Theorem ncanth 7372

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5281). Specifically, the identity function maps the universe onto its power class. Compare canth 7371 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3741): 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 4867). See also the remark in ru 3741 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 6862	. . 3 ⊢ I :V–1-1-onto→V
2		f1ofo 6829	. . 3 ⊢ ( I :V–1-1-onto→V → I :V–onto→V)
3	1, 2	ax-mp 5	. 2 ⊢ I :V–onto→V
4		pwv 4867	. . 3 ⊢ 𝒫 V = V
5		foeq3 6791	. . 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 3453  𝒫 cpw 4560   I cid 5553  –onto→wfo 6535  –1-1-onto→wf1o 6536

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 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544

This theorem is used by: (None)