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Theorem winalim2 10687
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 10683 . . . 4 (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
2 winainf 10685 . . . . 5 (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
3 cardalephex 10081 . . . . 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 231 . . 3 (𝐴 ∈ Inaccw → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
65adantr 482 . 2 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
7 df-rex 3072 . . 3 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ ∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)))
8 simprr 772 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 = (ℵ‘𝑥))
98eqcomd 2739 . . . . . 6 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (ℵ‘𝑥) = 𝐴)
10 simprl 770 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝑥 ∈ On)
11 onzsl 7830 . . . . . . . 8 (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
1210, 11sylib 217 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
13 simplr 768 . . . . . . . . . 10 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 ≠ ω)
14 fveq2 6888 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
15 aleph0 10057 . . . . . . . . . . . . . 14 (ℵ‘∅) = ω
1614, 15eqtrdi 2789 . . . . . . . . . . . . 13 (𝑥 = ∅ → (ℵ‘𝑥) = ω)
17 eqtr 2756 . . . . . . . . . . . . 13 ((𝐴 = (ℵ‘𝑥) ∧ (ℵ‘𝑥) = ω) → 𝐴 = ω)
1816, 17sylan2 594 . . . . . . . . . . . 12 ((𝐴 = (ℵ‘𝑥) ∧ 𝑥 = ∅) → 𝐴 = ω)
1918ex 414 . . . . . . . . . . 11 (𝐴 = (ℵ‘𝑥) → (𝑥 = ∅ → 𝐴 = ω))
2019necon3ad 2954 . . . . . . . . . 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 5150 . . . . . . . . . . . . . 14 (𝑧 = (ℵ‘𝑦) → (𝑧𝑤 ↔ (ℵ‘𝑦) ≺ 𝑤))
2423rexbidv 3179 . . . . . . . . . . . . 13 (𝑧 = (ℵ‘𝑦) → (∃𝑤𝐴 𝑧𝑤 ↔ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
25 elwina 10677 . . . . . . . . . . . . . . 15 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
2625simp3bi 1148 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw → ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
2726ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
28 onsuc 7794 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → suc 𝑦 ∈ On)
29 vex 3479 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
3029sucid 6443 . . . . . . . . . . . . . . . 16 𝑦 ∈ suc 𝑦
31 alephord2i 10068 . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
34 simplrr 777 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘𝑥))
35 fveq2 6888 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
3635ad2antll 728 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
3734, 36eqtrd 2773 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘suc 𝑦))
3833, 37eleqtrrd 2837 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ 𝐴)
3924, 27, 38rspcdva 3613 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
4039expr 458 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
41 iscard 9966 . . . . . . . . . . . . . . . . . . 19 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑤𝐴 𝑤𝐴))
4241simprbi 498 . . . . . . . . . . . . . . . . . 18 ((card‘𝐴) = 𝐴 → ∀𝑤𝐴 𝑤𝐴)
43 rsp 3245 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝐴 𝑤𝐴 → (𝑤𝐴𝑤𝐴))
441, 42, 433syl 18 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Inaccw → (𝑤𝐴𝑤𝐴))
4544ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤𝐴))
4637breq2d 5159 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤 ≺ (ℵ‘suc 𝑦)))
4745, 46sylibd 238 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤 ≺ (ℵ‘suc 𝑦)))
48 alephnbtwn2 10063 . . . . . . . . . . . . . . . 16 ¬ ((ℵ‘𝑦) ≺ 𝑤𝑤 ≺ (ℵ‘suc 𝑦))
49 pm3.21 473 . . . . . . . . . . . . . . . 16 (𝑤 ≺ (ℵ‘suc 𝑦) → ((ℵ‘𝑦) ≺ 𝑤 → ((ℵ‘𝑦) ≺ 𝑤𝑤 ≺ (ℵ‘suc 𝑦))))
5048, 49mtoi 198 . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 13 (((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) ∧ 𝑤𝐴) → ¬ (ℵ‘𝑦) ≺ 𝑤)
5352nrexdv 3150 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ¬ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
5453expr 458 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ¬ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
5540, 54pm2.65d 195 . . . . . . . . . 10 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → ¬ 𝑥 = suc 𝑦)
5655nrexdv 3150 . . . . . . . . 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 486 . . . . . . . . 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 1429 . . . . . . 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 513 . . . . 5 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
6362ex 414 . . . 4 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ((𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
6463eximdv 1921 . . 3 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → (∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
657, 64biimtrid 241 . 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 205  wa 397  w3o 1087   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  wrex 3071  Vcvv 3475  wss 3947  c0 4321   class class class wbr 5147  Oncon0 6361  Lim wlim 6362  suc csuc 6363  cfv 6540  ωcom 7850  csdm 8934  cardccrd 9926  cale 9927  cfccf 9928  Inaccwcwina 10673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-oi 9501  df-har 9548  df-card 9930  df-aleph 9931  df-cf 9932  df-wina 10675
This theorem is referenced by:  winafp  10688
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