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Theorem acacni 8575
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 463 . . . 4  |-  ( (CHOICE  /\  A  e.  V )  ->  A  e.  V )
2 vex 3050 . . . . 5  |-  x  e. 
_V
3 simpl 459 . . . . . 6  |-  ( (CHOICE  /\  A  e.  V )  -> CHOICE )
4 dfac10 8572 . . . . . 6  |-  (CHOICE  <->  dom  card  =  _V )
53, 4sylib 200 . . . . 5  |-  ( (CHOICE  /\  A  e.  V )  ->  dom  card  =  _V )
62, 5syl5eleqr 2538 . . . 4  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e.  dom  card )
7 numacn 8485 . . . 4  |-  ( A  e.  V  ->  (
x  e.  dom  card  ->  x  e. AC  A ) )
81, 6, 7sylc 62 . . 3  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e. AC  A )
92a1i 11 . . 3  |-  ( (CHOICE  /\  A  e.  V )  ->  x  e.  _V )
108, 92thd 244 . 2  |-  ( (CHOICE  /\  A  e.  V )  ->  ( x  e. AC  A  <->  x  e.  _V ) )
1110eqrdv 2451 1  |-  ( (CHOICE  /\  A  e.  V )  -> AC  A  =  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   _Vcvv 3047   dom cdm 4837   cardccrd 8374  AC wacn 8377  CHOICEwac 8551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-card 8378  df-acn 8381  df-ac 8552
This theorem is referenced by:  dfacacn  8576  dfac13  8577  ptcls  20643  dfac14  20645
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