Theorem ncanth 7353

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5272). Specifically, the identity function maps the universe onto its power class. Compare canth 7352 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3745): 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 4864). See also the remark in ru 3745 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

Step	Hyp	Ref	Expression
1		f1ovi 6849	. . 3 ⊢ I :V–1-1-onto→V
2		f1ofo 6816	. . 3 ⊢ ( I :V–1-1-onto→V → I :V–onto→V)
3	1, 2	ax-mp 5	. 2 ⊢ I :V–onto→V
4		pwv 4864	. . 3 ⊢ 𝒫 V = V
5		foeq3 6778	. . 3 ⊢ (𝒫 V = V → ( I :V–onto→𝒫 V ↔ I :V–onto→V))
6	4, 5	ax-mp 5	. 2 ⊢ ( I :V–onto→𝒫 V ↔ I :V–onto→V)
7	3, 6	mpbir 233	1 ⊢ I :V–onto→𝒫 V

Syntax hints:   ↔ wb 208   = wceq 1562  Vcvv 3456  𝒫 cpw 4557   I cid 5543  –onto→wfo 6521  –1-1-onto→wf1o 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530

This theorem is referenced by: (None)