Theorem ncanth 7375

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5274). Specifically, the identity function maps the universe onto its power class. Compare canth 7374 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3738): 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 3738 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 6865	. . 3 ⊢ I :V–1-1-onto→V
2		f1ofo 6832	. . 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 6794	. . 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 234	1 ⊢ I :V–onto→𝒫 V

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  Vcvv 3451  𝒫 cpw 4557   I cid 5545  –onto→wfo 6536  –1-1-onto→wf1o 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545

This theorem is used by: (None)