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Theorem acnnum 7897
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 4351 . . . . . . 7  |-  ( X  e. AC  ~P X  ->  ~P X  e.  _V )
2 difss 3442 . . . . . . 7  |-  ( ~P X  \  { (/) } )  C_  ~P X
3 ssdomg 7120 . . . . . . 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 7896 . . . . . 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 3895 . . . . . . 7  |-  ( x  e.  ( ~P X  \  { (/) } )  <->  ( x  e.  ~P X  /\  x  =/=  (/) ) )
8 elpwi 3775 . . . . . . . 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 2739 . . . . 5  |-  A. x  e.  ( ~P X  \  { (/) } ) ( x  C_  X  /\  x  =/=  (/) )
12 acni2 7891 . . . . 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 2702 . . . . . 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 7875 . . 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 4351 . . 3  |-  ( X  e.  dom  card  ->  ~P X  e.  _V )
25 numacn 7894 . . 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 1721    =/= wne 2575   A.wral 2674   _Vcvv 2924    \ cdif 3285    C_ wss 3288   (/)c0 3596   ~Pcpw 3767   {csn 3782   class class class wbr 4180   dom cdm 4845   -->wf 5417   ` cfv 5421    ~<_ cdom 7074   cardccrd 7786  AC wacn 7789
This theorem is referenced by:  dfac13  7986
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-1o 6691  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-fin 7080  df-card 7790  df-acn 7793
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