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Theorem canth2 7262
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 6541. (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
Dummy variables  x  y  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 canth2.1 . . 3  |-  A  e. 
_V
21pwex 4384 . . 3  |-  ~P A  e.  _V
3 snelpwi 4411 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2961 . . . . . . 7  |-  x  e. 
_V
54sneqr 3968 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3827 . . . . . 6  |-  ( x  =  y  ->  { x }  =  { y } )
75, 6impbii 182 . . . . 5  |-  ( { x }  =  {
y }  <->  x  =  y )
87a1i 11 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( { x }  =  { y }  <->  x  =  y ) )
93, 8dom3 7153 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 655 . 2  |-  A  ~<_  ~P A
111canth 6541 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5683 . . . . 5  |-  ( f : A -1-1-onto-> ~P A  ->  f : A -onto-> ~P A )
1311, 12mto 170 . . . 4  |-  -.  f : A -1-1-onto-> ~P A
1413nex 1565 . . 3  |-  -.  E. f  f : A -1-1-onto-> ~P A
15 bren 7119 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 292 . 2  |-  -.  A  ~~  ~P A
17 brsdom 7132 . 2  |-  ( A 
~<  ~P A  <->  ( A  ~<_  ~P A  /\  -.  A  ~~  ~P A ) )
1810, 16, 17mpbir2an 888 1  |-  A  ~<  ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958   ~Pcpw 3801   {csn 3816   class class class wbr 4214   -onto->wfo 5454   -1-1-onto->wf1o 5455    ~~ cen 7108    ~<_ cdom 7109    ~< csdm 7110
This theorem is referenced by:  canth2g  7263  r1sdom  7702  alephsucpw2  7994  dfac13  8024  pwsdompw  8086  numthcor  8376  alephexp1  8456  pwcfsdom  8460  cfpwsdom  8461  gchhar  8548  gchac  8550  inawinalem  8566  tskcard  8658  gruina  8695  grothac  8707  rpnnen  12828  rexpen  12829  rucALT  12831  rectbntr0  18865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-en 7112  df-dom 7113  df-sdom 7114
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