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Theorem winacard 10716
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 10710 . 2 (𝐴 ∈ Inaccw ↔ (𝐴 β‰  βˆ… ∧ (cfβ€˜π΄) = 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐴 π‘₯ β‰Ί 𝑦))
2 cardcf 10276 . . . 4 (cardβ€˜(cfβ€˜π΄)) = (cfβ€˜π΄)
3 fveq2 6897 . . . 4 ((cfβ€˜π΄) = 𝐴 β†’ (cardβ€˜(cfβ€˜π΄)) = (cardβ€˜π΄))
4 id 22 . . . 4 ((cfβ€˜π΄) = 𝐴 β†’ (cfβ€˜π΄) = 𝐴)
52, 3, 43eqtr3a 2792 . . 3 ((cfβ€˜π΄) = 𝐴 β†’ (cardβ€˜π΄) = 𝐴)
653ad2ant2 1132 . 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 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067  βˆ…c0 4323   class class class wbr 5148  β€˜cfv 6548   β‰Ί csdm 8963  cardccrd 9959  cfccf 9961  Inaccwcwina 10706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-er 8725  df-en 8965  df-card 9963  df-cf 9965  df-wina 10708
This theorem is referenced by:  winalim  10719  winalim2  10720  gchina  10723  inar1  10799
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