dflim5 - Mathbox for Richard Penner

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Theorem dflim5 43362

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 6362	. . . . 5 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordeleqon 7710	. . . . . . 7 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3	2	biimpi 216	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
4	3	orcomd 871	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = On ∨ 𝐴 ∈ On))
5	1, 4	syl 17	. . . 4 ⊢ (Lim 𝐴 → (𝐴 = On ∨ 𝐴 ∈ On))
6	5	pm4.71ri 560	. . 3 ⊢ (Lim 𝐴 ↔ ((𝐴 = On ∨ 𝐴 ∈ On) ∧ Lim 𝐴))
7		andir 1010	. . 3 ⊢ (((𝐴 = On ∨ 𝐴 ∈ On) ∧ Lim 𝐴) ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
8	6, 7	bitri 275	. 2 ⊢ (Lim 𝐴 ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
9		limon 7761	. . . . 5 ⊢ Lim On
10		limeq 6313	. . . . 5 ⊢ (𝐴 = On → (Lim 𝐴 ↔ Lim On))
11	9, 10	mpbiri 258	. . . 4 ⊢ (𝐴 = On → Lim 𝐴)
12	11	pm4.71i 559	. . 3 ⊢ (𝐴 = On ↔ (𝐴 = On ∧ Lim 𝐴))
13	12	orbi1i 913	. 2 ⊢ ((𝐴 = On ∨ (𝐴 ∈ On ∧ Lim 𝐴)) ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
14		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → 𝐴 ∈ On)
15		omelon 9531	. . . . . . . 8 ⊢ ω ∈ On
16	15	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ On → ω ∈ On)
17		id 22	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ On)
18		peano1 7814	. . . . . . . . 9 ⊢ ∅ ∈ ω
19	18	ne0ii 4289	. . . . . . . 8 ⊢ ω ≠ ∅
20	19	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ On → ω ≠ ∅)
21	16, 17, 20	3jca 1128	. . . . . 6 ⊢ (𝐴 ∈ On → (ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅))
22		omeulem1 8492	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ ω ((ω ·_o 𝑥) +_o 𝑦) = 𝐴)
23	14, 21, 22	3syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ∃𝑥 ∈ On ∃𝑦 ∈ ω ((ω ·_o 𝑥) +_o 𝑦) = 𝐴)
24		limeq 6313	. . . . . . . . . . . . . . . 16 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim 𝐴))
25	24	biimprd 248	. . . . . . . . . . . . . . 15 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (Lim 𝐴 → Lim ((ω ·_o 𝑥) +_o 𝑦)))
26		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → 𝑦 ∈ ω)
27		nnlim 7805	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ω → ¬ Lim 𝑦)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → ¬ Lim 𝑦)
29		on0eln0 6358	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
30	29	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
31	30	necon1bd 2946	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ On → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
32	31	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ ∅ ∈ 𝑥 → 𝑥 = ∅))
33	32	imp 406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
34	33, 26	jca 511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → (𝑥 = ∅ ∧ 𝑦 ∈ ω))
35		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → 𝑥 = ∅)
36	35	oveq2d 7357	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·_o 𝑥) = (ω ·_o ∅))
37		om0 8427	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ω ∈ On → (ω ·_o ∅) = ∅)
38	15, 37	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·_o ∅) = ∅)
39	36, 38	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·_o 𝑥) = ∅)
40	39	oveq1d 7356	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → ((ω ·_o 𝑥) +_o 𝑦) = (∅ +_o 𝑦))
41		nna0r 8519	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ω → (∅ +_o 𝑦) = 𝑦)
42	41	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (∅ +_o 𝑦) = 𝑦)
43	40, 42	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → ((ω ·_o 𝑥) +_o 𝑦) = 𝑦)
44		limeq 6313	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = 𝑦 → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim 𝑦))
45	34, 43, 44	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim 𝑦))
46	28, 45	mtbird 325	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦))
47	46	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ ∅ ∈ 𝑥 → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦)))
48		ovex 7374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ω ·_o 𝑥) +_o ∪ 𝑦) ∈ V
49		nlimsucg 7767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ω ·_o 𝑥) +_o ∪ 𝑦) ∈ V → ¬ Lim suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
50	48, 49	mp1i 13	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ¬ Lim suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
51		nnord 7799	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ω → Ord 𝑦)
52		orduniorsuc 7755	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Ord 𝑦 → (𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦))
53	51, 52	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ω → (𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦))
54		3ianor 1106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ (Ord 𝑦 ∧ 𝑦 ≠ ∅ ∧ 𝑦 = ∪ 𝑦) ↔ (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = ∪ 𝑦))
55		df-lim 6306	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (Lim 𝑦 ↔ (Ord 𝑦 ∧ 𝑦 ≠ ∅ ∧ 𝑦 = ∪ 𝑦))
56	54, 55	xchnxbir 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ Lim 𝑦 ↔ (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = ∪ 𝑦))
57	27, 56	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ω → (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = ∪ 𝑦))
58	51	pm2.24d 151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ ω → (¬ Ord 𝑦 → (𝑦 = ∪ 𝑦 → 𝑦 = ∅)))
59		nne 2932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
60	59	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑦 ≠ ∅ → 𝑦 = ∅)
61	60	a1i13 27	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ ω → (¬ 𝑦 ≠ ∅ → (𝑦 = ∪ 𝑦 → 𝑦 = ∅)))
62		pm2.21 123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑦 = ∪ 𝑦 → (𝑦 = ∪ 𝑦 → 𝑦 = ∅))
63	62	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∪ 𝑦 → (𝑦 = ∪ 𝑦 → 𝑦 = ∅)))
64	58, 61, 63	3jaod 1431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ω → ((¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = ∪ 𝑦) → (𝑦 = ∪ 𝑦 → 𝑦 = ∅)))
65	57, 64	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ω → (𝑦 = ∪ 𝑦 → 𝑦 = ∅))
66	65	orim1d 967	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ω → ((𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦) → (𝑦 = ∅ ∨ 𝑦 = suc ∪ 𝑦)))
67	53, 66	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦 = suc ∪ 𝑦))
68	67	ord 864	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 = suc ∪ 𝑦))
69	68	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ 𝑦 = ∅ → 𝑦 = suc ∪ 𝑦))
70	69	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → 𝑦 = suc ∪ 𝑦)
71	70	oveq2d 7357	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·_o 𝑥) +_o 𝑦) = ((ω ·_o 𝑥) +_o suc ∪ 𝑦))
72		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
73	72	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → 𝑥 ∈ On)
74		omcl 8446	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ·_o 𝑥) ∈ On)
75	15, 73, 74	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → (ω ·_o 𝑥) ∈ On)
76		nnon 7797	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
77		onuni 7716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ On → ∪ 𝑦 ∈ On)
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ω → ∪ 𝑦 ∈ On)
79	78	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → ∪ 𝑦 ∈ On)
80	79	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ∪ 𝑦 ∈ On)
81		oasuc 8434	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ω ·_o 𝑥) ∈ On ∧ ∪ 𝑦 ∈ On) → ((ω ·_o 𝑥) +_o suc ∪ 𝑦) = suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
82	75, 80, 81	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·_o 𝑥) +_o suc ∪ 𝑦) = suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
83	71, 82	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·_o 𝑥) +_o 𝑦) = suc ((ω ·_o 𝑥) +_o ∪ 𝑦))
84		limeq 6313	. . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → (Lim ((ω ·_o 𝑥) +_o 𝑦) ↔ Lim suc ((ω ·_o 𝑥) +_o ∪ 𝑦)))
86	50, 85	mtbird 325	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦))
87	86	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ 𝑦 = ∅ → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦)))
88	47, 87	jaod 859	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → ((¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅) → ¬ Lim ((ω ·_o 𝑥) +_o 𝑦)))
89	88	con2d 134	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim ((ω ·_o 𝑥) +_o 𝑦) → ¬ (¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅)))
90		anor 984	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ∈ 𝑥 ∧ 𝑦 = ∅) ↔ ¬ (¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅))
91	89, 90	imbitrrdi 252	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim ((ω ·_o 𝑥) +_o 𝑦) → (∅ ∈ 𝑥 ∧ 𝑦 = ∅)))
92	25, 91	syl9 77	. . . . . . . . . . . . . 14 ⊢ (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim 𝐴 → (∅ ∈ 𝑥 ∧ 𝑦 = ∅))))
93	92	com13 88	. . . . . . . . . . . . 13 ⊢ (Lim 𝐴 → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (∅ ∈ 𝑥 ∧ 𝑦 = ∅))))
94	93	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (((ω ·_o 𝑥) +_o 𝑦) = 𝐴 → (∅ ∈ 𝑥 ∧ 𝑦 = ∅))))
95	94	3imp 1110	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → (∅ ∈ 𝑥 ∧ 𝑦 = ∅))
96		simp2 1137	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ ω))
97	96, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) → 𝑥 ∈ On)
98		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((∅ ∈ 𝑥 ∧ 𝑦 = ∅) → ∅ ∈ 𝑥)
99	97, 98	anim12i 613	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → (𝑥 ∈ On ∧ ∅ ∈ 𝑥))
100		ondif1 8411	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1_o) ↔ (𝑥 ∈ On ∧ ∅ ∈ 𝑥))
101	99, 100	sylibr 234	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·_o 𝑥) +_o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥 ∧ 𝑦 = ∅)) → 𝑥 ∈ (On ∖ 1_o))
102		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ 𝑥 ∧ 𝑦 = ∅) → 𝑦 = ∅)
103	102	oveq2d 7357	. . . . . . . . . . . . . 14 ⊢ ((∅ ∈ 𝑥 ∧ 𝑦 = ∅) → ((ω ·_o 𝑥) +_o 𝑦) = ((ω ·_o 𝑥) +_o ∅))
104	103	adantl 481	. . . . . . . . . . . . 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 1194	. . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (ω ·_o 𝑥) ∈ On)
107		oa0 8426	. . . . . . . . . . . . . . 15 ⊢ ((ω ·_o 𝑥) ∈ On → ((ω ·_o 𝑥) +_o ∅) = (ω ·_o 𝑥))
108	96, 106, 107	3syl 18	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 511	. . . . . . . . . . 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 687	. . . . . . . . . 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 1119	. . . . . . . . 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 452	. . . . . . . 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 3131	. . . . . . 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 453	. . . . . 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 3142	. . . . 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 15	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥))
119		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → 𝐴 = (ω ·_o 𝑥))
120		eldifi 4076	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 1_o) → 𝑥 ∈ On)
121	15, 120, 74	sylancr 587	. . . . . . . 8 ⊢ (𝑥 ∈ (On ∖ 1_o) → (ω ·_o 𝑥) ∈ On)
122	121	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → (ω ·_o 𝑥) ∈ On)
123	119, 122	eqeltrd 2831	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → 𝐴 ∈ On)
124		limom 7807	. . . . . . . . . . 11 ⊢ Lim ω
125	15, 124	pm3.2i 470	. . . . . . . . . 10 ⊢ (ω ∈ On ∧ Lim ω)
126		omlimcl2 43275	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑥) → Lim (ω ·_o 𝑥))
127	125, 126	mpanl2 701	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → Lim (ω ·_o 𝑥))
128	100, 127	sylbi 217	. . . . . . . 8 ⊢ (𝑥 ∈ (On ∖ 1_o) → Lim (ω ·_o 𝑥))
129	128	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → Lim (ω ·_o 𝑥))
130		limeq 6313	. . . . . . . 8 ⊢ (𝐴 = (ω ·_o 𝑥) → (Lim 𝐴 ↔ Lim (ω ·_o 𝑥)))
131	130	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → (Lim 𝐴 ↔ Lim (ω ·_o 𝑥)))
132	129, 131	mpbird 257	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → Lim 𝐴)
133	123, 132	jca 511	. . . . 5 ⊢ ((𝑥 ∈ (On ∖ 1_o) ∧ 𝐴 = (ω ·_o 𝑥)) → (𝐴 ∈ On ∧ Lim 𝐴))
134	133	rexlimiva 3125	. . . 4 ⊢ (∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥) → (𝐴 ∈ On ∧ Lim 𝐴))
135	118, 134	impbii 209	. . 3 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) ↔ ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥))
136	135	orbi2i 912	. 2 ⊢ ((𝐴 = On ∨ (𝐴 ∈ On ∧ Lim 𝐴)) ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥)))
137	8, 13, 136	3bitr2i 299	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 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 Vcvv 3436 ∖ cdif 3894 ∅c0 4278 ∪ cuni 4854 Ord word 6300 Oncon0 6301 Lim wlim 6302 suc csuc 6303 (class class class)co 7341 ωcom 7791 1_oc1o 8373 +_o coa 8377 ·_o comu 8378

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pr 5365 ax-un 7663 ax-inf2 9526

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-oadd 8384 df-omul 8385

This theorem is referenced by: (None)

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