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Theorem canth2 7014
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 6294. (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 4193 . . 3  |-  ~P A  e.  _V
3 snelpwi 4220 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2791 . . . . . . 7  |-  x  e. 
_V
54sneqr 3780 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3651 . . . . . 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 6905 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 653 . 2  |-  A  ~<_  ~P A
111canth 6294 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5479 . . . . 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 1542 . . 3  |-  -.  E. f  f : A -1-1-onto-> ~P A
15 bren 6871 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 290 . 2  |-  -.  A  ~~  ~P A
17 brsdom 6884 . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   {csn 3640   class class class wbr 4023   -onto->wfo 5253   -1-1-onto->wf1o 5254    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  canth2g  7015  r1sdom  7446  alephsucpw2  7738  dfac13  7768  pwsdompw  7830  numthcor  8121  alephexp1  8201  pwcfsdom  8205  cfpwsdom  8206  gchhar  8293  gchac  8295  inawinalem  8311  tskcard  8403  gruina  8440  grothac  8452  rpnnen  12505  rexpen  12506  rucALT  12508  rectbntr0  18337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-en 6864  df-dom 6865  df-sdom 6866
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