Theorem winalim2 10710

Description: A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)

Assertion

Ref	Expression
winalim2	⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem winalim2

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		winacard 10706	. . . 4 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
2		winainf 10708	. . . . 5 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
3		cardalephex 10104	. . . . 5 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
4	2, 3	syl 17	. . . 4 ⊢ (𝐴 ∈ Inaccw → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
5	1, 4	mpbid 232	. . 3 ⊢ (𝐴 ∈ Inaccw → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
6	5	adantr 480	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
7		df-rex 3061	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ ∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)))
8		simprr 772	. . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 = (ℵ‘𝑥))
9	8	eqcomd 2741	. . . . . 6 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (ℵ‘𝑥) = 𝐴)
10		simprl 770	. . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝑥 ∈ On)
11		onzsl 7841	. . . . . . . 8 ⊢ (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
12	10, 11	sylib 218	. . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
13		simplr 768	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 ≠ ω)
14		fveq2 6876	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
15		aleph0 10080	. . . . . . . . . . . . . 14 ⊢ (ℵ‘∅) = ω
16	14, 15	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = ω)
17		eqtr 2755	. . . . . . . . . . . . 13 ⊢ ((𝐴 = (ℵ‘𝑥) ∧ (ℵ‘𝑥) = ω) → 𝐴 = ω)
18	16, 17	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝐴 = (ℵ‘𝑥) ∧ 𝑥 = ∅) → 𝐴 = ω)
19	18	ex 412	. . . . . . . . . . 11 ⊢ (𝐴 = (ℵ‘𝑥) → (𝑥 = ∅ → 𝐴 = ω))
20	19	necon3ad 2945	. . . . . . . . . 10 ⊢ (𝐴 = (ℵ‘𝑥) → (𝐴 ≠ ω → ¬ 𝑥 = ∅))
21	8, 13, 20	sylc 65	. . . . . . . . 9 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ 𝑥 = ∅)
22	21	pm2.21d 121	. . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ → Lim 𝑥))
23		breq1 5122	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ≺ 𝑤 ↔ (ℵ‘𝑦) ≺ 𝑤))
24	23	rexbidv 3164	. . . . . . . . . . . . 13 ⊢ (𝑧 = (ℵ‘𝑦) → (∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤 ↔ ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤))
25		elwina 10700	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤))
26	25	simp3bi 1147	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inaccw → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤)
27	26	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤)
28		onsuc 7805	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
29		vex 3463	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
30	29	sucid 6436	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ suc 𝑦
31		alephord2i 10091	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑦 ∈ On → (𝑦 ∈ suc 𝑦 → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦)))
32	28, 30, 31	mpisyl 21	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
33	32	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
34		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘𝑥))
35		fveq2 6876	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
36	35	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
37	34, 36	eqtrd 2770	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘suc 𝑦))
38	33, 37	eleqtrrd 2837	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ 𝐴)
39	24, 27, 38	rspcdva 3602	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤)
40	39	expr 456	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤))
41		iscard 9989	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴))
42	41	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘𝐴) = 𝐴 → ∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴)
43		rsp 3230	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴 → (𝑤 ∈ 𝐴 → 𝑤 ≺ 𝐴))
44	1, 42, 43	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Inaccw → (𝑤 ∈ 𝐴 → 𝑤 ≺ 𝐴))
45	44	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ∈ 𝐴 → 𝑤 ≺ 𝐴))
46	37	breq2d 5131	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ≺ 𝐴 ↔ 𝑤 ≺ (ℵ‘suc 𝑦)))
47	45, 46	sylibd 239	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ∈ 𝐴 → 𝑤 ≺ (ℵ‘suc 𝑦)))
48		alephnbtwn2 10086	. . . . . . . . . . . . . . . 16 ⊢ ¬ ((ℵ‘𝑦) ≺ 𝑤 ∧ 𝑤 ≺ (ℵ‘suc 𝑦))
49		pm3.21 471	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ≺ (ℵ‘suc 𝑦) → ((ℵ‘𝑦) ≺ 𝑤 → ((ℵ‘𝑦) ≺ 𝑤 ∧ 𝑤 ≺ (ℵ‘suc 𝑦))))
50	48, 49	mtoi 199	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ≺ (ℵ‘suc 𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑤)
51	47, 50	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ∈ 𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑤))
52	51	imp 406	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) ∧ 𝑤 ∈ 𝐴) → ¬ (ℵ‘𝑦) ≺ 𝑤)
53	52	nrexdv 3135	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ¬ ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤)
54	53	expr 456	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ¬ ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤))
55	40, 54	pm2.65d 196	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → ¬ 𝑥 = suc 𝑦)
56	55	nrexdv 3135	. . . . . . . . 9 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ ∃𝑦 ∈ On 𝑥 = suc 𝑦)
57	56	pm2.21d 121	. . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (∃𝑦 ∈ On 𝑥 = suc 𝑦 → Lim 𝑥))
58		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → Lim 𝑥)
59	58	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((𝑥 ∈ V ∧ Lim 𝑥) → Lim 𝑥))
60	22, 57, 59	3jaod 1431	. . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim 𝑥))
61	12, 60	mpd 15	. . . . . 6 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → Lim 𝑥)
62	9, 61	jca 511	. . . . 5 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
63	62	ex 412	. . . 4 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ((𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
64	63	eximdv 1917	. . 3 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
65	7, 64	biimtrid 242	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
66	6, 65	mpd 15	1 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119  Oncon0 6352  Lim wlim 6353  suc csuc 6354  ‘cfv 6531  ωcom 7861   ≺ csdm 8958  cardccrd 9949  ℵcale 9950  cfccf 9951  Inacc_wcwina 10696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-oi 9524  df-har 9571  df-card 9953  df-aleph 9954  df-cf 9955  df-wina 10698

This theorem is referenced by:  winafp  10711