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Theorem acacni 8511
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 461 . . . 4  |-  ( (CHOICE  /\  A  e.  V )  ->  A  e.  V )
2 vex 3111 . . . . 5  |-  x  e. 
_V
3 simpl 457 . . . . . 6  |-  ( (CHOICE  /\  A  e.  V )  -> CHOICE )
4 dfac10 8508 . . . . . 6  |-  (CHOICE  <->  dom  card  =  _V )
53, 4sylib 196 . . . . 5  |-  ( (CHOICE  /\  A  e.  V )  ->  dom  card  =  _V )
62, 5syl5eleqr 2557 . . . 4  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e.  dom  card )
7 numacn 8421 . . . 4  |-  ( A  e.  V  ->  (
x  e.  dom  card  ->  x  e. AC  A ) )
81, 6, 7sylc 60 . . 3  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e. AC  A )
92a1i 11 . . 3  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e.  _V )
108, 92thd 240 . 2  |-  ( (CHOICE  /\  A  e.  V )  ->  ( x  e. AC  A  <->  x  e.  _V ) )
1110eqrdv 2459 1  |-  ( (CHOICE  /\  A  e.  V )  -> AC  A  =  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108   dom cdm 4994   cardccrd 8307  AC wacn 8310  CHOICEwac 8487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-recs 7034  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-card 8311  df-acn 8314  df-ac 8488
This theorem is referenced by:  dfacacn  8512  dfac13  8513  ptcls  19847  dfac14  19849
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