Theorem oalimcl 8353

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 6314	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		oacl 8327	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3		eloni 6261	. . . 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 592	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +o 𝐵))
6		0ellim 6313	. . . . . 6 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
7		n0i 4264	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
8	6, 7	syl 17	. . . . 5 ⊢ (Lim 𝐵 → ¬ 𝐵 = ∅)
9	8	ad2antll 725	. . . 4 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10		oa00 8352	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11		simpr 484	. . . . . . 7 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
12	10, 11	syl6bi 252	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ → 𝐵 = ∅))
13	12	con3d 152	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
14	1, 13	sylan2 592	. . . 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 3426	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
17	16	sucid 6330	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
18		oalim 8324	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
19		eqeq1 2742	. . . . . . . . . . . 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 243	. . . . . . . . . . 11 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥)))
21	20	imp 406	. . . . . . . . . 10 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
22	17, 21	eleqtrid 2845	. . . . . . . . 9 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
23		eliun 4925	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +o 𝑥))
24	22, 23	sylib 217	. . . . . . . 8 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +o 𝑥))
25		onelon 6276	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
26	1, 25	sylan 579	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
27		onnbtwn 6342	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
28		imnan 399	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
29	27, 28	sylibr 233	. . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34		oacl 8327	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
35		eloni 6261	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 +o 𝑥) ∈ On → Ord (𝐴 +o 𝑥))
36		ordsucelsuc 7644	. . . . . . . . . . . . . . . . . . . . . 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 8316	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
39	38	eleq2d 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +o 𝑥)))
40	37, 39	bitr4d 281	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
41	26, 40	sylan2 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
42		eleq1 2826	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
43	42	bicomd 222	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
44	41, 43	sylan9bbr 510	. . . . . . . . . . . . . . . . . 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 7639	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
47	26, 46	sylib 217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → suc 𝑥 ∈ On)
48	45, 47	jca 511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49		oaord 8340	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
50	49	3expa 1116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
51	48, 50	sylan 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
52	51	ancoms 458	. . . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ 𝐵 ∈ suc 𝑥))
55	54	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) → 𝐵 ∈ suc 𝑥))
56	55	exp32 420	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → (𝐴 ∈ On → (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ (𝐴 +o 𝑥) → 𝐵 ∈ suc 𝑥))))
57	56	com4l 92	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ (𝐴 +o 𝑥) → ((𝐴 +o 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥))))
58	57	imp32 418	. . . . . . . . . . . . 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 197	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
60	59	exp44 437	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝑥 ∈ 𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦))))
61	60	imp 406	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ 𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦)))
62	61	rexlimdv 3211	. . . . . . . . 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 413	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ((𝐴 +o 𝐵) = suc 𝑦 → ¬ (𝐴 +o 𝐵) = suc 𝑦))
66	65	pm2.01d 189	. . . . 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 3197	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)
69		ioran 980	. . 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 582	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦))
71		dflim3 7669	. 2 ⊢ (Lim (𝐴 +o 𝐵) ↔ (Ord (𝐴 +o 𝐵) ∧ ¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)))
72	5, 70, 71	sylanbrc 582	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +o 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  ∅c0 4253  ∪ ciun 4921  Ord word 6250  Oncon0 6251  Lim wlim 6252  suc csuc 6253  (class class class)co 7255   +_o coa 8264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-oadd 8271

This theorem is referenced by:  oaass  8354  odi  8372  wunex3  10428