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

Theorem canth 6235
 Description: No set is equinumerous to its power set (Cantor's theorem), i.e. no function can map it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 7662. Note that must be a set: this theorem does not hold when is too large to be a set; see ncanth 6236 for a counterexample. (Use nex 1605 if you want the form .) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
Hypothesis
Ref Expression
canth.1
Assertion
Ref Expression
canth

Proof of Theorem canth
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3580 . . . 4
2 canth.1 . . . . 5
32elpw2 4606 . . . 4
41, 3mpbir 209 . . 3
5 forn 5791 . . 3
64, 5syl5eleqr 2557 . 2
7 id 22 . . . . . . . . . 10
8 fveq2 5859 . . . . . . . . . 10
97, 8eleq12d 2544 . . . . . . . . 9
109notbid 294 . . . . . . . 8
1110elrab 3256 . . . . . . 7
1211baibr 899 . . . . . 6
13 nbbn 358 . . . . . 6
1412, 13sylib 196 . . . . 5
15 eleq2 2535 . . . . 5
1614, 15nsyl 121 . . . 4
1716nrex 2914 . . 3
18 fofn 5790 . . . 4
19 fvelrnb 5908 . . . 4
2018, 19syl 16 . . 3
2117, 20mtbiri 303 . 2
226, 21pm2.65i 173 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 184   wceq 1374   wcel 1762  wrex 2810  crab 2813  cvv 3108   wss 3471  cpw 4005   crn 4995   wfn 5576  wfo 5579  cfv 5581 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589 This theorem is referenced by:  canth2  7662  canthwdom  7996
 Copyright terms: Public domain W3C validator