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Theorem acacni 7766
Description: A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acacni  |-  ( (CHOICE  /\  A  e.  V )  -> AC  A  =  _V )

Proof of Theorem acacni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . 4  |-  ( (CHOICE  /\  A  e.  V )  ->  A  e.  V )
2 vex 2791 . . . . 5  |-  x  e. 
_V
3 simpl 443 . . . . . 6  |-  ( (CHOICE  /\  A  e.  V )  -> CHOICE )
4 dfac10 7763 . . . . . 6  |-  (CHOICE  <->  dom  card  =  _V )
53, 4sylib 188 . . . . 5  |-  ( (CHOICE  /\  A  e.  V )  ->  dom  card  =  _V )
62, 5syl5eleqr 2370 . . . 4  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e.  dom  card )
7 numacn 7676 . . . 4  |-  ( A  e.  V  ->  (
x  e.  dom  card  ->  x  e. AC  A ) )
81, 6, 7sylc 56 . . 3  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e. AC  A )
92a1i 10 . . 3  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e.  _V )
108, 92thd 231 . 2  |-  ( (CHOICE  /\  A  e.  V )  ->  ( x  e. AC  A  <->  x  e.  _V ) )
1110eqrdv 2281 1  |-  ( (CHOICE  /\  A  e.  V )  -> AC  A  =  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   dom cdm 4689   cardccrd 7568  AC wacn 7571  CHOICEwac 7742
This theorem is referenced by:  dfacacn  7767  dfac13  7768  ptcls  17310  dfac14  17312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-card 7572  df-acn 7575  df-ac 7743
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