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Theorem winalim2 10117
 Description: A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
winalim2 ((𝐴 ∈ Inaccw𝐴 ≠ 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.
StepHypRef Expression
1 winacard 10113 . . . 4 (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
2 winainf 10115 . . . . 5 (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
3 cardalephex 9515 . . . . 5 (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
42, 3syl 17 . . . 4 (𝐴 ∈ Inaccw → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
51, 4mpbid 234 . . 3 (𝐴 ∈ Inaccw → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
65adantr 483 . 2 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
7 df-rex 3144 . . 3 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ ∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)))
8 simprr 771 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 = (ℵ‘𝑥))
98eqcomd 2827 . . . . . 6 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (ℵ‘𝑥) = 𝐴)
10 simprl 769 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝑥 ∈ On)
11 onzsl 7560 . . . . . . . 8 (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
1210, 11sylib 220 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
13 simplr 767 . . . . . . . . . 10 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 ≠ ω)
14 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
15 aleph0 9491 . . . . . . . . . . . . . 14 (ℵ‘∅) = ω
1614, 15syl6eq 2872 . . . . . . . . . . . . 13 (𝑥 = ∅ → (ℵ‘𝑥) = ω)
17 eqtr 2841 . . . . . . . . . . . . 13 ((𝐴 = (ℵ‘𝑥) ∧ (ℵ‘𝑥) = ω) → 𝐴 = ω)
1816, 17sylan2 594 . . . . . . . . . . . 12 ((𝐴 = (ℵ‘𝑥) ∧ 𝑥 = ∅) → 𝐴 = ω)
1918ex 415 . . . . . . . . . . 11 (𝐴 = (ℵ‘𝑥) → (𝑥 = ∅ → 𝐴 = ω))
2019necon3ad 3029 . . . . . . . . . 10 (𝐴 = (ℵ‘𝑥) → (𝐴 ≠ ω → ¬ 𝑥 = ∅))
218, 13, 20sylc 65 . . . . . . . . 9 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ 𝑥 = ∅)
2221pm2.21d 121 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ → Lim 𝑥))
23 breq1 5068 . . . . . . . . . . . . . 14 (𝑧 = (ℵ‘𝑦) → (𝑧𝑤 ↔ (ℵ‘𝑦) ≺ 𝑤))
2423rexbidv 3297 . . . . . . . . . . . . 13 (𝑧 = (ℵ‘𝑦) → (∃𝑤𝐴 𝑧𝑤 ↔ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
25 elwina 10107 . . . . . . . . . . . . . . 15 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
2625simp3bi 1143 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw → ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
2726ad3antrrr 728 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
28 suceloni 7527 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → suc 𝑦 ∈ On)
29 vex 3497 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
3029sucid 6269 . . . . . . . . . . . . . . . 16 𝑦 ∈ suc 𝑦
31 alephord2i 9502 . . . . . . . . . . . . . . . 16 (suc 𝑦 ∈ On → (𝑦 ∈ suc 𝑦 → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦)))
3228, 30, 31mpisyl 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ On → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
3332ad2antrl 726 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
34 simplrr 776 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘𝑥))
35 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
3635ad2antll 727 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
3734, 36eqtrd 2856 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘suc 𝑦))
3833, 37eleqtrrd 2916 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ 𝐴)
3924, 27, 38rspcdva 3624 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
4039expr 459 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
41 iscard 9403 . . . . . . . . . . . . . . . . . . 19 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑤𝐴 𝑤𝐴))
4241simprbi 499 . . . . . . . . . . . . . . . . . 18 ((card‘𝐴) = 𝐴 → ∀𝑤𝐴 𝑤𝐴)
43 rsp 3205 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝐴 𝑤𝐴 → (𝑤𝐴𝑤𝐴))
441, 42, 433syl 18 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Inaccw → (𝑤𝐴𝑤𝐴))
4544ad3antrrr 728 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤𝐴))
4637breq2d 5077 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤 ≺ (ℵ‘suc 𝑦)))
4745, 46sylibd 241 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤 ≺ (ℵ‘suc 𝑦)))
48 alephnbtwn2 9497 . . . . . . . . . . . . . . . 16 ¬ ((ℵ‘𝑦) ≺ 𝑤𝑤 ≺ (ℵ‘suc 𝑦))
49 pm3.21 474 . . . . . . . . . . . . . . . 16 (𝑤 ≺ (ℵ‘suc 𝑦) → ((ℵ‘𝑦) ≺ 𝑤 → ((ℵ‘𝑦) ≺ 𝑤𝑤 ≺ (ℵ‘suc 𝑦))))
5048, 49mtoi 201 . . . . . . . . . . . . . . 15 (𝑤 ≺ (ℵ‘suc 𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑤)
5147, 50syl6 35 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑤))
5251imp 409 . . . . . . . . . . . . 13 (((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) ∧ 𝑤𝐴) → ¬ (ℵ‘𝑦) ≺ 𝑤)
5352nrexdv 3270 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ¬ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
5453expr 459 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ¬ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
5540, 54pm2.65d 198 . . . . . . . . . 10 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → ¬ 𝑥 = suc 𝑦)
5655nrexdv 3270 . . . . . . . . 9 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ ∃𝑦 ∈ On 𝑥 = suc 𝑦)
5756pm2.21d 121 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (∃𝑦 ∈ On 𝑥 = suc 𝑦 → Lim 𝑥))
58 simpr 487 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → Lim 𝑥)
5958a1i 11 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((𝑥 ∈ V ∧ Lim 𝑥) → Lim 𝑥))
6022, 57, 593jaod 1424 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim 𝑥))
6112, 60mpd 15 . . . . . 6 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → Lim 𝑥)
629, 61jca 514 . . . . 5 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
6362ex 415 . . . 4 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ((𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
6463eximdv 1914 . . 3 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → (∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
657, 64syl5bi 244 . 2 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
666, 65mpd 15 1 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ w3o 1082   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ⊆ wss 3935  ∅c0 4290   class class class wbr 5065  Oncon0 6190  Lim wlim 6191  suc csuc 6192  ‘cfv 6354  ωcom 7579   ≺ csdm 8507  cardccrd 9363  ℵcale 9364  cfccf 9365  Inaccwcwina 10103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-oi 8973  df-har 9021  df-card 9367  df-aleph 9368  df-cf 9369  df-wina 10105 This theorem is referenced by:  winafp  10118
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