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Theorem canth 6235
Description: No set  A is equinumerous to its power set (Cantor's theorem), i.e. no function can map  A it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 7668. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6236 for a counterexample. (Use nex 1612 if you want the form  -.  E. f f : A -onto-> ~P A.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
Hypothesis
Ref Expression
canth.1  |-  A  e. 
_V
Assertion
Ref Expression
canth  |-  -.  F : A -onto-> ~P A

Proof of Theorem canth
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3567 . . . 4  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A
2 canth.1 . . . . 5  |-  A  e. 
_V
32elpw2 4597 . . . 4  |-  ( { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ~P A  <->  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A )
41, 3mpbir 209 . . 3  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ~P A
5 forn 5784 . . 3  |-  ( F : A -onto-> ~P A  ->  ran  F  =  ~P A )
64, 5syl5eleqr 2536 . 2  |-  ( F : A -onto-> ~P A  ->  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ran  F
)
7 id 22 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
8 fveq2 5852 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
97, 8eleq12d 2523 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  ( F `
 x )  <->  y  e.  ( F `  y ) ) )
109notbid 294 . . . . . . . 8  |-  ( x  =  y  ->  ( -.  x  e.  ( F `  x )  <->  -.  y  e.  ( F `
 y ) ) )
1110elrab 3241 . . . . . . 7  |-  ( y  e.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  <->  ( y  e.  A  /\  -.  y  e.  ( F `  y
) ) )
1211baibr 902 . . . . . 6  |-  ( y  e.  A  ->  ( -.  y  e.  ( F `  y )  <->  y  e.  { x  e.  A  |  -.  x  e.  ( F `  x
) } ) )
13 nbbn 358 . . . . . 6  |-  ( ( -.  y  e.  ( F `  y )  <-> 
y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) } )  <->  -.  ( y  e.  ( F `  y )  <-> 
y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) } ) )
1412, 13sylib 196 . . . . 5  |-  ( y  e.  A  ->  -.  ( y  e.  ( F `  y )  <-> 
y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) } ) )
15 eleq2 2514 . . . . 5  |-  ( ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) }  ->  (
y  e.  ( F `
 y )  <->  y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) } ) )
1614, 15nsyl 121 . . . 4  |-  ( y  e.  A  ->  -.  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )
1716nrex 2896 . . 3  |-  -.  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x ) }
18 fofn 5783 . . . 4  |-  ( F : A -onto-> ~P A  ->  F  Fn  A )
19 fvelrnb 5901 . . . 4  |-  ( F  Fn  A  ->  ( { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } ) )
2018, 19syl 16 . . 3  |-  ( F : A -onto-> ~P A  ->  ( { x  e.  A  |  -.  x  e.  ( F `  x
) }  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } ) )
2117, 20mtbiri 303 . 2  |-  ( F : A -onto-> ~P A  ->  -.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  e.  ran  F )
226, 21pm2.65i 173 1  |-  -.  F : A -onto-> ~P A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1381    e. wcel 1802   E.wrex 2792   {crab 2795   _Vcvv 3093    C_ wss 3458   ~Pcpw 3993   ran crn 4986    Fn wfn 5569   -onto->wfo 5572   ` cfv 5574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fo 5580  df-fv 5582
This theorem is referenced by:  canth2  7668  canthwdom  8003
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