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Theorem acnnum 8243
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 4497 . . . . . . 7  |-  ( X  e. AC  ~P X  ->  ~P X  e.  _V )
2 difss 3504 . . . . . . 7  |-  ( ~P X  \  { (/) } )  C_  ~P X
3 ssdomg 7376 . . . . . . 7  |-  ( ~P X  e.  _V  ->  ( ( ~P X  \  { (/) } )  C_  ~P X  ->  ( ~P X  \  { (/) } )  ~<_  ~P X ) )
41, 2, 3mpisyl 18 . . . . . 6  |-  ( X  e. AC  ~P X  ->  ( ~P X  \  { (/) } )  ~<_  ~P X )
5 acndom 8242 . . . . . 6  |-  ( ( ~P X  \  { (/)
} )  ~<_  ~P X  ->  ( X  e. AC  ~P X  ->  X  e. AC  ( ~P X  \  { (/) } ) ) )
64, 5mpcom 36 . . . . 5  |-  ( X  e. AC  ~P X  ->  X  e. AC  ( ~P X  \  { (/) } ) )
7 eldifsn 4021 . . . . . . 7  |-  ( x  e.  ( ~P X  \  { (/) } )  <->  ( x  e.  ~P X  /\  x  =/=  (/) ) )
8 elpwi 3890 . . . . . . . 8  |-  ( x  e.  ~P X  ->  x  C_  X )
98anim1i 568 . . . . . . 7  |-  ( ( x  e.  ~P X  /\  x  =/=  (/) )  -> 
( x  C_  X  /\  x  =/=  (/) ) )
107, 9sylbi 195 . . . . . 6  |-  ( x  e.  ( ~P X  \  { (/) } )  -> 
( x  C_  X  /\  x  =/=  (/) ) )
1110rgen 2802 . . . . 5  |-  A. x  e.  ( ~P X  \  { (/) } ) ( x  C_  X  /\  x  =/=  (/) )
12 acni2 8237 . . . . 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 662 . . . 4  |-  ( X  e. AC  ~P X  ->  E. f
( f : ( ~P X  \  { (/)
} ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x )  e.  x
) )
14 simpr 461 . . . . . 6  |-  ( ( f : ( ~P X  \  { (/) } ) --> X  /\  A. x  e.  ( ~P X  \  { (/) } ) ( f `  x
)  e.  x )  ->  A. x  e.  ( ~P X  \  { (/)
} ) ( f `
 x )  e.  x )
157imbi1i 325 . . . . . . . 8  |-  ( ( x  e.  ( ~P X  \  { (/) } )  ->  ( f `  x )  e.  x
)  <->  ( ( x  e.  ~P X  /\  x  =/=  (/) )  ->  (
f `  x )  e.  x ) )
16 impexp 446 . . . . . . . 8  |-  ( ( ( x  e.  ~P X  /\  x  =/=  (/) )  -> 
( f `  x
)  e.  x )  <-> 
( x  e.  ~P X  ->  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
1715, 16bitri 249 . . . . . . 7  |-  ( ( x  e.  ( ~P X  \  { (/) } )  ->  ( f `  x )  e.  x
)  <->  ( x  e. 
~P X  ->  (
x  =/=  (/)  ->  (
f `  x )  e.  x ) ) )
1817ralbii2 2764 . . . . . 6  |-  ( A. x  e.  ( ~P X  \  { (/) } ) ( f `  x
)  e.  x  <->  A. x  e.  ~P  X ( x  =/=  (/)  ->  ( f `  x )  e.  x
) )
1914, 18sylib 196 . . . . 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 1625 . . . 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 8221 . . 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 4497 . . 3  |-  ( X  e.  dom  card  ->  ~P X  e.  _V )
25 numacn 8240 . . 3  |-  ( ~P X  e.  _V  ->  ( X  e.  dom  card  ->  X  e. AC  ~P X ) )
2624, 25mpcom 36 . 2  |-  ( X  e.  dom  card  ->  X  e. AC  ~P X )
2723, 26impbii 188 1  |-  ( X  e. AC  ~P X  <->  X  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1586    e. wcel 1756    =/= wne 2620   A.wral 2736   _Vcvv 2993    \ cdif 3346    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   {csn 3898   class class class wbr 4313   dom cdm 4861   -->wf 5435   ` cfv 5439    ~<_ cdom 7329   cardccrd 8126  AC wacn 8129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-1o 6941  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-fin 7335  df-card 8130  df-acn 8133
This theorem is referenced by:  dfac13  8332
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