Theorem winalim2 10609

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 10605	. . . 4 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
2		winainf 10607	. . . . 5 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
3		cardalephex 10003	. . . . 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 3054	. . 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 2735	. . . . . 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 7786	. . . . . . . 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 6826	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
15		aleph0 9979	. . . . . . . . . . . . . 14 ⊢ (ℵ‘∅) = ω
16	14, 15	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = ω)
17		eqtr 2749	. . . . . . . . . . . . 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 2938	. . . . . . . . . 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 5098	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ≺ 𝑤 ↔ (ℵ‘𝑦) ≺ 𝑤))
24	23	rexbidv 3153	. . . . . . . . . . . . 13 ⊢ (𝑧 = (ℵ‘𝑦) → (∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤 ↔ ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤))
25		elwina 10599	. . . . . . . . . . . . . . 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 7751	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
29		vex 3442	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
30	29	sucid 6395	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ suc 𝑦
31		alephord2i 9990	. . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘suc 𝑦))
38	33, 37	eleqtrrd 2831	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ 𝐴)
39	24, 27, 38	rspcdva 3580	. . . . . . . . . . . 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 9890	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴))
42	41	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘𝐴) = 𝐴 → ∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴)
43		rsp 3217	. . . . . . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . . 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 9985	. . . . . . . . . . . . . . . 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 3124	. . . . . . . . . . . 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 3124	. . . . . . . . 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 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3438   ⊆ wss 3905  ∅c0 4286   class class class wbr 5095  Oncon0 6311  Lim wlim 6312  suc csuc 6313  ‘cfv 6486  ωcom 7806   ≺ csdm 8878  cardccrd 9850  ℵcale 9851  cfccf 9852  Inacc_wcwina 10595

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-oi 9421  df-har 9468  df-card 9854  df-aleph 9855  df-cf 9856  df-wina 10597

This theorem is referenced by:  winafp  10610