Theorem ncanth 6760

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4936). Specifically, the identity function maps the universe onto its power class. Compare canth 6759 that works for sets. See also the remark in ru 3563 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.)

Assertion

Ref	Expression
ncanth	⊢ I :V–onto→𝒫 V

Proof of Theorem ncanth

Step	Hyp	Ref	Expression
1		f1ovi 6324	. . 3 ⊢ I :V–1-1-onto→V
2		pwv 4573	. . . 4 ⊢ 𝒫 V = V
3		f1oeq3 6278	. . . 4 ⊢ (𝒫 V = V → ( I :V–1-1-onto→𝒫 V ↔ I :V–1-1-onto→V))
4	2, 3	ax-mp 5	. . 3 ⊢ ( I :V–1-1-onto→𝒫 V ↔ I :V–1-1-onto→V)
5	1, 4	mpbir 221	. 2 ⊢ I :V–1-1-onto→𝒫 V
6		f1ofo 6293	. 2 ⊢ ( I :V–1-1-onto→𝒫 V → I :V–onto→𝒫 V)
7	5, 6	ax-mp 5	1 ⊢ I :V–onto→𝒫 V

Syntax hints:   ↔ wb 196   = wceq 1620  Vcvv 3328  𝒫 cpw 4290   I cid 5161  –onto→wfo 6035  –1-1-onto→wf1o 6036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044

This theorem is referenced by: (None)