MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  canthwdom Structured version   Unicode version

Theorem canthwdom 8008
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7672, equivalent to canth 6239). (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 4606 . . . . 5  |-  (/)  e.  ~P A
2 ne0i 3776 . . . . 5  |-  ( (/)  e.  ~P A  ->  ~P A  =/=  (/) )
31, 2mp1i 12 . . . 4  |-  ( ~P A  ~<_*  A  ->  ~P A  =/=  (/) )
4 brwdomn0 7998 . . . 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 7995 . . . . 5  |-  Rel  ~<_*
87brrelex2i 5031 . . . 4  |-  ( ~P A  ~<_*  A  ->  A  e.  _V )
9 foeq2 5782 . . . . . . 7  |-  ( x  =  A  ->  (
f : x -onto-> ~P x  <->  f : A -onto-> ~P x ) )
10 pweq 4000 . . . . . . . 8  |-  ( x  =  A  ->  ~P x  =  ~P A
)
11 foeq3 5783 . . . . . . . 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 3098 . . . . . 6  |-  x  e. 
_V
1615canth 6239 . . . . 5  |-  -.  f : x -onto-> ~P x
1714, 16vtoclg 3153 . . . 4  |-  ( A  e.  _V  ->  -.  f : A -onto-> ~P A
)
188, 17syl 16 . . 3  |-  ( ~P A  ~<_*  A  ->  -.  f : A -onto-> ~P A )
1918nexdv 1870 . 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 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   _Vcvv 3095   (/)c0 3770   ~Pcpw 3997   class class class wbr 4437   -onto->wfo 5576    ~<_* cwdom 7986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-wdom 7988
This theorem is referenced by:  pwcdadom  8599
  Copyright terms: Public domain W3C validator