Theorem omlimcl 8559

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 6423	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		omcl 8517	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
3		eloni 6367	. . . . 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 604	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ·o 𝐵))
6	5	adantr 485	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord (𝐴 ·o 𝐵))
7		0ellim 6422	. . . . . . . 8 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
8		n0i 4301	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
9	7, 8	syl 18	. . . . . . 7 ⊢ (Lim 𝐵 → ¬ 𝐵 = ∅)
10		n0i 4301	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → ¬ 𝐴 = ∅)
11	9, 10	anim12ci 625	. . . . . 6 ⊢ ((Lim 𝐵 ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
12	11	adantll 726	. . . . 5 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
13	12	adantll 726	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
14		om00 8556	. . . . . . . 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 604	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
19	18	adantr 485	. . . 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 3467	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	sucid 6442	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
23		omlim 8514	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
24		eqeq1 2773	. . . . . . . . . . . 12 ⊢ ((𝐴 ·o 𝐵) = suc 𝑦 → ((𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥) ↔ suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥)))
25	24	biimpac 483	. . . . . . . . . . 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 591	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
27	22, 26	eleqtrid 2875	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
28		eliun 4961	. . . . . . . . 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 727	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
31		onelon 6382	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
32	1, 31	sylan 591	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
33		onnbtwn 6454	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
34		imnan 404	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 772	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ¬ 𝐵 ∈ suc 𝑥)
40		simpl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ On)
41	40, 31	jca 520	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
42	1, 41	sylan 591	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
43	42	anim2i 628	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
44	43	anassrs 472	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
45		omcl 8517	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
46		eloni 6367	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·o 𝑥) ∈ On → Ord (𝐴 ·o 𝑥))
47		ordsucelsuc 7814	. . . . . . . . . . . . . . . . . . . . . 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 8512	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·o 𝑥) ∈ On → ((𝐴 ·o 𝑥) +o 1o) = suc (𝐴 ·o 𝑥))
50	49	eleq2d 2855	. . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
54		eloni 6367	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ On → Ord 𝐴)
55		ordgt0ge1 8474	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
56	54, 55	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
57	56	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
58		1on 8462	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1o ∈ On
59		oaword 8530	. . . . . . . . . . . . . . . . . . . . . . . 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 1474	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (1o ⊆ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
61	45, 60	syldan 602	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴))
64		omsuc 8507	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
65	64	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
66	63, 65	sseqtrrd 3982	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) +o 1o) ⊆ (𝐴 ·o suc 𝑥))
67	66	sseld 3944	. . . . . . . . . . . . . . . . . 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 2857	. . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . 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 7809	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
75		omord 8549	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥)))
76		simpl 487	. . . . . . . . . . . . . . . . . . 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 1429	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
79	78	3comr 1141	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
80	79	3expb 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
81	80	adantr 485	. . . . . . . . . . . . . 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 591	. . . . . . . . . . . 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 664	. . . . . . . . . . 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 411	. . . . . . . . . 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 3174	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥) → ¬ (𝐴 ·o 𝐵) = suc 𝑦))
88	87	adantr 485	. . . . . . 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 810	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
91	90	adantr 485	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ On) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
92	91	nrexdv 3166	. . 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 594	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦))
95		dflim3 7839	. 2 ⊢ (Lim (𝐴 ·o 𝐵) ↔ (Ord (𝐴 ·o 𝐵) ∧ ¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦)))
96	6, 94, 95	sylanbrc 594	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃wrex 3095   ⊆ wss 3913  ∅c0 4294  ∪ ciun 4957  Ord word 6356  Oncon0 6357  Lim wlim 6358  suc csuc 6359  (class class class)co 7408  1_oc1o 8442   +_o coa 8446   ·_o comu 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-omul 8454

This theorem is referenced by:  odi  8560  omass  8561  omlimcl2  43854  omlim2  43911