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Theorem ncanth 6526
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4328). Specifically, the identity function maps the universe onto its power class. Compare canth 6525 that works for sets. See also the remark in ru 3147 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 5700 . . 3  |-  _I  : _V
-1-1-onto-> _V
2 pwv 4001 . . . 4  |-  ~P _V  =  _V
3 f1oeq3 5653 . . . 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 5667 . 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 1652   _Vcvv 2943   ~Pcpw 3786    _I cid 4480   -onto->wfo 5438   -1-1-onto->wf1o 5439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-br 4200  df-opab 4254  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447
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