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Theorem ncanth 6179
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4049). Specifically, the identity function maps the universe onto its power class. Compare canth 6178 that works for sets. See also the remark in ru 2920 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 5369 . . 3  |-  _I  : _V
-1-1-onto-> _V
2 pwv 3726 . . . 4  |-  ~P _V  =  _V
3 f1oeq3 5322 . . . 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 5336 . 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 1619   _Vcvv 2727   ~Pcpw 3530    _I cid 4197   -onto->wfo 4590   -1-1-onto->wf1o 4591
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607
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