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Theorem winacard 10091
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 10085 . 2 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
2 cardcf 9651 . . . 4 (card‘(cf‘𝐴)) = (cf‘𝐴)
3 fveq2 6643 . . . 4 ((cf‘𝐴) = 𝐴 → (card‘(cf‘𝐴)) = (card‘𝐴))
4 id 22 . . . 4 ((cf‘𝐴) = 𝐴 → (cf‘𝐴) = 𝐴)
52, 3, 43eqtr3a 2880 . . 3 ((cf‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴)
653ad2ant2 1131 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → (card‘𝐴) = 𝐴)
71, 6sylbi 220 1 (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  wne 3007  wral 3126  wrex 3127  c0 4266   class class class wbr 5039  cfv 6328  csdm 8483  cardccrd 9340  cfccf 9342  Inaccwcwina 10081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6167  df-on 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-er 8264  df-en 8485  df-card 9344  df-cf 9346  df-wina 10083
This theorem is referenced by:  winalim  10094  winalim2  10095  gchina  10098  inar1  10174
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