Theorem ncanth 7300

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5267). Specifically, the identity function maps the universe onto its power class. Compare canth 7299 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3733): 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 4857). See also the remark in ru 3733 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 6815	. . 3 ⊢ I :V–1-1-onto→V
2		f1ofo 6783	. . 3 ⊢ ( I :V–1-1-onto→V → I :V–onto→V)
3	1, 2	ax-mp 5	. 2 ⊢ I :V–onto→V
4		pwv 4857	. . 3 ⊢ 𝒫 V = V
5		foeq3 6746	. . 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 230	1 ⊢ I :V–onto→𝒫 V

Syntax hints:   ↔ wb 205   = wceq 1541  Vcvv 3443  𝒫 cpw 4555   I cid 5524  –onto→wfo 6486  –1-1-onto→wf1o 6487

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2708  ax-sep 5251  ax-nul 5258  ax-pr 5379

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-br 5101  df-opab 5163  df-id 5525  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495

This theorem is referenced by: (None)