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Theorem acnnum 7679
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 4194 . . . . . . 7  |-  ( X  e. AC  ~P X  ->  ~P X  e.  _V )
2 difss 3303 . . . . . . 7  |-  ( ~P X  \  { (/) } )  C_  ~P X
3 ssdomg 6907 . . . . . . 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 7678 . . . . . 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 3749 . . . . . . 7  |-  ( x  e.  ( ~P X  \  { (/) } )  <->  ( x  e.  ~P X  /\  x  =/=  (/) ) )
8 elpwi 3633 . . . . . . . 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 2608 . . . . 5  |-  A. x  e.  ( ~P X  \  { (/) } ) ( x  C_  X  /\  x  =/=  (/) )
12 acni2 7673 . . . . 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 2571 . . . . . 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 1563 . . . 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 7657 . . 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 4194 . . 3  |-  ( X  e.  dom  card  ->  ~P X  e.  _V )
25 numacn 7676 . . 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 1528    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023   dom cdm 4689   -->wf 5251   ` cfv 5255    ~<_ cdom 6861   cardccrd 7568  AC wacn 7571
This theorem is referenced by:  dfac13  7768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-1o 6479  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-fin 6867  df-card 7572  df-acn 7575
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