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Theorem acnnum 7868
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 4326 . . . . . . 7  |-  ( X  e. AC  ~P X  ->  ~P X  e.  _V )
2 difss 3419 . . . . . . 7  |-  ( ~P X  \  { (/) } )  C_  ~P X
3 ssdomg 7091 . . . . . . 7  |-  ( ~P X  e.  _V  ->  ( ( ~P X  \  { (/) } )  C_  ~P X  ->  ( ~P X  \  { (/) } )  ~<_  ~P X ) )
41, 2, 3ee10 1382 . . . . . 6  |-  ( X  e. AC  ~P X  ->  ( ~P X  \  { (/) } )  ~<_  ~P X )
5 acndom 7867 . . . . . 6  |-  ( ( ~P X  \  { (/)
} )  ~<_  ~P X  ->  ( X  e. AC  ~P X  ->  X  e. AC  ( ~P X  \  { (/) } ) ) )
64, 5mpcom 34 . . . . 5  |-  ( X  e. AC  ~P X  ->  X  e. AC  ( ~P X  \  { (/) } ) )
7 eldifsn 3872 . . . . . . 7  |-  ( x  e.  ( ~P X  \  { (/) } )  <->  ( x  e.  ~P X  /\  x  =/=  (/) ) )
8 elpwi 3752 . . . . . . . 8  |-  ( x  e.  ~P X  ->  x  C_  X )
98anim1i 552 . . . . . . 7  |-  ( ( x  e.  ~P X  /\  x  =/=  (/) )  -> 
( x  C_  X  /\  x  =/=  (/) ) )
107, 9sylbi 188 . . . . . 6  |-  ( x  e.  ( ~P X  \  { (/) } )  -> 
( x  C_  X  /\  x  =/=  (/) ) )
1110rgen 2716 . . . . 5  |-  A. x  e.  ( ~P X  \  { (/) } ) ( x  C_  X  /\  x  =/=  (/) )
12 acni2 7862 . . . . 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 644 . . . 4  |-  ( X  e. AC  ~P X  ->  E. f
( f : ( ~P X  \  { (/)
} ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x )  e.  x
) )
14 simpr 448 . . . . . 6  |-  ( ( f : ( ~P X  \  { (/) } ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x
)  e.  x )  ->  A. x  e.  ( ~P X  \  { (/)
} ) ( f `
 x )  e.  x )
157imbi1i 316 . . . . . . . 8  |-  ( ( x  e.  ( ~P X  \  { (/) } )  ->  ( f `  x )  e.  x
)  <->  ( ( x  e.  ~P X  /\  x  =/=  (/) )  ->  (
f `  x )  e.  x ) )
16 impexp 434 . . . . . . . 8  |-  ( ( ( x  e.  ~P X  /\  x  =/=  (/) )  -> 
( f `  x
)  e.  x )  <-> 
( x  e.  ~P X  ->  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
1715, 16bitri 241 . . . . . . 7  |-  ( ( x  e.  ( ~P X  \  { (/) } )  ->  ( f `  x )  e.  x
)  <->  ( x  e. 
~P X  ->  (
x  =/=  (/)  ->  (
f `  x )  e.  x ) ) )
1817ralbii2 2679 . . . . . 6  |-  ( A. x  e.  ( ~P X  \  { (/) } ) ( f `  x
)  e.  x  <->  A. x  e.  ~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
) )
1914, 18sylib 189 . . . . 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 1582 . . . 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 16 . . 3  |-  ( X  e. AC  ~P X  ->  E. f A. x  e.  ~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
) )
22 dfac8a 7846 . . 3  |-  ( X  e. AC  ~P X  ->  ( E. f A. x  e. 
~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
)  ->  X  e.  dom  card ) )
2321, 22mpd 15 . 2  |-  ( X  e. AC  ~P X  ->  X  e.  dom  card )
24 pwexg 4326 . . 3  |-  ( X  e.  dom  card  ->  ~P X  e.  _V )
25 numacn 7865 . . 3  |-  ( ~P X  e.  _V  ->  ( X  e.  dom  card  ->  X  e. AC  ~P X ) )
2624, 25mpcom 34 . 2  |-  ( X  e.  dom  card  ->  X  e. AC  ~P X )
2723, 26impbii 181 1  |-  ( X  e. AC  ~P X  <->  X  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    e. wcel 1717    =/= wne 2552   A.wral 2651   _Vcvv 2901    \ cdif 3262    C_ wss 3265   (/)c0 3573   ~Pcpw 3744   {csn 3759   class class class wbr 4155   dom cdm 4820   -->wf 5392   ` cfv 5396    ~<_ cdom 7045   cardccrd 7757  AC wacn 7760
This theorem is referenced by:  dfac13  7957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-1o 6662  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-fin 7051  df-card 7761  df-acn 7764
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