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Theorem canth2 6947
Description: Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 6225. (Contributed by NM, 7-Aug-1994.)
Hypothesis
Ref Expression
canth2.1  |-  A  e. 
_V
Assertion
Ref Expression
canth2  |-  A  ~<  ~P A

Proof of Theorem canth2
StepHypRef Expression
1 canth2.1 . . 3  |-  A  e. 
_V
21pwex 4131 . . 3  |-  ~P A  e.  _V
3 snelpwi 4158 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2743 . . . . . . 7  |-  x  e. 
_V
54sneqr 3721 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3592 . . . . . 6  |-  ( x  =  y  ->  { x }  =  { y } )
75, 6impbii 182 . . . . 5  |-  ( { x }  =  {
y }  <->  x  =  y )
87a1i 12 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( { x }  =  { y }  <->  x  =  y ) )
93, 8dom3 6838 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 656 . 2  |-  A  ~<_  ~P A
111canth 6225 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5382 . . . . 5  |-  ( f : A -1-1-onto-> ~P A  ->  f : A -onto-> ~P A )
1311, 12mto 169 . . . 4  |-  -.  f : A -1-1-onto-> ~P A
1413nex 1587 . . 3  |-  -.  E. f  f : A -1-1-onto-> ~P A
15 bren 6804 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 292 . 2  |-  -.  A  ~~  ~P A
17 brsdom 6817 . 2  |-  ( A 
~<  ~P A  <->  ( A  ~<_  ~P A  /\  -.  A  ~~  ~P A ) )
1810, 16, 17mpbir2an 891 1  |-  A  ~<  ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2740   ~Pcpw 3566   {csn 3581   class class class wbr 3963   -onto->wfo 4636   -1-1-onto->wf1o 4637    ~~ cen 6793    ~<_ cdom 6794    ~< csdm 6795
This theorem is referenced by:  canth2g  6948  r1sdom  7379  alephsucpw2  7671  dfac13  7701  pwsdompw  7763  numthcor  8054  alephexp1  8134  pwcfsdom  8138  cfpwsdom  8139  gchhar  8226  gchac  8228  inawinalem  8244  tskcard  8336  gruina  8373  grothac  8385  rpnnen  12432  rexpen  12433  rucALT  12435  rectbntr0  18264
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-en 6797  df-dom 6798  df-sdom 6799
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