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Theorem ncanth 6255
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4590). Specifically, the identity function maps the universe onto its power class. Compare canth 6254 that works for sets. See also the remark in ru 3326 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

Proof of Theorem ncanth
StepHypRef Expression
1 f1ovi 5857 . . 3
2 pwv 4248 . . . 4
3 f1oeq3 5814 . . . 4
42, 3ax-mp 5 . . 3
51, 4mpbir 209 . 2
6 f1ofo 5828 . 2
75, 6ax-mp 5 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  =wceq 1395   cvv 3109  ~Pcpw 4012   cid 4795  -onto->wfo 5591  -1-1-onto->wf1o 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600
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