Theorem oalimcl 8550

Description: The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. Lemma 3.4 of [Schloeder] p. 7. (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 6427	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		oacl 8525	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
3		eloni 6371	. . . 4 ⊢ ((𝐴 +o 𝐵) ∈ On → Ord (𝐴 +o 𝐵))
4	2, 3	syl 18	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 +o 𝐵))
5	1, 4	sylan2 605	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 +o 𝐵))
6		0ellim 6426	. . . . . 6 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
7		n0i 4289	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
8	6, 7	syl 18	. . . . 5 ⊢ (Lim 𝐵 → ¬ 𝐵 = ∅)
9	8	ad2antll 742	. . . 4 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ 𝐵 = ∅)
10		oa00 8549	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
11		simpr 490	. . . . . . 7 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐵 = ∅)
12	10, 11	biimtrdi 256	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ → 𝐵 = ∅))
13	12	con3d 153	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
14	1, 13	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (¬ 𝐵 = ∅ → ¬ (𝐴 +o 𝐵) = ∅))
15	9, 14	mpd 16	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +o 𝐵) = ∅)
16		vex 3457	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
17	16	sucid 6446	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
18		oalim 8522	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
19		eqeq1 2766	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥) ↔ suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥)))
20	18, 19	imbitrid 247	. . . . . . . . . . 11 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥)))
21	20	imp 412	. . . . . . . . . 10 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → suc 𝑦 = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
22	17, 21	eleqtrid 2868	. . . . . . . . 9 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥))
23		eliun 4958	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +o 𝑥))
24	22, 23	sylib 221	. . . . . . . 8 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +o 𝑥))
25		onelon 6386	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
26	1, 25	sylan 592	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
27		onnbtwn 6458	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
28		imnan 405	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝑥))
29	27, 28	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (𝑥 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝑥))
30	29	com12 33	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
31	30	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
32	26, 31	mpd 16	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → ¬ 𝐵 ∈ suc 𝑥)
33	32	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ 𝐵 ∈ suc 𝑥)
34		oacl 8525	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
35		eloni 6371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 +o 𝑥) ∈ On → Ord (𝐴 +o 𝑥))
36		ordsucelsuc 7821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Ord (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +o 𝑥)))
37	34, 35, 36	3syl 19	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +o 𝑥)))
38		oasuc 8514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
39	38	eleq2d 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 +o 𝑥)))
40	37, 39	bitr4d 285	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
41	26, 40	sylan2 605	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
42		eleq1 2850	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → ((𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 +o suc 𝑥)))
43	42	bicomd 226	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 +o suc 𝑥) ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
44	41, 43	sylan9bbr 520	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ On)
46		onsucb 7816	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
47	26, 46	sylib 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → suc 𝑥 ∈ On)
48	45, 47	jca 521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐵 ∈ On ∧ suc 𝑥 ∈ On))
49		oaord 8537	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
50	49	3expa 1136	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∈ On ∧ suc 𝑥 ∈ On) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
51	48, 50	sylan 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
52	51	ancoms 464	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵)) → (𝐵 ∈ suc 𝑥 ↔ (𝐴 +o 𝐵) ∈ (𝐴 +o suc 𝑥)))
53	52	adantl 487	. . . . . . . . . . . . . . . . . 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 285	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) ↔ 𝐵 ∈ suc 𝑥))
55	54	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵))) → (𝑦 ∈ (𝐴 +o 𝑥) → 𝐵 ∈ suc 𝑥))
56	55	exp32 426	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +o 𝐵) = suc 𝑦 → (𝐴 ∈ On → (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ (𝐴 +o 𝑥) → 𝐵 ∈ suc 𝑥))))
57	56	com4l 93	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ (𝐴 +o 𝑥) → ((𝐴 +o 𝐵) = suc 𝑦 → 𝐵 ∈ suc 𝑥))))
58	57	imp32 424	. . . . . . . . . . . . 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 201	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 +o 𝑥))) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
60	59	exp44 443	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝑥 ∈ 𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦))))
61	60	imp 412	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ 𝐵 → (𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦)))
62	61	rexlimdv 3163	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 +o 𝑥) → ¬ (𝐴 +o 𝐵) = suc 𝑦))
63	62	adantl 487	. . . . . . . 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 16	. . . . . . 7 ⊢ (((𝐴 +o 𝐵) = suc 𝑦 ∧ (𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵))) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
65	64	expcom 419	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ((𝐴 +o 𝐵) = suc 𝑦 → ¬ (𝐴 +o 𝐵) = suc 𝑦))
66	65	pm2.01d 192	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
67	66	adantr 486	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ On) → ¬ (𝐴 +o 𝐵) = suc 𝑦)
68	67	nrexdv 3159	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)
69		ioran 999	. . 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 595	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦))
71		dflim3 7846	. 2 ⊢ (Lim (𝐴 +o 𝐵) ↔ (Ord (𝐴 +o 𝐵) ∧ ¬ ((𝐴 +o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 +o 𝐵) = suc 𝑦)))
72	5, 70, 71	sylanbrc 595	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +o 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  ∅c0 4282  ∪ ciun 4954  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7416   +_o coa 8455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-oadd 8462

This theorem is used by:  oaass  8551  odi  8569  wunex3  10753  omlimcl2  44070  oalim2cl  44117