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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  ℵcale 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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