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Theorem canthwdom 8004
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7670, equivalent to canth 6241). (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 4616 . . . . 5  |-  (/)  e.  ~P A
2 ne0i 3791 . . . . 5  |-  ( (/)  e.  ~P A  ->  ~P A  =/=  (/) )
31, 2mp1i 12 . . . 4  |-  ( ~P A  ~<_*  A  ->  ~P A  =/=  (/) )
4 brwdomn0 7994 . . . 4  |-  ( ~P A  =/=  (/)  ->  ( ~P A  ~<_*  A  <->  E. f  f : A -onto-> ~P A ) )
53, 4syl 16 . . 3  |-  ( ~P A  ~<_*  A  ->  ( ~P A  ~<_*  A  <->  E. f  f : A -onto-> ~P A ) )
65ibi 241 . 2  |-  ( ~P A  ~<_*  A  ->  E. f 
f : A -onto-> ~P A )
7 relwdom 7991 . . . . 5  |-  Rel  ~<_*
87brrelex2i 5040 . . . 4  |-  ( ~P A  ~<_*  A  ->  A  e.  _V )
9 foeq2 5791 . . . . . . 7  |-  ( x  =  A  ->  (
f : x -onto-> ~P x  <->  f : A -onto-> ~P x ) )
10 pweq 4013 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
11 foeq3 5792 . . . . . . . 8  |-  ( ~P x  =  ~P A  ->  ( f : A -onto-> ~P x  <->  f : A -onto-> ~P A ) )
1210, 11syl 16 . . . . . . 7  |-  ( x  =  A  ->  (
f : A -onto-> ~P x 
<->  f : A -onto-> ~P A ) )
139, 12bitrd 253 . . . . . 6  |-  ( x  =  A  ->  (
f : x -onto-> ~P x  <->  f : A -onto-> ~P A ) )
1413notbid 294 . . . . 5  |-  ( x  =  A  ->  ( -.  f : x -onto-> ~P x  <->  -.  f : A -onto-> ~P A ) )
15 vex 3116 . . . . . 6  |-  x  e. 
_V
1615canth 6241 . . . . 5  |-  -.  f : x -onto-> ~P x
1714, 16vtoclg 3171 . . . 4  |-  ( A  e.  _V  ->  -.  f : A -onto-> ~P A
)
188, 17syl 16 . . 3  |-  ( ~P A  ~<_*  A  ->  -.  f : A -onto-> ~P A )
1918nexdv 1832 . 2  |-  ( ~P A  ~<_*  A  ->  -.  E. f 
f : A -onto-> ~P A )
206, 19pm2.65i 173 1  |-  -.  ~P A  ~<_*  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447   -onto->wfo 5585    ~<_* cwdom 7982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-fo 5593  df-fv 5595  df-wdom 7984
This theorem is referenced by:  pwcdadom  8595
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