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Theorem ncanth 6649
 Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4829). Specifically, the identity function maps the universe onto its power class. Compare canth 6648 that works for sets. See also the remark in ru 3467 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 6213 . . 3 I :V–1-1-onto→V
2 pwv 4465 . . . 4 𝒫 V = V
3 f1oeq3 6167 . . . 4 (𝒫 V = V → ( I :V–1-1-onto→𝒫 V ↔ I :V–1-1-onto→V))
42, 3ax-mp 5 . . 3 ( I :V–1-1-onto→𝒫 V ↔ I :V–1-1-onto→V)
51, 4mpbir 221 . 2 I :V–1-1-onto→𝒫 V
6 f1ofo 6182 . 2 ( I :V–1-1-onto→𝒫 V → I :V–onto→𝒫 V)
75, 6ax-mp 5 1 I :V–onto→𝒫 V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1523  Vcvv 3231  𝒫 cpw 4191   I cid 5052  –onto→wfo 5924  –1-1-onto→wf1o 5925 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933 This theorem is referenced by: (None)
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