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

Theorem ncanth 6499
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4301). Specifically, the identity function maps the universe onto its power class. Compare canth 6498 that works for sets. See also the remark in ru 3120 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 5673 . . 3  |-  _I  : _V
-1-1-onto-> _V
2 pwv 3974 . . . 4  |-  ~P _V  =  _V
3 f1oeq3 5626 . . . 4  |-  ( ~P _V  =  _V  ->  (  _I  : _V -1-1-onto-> ~P _V  <->  _I  : _V -1-1-onto-> _V ) )
42, 3ax-mp 8 . . 3  |-  (  _I  : _V -1-1-onto-> ~P _V  <->  _I  : _V -1-1-onto-> _V )
51, 4mpbir 201 . 2  |-  _I  : _V
-1-1-onto-> ~P _V
6 f1ofo 5640 . 2  |-  (  _I  : _V -1-1-onto-> ~P _V  ->  _I  : _V -onto-> ~P _V )
75, 6ax-mp 8 1  |-  _I  : _V -onto-> ~P _V
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   _Vcvv 2916   ~Pcpw 3759    _I cid 4453   -onto->wfo 5411   -1-1-onto->wf1o 5412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
  Copyright terms: Public domain W3C validator