Theorem omlimcl 8505

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 6382	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		omcl 8463	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
3		eloni 6327	. . . . 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 593	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ·o 𝐵))
6	5	adantr 480	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord (𝐴 ·o 𝐵))
7		0ellim 6381	. . . . . . . 8 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
8		n0i 4292	. . . . . . . 8 ⊢ (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
9	7, 8	syl 17	. . . . . . 7 ⊢ (Lim 𝐵 → ¬ 𝐵 = ∅)
10		n0i 4292	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → ¬ 𝐴 = ∅)
11	9, 10	anim12ci 614	. . . . . 6 ⊢ ((Lim 𝐵 ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
12	11	adantll 714	. . . . 5 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
13	12	adantll 714	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
14		om00 8502	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
15	14	notbid 318	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅)))
16		ioran 985	. . . . . . 7 ⊢ (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
17	15, 16	bitrdi 287	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
18	1, 17	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
19	18	adantr 480	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
20	13, 19	mpbird 257	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·o 𝐵) = ∅)
21		vex 3444	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	sucid 6401	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
23		omlim 8460	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
24		eqeq1 2740	. . . . . . . . . . . 12 ⊢ ((𝐴 ·o 𝐵) = suc 𝑦 → ((𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥) ↔ suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥)))
25	24	biimpac 478	. . . . . . . . . . 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 580	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
27	22, 26	eleqtrid 2842	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥))
28		eliun 4950	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 ·o 𝑥) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
29	27, 28	sylib 218	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
30	29	adantlr 715	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
31		onelon 6342	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
32	1, 31	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
33		onnbtwn 6413	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
34		imnan 399	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
35	33, 34	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → (𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥))
36	35	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
37	36	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
38	32, 37	mpd 15	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → ¬ 𝐵 ∈ suc 𝑥)
39	38	ad5ant24 760	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ¬ 𝐵 ∈ suc 𝑥)
40		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ On)
41	40, 31	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
42	1, 41	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
43	42	anim2i 617	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
44	43	anassrs 467	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
45		omcl 8463	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
46		eloni 6327	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·o 𝑥) ∈ On → Ord (𝐴 ·o 𝑥))
47		ordsucelsuc 7764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Ord (𝐴 ·o 𝑥) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
48	46, 47	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ·o 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
49		oa1suc 8458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ·o 𝑥) ∈ On → ((𝐴 ·o 𝑥) +o 1o) = suc (𝐴 ·o 𝑥))
50	49	eleq2d 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ·o 𝑥) ∈ On → (suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
51	48, 50	bitr4d 282	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ·o 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
52	45, 51	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
53	52	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
54		eloni 6327	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ On → Ord 𝐴)
55		ordgt0ge1 8420	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
57	56	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
58		1on 8409	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1o ∈ On
59		oaword 8476	. . . . . . . . . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (1o ⊆ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
61	45, 60	syldan 591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (1o ⊆ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
62	57, 61	bitrd 279	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
63	62	biimpa 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴))
64		omsuc 8453	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
65	64	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
66	63, 65	sseqtrrd 3971	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) +o 1o) ⊆ (𝐴 ·o suc 𝑥))
67	66	sseld 3932	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o) → suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
68	53, 67	sylbid 240	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
69		eleq1 2824	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ·o 𝐵) = suc 𝑦 → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
70	69	biimprd 248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ·o 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 ·o suc 𝑥) → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥)))
71	68, 70	syl9 77	. . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . 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 720	. . . . . . . . . . . . . 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 7759	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
75		omord 8495	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥)))
76		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) → 𝐵 ∈ suc 𝑥)
77	75, 76	biimtrrdi 254	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
78	74, 77	syl3an2b 1406	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
79	78	3comr 1125	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
80	79	3expb 1120	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
81	80	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
82	73, 81	syl6d 75	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥)))
83	44, 82	sylan 580	. . . . . . . . . . . 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 652	. . . . . . . . . . 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 406	. . . . . . . . . 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 198	. . . . . . . . 9 ⊢ (((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
87	86	rexlimdva2 3139	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ·o 𝑥) → ¬ (𝐴 ·o 𝐵) = suc 𝑦))
88	87	adantr 480	. . . . . . 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 15	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
90	89	pm2.01da 798	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
91	90	adantr 480	. . . 4 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ On) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
92	91	nrexdv 3131	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦)
93		ioran 985	. . 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 583	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦))
95		dflim3 7789	. 2 ⊢ (Lim (𝐴 ·o 𝐵) ↔ (Ord (𝐴 ·o 𝐵) ∧ ¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦)))
96	6, 94, 95	sylanbrc 583	1 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ⊆ wss 3901  ∅c0 4285  ∪ ciun 4946  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7358  1_oc1o 8390   +_o coa 8394   ·_o comu 8395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-omul 8402

This theorem is referenced by:  odi  8506  omass  8507  omlimcl2  43484  omlim2  43541