MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  winacard Structured version   Visualization version   GIF version

Theorem winacard 10449
Description: A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
winacard (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)

Proof of Theorem winacard
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elwina 10443 . 2 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
2 cardcf 10009 . . . 4 (card‘(cf‘𝐴)) = (cf‘𝐴)
3 fveq2 6771 . . . 4 ((cf‘𝐴) = 𝐴 → (card‘(cf‘𝐴)) = (card‘𝐴))
4 id 22 . . . 4 ((cf‘𝐴) = 𝐴 → (cf‘𝐴) = 𝐴)
52, 3, 43eqtr3a 2804 . . 3 ((cf‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴)
653ad2ant2 1133 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → (card‘𝐴) = 𝐴)
71, 6sylbi 216 1 (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1542  wcel 2110  wne 2945  wral 3066  wrex 3067  c0 4262   class class class wbr 5079  cfv 6432  csdm 8715  cardccrd 9694  cfccf 9696  Inaccwcwina 10439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6268  df-on 6269  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-er 8481  df-en 8717  df-card 9698  df-cf 9700  df-wina 10441
This theorem is referenced by:  winalim  10452  winalim2  10453  gchina  10456  inar1  10532
  Copyright terms: Public domain W3C validator