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Theorem canth2 7690
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 6254. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)
Hypothesis
Ref Expression
canth2.1
Assertion
Ref Expression
canth2

Proof of Theorem canth2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 canth2.1 . . 3
21pwex 4635 . . 3
3 snelpwi 4697 . . . 4
4 vex 3112 . . . . . . 7
54sneqr 4197 . . . . . 6
6 sneq 4039 . . . . . 6
75, 6impbii 188 . . . . 5
87a1i 11 . . . 4
93, 8dom3 7579 . . 3
101, 2, 9mp2an 672 . 2
111canth 6254 . . . . 5
12 f1ofo 5828 . . . . 5
1311, 12mto 176 . . . 4
1413nex 1627 . . 3
15 bren 7545 . . 3
1614, 15mtbir 299 . 2
17 brsdom 7558 . 2
1810, 16, 17mpbir2an 920 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  ~Pcpw 4012  {csn 4029   class class class wbr 4452  -onto->wfo 5591  -1-1-onto->wf1o 5592   cen 7533   cdom 7534   csdm 7535
This theorem is referenced by:  canth2g  7691  r1sdom  8213  alephsucpw2  8513  dfac13  8543  pwsdompw  8605  numthcor  8895  alephexp1  8975  pwcfsdom  8979  cfpwsdom  8980  gchhar  9078  gchac  9080  inawinalem  9088  tskcard  9180  gruina  9217  grothac  9229  rpnnen  13960  rexpen  13961  rucALT  13963  rectbntr0  21337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-en 7537  df-dom 7538  df-sdom 7539
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