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Theorem canth 5807
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 1493 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 3232 . . . 4  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  C_  A
31, 2elpwi2 4144 . . 3  |-  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ~P A
4 forn 5423 . . 3  |-  ( F : A -onto-> ~P A  ->  ran  F  =  ~P A )
53, 4eleqtrrid 2260 . 2  |-  ( F : A -onto-> ~P A  ->  { x  e.  A  |  -.  x  e.  ( F `  x ) }  e.  ran  F
)
6 pm5.19 701 . . . . . 6  |-  -.  (
y  e.  ( F `
 y )  <->  -.  y  e.  ( F `  y
) )
7 eleq2 2234 . . . . . . 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 5496 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
108, 9eleq12d 2241 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  ( F `
 x )  <->  y  e.  ( F `  y ) ) )
1110notbid 662 . . . . . . . 8  |-  ( x  =  y  ->  ( -.  x  e.  ( F `  x )  <->  -.  y  e.  ( F `
 y ) ) )
1211elrab3 2887 . . . . . . 7  |-  ( y  e.  A  ->  (
y  e.  { x  e.  A  |  -.  x  e.  ( F `  x ) }  <->  -.  y  e.  ( F `  y
) ) )
137, 12sylan9bbr 460 . . . . . 6  |-  ( ( y  e.  A  /\  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )  -> 
( y  e.  ( F `  y )  <->  -.  y  e.  ( F `  y )
) )
146, 13mto 657 . . . . 5  |-  -.  (
y  e.  A  /\  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )
1514imnani 686 . . . 4  |-  ( y  e.  A  ->  -.  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x
) } )
1615nrex 2562 . . 3  |-  -.  E. y  e.  A  ( F `  y )  =  { x  e.  A  |  -.  x  e.  ( F `  x ) }
17 fofn 5422 . . . 4  |-  ( F : A -onto-> ~P A  ->  F  Fn  A )
18 fvelrnb 5544 . . . 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 670 . 2  |-  ( F : A -onto-> ~P A  ->  -.  { x  e.  A  |  -.  x  e.  ( F `  x
) }  e.  ran  F )
215, 20pm2.65i 634 1  |-  -.  F : A -onto-> ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   E.wrex 2449   {crab 2452   _Vcvv 2730   ~Pcpw 3566   ran crn 4612    Fn wfn 5193   -onto->wfo 5196   ` cfv 5198
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206
This theorem is referenced by: (None)
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