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Theorem canthwdom 7309
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7030, equivalent to canth 6310). (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 4196 . . . . 5  |-  (/)  e.  ~P A
2 ne0i 3474 . . . . 5  |-  ( (/)  e.  ~P A  ->  ~P A  =/=  (/) )
31, 2mp1i 11 . . . 4  |-  ( ~P A  ~<_*  A  ->  ~P A  =/=  (/) )
4 brwdomn0 7299 . . . 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 7296 . . . . 5  |-  Rel  ~<_*
87brrelex2i 4746 . . . 4  |-  ( ~P A  ~<_*  A  ->  A  e.  _V )
9 foeq2 5464 . . . . . . 7  |-  ( x  =  A  ->  (
f : x -onto-> ~P x  <->  f : A -onto-> ~P x ) )
10 pweq 3641 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
11 foeq3 5465 . . . . . . . 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 2804 . . . . . 6  |-  x  e. 
_V
1615canth 6310 . . . . 5  |-  -.  f : x -onto-> ~P x
1714, 16vtoclg 2856 . . . 4  |-  ( A  e.  _V  ->  -.  f : A -onto-> ~P A
)
188, 17syl 15 . . 3  |-  ( ~P A  ~<_*  A  ->  -.  f : A -onto-> ~P A )
1918nexdv 1869 . 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   ~Pcpw 3638   class class class wbr 4039   -onto->wfo 5269    ~<_* cwdom 7287
This theorem is referenced by:  pwcdadom  7858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-wdom 7289
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