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

Theorem winacard 9552
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 9546 . 2 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
2 cardcf 9112 . . . 4 (card‘(cf‘𝐴)) = (cf‘𝐴)
3 fveq2 6229 . . . 4 ((cf‘𝐴) = 𝐴 → (card‘(cf‘𝐴)) = (card‘𝐴))
4 id 22 . . . 4 ((cf‘𝐴) = 𝐴 → (cf‘𝐴) = 𝐴)
52, 3, 43eqtr3a 2709 . . 3 ((cf‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴)
653ad2ant2 1103 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → (card‘𝐴) = 𝐴)
71, 6sylbi 207 1 (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  c0 3948   class class class wbr 4685  cfv 5926  csdm 7996  cardccrd 8799  cfccf 8801  Inaccwcwina 9542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-card 8803  df-cf 8805  df-wina 9544
This theorem is referenced by:  winalim  9555  winalim2  9556  gchina  9559  inar1  9635
  Copyright terms: Public domain W3C validator