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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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