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Theorem canthwdom 7293
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7014, equivalent to canth 6294). (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
canthwdom  |-  -.  ~P A  ~<_*  A

Proof of Theorem canthwdom
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 4180 . . . . 5  |-  (/)  e.  ~P A
2 ne0i 3461 . . . . 5  |-  ( (/)  e.  ~P A  ->  ~P A  =/=  (/) )
31, 2mp1i 11 . . . 4  |-  ( ~P A  ~<_*  A  ->  ~P A  =/=  (/) )
4 brwdomn0 7283 . . . 4  |-  ( ~P A  =/=  (/)  ->  ( ~P A  ~<_*  A  <->  E. f  f : A -onto-> ~P A ) )
53, 4syl 15 . . 3  |-  ( ~P A  ~<_*  A  ->  ( ~P A  ~<_*  A  <->  E. f  f : A -onto-> ~P A ) )
65ibi 232 . 2  |-  ( ~P A  ~<_*  A  ->  E. f 
f : A -onto-> ~P A )
7 relwdom 7280 . . . . 5  |-  Rel  ~<_*
87brrelex2i 4730 . . . 4  |-  ( ~P A  ~<_*  A  ->  A  e.  _V )
9 foeq2 5448 . . . . . . 7  |-  ( x  =  A  ->  (
f : x -onto-> ~P x  <->  f : A -onto-> ~P x ) )
10 pweq 3628 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
11 foeq3 5449 . . . . . . . 8  |-  ( ~P x  =  ~P A  ->  ( f : A -onto-> ~P x  <->  f : A -onto-> ~P A ) )
1210, 11syl 15 . . . . . . 7  |-  ( x  =  A  ->  (
f : A -onto-> ~P x 
<->  f : A -onto-> ~P A ) )
139, 12bitrd 244 . . . . . 6  |-  ( x  =  A  ->  (
f : x -onto-> ~P x  <->  f : A -onto-> ~P A ) )
1413notbid 285 . . . . 5  |-  ( x  =  A  ->  ( -.  f : x -onto-> ~P x  <->  -.  f : A -onto-> ~P A ) )
15 vex 2791 . . . . . 6  |-  x  e. 
_V
1615canth 6294 . . . . 5  |-  -.  f : x -onto-> ~P x
1714, 16vtoclg 2843 . . . 4  |-  ( A  e.  _V  ->  -.  f : A -onto-> ~P A
)
188, 17syl 15 . . 3  |-  ( ~P A  ~<_*  A  ->  -.  f : A -onto-> ~P A )
1918nexdv 1857 . 2  |-  ( ~P A  ~<_*  A  ->  -.  E. f 
f : A -onto-> ~P A )
206, 19pm2.65i 165 1  |-  -.  ~P A  ~<_*  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023   -onto->wfo 5253    ~<_* cwdom 7271
This theorem is referenced by:  pwcdadom  7842
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-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-wdom 7273
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