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

Theorem ncanth 6288
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4153). Specifically, the identity function maps the universe onto its power class. Compare canth 6287 that works for sets. See also the remark in ru 2991 about NF, in which Cantor's theorem fails for sets that are "too large." This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.)
Assertion
Ref Expression
ncanth  |-  _I  : _V -onto-> ~P _V

Proof of Theorem ncanth
StepHypRef Expression
1 f1ovi 5477 . . 3  |-  _I  : _V
-1-1-onto-> _V
2 pwv 3827 . . . 4  |-  ~P _V  =  _V
3 f1oeq3 5430 . . . 4  |-  ( ~P _V  =  _V  ->  (  _I  : _V -1-1-onto-> ~P _V  <->  _I  : _V -1-1-onto-> _V ) )
42, 3ax-mp 10 . . 3  |-  (  _I  : _V -1-1-onto-> ~P _V  <->  _I  : _V -1-1-onto-> _V )
51, 4mpbir 202 . 2  |-  _I  : _V
-1-1-onto-> ~P _V
6 f1ofo 5444 . 2  |-  (  _I  : _V -1-1-onto-> ~P _V  ->  _I  : _V -onto-> ~P _V )
75, 6ax-mp 10 1  |-  _I  : _V -onto-> ~P _V
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1624   _Vcvv 2789   ~Pcpw 3626    _I cid 4303   -onto->wfo 5219   -1-1-onto->wf1o 5220
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228
  Copyright terms: Public domain W3C validator