Theorem ncanth 7366

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5285). Specifically, the identity function maps the universe onto its power class. Compare canth 7365 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3752): 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 4873). See also the remark in ru 3752 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 4873	. . 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

Syntax hints:   ↔ wb 209   = wceq 1567  Vcvv 3463  𝒫 cpw 4567   I cid 5556  –onto→wfo 6535  –1-1-onto→wf1o 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544

This theorem is referenced by: (None)