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Theorem winacard 10686
Description: A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
winacard (𝐴 ∈ Inaccw β†’ (cardβ€˜π΄) = 𝐴)

Proof of Theorem winacard
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elwina 10680 . 2 (𝐴 ∈ Inaccw ↔ (𝐴 β‰  βˆ… ∧ (cfβ€˜π΄) = 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐴 π‘₯ β‰Ί 𝑦))
2 cardcf 10246 . . . 4 (cardβ€˜(cfβ€˜π΄)) = (cfβ€˜π΄)
3 fveq2 6884 . . . 4 ((cfβ€˜π΄) = 𝐴 β†’ (cardβ€˜(cfβ€˜π΄)) = (cardβ€˜π΄))
4 id 22 . . . 4 ((cfβ€˜π΄) = 𝐴 β†’ (cfβ€˜π΄) = 𝐴)
52, 3, 43eqtr3a 2790 . . 3 ((cfβ€˜π΄) = 𝐴 β†’ (cardβ€˜π΄) = 𝐴)
653ad2ant2 1131 . 2 ((𝐴 β‰  βˆ… ∧ (cfβ€˜π΄) = 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐴 π‘₯ β‰Ί 𝑦) β†’ (cardβ€˜π΄) = 𝐴)
71, 6sylbi 216 1 (𝐴 ∈ Inaccw β†’ (cardβ€˜π΄) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  βˆ…c0 4317   class class class wbr 5141  β€˜cfv 6536   β‰Ί csdm 8937  cardccrd 9929  cfccf 9931  Inaccwcwina 10676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8702  df-en 8939  df-card 9933  df-cf 9935  df-wina 10678
This theorem is referenced by:  winalim  10689  winalim2  10690  gchina  10693  inar1  10769
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