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Theorem canth 5779
Description: No set  A is equinumerous to its power set (Cantor's theorem), i.e., no function can map  A onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1480 if you want the form  -.  E. f
f : A -onto-> ~P A.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
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 canth.1 . . . 4  |-  A  e. 
_V
2 ssrab2 3213 . . . 4  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A
31, 2elpwi2 4120 . . 3  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ~P A
4 forn 5396 . . 3  |-  ( F : A -onto-> ~P A  ->  ran  F  =  ~P A )
53, 4eleqtrrid 2247 . 2  |-  ( F : A -onto-> ~P A  ->  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ran  F
)
6 pm5.19 696 . . . . . 6  |-  -.  (
y  e.  ( F `
 y )  <->  -.  y  e.  ( F `  y
) )
7 eleq2 2221 . . . . . . 7  |-  ( ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) }  ->  (
y  e.  ( F `
 y )  <->  y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) } ) )
8 id 19 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
9 fveq2 5469 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
108, 9eleq12d 2228 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  ( F `
 x )  <->  y  e.  ( F `  y ) ) )
1110notbid 657 . . . . . . . 8  |-  ( x  =  y  ->  ( -.  x  e.  ( F `  x )  <->  -.  y  e.  ( F `
 y ) ) )
1211elrab3 2869 . . . . . . 7  |-  ( y  e.  A  ->  (
y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) }  <->  -.  y  e.  ( F `  y
) ) )
137, 12sylan9bbr 459 . . . . . 6  |-  ( ( y  e.  A  /\  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )  -> 
( y  e.  ( F `  y )  <->  -.  y  e.  ( F `  y )
) )
146, 13mto 652 . . . . 5  |-  -.  (
y  e.  A  /\  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )
1514imnani 681 . . . 4  |-  ( y  e.  A  ->  -.  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )
1615nrex 2549 . . 3  |-  -.  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x ) }
17 fofn 5395 . . . 4  |-  ( F : A -onto-> ~P A  ->  F  Fn  A )
18 fvelrnb 5517 . . . 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
) } ) )
1917, 18syl 14 . . 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
) } ) )
2016, 19mtbiri 665 . 2  |-  ( F : A -onto-> ~P A  ->  -.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  e.  ran  F )
215, 20pm2.65i 629 1  |-  -.  F : A -onto-> ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   E.wrex 2436   {crab 2439   _Vcvv 2712   ~Pcpw 3543   ran crn 4588    Fn wfn 5166   -onto->wfo 5169   ` cfv 5171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-fo 5177  df-fv 5179
This theorem is referenced by: (None)
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