Theorem winalim2 10669

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 10665	. . . 4 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
2		winainf 10667	. . . . 5 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
3		cardalephex 10062	. . . . 5 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
4	2, 3	syl 18	. . . 4 ⊢ (𝐴 ∈ Inaccw → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
5	1, 4	mpbid 235	. . 3 ⊢ (𝐴 ∈ Inaccw → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
6	5	adantr 485	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
7		df-rex 3090	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ ∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)))
8		simprr 784	. . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 = (ℵ‘𝑥))
9	8	eqcomd 2771	. . . . . 6 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (ℵ‘𝑥) = 𝐴)
10		simprl 782	. . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝑥 ∈ On)
11		onzsl 7830	. . . . . . . 8 ⊢ (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
12	10, 11	sylib 221	. . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
13		simplr 780	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 ≠ ω)
14		fveq2 6871	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
15		aleph0 10038	. . . . . . . . . . . . . 14 ⊢ (ℵ‘∅) = ω
16	14, 15	eqtrdi 2816	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = ω)
17		eqtr 2785	. . . . . . . . . . . . 13 ⊢ ((𝐴 = (ℵ‘𝑥) ∧ (ℵ‘𝑥) = ω) → 𝐴 = ω)
18	16, 17	sylan2 604	. . . . . . . . . . . 12 ⊢ ((𝐴 = (ℵ‘𝑥) ∧ 𝑥 = ∅) → 𝐴 = ω)
19	18	ex 417	. . . . . . . . . . 11 ⊢ (𝐴 = (ℵ‘𝑥) → (𝑥 = ∅ → 𝐴 = ω))
20	19	necon3ad 2973	. . . . . . . . . 10 ⊢ (𝐴 = (ℵ‘𝑥) → (𝐴 ≠ ω → ¬ 𝑥 = ∅))
21	8, 13, 20	sylc 66	. . . . . . . . 9 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ 𝑥 = ∅)
22	21	pm2.21d 122	. . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ → Lim 𝑥))
23		breq1 5108	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ≺ 𝑤 ↔ (ℵ‘𝑦) ≺ 𝑤))
24	23	rexbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝑧 = (ℵ‘𝑦) → (∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤 ↔ ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤))
25		elwina 10659	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤))
26	25	simp3bi 1163	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inaccw → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤)
27	26	ad3antrrr 742	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤)
28		onsuc 7797	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
29		vex 3461	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
30	29	sucid 6434	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ suc 𝑦
31		alephord2i 10049	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑦 ∈ On → (𝑦 ∈ suc 𝑦 → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦)))
32	28, 30, 31	mpisyl 22	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
33	32	ad2antrl 740	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
34		simplrr 789	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘𝑥))
35		fveq2 6871	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
36	35	ad2antll 741	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
37	34, 36	eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘suc 𝑦))
38	33, 37	eleqtrrd 2868	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ 𝐴)
39	24, 27, 38	rspcdva 3585	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤)
40	39	expr 461	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤))
41		iscard 9949	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴))
42	41	simprbi 502	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘𝐴) = 𝐴 → ∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴)
43		rsp 3253	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴 → (𝑤 ∈ 𝐴 → 𝑤 ≺ 𝐴))
44	1, 42, 43	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Inaccw → (𝑤 ∈ 𝐴 → 𝑤 ≺ 𝐴))
45	44	ad3antrrr 742	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ∈ 𝐴 → 𝑤 ≺ 𝐴))
46	37	breq2d 5117	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ∈ 𝐴 → 𝑤 ≺ (ℵ‘suc 𝑦)))
48		alephnbtwn2 10044	. . . . . . . . . . . . . . . 16 ⊢ ¬ ((ℵ‘𝑦) ≺ 𝑤 ∧ 𝑤 ≺ (ℵ‘suc 𝑦))
49		pm3.21 476	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ≺ (ℵ‘suc 𝑦) → ((ℵ‘𝑦) ≺ 𝑤 → ((ℵ‘𝑦) ≺ 𝑤 ∧ 𝑤 ≺ (ℵ‘suc 𝑦))))
50	48, 49	mtoi 202	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ≺ (ℵ‘suc 𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑤)
51	47, 50	syl6 36	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ∈ 𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑤))
52	51	imp 411	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) ∧ 𝑤 ∈ 𝐴) → ¬ (ℵ‘𝑦) ≺ 𝑤)
53	52	nrexdv 3160	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ¬ ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤)
54	53	expr 461	. . . . . . . . . . 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 199	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → ¬ 𝑥 = suc 𝑦)
56	55	nrexdv 3160	. . . . . . . . 9 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ ∃𝑦 ∈ On 𝑥 = suc 𝑦)
57	56	pm2.21d 122	. . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (∃𝑦 ∈ On 𝑥 = suc 𝑦 → Lim 𝑥))
58		simpr 489	. . . . . . . . 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 1452	. . . . . . 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 16	. . . . . 6 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → Lim 𝑥)
62	9, 61	jca 520	. . . . 5 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
63	62	ex 417	. . . 4 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ((𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
64	63	eximdv 1940	. . 3 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
65	7, 64	biimtrid 245	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
66	6, 65	mpd 16	1 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ w3o 1100   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  ∅c0 4288   class class class wbr 5105  Oncon0 6350  Lim wlim 6351  suc csuc 6352  ‘cfv 6525  ωcom 7850   ≺ csdm 8930  cardccrd 9909  ℵcale 9910  cfccf 9911  Inacc_wcwina 10655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-oi 9460  df-har 9507  df-card 9913  df-aleph 9914  df-cf 9915  df-wina 10657

This theorem is referenced by:  winafp  10670