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Theorem winalim2 9804
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 9800 . . . 4 (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
2 winainf 9802 . . . . 5 (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
3 cardalephex 9197 . . . . 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 224 . . 3 (𝐴 ∈ Inaccw → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
65adantr 473 . 2 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
7 df-rex 3093 . . 3 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ ∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)))
8 simprr 790 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 = (ℵ‘𝑥))
98eqcomd 2803 . . . . . 6 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (ℵ‘𝑥) = 𝐴)
10 simprl 788 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝑥 ∈ On)
11 onzsl 7278 . . . . . . . 8 (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
1210, 11sylib 210 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
13 simplr 786 . . . . . . . . . 10 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 ≠ ω)
14 fveq2 6409 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
15 aleph0 9173 . . . . . . . . . . . . . 14 (ℵ‘∅) = ω
1614, 15syl6eq 2847 . . . . . . . . . . . . 13 (𝑥 = ∅ → (ℵ‘𝑥) = ω)
17 eqtr 2816 . . . . . . . . . . . . 13 ((𝐴 = (ℵ‘𝑥) ∧ (ℵ‘𝑥) = ω) → 𝐴 = ω)
1816, 17sylan2 587 . . . . . . . . . . . 12 ((𝐴 = (ℵ‘𝑥) ∧ 𝑥 = ∅) → 𝐴 = ω)
1918ex 402 . . . . . . . . . . 11 (𝐴 = (ℵ‘𝑥) → (𝑥 = ∅ → 𝐴 = ω))
2019necon3ad 2982 . . . . . . . . . 10 (𝐴 = (ℵ‘𝑥) → (𝐴 ≠ ω → ¬ 𝑥 = ∅))
218, 13, 20sylc 65 . . . . . . . . 9 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ 𝑥 = ∅)
2221pm2.21d 119 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ → Lim 𝑥))
23 breq1 4844 . . . . . . . . . . . . . 14 (𝑧 = (ℵ‘𝑦) → (𝑧𝑤 ↔ (ℵ‘𝑦) ≺ 𝑤))
2423rexbidv 3231 . . . . . . . . . . . . 13 (𝑧 = (ℵ‘𝑦) → (∃𝑤𝐴 𝑧𝑤 ↔ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
25 elwina 9794 . . . . . . . . . . . . . . 15 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
2625simp3bi 1178 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw → ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
2726ad3antrrr 722 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
28 suceloni 7245 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → suc 𝑦 ∈ On)
29 vex 3386 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
3029sucid 6018 . . . . . . . . . . . . . . . 16 𝑦 ∈ suc 𝑦
31 alephord2i 9184 . . . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
34 simplrr 797 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘𝑥))
35 fveq2 6409 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
3635ad2antll 721 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
3734, 36eqtrd 2831 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘suc 𝑦))
3833, 37eleqtrrd 2879 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ 𝐴)
3924, 27, 38rspcdva 3501 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
4039expr 449 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
41 iscard 9085 . . . . . . . . . . . . . . . . . . 19 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑤𝐴 𝑤𝐴))
4241simprbi 491 . . . . . . . . . . . . . . . . . 18 ((card‘𝐴) = 𝐴 → ∀𝑤𝐴 𝑤𝐴)
43 rsp 3108 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝐴 𝑤𝐴 → (𝑤𝐴𝑤𝐴))
441, 42, 433syl 18 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Inaccw → (𝑤𝐴𝑤𝐴))
4544ad3antrrr 722 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤𝐴))
4637breq2d 4853 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤 ≺ (ℵ‘suc 𝑦)))
4745, 46sylibd 231 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤 ≺ (ℵ‘suc 𝑦)))
48 alephnbtwn2 9179 . . . . . . . . . . . . . . . 16 ¬ ((ℵ‘𝑦) ≺ 𝑤𝑤 ≺ (ℵ‘suc 𝑦))
49 pm3.21 464 . . . . . . . . . . . . . . . 16 (𝑤 ≺ (ℵ‘suc 𝑦) → ((ℵ‘𝑦) ≺ 𝑤 → ((ℵ‘𝑦) ≺ 𝑤𝑤 ≺ (ℵ‘suc 𝑦))))
5048, 49mtoi 191 . . . . . . . . . . . . . . 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 396 . . . . . . . . . . . . 13 (((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) ∧ 𝑤𝐴) → ¬ (ℵ‘𝑦) ≺ 𝑤)
5352nrexdv 3179 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ¬ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
5453expr 449 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ¬ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
5540, 54pm2.65d 188 . . . . . . . . . 10 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → ¬ 𝑥 = suc 𝑦)
5655nrexdv 3179 . . . . . . . . 9 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ ∃𝑦 ∈ On 𝑥 = suc 𝑦)
5756pm2.21d 119 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (∃𝑦 ∈ On 𝑥 = suc 𝑦 → Lim 𝑥))
58 simpr 478 . . . . . . . . 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 1554 . . . . . . 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 508 . . . . 5 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
6362ex 402 . . . 4 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ((𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
6463eximdv 2013 . . 3 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → (∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
657, 64syl5bi 234 . 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 198  wa 385  w3o 1107   = wceq 1653  wex 1875  wcel 2157  wne 2969  wral 3087  wrex 3088  Vcvv 3383  wss 3767  c0 4113   class class class wbr 4841  Oncon0 5939  Lim wlim 5940  suc csuc 5941  cfv 6099  ωcom 7297  csdm 8192  cardccrd 9045  cale 9046  cfccf 9047  Inaccwcwina 9790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-inf2 8786
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-se 5270  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6837  df-om 7298  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-oi 8655  df-har 8703  df-card 9049  df-aleph 9050  df-cf 9051  df-wina 9792
This theorem is referenced by:  winafp  9805
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