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Theorem canth 6575
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 7296. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6576 for a counterexample. (Use nex 1565 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 3417 . . . 4  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A
2 canth.1 . . . . 5  |-  A  e. 
_V
32elpw2 4399 . . . 4  |-  ( { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ~P A  <->  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A )
41, 3mpbir 202 . . 3  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ~P A
5 forn 5691 . . 3  |-  ( F : A -onto-> ~P A  ->  ran  F  =  ~P A )
64, 5syl5eleqr 2530 . 2  |-  ( F : A -onto-> ~P A  ->  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ran  F
)
7 id 21 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
8 fveq2 5763 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
97, 8eleq12d 2511 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  ( F `
 x )  <->  y  e.  ( F `  y ) ) )
109notbid 287 . . . . . . . 8  |-  ( x  =  y  ->  ( -.  x  e.  ( F `  x )  <->  -.  y  e.  ( F `
 y ) ) )
1110elrab 3101 . . . . . . 7  |-  ( y  e.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  <->  ( y  e.  A  /\  -.  y  e.  ( F `  y
) ) )
1211baibr 874 . . . . . 6  |-  ( y  e.  A  ->  ( -.  y  e.  ( F `  y )  <->  y  e.  { x  e.  A  |  -.  x  e.  ( F `  x
) } ) )
13 nbbn 349 . . . . . 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 190 . . . . 5  |-  ( y  e.  A  ->  -.  ( y  e.  ( F `  y )  <-> 
y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) } ) )
15 eleq2 2504 . . . . 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 116 . . . 4  |-  ( y  e.  A  ->  -.  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )
1716nrex 2815 . . 3  |-  -.  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x ) }
18 fofn 5690 . . . 4  |-  ( F : A -onto-> ~P A  ->  F  Fn  A )
19 fvelrnb 5810 . . . 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 296 . 2  |-  ( F : A -onto-> ~P A  ->  -.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  e.  ran  F )
226, 21pm2.65i 168 1  |-  -.  F : A -onto-> ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    = wceq 1654    e. wcel 1728   E.wrex 2713   {crab 2716   _Vcvv 2965    C_ wss 3309   ~Pcpw 3828   ran crn 4914    Fn wfn 5484   -onto->wfo 5487   ` cfv 5489
This theorem is referenced by:  canth2  7296  canthwdom  7583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-fo 5495  df-fv 5497
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