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Theorem ncanth 6487
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4719). Specifically, the identity function maps the universe onto its power class. Compare canth 6486 that works for sets. See also the remark in ru 3400 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 6072 . . 3 I :V–1-1-onto→V
2 pwv 4365 . . . 4 𝒫 V = V
3 f1oeq3 6027 . . . 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 219 . 2 I :V–1-1-onto→𝒫 V
6 f1ofo 6042 . 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 194   = wceq 1474  Vcvv 3172  𝒫 cpw 4107   I cid 4938  ontowfo 5788  1-1-ontowf1o 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797
This theorem is referenced by: (None)
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