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Theorem winacard 10636
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 10630 . 2 (𝐴 ∈ Inaccw ↔ (𝐴 β‰  βˆ… ∧ (cfβ€˜π΄) = 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐴 π‘₯ β‰Ί 𝑦))
2 cardcf 10196 . . . 4 (cardβ€˜(cfβ€˜π΄)) = (cfβ€˜π΄)
3 fveq2 6846 . . . 4 ((cfβ€˜π΄) = 𝐴 β†’ (cardβ€˜(cfβ€˜π΄)) = (cardβ€˜π΄))
4 id 22 . . . 4 ((cfβ€˜π΄) = 𝐴 β†’ (cfβ€˜π΄) = 𝐴)
52, 3, 43eqtr3a 2797 . . 3 ((cfβ€˜π΄) = 𝐴 β†’ (cardβ€˜π΄) = 𝐴)
653ad2ant2 1135 . 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 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  βˆ…c0 4286   class class class wbr 5109  β€˜cfv 6500   β‰Ί csdm 8888  cardccrd 9879  cfccf 9881  Inaccwcwina 10626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-er 8654  df-en 8890  df-card 9883  df-cf 9885  df-wina 10628
This theorem is referenced by:  winalim  10639  winalim2  10640  gchina  10643  inar1  10719
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