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Theorem canth2 6982
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 6260. (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 4165 . . 3  |-  ~P A  e.  _V
3 snelpwi 4192 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2766 . . . . . . 7  |-  x  e. 
_V
54sneqr 3754 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3625 . . . . . 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 6873 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 656 . 2  |-  A  ~<_  ~P A
111canth 6260 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5417 . . . . 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 6839 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 292 . 2  |-  -.  A  ~~  ~P A
17 brsdom 6852 . 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 2763   ~Pcpw 3599   {csn 3614   class class class wbr 3997   -onto->wfo 4671   -1-1-onto->wf1o 4672    ~~ cen 6828    ~<_ cdom 6829    ~< csdm 6830
This theorem is referenced by:  canth2g  6983  r1sdom  7414  alephsucpw2  7706  dfac13  7736  pwsdompw  7798  numthcor  8089  alephexp1  8169  pwcfsdom  8173  cfpwsdom  8174  gchhar  8261  gchac  8263  inawinalem  8279  tskcard  8371  gruina  8408  grothac  8420  rpnnen  12467  rexpen  12468  rucALT  12470  rectbntr0  18299
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-en 6832  df-dom 6833  df-sdom 6834
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