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Theorem acnnum 7695
Description: A set  X which has choice sequences on it of length  ~P X is well-orderable (and hence has choice sequences of every length). (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acnnum  |-  ( X  e. AC  ~P X  <->  X  e.  dom  card )

Proof of Theorem acnnum
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4210 . . . . . . 7  |-  ( X  e. AC  ~P X  ->  ~P X  e.  _V )
2 difss 3316 . . . . . . 7  |-  ( ~P X  \  { (/) } )  C_  ~P X
3 ssdomg 6923 . . . . . . 7  |-  ( ~P X  e.  _V  ->  ( ( ~P X  \  { (/) } )  C_  ~P X  ->  ( ~P X  \  { (/) } )  ~<_  ~P X ) )
41, 2, 3ee10 1366 . . . . . 6  |-  ( X  e. AC  ~P X  ->  ( ~P X  \  { (/) } )  ~<_  ~P X )
5 acndom 7694 . . . . . 6  |-  ( ( ~P X  \  { (/)
} )  ~<_  ~P X  ->  ( X  e. AC  ~P X  ->  X  e. AC  ( ~P X  \  { (/) } ) ) )
64, 5mpcom 32 . . . . 5  |-  ( X  e. AC  ~P X  ->  X  e. AC  ( ~P X  \  { (/) } ) )
7 eldifsn 3762 . . . . . . 7  |-  ( x  e.  ( ~P X  \  { (/) } )  <->  ( x  e.  ~P X  /\  x  =/=  (/) ) )
8 elpwi 3646 . . . . . . . 8  |-  ( x  e.  ~P X  ->  x  C_  X )
98anim1i 551 . . . . . . 7  |-  ( ( x  e.  ~P X  /\  x  =/=  (/) )  -> 
( x  C_  X  /\  x  =/=  (/) ) )
107, 9sylbi 187 . . . . . 6  |-  ( x  e.  ( ~P X  \  { (/) } )  -> 
( x  C_  X  /\  x  =/=  (/) ) )
1110rgen 2621 . . . . 5  |-  A. x  e.  ( ~P X  \  { (/) } ) ( x  C_  X  /\  x  =/=  (/) )
12 acni2 7689 . . . . 5  |-  ( ( X  e. AC  ( ~P X  \  { (/) } )  /\  A. x  e.  ( ~P X  \  { (/) } ) ( x  C_  X  /\  x  =/=  (/) ) )  ->  E. f ( f : ( ~P X  \  { (/) } ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x )  e.  x ) )
136, 11, 12sylancl 643 . . . 4  |-  ( X  e. AC  ~P X  ->  E. f
( f : ( ~P X  \  { (/)
} ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x )  e.  x
) )
14 simpr 447 . . . . . 6  |-  ( ( f : ( ~P X  \  { (/) } ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x
)  e.  x )  ->  A. x  e.  ( ~P X  \  { (/)
} ) ( f `
 x )  e.  x )
157imbi1i 315 . . . . . . . 8  |-  ( ( x  e.  ( ~P X  \  { (/) } )  ->  ( f `  x )  e.  x
)  <->  ( ( x  e.  ~P X  /\  x  =/=  (/) )  ->  (
f `  x )  e.  x ) )
16 impexp 433 . . . . . . . 8  |-  ( ( ( x  e.  ~P X  /\  x  =/=  (/) )  -> 
( f `  x
)  e.  x )  <-> 
( x  e.  ~P X  ->  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
1715, 16bitri 240 . . . . . . 7  |-  ( ( x  e.  ( ~P X  \  { (/) } )  ->  ( f `  x )  e.  x
)  <->  ( x  e. 
~P X  ->  (
x  =/=  (/)  ->  (
f `  x )  e.  x ) ) )
1817ralbii2 2584 . . . . . 6  |-  ( A. x  e.  ( ~P X  \  { (/) } ) ( f `  x
)  e.  x  <->  A. x  e.  ~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
) )
1914, 18sylib 188 . . . . 5  |-  ( ( f : ( ~P X  \  { (/) } ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x
)  e.  x )  ->  A. x  e.  ~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
) )
2019eximi 1566 . . . 4  |-  ( E. f ( f : ( ~P X  \  { (/) } ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x )  e.  x )  ->  E. f A. x  e. 
~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
) )
2113, 20syl 15 . . 3  |-  ( X  e. AC  ~P X  ->  E. f A. x  e.  ~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
) )
22 dfac8a 7673 . . 3  |-  ( X  e. AC  ~P X  ->  ( E. f A. x  e. 
~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
)  ->  X  e.  dom  card ) )
2321, 22mpd 14 . 2  |-  ( X  e. AC  ~P X  ->  X  e.  dom  card )
24 pwexg 4210 . . 3  |-  ( X  e.  dom  card  ->  ~P X  e.  _V )
25 numacn 7692 . . 3  |-  ( ~P X  e.  _V  ->  ( X  e.  dom  card  ->  X  e. AC  ~P X ) )
2624, 25mpcom 32 . 2  |-  ( X  e.  dom  card  ->  X  e. AC  ~P X )
2723, 26impbii 180 1  |-  ( X  e. AC  ~P X  <->  X  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   class class class wbr 4039   dom cdm 4705   -->wf 5267   ` cfv 5271    ~<_ cdom 6877   cardccrd 7584  AC wacn 7587
This theorem is referenced by:  dfac13  7784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-1o 6495  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-fin 6883  df-card 7588  df-acn 7591
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