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Theorem winalim2 9469
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 9465 . . . 4 (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
2 winainf 9467 . . . . 5 (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
3 cardalephex 8864 . . . . 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 222 . . 3 (𝐴 ∈ Inaccw → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
65adantr 481 . 2 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
7 df-rex 2913 . . 3 (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) ↔ ∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)))
8 simprr 795 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 = (ℵ‘𝑥))
98eqcomd 2627 . . . . . 6 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (ℵ‘𝑥) = 𝐴)
10 simprl 793 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝑥 ∈ On)
11 onzsl 7000 . . . . . . . 8 (𝑥 ∈ On ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
1210, 11sylib 208 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)))
13 simplr 791 . . . . . . . . . 10 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → 𝐴 ≠ ω)
14 fveq2 6153 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
15 aleph0 8840 . . . . . . . . . . . . . 14 (ℵ‘∅) = ω
1614, 15syl6eq 2671 . . . . . . . . . . . . 13 (𝑥 = ∅ → (ℵ‘𝑥) = ω)
17 eqtr 2640 . . . . . . . . . . . . 13 ((𝐴 = (ℵ‘𝑥) ∧ (ℵ‘𝑥) = ω) → 𝐴 = ω)
1816, 17sylan2 491 . . . . . . . . . . . 12 ((𝐴 = (ℵ‘𝑥) ∧ 𝑥 = ∅) → 𝐴 = ω)
1918ex 450 . . . . . . . . . . 11 (𝐴 = (ℵ‘𝑥) → (𝑥 = ∅ → 𝐴 = ω))
2019necon3ad 2803 . . . . . . . . . 10 (𝐴 = (ℵ‘𝑥) → (𝐴 ≠ ω → ¬ 𝑥 = ∅))
218, 13, 20sylc 65 . . . . . . . . 9 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ 𝑥 = ∅)
2221pm2.21d 118 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (𝑥 = ∅ → Lim 𝑥))
23 suceloni 6967 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → suc 𝑦 ∈ On)
24 vex 3192 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
2524sucid 5768 . . . . . . . . . . . . . . . 16 𝑦 ∈ suc 𝑦
26 alephord2i 8851 . . . . . . . . . . . . . . . 16 (suc 𝑦 ∈ On → (𝑦 ∈ suc 𝑦 → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦)))
2723, 25, 26mpisyl 21 . . . . . . . . . . . . . . 15 (𝑦 ∈ On → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
2827ad2antrl 763 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ (ℵ‘suc 𝑦))
29 simplrr 800 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘𝑥))
30 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
3130ad2antll 764 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
3229, 31eqtrd 2655 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → 𝐴 = (ℵ‘suc 𝑦))
3328, 32eleqtrrd 2701 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (ℵ‘𝑦) ∈ 𝐴)
34 elwina 9459 . . . . . . . . . . . . . . 15 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
3534simp3bi 1076 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw → ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
3635ad3antrrr 765 . . . . . . . . . . . . 13 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
37 breq1 4621 . . . . . . . . . . . . . . 15 (𝑧 = (ℵ‘𝑦) → (𝑧𝑤 ↔ (ℵ‘𝑦) ≺ 𝑤))
3837rexbidv 3046 . . . . . . . . . . . . . 14 (𝑧 = (ℵ‘𝑦) → (∃𝑤𝐴 𝑧𝑤 ↔ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
3938rspcva 3296 . . . . . . . . . . . . 13 (((ℵ‘𝑦) ∈ 𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤) → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
4033, 36, 39syl2anc 692 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
4140expr 642 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
42 iscard 8752 . . . . . . . . . . . . . . . . . . 19 ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑤𝐴 𝑤𝐴))
4342simprbi 480 . . . . . . . . . . . . . . . . . 18 ((card‘𝐴) = 𝐴 → ∀𝑤𝐴 𝑤𝐴)
44 rsp 2924 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝐴 𝑤𝐴 → (𝑤𝐴𝑤𝐴))
451, 43, 443syl 18 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ Inaccw → (𝑤𝐴𝑤𝐴))
4645ad3antrrr 765 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤𝐴))
4732breq2d 4630 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤 ≺ (ℵ‘suc 𝑦)))
4846, 47sylibd 229 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴𝑤 ≺ (ℵ‘suc 𝑦)))
49 alephnbtwn2 8846 . . . . . . . . . . . . . . . 16 ¬ ((ℵ‘𝑦) ≺ 𝑤𝑤 ≺ (ℵ‘suc 𝑦))
50 pm3.21 464 . . . . . . . . . . . . . . . 16 (𝑤 ≺ (ℵ‘suc 𝑦) → ((ℵ‘𝑦) ≺ 𝑤 → ((ℵ‘𝑦) ≺ 𝑤𝑤 ≺ (ℵ‘suc 𝑦))))
5149, 50mtoi 190 . . . . . . . . . . . . . . 15 (𝑤 ≺ (ℵ‘suc 𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑤)
5248, 51syl6 35 . . . . . . . . . . . . . 14 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → (𝑤𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑤))
5352imp 445 . . . . . . . . . . . . 13 (((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) ∧ 𝑤𝐴) → ¬ (ℵ‘𝑦) ≺ 𝑤)
5453nrexdv 2996 . . . . . . . . . . . 12 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ (𝑦 ∈ On ∧ 𝑥 = suc 𝑦)) → ¬ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤)
5554expr 642 . . . . . . . . . . 11 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → (𝑥 = suc 𝑦 → ¬ ∃𝑤𝐴 (ℵ‘𝑦) ≺ 𝑤))
5641, 55pm2.65d 187 . . . . . . . . . 10 ((((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) ∧ 𝑦 ∈ On) → ¬ 𝑥 = suc 𝑦)
5756nrexdv 2996 . . . . . . . . 9 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ¬ ∃𝑦 ∈ On 𝑥 = suc 𝑦)
5857pm2.21d 118 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → (∃𝑦 ∈ On 𝑥 = suc 𝑦 → Lim 𝑥))
59 simpr 477 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → Lim 𝑥)
6059a1i 11 . . . . . . . 8 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((𝑥 ∈ V ∧ Lim 𝑥) → Lim 𝑥))
6122, 58, 603jaod 1389 . . . . . . 7 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦 ∨ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim 𝑥))
6212, 61mpd 15 . . . . . 6 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → Lim 𝑥)
639, 62jca 554 . . . . 5 (((𝐴 ∈ Inaccw𝐴 ≠ ω) ∧ (𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥))) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
6463ex 450 . . . 4 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ((𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
6564eximdv 1843 . . 3 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → (∃𝑥(𝑥 ∈ On ∧ 𝐴 = (ℵ‘𝑥)) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
667, 65syl5bi 232 . 2 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)))
676, 66mpd 15 1 ((𝐴 ∈ Inaccw𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3o 1035   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  wss 3559  c0 3896   class class class wbr 4618  Oncon0 5687  Lim wlim 5688  suc csuc 5689  cfv 5852  ωcom 7019  csdm 7905  cardccrd 8712  cale 8713  cfccf 8714  Inaccwcwina 9455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-oi 8366  df-har 8414  df-card 8716  df-aleph 8717  df-cf 8718  df-wina 9457
This theorem is referenced by:  winafp  9470
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