Theorem winalim2 10651

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 10647	. . . 4 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
2		winainf 10649	. . . . 5 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
3		cardalephex 10043	. . . . 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 234	. . 3 ⊢ (𝐴 ∈ Inaccw → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
6	5	adantr 484	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
7		df-rex 3086	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ ∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)))
8		simprr 782	. . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 = (ℵ‘𝑥))
9	8	eqcomd 2767	. . . . . 6 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (ℵ‘𝑥) = 𝐴)
10		simprl 780	. . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝑥 ∈ On)
11		onzsl 7822	. . . . . . . 8 ⊢ (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
12	10, 11	sylib 220	. . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
13		simplr 778	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 ≠ ω)
14		fveq2 6863	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
15		aleph0 10019	. . . . . . . . . . . . . 14 ⊢ (ℵ‘∅) = ω
16	14, 15	eqtrdi 2812	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = ω)
17		eqtr 2781	. . . . . . . . . . . . 13 ⊢ ((𝐴 = (ℵ‘𝑥) ∧ (ℵ‘𝑥) = ω) → 𝐴 = ω)
18	16, 17	sylan2 602	. . . . . . . . . . . 12 ⊢ ((𝐴 = (ℵ‘𝑥) ∧ 𝑥 = ∅) → 𝐴 = ω)
19	18	ex 416	. . . . . . . . . . 11 ⊢ (𝐴 = (ℵ‘𝑥) → (𝑥 = ∅ → 𝐴 = ω))
20	19	necon3ad 2969	. . . . . . . . . 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 5102	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ≺ 𝑤 ↔ (ℵ‘𝑦) ≺ 𝑤))
24	23	rexbidv 3185	. . . . . . . . . . . . 13 ⊢ (𝑧 = (ℵ‘𝑦) → (∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤 ↔ ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤))
25		elwina 10641	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤))
26	25	simp3bi 1159	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inaccw → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤)
27	26	ad3antrrr 740	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ≺ 𝑤)
28		onsuc 7789	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
29		vex 3457	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
30	29	sucid 6426	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ suc 𝑦
31		alephord2i 10030	. . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
34		simplrr 787	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘𝑥))
35		fveq2 6863	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
36	35	ad2antll 739	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
37	34, 36	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘suc 𝑦))
38	33, 37	eleqtrrd 2864	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ 𝐴)
39	24, 27, 38	rspcdva 3582	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤)
40	39	expr 460	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤))
41		iscard 9930	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴))
42	41	simprbi 501	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘𝐴) = 𝐴 → ∀𝑤 ∈ 𝐴 𝑤 ≺ 𝐴)
43		rsp 3249	. . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ∈ 𝐴 → 𝑤 ≺ 𝐴))
46	37	breq2d 5111	. . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤 ∈ 𝐴 → 𝑤 ≺ (ℵ‘suc 𝑦)))
48		alephnbtwn2 10025	. . . . . . . . . . . . . . . 16 ⊢ ¬ ((ℵ‘𝑦) ≺ 𝑤 ∧ 𝑤 ≺ (ℵ‘suc 𝑦))
49		pm3.21 475	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ≺ (ℵ‘suc 𝑦) → ((ℵ‘𝑦) ≺ 𝑤 → ((ℵ‘𝑦) ≺ 𝑤 ∧ 𝑤 ≺ (ℵ‘suc 𝑦))))
50	48, 49	mtoi 201	. . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) ∧ 𝑤 ∈ 𝐴) → ¬ (ℵ‘𝑦) ≺ 𝑤)
53	52	nrexdv 3156	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ¬ ∃𝑤 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑤)
54	53	expr 460	. . . . . . . . . . 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 198	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → ¬ 𝑥 = suc 𝑦)
56	55	nrexdv 3156	. . . . . . . . 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 488	. . . . . . . . 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 1448	. . . . . . 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 519	. . . . 5 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
63	62	ex 416	. . . 4 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ((𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
64	63	eximdv 1936	. . 3 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
65	7, 64	biimtrid 244	. 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 208   ∧ wa 399   ∨ w3o 1096   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ⊆ wss 3904  ∅c0 4285   class class class wbr 5099  Oncon0 6342  Lim wlim 6343  suc csuc 6344  ‘cfv 6517  ωcom 7842   ≺ csdm 8922  cardccrd 9890  ℵcale 9891  cfccf 9892  Inacc_wcwina 10637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-oi 9455  df-har 9502  df-card 9894  df-aleph 9895  df-cf 9896  df-wina 10639

This theorem is referenced by:  winafp  10652