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Theorem canth2 7030
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 6310. (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 4209 . . 3  |-  ~P A  e.  _V
3 snelpwi 4236 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2804 . . . . . . 7  |-  x  e. 
_V
54sneqr 3796 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3664 . . . . . 6  |-  ( x  =  y  ->  { x }  =  { y } )
75, 6impbii 180 . . . . 5  |-  ( { x }  =  {
y }  <->  x  =  y )
87a1i 10 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( { x }  =  { y }  <->  x  =  y ) )
93, 8dom3 6921 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 653 . 2  |-  A  ~<_  ~P A
111canth 6310 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5495 . . . . 5  |-  ( f : A -1-1-onto-> ~P A  ->  f : A -onto-> ~P A )
1311, 12mto 167 . . . 4  |-  -.  f : A -1-1-onto-> ~P A
1413nex 1545 . . 3  |-  -.  E. f  f : A -1-1-onto-> ~P A
15 bren 6887 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 290 . 2  |-  -.  A  ~~  ~P A
17 brsdom 6900 . 2  |-  ( A 
~<  ~P A  <->  ( A  ~<_  ~P A  /\  -.  A  ~~  ~P A ) )
1810, 16, 17mpbir2an 886 1  |-  A  ~<  ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   ~Pcpw 3638   {csn 3653   class class class wbr 4039   -onto->wfo 5269   -1-1-onto->wf1o 5270    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  canth2g  7031  r1sdom  7462  alephsucpw2  7754  dfac13  7784  pwsdompw  7846  numthcor  8137  alephexp1  8217  pwcfsdom  8221  cfpwsdom  8222  gchhar  8309  gchac  8311  inawinalem  8327  tskcard  8419  gruina  8456  grothac  8468  rpnnen  12521  rexpen  12522  rucALT  12524  rectbntr0  18353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-en 6880  df-dom 6881  df-sdom 6882
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