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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 7662. 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 1605 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 3580 . . . 4  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A
2 canth.1 . . . . 5  |-  A  e. 
_V
32elpw2 4606 . . . 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 5791 . . 3  |-  ( F : A -onto-> ~P A  ->  ran  F  =  ~P A )
64, 5syl5eleqr 2557 . 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 5859 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
97, 8eleq12d 2544 . . . . . . . . 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 3256 . . . . . . 7  |-  ( y  e.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  <->  ( y  e.  A  /\  -.  y  e.  ( F `  y
) ) )
1211baibr 899 . . . . . 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 2535 . . . . 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 2914 . . 3  |-  -.  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x ) }
18 fofn 5790 . . . 4  |-  ( F : A -onto-> ~P A  ->  F  Fn  A )
19 fvelrnb 5908 . . . 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 1374    e. wcel 1762   E.wrex 2810   {crab 2813   _Vcvv 3108    C_ wss 3471   ~Pcpw 4005   ran crn 4995    Fn wfn 5576   -onto->wfo 5579   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589
This theorem is referenced by:  canth2  7662  canthwdom  7996
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