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Theorem ncanth 6509
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4724). Specifically, the identity function maps the universe onto its power class. Compare canth 6508 that works for sets. See also the remark in ru 3401 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 6087 . . 3 I :V–1-1-onto→V
2 pwv 4371 . . . 4 𝒫 V = V
3 f1oeq3 6042 . . . 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 220 . 2 I :V–1-1-onto→𝒫 V
6 f1ofo 6057 . 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 195   = wceq 1475  Vcvv 3173  𝒫 cpw 4108   I cid 4948  ontowfo 5802  1-1-ontowf1o 5803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811
This theorem is referenced by: (None)
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