MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  canth2 Unicode version

Theorem canth2 6896
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 6175. (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 4084 . . 3  |-  ~P A  e.  _V
3 snelpwi 4111 . . . 4  |-  ( x  e.  A  ->  { x }  e.  ~P A
)
4 vex 2728 . . . . . . 7  |-  x  e. 
_V
54sneqr 3677 . . . . . 6  |-  ( { x }  =  {
y }  ->  x  =  y )
6 sneq 3552 . . . . . 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 6788 . . 3  |-  ( ( A  e.  _V  /\  ~P A  e.  _V )  ->  A  ~<_  ~P A
)
101, 2, 9mp2an 656 . 2  |-  A  ~<_  ~P A
111canth 6175 . . . . 5  |-  -.  f : A -onto-> ~P A
12 f1ofo 5333 . . . . 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 6754 . . 3  |-  ( A 
~~  ~P A  <->  E. f 
f : A -1-1-onto-> ~P A
)
1614, 15mtbir 292 . 2  |-  -.  A  ~~  ~P A
17 brsdom 6767 . 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 2725   ~Pcpw 3527   {csn 3541   class class class wbr 3917   -onto->wfo 4587   -1-1-onto->wf1o 4588    ~~ cen 6743    ~<_ cdom 6744    ~< csdm 6745
This theorem is referenced by:  canth2g  6897  r1sdom  7327  alephsucpw2  7619  dfac13  7649  pwsdompw  7711  numthcor  8002  alephexp1  8078  pwcfsdom  8082  cfpwsdom  8083  gchhar  8170  gchac  8172  inawinalem  8188  tskcard  8280  gruina  8317  grothac  8329  rpnnen  12341  rexpen  12342  rucALT  12344  rectbntr0  18131
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 4035  ax-nul 4043  ax-pow 4079  ax-pr 4105  ax-un 4400
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 2511  df-rex 2512  df-rab 2514  df-v 2727  df-sbc 2920  df-csb 3007  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-nul 3360  df-if 3468  df-pw 3529  df-sn 3547  df-pr 3548  df-op 3550  df-uni 3725  df-br 3918  df-opab 3972  df-mpt 3973  df-id 4199  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 4599  df-fn 4600  df-f 4601  df-f1 4602  df-fo 4603  df-f1o 4604  df-fv 4605  df-en 6747  df-dom 6748  df-sdom 6749
  Copyright terms: Public domain W3C validator