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Theorem ncanth 6295
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4152). Specifically, the identity function maps the universe onto its power class. Compare canth 6294 that works for sets. See also the remark in ru 2990 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 5512 . . 3  |-  _I  : _V
-1-1-onto-> _V
2 pwv 3826 . . . 4  |-  ~P _V  =  _V
3 f1oeq3 5465 . . . 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 200 . 2  |-  _I  : _V
-1-1-onto-> ~P _V
6 f1ofo 5479 . 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 176    = wceq 1623   _Vcvv 2788   ~Pcpw 3625    _I cid 4304   -onto->wfo 5253   -1-1-onto->wf1o 5254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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