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Theorem canth2 7009
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 6287. (Contributed by NM, 7-Aug-1994.)
Hypothesis
Ref Expression
canth2.1  |-  A  e. 
_V
Assertion
Ref Expression
canth2  |-  A  ~<  ~P A
Dummy variables  x  y  f are mutually distinct and distinct from all other variables.

Proof of Theorem canth2
StepHypRef Expression
1 canth2.1 . . 3  |-  A  e. 
_V
21pwex 4192 . . 3  |-  ~P A  e.  _V
3 snelpwi 4219 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2792 . . . . . . 7  |-  x  e. 
_V
54sneqr 3781 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3652 . . . . . 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 6900 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 655 . 2  |-  A  ~<_  ~P A
111canth 6287 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5444 . . . . 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 1543 . . 3  |-  -.  E. f  f : A -1-1-onto-> ~P A
15 bren 6866 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 292 . 2  |-  -.  A  ~~  ~P A
17 brsdom 6879 . 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 5    <-> wb 178    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   _Vcvv 2789   ~Pcpw 3626   {csn 3641   class class class wbr 4024   -onto->wfo 5219   -1-1-onto->wf1o 5220    ~~ cen 6855    ~<_ cdom 6856    ~< csdm 6857
This theorem is referenced by:  canth2g  7010  r1sdom  7441  alephsucpw2  7733  dfac13  7763  pwsdompw  7825  numthcor  8116  alephexp1  8196  pwcfsdom  8200  cfpwsdom  8201  gchhar  8288  gchac  8290  inawinalem  8306  tskcard  8398  gruina  8435  grothac  8447  rpnnen  12499  rexpen  12500  rucALT  12502  rectbntr0  18331
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-en 6859  df-dom 6860  df-sdom 6861
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