dflim5 - Mathbox for Richard Penner

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Theorem dflim5 43941

Description: A limit ordinal is either the proper class of ordinals or some nonzero product with omega. (Contributed by RP, 8-Jan-2025.)

**Assertion**
Ref	Expression
dflim5	⊢ (Lim 𝐴 ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥)))

Distinct variable group: 𝑥,𝐴

Proof of Theorem dflim5 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limord 6419	. . . . 5 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordeleqon 7777	. . . . . . 7 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3	2	biimpi 219	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
4	3	orcomd 884	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = On ∨ 𝐴 ∈ On))
5	1, 4	syl 18	. . . 4 ⊢ (Lim 𝐴 → (𝐴 = On ∨ 𝐴 ∈ On))
6	5	pm4.71ri 569	. . 3 ⊢ (Lim 𝐴 ↔ ((𝐴 = On ∨ 𝐴 ∈ On) ∧ Lim 𝐴))
7		andir 1024	. . 3 ⊢ (((𝐴 = On ∨ 𝐴 ∈ On) ∧ Lim 𝐴) ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
8	6, 7	bitri 278	. 2 ⊢ (Lim 𝐴 ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
9		limon 7828	. . . . 5 ⊢ Lim On
10		limeq 6369	. . . . 5 ⊢ (𝐴 = On → (Lim 𝐴 ↔ Lim On))
11	9, 10	mpbiri 261	. . . 4 ⊢ (𝐴 = On → Lim 𝐴)
12	11	pm4.71i 568	. . 3 ⊢ (𝐴 = On ↔ (𝐴 = On ∧ Lim 𝐴))
13	12	orbi1i 926	. 2 ⊢ ((𝐴 = On ∨ (𝐴 ∈ On ∧ Lim 𝐴)) ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
14		simpl 487	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → 𝐴 ∈ On)
15		omelon 9611	. . . . . . . 8 ⊢ ω ∈ On
16	15	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ On → ω ∈ On)
17		id 23	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ On)
18		peano1 7881	. . . . . . . . 9 ⊢ ∅ ∈ ω
19	18	ne0ii 4305	. . . . . . . 8 ⊢ ω ≠ ∅
20	19	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ On → ω ≠ ∅)
21	16, 17, 20	3jca 1144	. . . . . 6 ⊢ (𝐴 ∈ On → (ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅))
22		omeulem1 8563	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ ω ((ω ·_o 𝑥) +_o 𝑦) = 𝐴)
23	14, 21, 22	3syl 19	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ∃𝑥 ∈ On ∃𝑦 ∈ ω ((ω ·_o 𝑥) +_o 𝑦) = 𝐴)
24		limeq 6369	. . . . . . . . . . . . . . . 16 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim 𝐴))
25	24	biimprd 251	. . . . . . . . . . . . . . 15 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (Lim 𝐴 → Lim ((ω ·_o 𝑥) +_o 𝑦)))
26		simplr 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → 𝑦 ∈ ω)
27		nnlim 7872	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ω → ¬ Lim 𝑦)
28	26, 27	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → ¬ Lim 𝑦)
29		on0eln0 6415	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
30	29	biimprd 251	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
31	30	necon1bd 2982	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ On → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
32	31	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
33	32	imp 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
34	33, 26	jca 520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → (𝑥 = ∅ ∧ 𝑦 ∈ ω))
35		simpl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → 𝑥 = ∅)
36	35	oveq2d 7424	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·_o 𝑥) = (ω ·_o ∅))
37		om0 8498	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ω ∈ On → (ω ·_o ∅) = ∅)
38	15, 37	mp1i 14	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·_o ∅) = ∅)
39	36, 38	eqtrd 2804	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·_o 𝑥) = ∅)
40	39	oveq1d 7423	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → ((ω ·_o 𝑥) +_o 𝑦) = (∅ +_o 𝑦))
41		nna0r 8591	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ω → (∅ +_o 𝑦) = 𝑦)
42	41	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (∅ +_o 𝑦) = 𝑦)
43	40, 42	eqtrd 2804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → ((ω ·_o 𝑥) +_o 𝑦) = 𝑦)
44		limeq 6369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = 𝑦 → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim 𝑦))
45	34, 43, 44	3syl 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim 𝑦))
46	28, 45	mtbird 328	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦))
47	46	ex 417	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ ∅ ∈ 𝑥 → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦)))
48		ovex 7441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ω ·_o 𝑥) +_o ∪ 𝑦) ∈ V
49		nlimsucg 7834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ω ·_o 𝑥) +_o ∪ 𝑦) ∈ V → ¬ Lim suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
50	48, 49	mp1i 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ¬ Lim suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
51		nnord 7866	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ω → Ord 𝑦)
52		orduniorsuc 7822	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Ord 𝑦 → (𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦))
53	51, 52	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ω → (𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦))
54		3ianor 1122	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ (Ord 𝑦 ∧ 𝑦 ≠ ∅ ∧ 𝑦 = ∪ 𝑦) ↔ (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = ∪ 𝑦))
55		df-lim 6362	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (Lim 𝑦 ↔ (Ord 𝑦 ∧ 𝑦 ≠ ∅ ∧ 𝑦 = ∪ 𝑦))
56	54, 55	xchnxbir 336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ Lim 𝑦 ↔ (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = ∪ 𝑦))
57	27, 56	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ω → (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = ∪ 𝑦))
58	51	pm2.24d 152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ ω → (¬ Ord 𝑦 → (𝑦 = ∪ 𝑦 → 𝑦 = ∅)))
59		nne 2968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
60	59	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑦 ≠ ∅ → 𝑦 = ∅)
61	60	a1i13 28	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ ω → (¬ 𝑦 ≠ ∅ → (𝑦 = ∪ 𝑦 → 𝑦 = ∅)))
62		pm2.21 124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑦 = ∪ 𝑦 → (𝑦 = ∪ 𝑦 → 𝑦 = ∅))
63	62	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∪ 𝑦 → (𝑦 = ∪ 𝑦 → 𝑦 = ∅)))
64	58, 61, 63	3jaod 1454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ω → ((¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = ∪ 𝑦) → (𝑦 = ∪ 𝑦 → 𝑦 = ∅)))
65	57, 64	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ω → (𝑦 = ∪ 𝑦 → 𝑦 = ∅))
66	65	orim1d 981	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ω → ((𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦) → (𝑦 = ∅ ∨ 𝑦 = suc ∪ 𝑦)))
67	53, 66	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦 = suc ∪ 𝑦))
68	67	ord 877	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 = suc ∪ 𝑦))
69	68	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ 𝑦 = ∅ → 𝑦 = suc ∪ 𝑦))
70	69	imp 411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → 𝑦 = suc ∪ 𝑦)
71	70	oveq2d 7424	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·_o 𝑥) +_o 𝑦) = ((ω ·_o 𝑥) +_o suc ∪ 𝑦))
72		simpl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
73	72	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → 𝑥 ∈ On)
74		omcl 8517	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ·_o 𝑥) ∈ On)
75	15, 73, 74	sylancr 598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → (ω ·_o 𝑥) ∈ On)
76		nnon 7864	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
77		onuni 7783	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ On → ∪ 𝑦 ∈ On)
78	76, 77	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ω → ∪ 𝑦 ∈ On)
79	78	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → ∪ 𝑦 ∈ On)
80	79	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ∪ 𝑦 ∈ On)
81		oasuc 8505	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ω ·_o 𝑥) ∈ On ∧ ∪ 𝑦 ∈ On) → ((ω ·_o 𝑥) +_o suc ∪ 𝑦) = suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
82	75, 80, 81	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·_o 𝑥) +_o suc ∪ 𝑦) = suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
83	71, 82	eqtrd 2804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·_o 𝑥) +_o 𝑦) = suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
84		limeq 6369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = suc ((ω ·_o 𝑥) +_o ∪ 𝑦) → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim suc ((ω ·_o 𝑥) +_o ∪ 𝑦)))
85	83, 84	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim suc ((ω ·_o 𝑥) +_o ∪ 𝑦)))
86	50, 85	mtbird 328	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦))
87	86	ex 417	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ 𝑦 = ∅ → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦)))
88	47, 87	jaod 872	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → ((¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅) → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦)))
89	88	con2d 135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim ((ω ·_o 𝑥) +_o 𝑦) → ¬ (¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅)))
90		anor 998	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ∈ 𝑥 ∧ 𝑦 = ∅) ↔ ¬ (¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅))
91	89, 90	imbitrrdi 255	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim ((ω ·_o 𝑥) +_o 𝑦) → (∅ ∈ 𝑥 ∧ 𝑦 = ∅)))
92	25, 91	syl9 78	. . . . . . . . . . . . . 14 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim 𝐴 → (∅ ∈ 𝑥 ∧ 𝑦 = ∅))))
93	92	com13 89	. . . . . . . . . . . . 13 ⊢ (Lim 𝐴 → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (∅ ∈ 𝑥 ∧ 𝑦 = ∅))))
94	93	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (∅ ∈ 𝑥 ∧ 𝑦 = ∅))))
95	94	3imp 1126	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → (∅ ∈ 𝑥 ∧ 𝑦 = ∅))
96		simp2 1153	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ ω))
97	96, 72	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → 𝑥 ∈ On)
98		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((∅ ∈ 𝑥 ∧ 𝑦 = ∅) → ∅ ∈ 𝑥)
99	97, 98	anim12i 624	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → (𝑥 ∈ On ∧ ∅ ∈ 𝑥))
100		ondif1 8482	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1_o) ↔ (𝑥 ∈ On ∧ ∅ ∈ 𝑥))
101	99, 100	sylibr 237	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → 𝑥 ∈ (On ∖ 1_o))
102		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ 𝑥 ∧ 𝑦 = ∅) → 𝑦 = ∅)
103	102	oveq2d 7424	. . . . . . . . . . . . . 14 ⊢ ((∅ ∈ 𝑥 ∧ 𝑦 = ∅) → ((ω ·_o 𝑥) +_o 𝑦) = ((ω ·_o 𝑥) +_o ∅))
104	103	adantl 486	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → ((ω ·_o 𝑥) +_o 𝑦) = ((ω ·_o 𝑥) +_o ∅))
105		simpl3 1210	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → ((ω ·_o 𝑥) +_o 𝑦) = 𝐴)
106	15, 72, 74	sylancr 598	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (ω ·_o 𝑥) ∈ On)
107		oa0 8497	. . . . . . . . . . . . . . 15 ⊢ ((ω ·_o 𝑥) ∈ On → ((ω ·_o 𝑥) +_o ∅) = (ω ·_o 𝑥))
108	96, 106, 107	3syl 19	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → ((ω ·_o 𝑥) +_o ∅) = (ω ·_o 𝑥))
109	108	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → ((ω ·_o 𝑥) +_o ∅) = (ω ·_o 𝑥))
110	104, 105, 109	3eqtr3d 2812	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → 𝐴 = (ω ·_o 𝑥))
111	101, 110	jca 520	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → (𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)))
112	95, 111	mpdan 699	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → (𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)))
113	112	3exp 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)))))
114	113	expdimp 457	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ 𝑥 ∈ On) → (𝑦 ∈ ω → (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)))))
115	114	rexlimdv 3170	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ 𝑥 ∈ On) → (∃𝑦 ∈ ω ((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥))))
116	115	expimpd 458	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ((𝑥 ∈ On ∧ ∃𝑦 ∈ ω ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → (𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥))))
117	116	reximdv2 3181	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (∃𝑥 ∈ On ∃𝑦 ∈ ω ((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥)))
118	23, 117	mpd 16	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥))
119		simpr 489	. . . . . . 7 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → 𝐴 = (ω ·_o 𝑥))
120		eldifi 4093	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 1_o) → 𝑥 ∈ On)
121	15, 120, 74	sylancr 598	. . . . . . . 8 ⊢ (𝑥 ∈ (On ∖ 1_o) → (ω ·_o 𝑥) ∈ On)
122	121	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → (ω ·_o 𝑥) ∈ On)
123	119, 122	eqeltrd 2869	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → 𝐴 ∈ On)
124		limom 7874	. . . . . . . . . . 11 ⊢ Lim ω
125	15, 124	pm3.2i 475	. . . . . . . . . 10 ⊢ (ω ∈ On ∧ Lim ω)
126		omlimcl2 43854	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑥) → Lim (ω ·_o 𝑥))
127	125, 126	mpanl2 713	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → Lim (ω ·_o 𝑥))
128	100, 127	sylbi 220	. . . . . . . 8 ⊢ (𝑥 ∈ (On ∖ 1_o) → Lim (ω ·_o 𝑥))
129	128	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → Lim (ω ·_o 𝑥))
130		limeq 6369	. . . . . . . 8 ⊢ (𝐴 = (ω ·_o 𝑥) → (Lim 𝐴 ↔ Lim (ω ·_o 𝑥)))
131	130	adantl 486	. . . . . . 7 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → (Lim 𝐴 ↔ Lim (ω ·_o 𝑥)))
132	129, 131	mpbird 260	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → Lim 𝐴)
133	123, 132	jca 520	. . . . 5 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → (𝐴 ∈ On ∧ Lim 𝐴))
134	133	rexlimiva 3164	. . . 4 ⊢ (∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥) → (𝐴 ∈ On ∧ Lim 𝐴))
135	118, 134	impbii 212	. . 3 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) ↔ ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥))
136	135	orbi2i 925	. 2 ⊢ ((𝐴 = On ∨ (𝐴 ∈ On ∧ Lim 𝐴)) ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥)))
137	8, 13, 136	3bitr2i 302	1 ⊢ (Lim 𝐴 ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1100 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∃wrex 3095 Vcvv 3463 ∖ cdif 3910 ∅c0 4294 ∪ cuni 4873 Ord word 6356 Oncon0 6357 Lim wlim 6358 suc csuc 6359 (class class class)co 7408 ωcom 7858 1_oc1o 8442 +_o coa 8446 ·_o comu 8447

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pr 5402 ax-un 7730 ax-inf2 9606

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-omul 8454

This theorem is referenced by: (None)

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