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Theorem ncanth 7105
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5205). Specifically, the identity function maps the universe onto its power class. Compare canth 7104 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3757): 𝒫 V, being a class, cannot contain proper classes, so it is no larger than V, which is why the identity function "succeeds" in being surjective onto 𝒫 V (see pwv 4821). See also the remark in ru 3757 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.) (Proof shortened by BJ, 29-Dec-2023.)

Assertion
Ref Expression
ncanth I :V–onto→𝒫 V

Proof of Theorem ncanth
StepHypRef Expression
1 f1ovi 6644 . . 3 I :V–1-1-onto→V
2 f1ofo 6613 . . 3 ( I :V–1-1-onto→V → I :V–onto→V)
31, 2ax-mp 5 . 2 I :V–onto→V
4 pwv 4821 . . 3 𝒫 V = V
5 foeq3 6579 . . 3 (𝒫 V = V → ( I :V–onto→𝒫 V ↔ I :V–onto→V))
64, 5ax-mp 5 . 2 ( I :V–onto→𝒫 V ↔ I :V–onto→V)
73, 6mpbir 234 1 I :V–onto→𝒫 V
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  Vcvv 3480  𝒫 cpw 4522   I cid 5446  ontowfo 6341  1-1-ontowf1o 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350
This theorem is referenced by: (None)
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