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Theorem canth 6178
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 6899. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6179 for a counterexample. (Use nex 1587 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
StepHypRef Expression
1 ssrab2 3179 . . . 4  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A
2 canth.1 . . . . 5  |-  A  e. 
_V
32elpw2 4064 . . . 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 5311 . . 3  |-  ( F : A -onto-> ~P A  ->  ran  F  =  ~P A )
64, 5syl5eleqr 2340 . 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 5377 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
97, 8eleq12d 2321 . . . . . . . . 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 2860 . . . . . . 7  |-  ( y  e.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  <->  ( y  e.  A  /\  -.  y  e.  ( F `  y
) ) )
1211baibr 877 . . . . . 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 2314 . . . . 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 2607 . . 3  |-  -.  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x ) }
18 fofn 5310 . . . 4  |-  ( F : A -onto-> ~P A  ->  F  Fn  A )
19 fvelrnb 5422 . . . 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 17 . . 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 167 1  |-  -.  F : A -onto-> ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    = wceq 1619    e. wcel 1621   E.wrex 2510   {crab 2512   _Vcvv 2727    C_ wss 3078   ~Pcpw 3530   ran crn 4581    Fn wfn 4587   -onto->wfo 4590   ` cfv 4592
This theorem is referenced by:  canth2  6899  canthwdom  7177
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fo 4606  df-fv 4608
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