Theorem oalimcl 7527

Description: The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. (Contributed by NM, 8-Dec-2004.)

Assertion

Ref	Expression
oalimcl	⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +𝑜 𝐵))

Proof of Theorem oalimcl

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

Step	Hyp	Ref	Expression
1		limelon 5705	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		oacl 7502	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
3		eloni 5650	. . . 4 ⊢ ((𝐴 +𝑜 𝐵) ∈ On → Ord (𝐴 +𝑜 𝐵))
4	2, 3	syl 17	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +𝑜 𝐵))
5	1, 4	sylan2 490	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +𝑜 𝐵))
6		0ellim 5704	. . . . . 6 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
7		n0i 3879	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
8	6, 7	syl 17	. . . . 5 ⊢ (Lim 𝐵 → ¬ 𝐵 = ∅)
9	8	ad2antll 761	. . . 4 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10		oa00 7526	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11		simpr 476	. . . . . . 7 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
12	10, 11	syl6bi 242	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ → 𝐵 = ∅))
13	12	con3d 147	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
14	1, 13	sylan2 490	. . . 4 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +𝑜 𝐵) = ∅))
15	9, 14	mpd 15	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = ∅)
16		vex 3176	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
17	16	sucid 5721	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
18		oalim 7499	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +𝑜 𝑥))
19		eqeq1 2614	. . . . . . . . . . . 12 ⊢ ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 +𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +𝑜 𝑥) ↔ suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 +𝑜 𝑥)))
20	18, 19	syl5ib 233	. . . . . . . . . . 11 ⊢ ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 +𝑜 𝑥)))
21	20	imp 444	. . . . . . . . . 10 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 +𝑜 𝑥))
22	17, 21	syl5eleq 2694	. . . . . . . . 9 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 +𝑜 𝑥))
23		eliun 4460	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 +𝑜 𝑥) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥))
24	22, 23	sylib 207	. . . . . . . 8 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥))
25		onelon 5665	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
26	1, 25	sylan 487	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
27		onnbtwn 5735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
28		imnan 437	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
29	27, 28	sylibr 223	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥))
30	29	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
31	30	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
32	26, 31	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → ¬ 𝐵 ∈ suc 𝑥)
33	32	ad2antrl 760	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34		oacl 7502	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On)
35		eloni 5650	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 +𝑜 𝑥) ∈ On → Ord (𝐴 +𝑜 𝑥))
36		ordsucelsuc 6914	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Ord (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +𝑜 𝑥)))
37	34, 35, 36	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +𝑜 𝑥)))
38		oasuc 7491	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
39	38	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +𝑜 𝑥)))
40	37, 39	bitr4d 270	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
41	26, 40	sylan2 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
42		eleq1 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 +𝑜 𝐵) = suc 𝑦 → ((𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥)))
43	42	bicomd 212	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 +𝑜 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +𝑜 suc 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
44	41, 43	sylan9bbr 733	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
45	1	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ On)
46		sucelon 6909	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
47	26, 46	sylib 207	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → suc 𝑥 ∈ On)
48	45, 47	jca 553	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49		oaord 7514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
50	49	3expa 1257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
51	48, 50	sylan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
52	51	ancoms 468	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
53	52	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵))) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +𝑜 𝐵) ∈ (𝐴 +𝑜 suc 𝑥)))
54	44, 53	bitr4d 270	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐵 ∈ suc 𝑥))
55	54	biimpd 218	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵))) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → 𝐵 ∈ suc 𝑥))
56	55	exp32 629	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +𝑜 𝐵) = suc 𝑦 → (𝐴 ∈ On → (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → 𝐵 ∈ suc 𝑥))))
57	56	com4l 90	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ((𝐴 +𝑜 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥))))
58	57	imp32 448	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ((𝐴 +𝑜 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥))
59	33, 58	mtod 188	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 +𝑜 𝑥))) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
60	59	exp44 639	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝑥 ∈ 𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))))
61	60	imp 444	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ 𝐵 → (𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)))
62	61	rexlimdv 3012	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
63	62	adantl 481	. . . . . . . 8 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +𝑜 𝑥) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
64	24, 63	mpd 15	. . . . . . 7 ⊢ (((𝐴 +𝑜 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
65	64	expcom 450	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ((𝐴 +𝑜 𝐵) = suc 𝑦 → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦))
66	65	pm2.01d 180	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
67	66	adantr 480	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ On) → ¬ (𝐴 +𝑜 𝐵) = suc 𝑦)
68	67	nrexdv 2984	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)
69		ioran 510	. . 3 ⊢ (¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦) ↔ (¬ (𝐴 +𝑜 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
70	15, 68, 69	sylanbrc 695	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦))
71		dflim3 6939	. 2 ⊢ (Lim (𝐴 +𝑜 𝐵) ↔ (Ord (𝐴 +𝑜 𝐵) ∧ ¬ ((𝐴 +𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +𝑜 𝐵) = suc 𝑦)))
72	5, 70, 71	sylanbrc 695	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +𝑜 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∅c0 3874  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451

This theorem is referenced by:  oaass  7528  odi  7546  wunex3  9442