ 𝑧 ⊆ 𝐵))
65	63, 64	sylibrd 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵 ∈ On → (𝑧 ∈ 𝐵 → 𝑧 ⊆ (𝐵 +o ∅)))
66	65	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ 𝐵) → 𝑧 ⊆ (𝐵 +o ∅))
67		oveq2 7378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → (𝐵 +o 𝑦) = (𝐵 +o ∅))
68	67	sseq2d 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +o 𝑦) ↔ 𝑧 ⊆ (𝐵 +o ∅)))
69	68	rspcev 3578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∅ ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +o ∅)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
70	62, 66, 69	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Lim 𝑥 ∧ (𝐵 ∈ On ∧ 𝑧 ∈ 𝐵)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
71	70	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
72	71	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
73	72	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧 ∈ 𝐵 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
74		oawordex 8496	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵 ⊆ 𝑧 ↔ ∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧))
75	74	ad2ant2l 747	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵 ⊆ 𝑧 ↔ ∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧))
76		oaord 8486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
77	76	3expb 1121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦 ∈ 𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
78		eleq1 2825	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐵 +o 𝑦) = 𝑧 → ((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
79	77, 78	sylan9bb 509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
80	79	an32s 653	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
81	80	biimpar 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑦 ∈ 𝑥)
82		eqimss2 3995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐵 +o 𝑦) = 𝑧 → 𝑧 ⊆ (𝐵 +o 𝑦))
83	82	ad3antlr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ⊆ (𝐵 +o 𝑦))
84	81, 83	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +o 𝑦)))
85	84	anasss 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥))) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +o 𝑦)))
86	85	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦 ∈ 𝑥 ∧ 𝑧 ⊆ (𝐵 +o 𝑦))))
87	86	reximdv2 3148	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
88	87	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵 ⊆ 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
90	89	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝐵 ⊆ 𝑧 → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
91		eloni 6337	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ On → Ord 𝑧)
92		eloni 6337	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐵 ∈ On → Ord 𝐵)
93		ordtri2or 6427	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Ord 𝑧 ∧ Ord 𝐵) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
94	91, 92, 93	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
95	94	ad2ant2l 747	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
96	95	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧 ∈ 𝐵 ∨ 𝐵 ⊆ 𝑧))
97	73, 90, 96	mpjaod 861	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
98	97	exp45 438	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Lim 𝑥 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))))
99	98	imp 406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
100	99	adantld 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ On → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
101	100	imp32 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦 ∈ 𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
102		simplrr 778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ On)
103		onelon 6352	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
104	103, 30	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ 𝑥)) → (𝐵 +o 𝑦) ∈ On)
105	104	exp32 420	. . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ 𝑥) → (𝐵 +o 𝑦) ∈ On)
108	107	ad4ant24 755	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → (𝐵 +o 𝑦) ∈ On)
109		simpll 767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On) → 𝐴 ∈ On)
110	109	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦 ∈ 𝑥) → 𝐴 ∈ On)
111		oaword 8488	. . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . 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 3160	. . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . 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 431	. . . . . . . . . . . . . 14 ⊢ (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))))
120	119	imp31 417	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
121	120	ralrimiv 3129	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑦 ∈ 𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))
122		iunss2 5007	. . . . . . . . . . . 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 458	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ 𝐵 ∈ On)) → ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)))
125		oaordi 8485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 → (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
126	125	anim1d 612	. . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐵 +o 𝑦) → (𝐴 +o 𝑧) = (𝐴 +o (𝐵 +o 𝑦)))
128	127	eleq2d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐵 +o 𝑦) → (𝑤 ∈ (𝐴 +o 𝑧) ↔ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))))
129	128	rspcev 3578	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝑥 → (𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))))
132	131	rexlimdv 3137	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦 ∈ 𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧)))
133		eliun 4952	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) ↔ ∃𝑦 ∈ 𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)))
134		eliun 4952	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ↔ ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))
135	132, 133, 134	3imtr4g 296	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑤 ∈ ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) → 𝑤 ∈ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧)))
136	135	ssrdv 3941	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
137	54, 136	sylan 581	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
138	137	adantl 481	. . . . . . . . . 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 3953	. . . . . . . . 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 650	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o (𝐵 +o 𝑦)))
142	141	adantr 480	. . . . . 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 419	. . . 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 7819	. . 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 1118	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  ∪ ciun 4948  Ord word 6326  Oncon0 6327  Lim wlim 6328  suc csuc 6329  (class class class)co 7370   +_o coa 8406

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-oadd 8413

This theorem is referenced by:  odi  8518  oaabs  8588  oaabs2  8589  oaabsb  43680  omabs2  43718  ofoaass  43746  naddwordnexlem3  43785  naddwordnexlem4  43787