Theorem omlimcl 8624

Description: The product of any nonzero ordinal with a limit ordinal is a limit ordinal. Proposition 8.24 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)

Assertion

Ref	Expression
omlimcl	⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝐵))

Proof of Theorem omlimcl

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

Step	Hyp	Ref	Expression
1		limelon 6454	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		omcl 8582	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
3		eloni 6398	. . . . 5 ⊢ ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
4	2, 3	syl 18	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·o 𝐵))
5	1, 4	sylan2 605	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ·o 𝐵))
6	5	adantr 486	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord (𝐴 ·o 𝐵))
7		0ellim 6453	. . . . . . . 8 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
8		n0i 4289	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
9	7, 8	syl 18	. . . . . . 7 ⊢ (Lim 𝐵 → ¬ 𝐵 = ∅)
10		n0i 4289	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → ¬ 𝐴 = ∅)
11	9, 10	anim12ci 626	. . . . . 6 ⊢ ((Lim 𝐵 ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
12	11	adantll 727	. . . . 5 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
13	12	adantll 727	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
14		om00 8621	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
15	14	notbid 321	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅)))
16		ioran 999	. . . . . . 7 ⊢ (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
17	15, 16	bitrdi 290	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
18	1, 17	sylan2 605	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
19	18	adantr 486	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
20	13, 19	mpbird 260	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·o 𝐵) = ∅)
21		vex 3458	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	sucid 6473	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
23		omlim 8579	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
24		eqeq1 2767	. . . . . . . . . . . 12 ⊢ ((𝐴 ·o 𝐵) = suc 𝑦 → ((𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥) ↔ suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥)))
25	24	biimpac 484	. . . . . . . . . . 11 ⊢ (((𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
26	23, 25	sylan 592	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
27	22, 26	eleqtrid 2870	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
28		eliun 4960	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
29	27, 28	sylib 221	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
30	29	adantlr 728	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
31		onelon 6413	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
32	1, 31	sylan 592	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
33		onnbtwn 6485	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
34		imnan 405	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
35	33, 34	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → (𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥))
36	35	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
37	36	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
38	32, 37	mpd 16	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → ¬ 𝐵 ∈ suc 𝑥)
39	38	ad5ant24 773	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ¬ 𝐵 ∈ suc 𝑥)
40		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ On)
41	40, 31	jca 521	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
42	1, 41	sylan 592	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
43	42	anim2i 629	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
44	43	anassrs 473	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
45		omcl 8582	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
46		eloni 6398	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·o 𝑥) ∈ On → Ord (𝐴 ·o 𝑥))
47		ordsucelsuc 7867	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Ord (𝐴 ·o 𝑥) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
48	46, 47	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ·o 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
49		oa1suc 8577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·o 𝑥) ∈ On → ((𝐴 ·o 𝑥) +o 1o) = suc (𝐴 ·o 𝑥))
50	49	eleq2d 2850	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ·o 𝑥) ∈ On → (suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
51	48, 50	bitr4d 285	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ·o 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
52	45, 51	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
53	52	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
54		eloni 6398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ On → Ord 𝐴)
55		ordgt0ge1 8539	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
56	54, 55	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
57	56	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
58		1on 8527	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1o ∈ On
59		oaword 8595	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1o ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (1o ⊆ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
60	58, 59	mp3an1 1478	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (1o ⊆ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
61	45, 60	syldan 603	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (1o ⊆ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
62	57, 61	bitrd 282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
63	62	biimpa 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴))
64		omsuc 8572	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
65	64	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
66	63, 65	sseqtrrd 3950	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) +o 1o) ⊆ (𝐴 ·o suc 𝑥))
67	66	sseld 3912	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o) → suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
68	53, 67	sylbid 243	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
69		eleq1 2852	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ·o 𝐵) = suc 𝑦 → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
70	69	biimprd 251	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ·o 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 ·o suc 𝑥) → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥)))
71	68, 70	syl9 78	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = suc 𝑦 → (𝑦 ∈ (𝐴 ·o 𝑥) → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥))))
72	71	com23 87	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥))))
73	72	adantlrl 733	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥))))
74		onsucb 7862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
75		omord 8614	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥)))
76		simpl 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) → 𝐵 ∈ suc 𝑥)
77	75, 76	biimtrrdi 257	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
78	74, 77	syl3an2b 1432	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
79	78	3comr 1143	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
80	79	3expb 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
81	80	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
82	73, 81	syl6d 76	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥)))
83	44, 82	sylan 592	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑥 ∈ 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥)))
84	83	an32s 665	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥)))
85	84	imp 412	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ((𝐴 ·o 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥))
86	39, 85	mtod 201	. . . . . . . . 9 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
87	86	rexlimdva2 3169	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥) → ¬ (𝐴 ·o 𝐵) = suc 𝑦))
88	87	adantr 486	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥) → ¬ (𝐴 ·o 𝐵) = suc 𝑦))
89	30, 88	mpd 16	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
90	89	pm2.01da 811	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
91	90	adantr 486	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ On) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
92	91	nrexdv 3161	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦)
93		ioran 999	. . 3 ⊢ (¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦) ↔ (¬ (𝐴 ·o 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦))
94	20, 92, 93	sylanbrc 595	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦))
95		dflim3 7892	. 2 ⊢ (Lim (𝐴 ·o 𝐵) ↔ (Ord (𝐴 ·o 𝐵) ∧ ¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦)))
96	6, 94, 95	sylanbrc 595	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  ∪ ciun 4956  Ord word 6387  Oncon0 6388  Lim wlim 6389  suc csuc 6390  (class class class)co 7451  1_oc1o 8507   +_o coa 8511   ·_o comu 8512

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-oadd 8518  df-omul 8519

This theorem is used by:  odi  8625  omass  8626  omlimcl2  44505  omlim2  44562