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Theorem canth 6498
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 7219. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6499 for a counterexample. (Use nex 1561 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 3388 . . . 4  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A
2 canth.1 . . . . 5  |-  A  e. 
_V
32elpw2 4324 . . . 4  |-  ( { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ~P A  <->  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A )
41, 3mpbir 201 . . 3  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ~P A
5 forn 5615 . . 3  |-  ( F : A -onto-> ~P A  ->  ran  F  =  ~P A )
64, 5syl5eleqr 2491 . 2  |-  ( F : A -onto-> ~P A  ->  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ran  F
)
7 id 20 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
8 fveq2 5687 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
97, 8eleq12d 2472 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  ( F `
 x )  <->  y  e.  ( F `  y ) ) )
109notbid 286 . . . . . . . 8  |-  ( x  =  y  ->  ( -.  x  e.  ( F `  x )  <->  -.  y  e.  ( F `
 y ) ) )
1110elrab 3052 . . . . . . 7  |-  ( y  e.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  <->  ( y  e.  A  /\  -.  y  e.  ( F `  y
) ) )
1211baibr 873 . . . . . 6  |-  ( y  e.  A  ->  ( -.  y  e.  ( F `  y )  <->  y  e.  { x  e.  A  |  -.  x  e.  ( F `  x
) } ) )
13 nbbn 348 . . . . . 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 189 . . . . 5  |-  ( y  e.  A  ->  -.  ( y  e.  ( F `  y )  <-> 
y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) } ) )
15 eleq2 2465 . . . . 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 115 . . . 4  |-  ( y  e.  A  ->  -.  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )
1716nrex 2768 . . 3  |-  -.  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x ) }
18 fofn 5614 . . . 4  |-  ( F : A -onto-> ~P A  ->  F  Fn  A )
19 fvelrnb 5733 . . . 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 295 . 2  |-  ( F : A -onto-> ~P A  ->  -.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  e.  ran  F )
226, 21pm2.65i 167 1  |-  -.  F : A -onto-> ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   ran crn 4838    Fn wfn 5408   -onto->wfo 5411   ` cfv 5413
This theorem is referenced by:  canth2  7219  canthwdom  7503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421
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