Theorem omlimcl 7658

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 5788	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		omcl 7616	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
3		eloni 5733	. . . . 5 ⊢ ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
4	2, 3	syl 17	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·𝑜 𝐵))
5	1, 4	sylan2 491	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ·𝑜 𝐵))
6	5	adantr 481	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord (𝐴 ·𝑜 𝐵))
7		0ellim 5787	. . . . . . . 8 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
8		n0i 3920	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
9	7, 8	syl 17	. . . . . . 7 ⊢ (Lim 𝐵 → ¬ 𝐵 = ∅)
10		n0i 3920	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → ¬ 𝐴 = ∅)
11	9, 10	anim12ci 591	. . . . . 6 ⊢ ((Lim 𝐵 ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
12	11	adantll 750	. . . . 5 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
13	12	adantll 750	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
14		om00 7655	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
15	14	notbid 308	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅)))
16		ioran 511	. . . . . . 7 ⊢ (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
17	15, 16	syl6bb 276	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
18	1, 17	sylan2 491	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
19	18	adantr 481	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
20	13, 19	mpbird 247	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·𝑜 𝐵) = ∅)
21		vex 3203	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	sucid 5804	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
23		omlim 7613	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥))
24		eqeq1 2626	. . . . . . . . . . . 12 ⊢ ((𝐴 ·𝑜 𝐵) = suc 𝑦 → ((𝐴 ·𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥)))
25	24	biimpac 503	. . . . . . . . . . 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 488	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥))
27	22, 26	syl5eleq 2707	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥))
28		eliun 4524	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 ·𝑜 𝑥) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
29	27, 28	sylib 208	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
30	29	adantlr 751	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
31		onelon 5748	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
32	1, 31	sylan 488	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
33		onnbtwn 5818	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
34		imnan 438	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
35	33, 34	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥))
36	35	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
37	36	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
38	32, 37	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → ¬ 𝐵 ∈ suc 𝑥)
39	38	adantll 750	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑥 ∈ 𝐵) → ¬ 𝐵 ∈ suc 𝑥)
40	39	adantlr 751	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ¬ 𝐵 ∈ suc 𝑥)
41	40	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ¬ 𝐵 ∈ suc 𝑥)
42		simpl 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ On)
43	42, 31	jca 554	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
44	1, 43	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
45	44	anim2i 593	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
46	45	anassrs 680	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
47		omcl 7616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
48		eloni 5733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ·𝑜 𝑥) ∈ On → Ord (𝐴 ·𝑜 𝑥))
49		ordsucelsuc 7022	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord (𝐴 ·𝑜 𝑥) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·𝑜 𝑥)))
50	48, 49	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·𝑜 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·𝑜 𝑥)))
51		oa1suc 7611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ·𝑜 𝑥) ∈ On → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) = suc (𝐴 ·𝑜 𝑥))
52	51	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·𝑜 𝑥) ∈ On → (suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ↔ suc 𝑦 ∈ suc (𝐴 ·𝑜 𝑥)))
53	50, 52	bitr4d 271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ·𝑜 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜)))
54	47, 53	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜)))
55	54	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜)))
56		eloni 5733	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ On → Ord 𝐴)
57		ordgt0ge1 7577	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴))
58	56, 57	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴))
59	58	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴))
60		1on 7567	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1𝑜 ∈ On
61		oaword 7629	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1𝑜 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (1𝑜 ⊆ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
62	60, 61	mp3an1 1411	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (1𝑜 ⊆ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
63	47, 62	syldan 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (1𝑜 ⊆ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
64	59, 63	bitrd 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
65	64	biimpa 501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
66		omsuc 7606	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
67	66	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
68	65, 67	sseqtr4d 3642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ (𝐴 ·𝑜 suc 𝑥))
69	68	sseld 3602	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) → suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
70	55, 69	sylbid 230	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
71		eleq1 2689	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ·𝑜 𝐵) = suc 𝑦 → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
72	71	biimprd 238	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥) → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥)))
73	70, 72	syl9 77	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥))))
74	73	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥))))
75	74	adantlrl 756	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥))))
76		sucelon 7017	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
77		omord 7648	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥)))
78		simpl 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) → 𝐵 ∈ suc 𝑥)
79	77, 78	syl6bir 244	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
80	76, 79	syl3an2b 1363	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
81	80	3comr 1273	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
82	81	3expb 1266	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
83	82	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
84	75, 83	syl6d 75	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥)))
85	46, 84	sylan 488	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑥 ∈ 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥)))
86	85	an32s 846	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥)))
87	86	imp 445	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥))
88	41, 87	mtod 189	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
89	88	exp31 630	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝑥 ∈ 𝐵 → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)))
90	89	rexlimdv 3030	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦))
91	90	adantr 481	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦))
92	30, 91	mpd 15	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
93	92	pm2.01da 458	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
94	93	adantr 481	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ On) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
95	94	nrexdv 3001	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦)
96		ioran 511	. . 3 ⊢ (¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦) ↔ (¬ (𝐴 ·𝑜 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦))
97	20, 95, 96	sylanbrc 698	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦))
98		dflim3 7047	. 2 ⊢ (Lim (𝐴 ·𝑜 𝐵) ↔ (Ord (𝐴 ·𝑜 𝐵) ∧ ¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦)))
99	6, 97, 98	sylanbrc 698	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·𝑜 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650  1_𝑜c1o 7553   +_𝑜 coa 7557   ·_𝑜 comu 7558

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-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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565

This theorem is referenced by:  odi  7659  omass  7660