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Theorem canth2 7219
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 6498. (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 4342 . . 3  |-  ~P A  e.  _V
3 snelpwi 4369 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2919 . . . . . . 7  |-  x  e. 
_V
54sneqr 3926 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3785 . . . . . 6  |-  ( x  =  y  ->  { x }  =  { y } )
75, 6impbii 181 . . . . 5  |-  ( { x }  =  {
y }  <->  x  =  y )
87a1i 11 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( { x }  =  { y }  <->  x  =  y ) )
93, 8dom3 7110 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 654 . 2  |-  A  ~<_  ~P A
111canth 6498 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5640 . . . . 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 1561 . . 3  |-  -.  E. f  f : A -1-1-onto-> ~P A
15 bren 7076 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 291 . 2  |-  -.  A  ~~  ~P A
17 brsdom 7089 . 2  |-  ( A 
~<  ~P A  <->  ( A  ~<_  ~P A  /\  -.  A  ~~  ~P A ) )
1810, 16, 17mpbir2an 887 1  |-  A  ~<  ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916   ~Pcpw 3759   {csn 3774   class class class wbr 4172   -onto->wfo 5411   -1-1-onto->wf1o 5412    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067
This theorem is referenced by:  canth2g  7220  r1sdom  7656  alephsucpw2  7948  dfac13  7978  pwsdompw  8040  numthcor  8330  alephexp1  8410  pwcfsdom  8414  cfpwsdom  8415  gchhar  8502  gchac  8504  inawinalem  8520  tskcard  8612  gruina  8649  grothac  8661  rpnnen  12781  rexpen  12782  rucALT  12784  rectbntr0  18816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-en 7069  df-dom 7070  df-sdom 7071
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