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Theorem canth2 6899
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 6178. (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
StepHypRef Expression
1 canth2.1 . . 3  |-  A  e. 
_V
21pwex 4087 . . 3  |-  ~P A  e.  _V
3 snelpwi 4114 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2730 . . . . . . 7  |-  x  e. 
_V
54sneqr 3680 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3555 . . . . . 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 6791 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 656 . 2  |-  A  ~<_  ~P A
111canth 6178 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5336 . . . . 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 1587 . . 3  |-  -.  E. f  f : A -1-1-onto-> ~P A
15 bren 6757 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 292 . 2  |-  -.  A  ~~  ~P A
17 brsdom 6770 . 2  |-  ( A 
~<  ~P A  <->  ( A  ~<_  ~P A  /\  -.  A  ~~  ~P A ) )
1810, 16, 17mpbir2an 891 1  |-  A  ~<  ~P A
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2727   ~Pcpw 3530   {csn 3544   class class class wbr 3920   -onto->wfo 4590   -1-1-onto->wf1o 4591    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748
This theorem is referenced by:  canth2g  6900  r1sdom  7330  alephsucpw2  7622  dfac13  7652  pwsdompw  7714  numthcor  8005  alephexp1  8081  pwcfsdom  8085  cfpwsdom  8086  gchhar  8173  gchac  8175  inawinalem  8191  tskcard  8283  gruina  8320  grothac  8332  rpnnen  12379  rexpen  12380  rucALT  12382  rectbntr0  18169
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-en 6750  df-dom 6751  df-sdom 6752
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