Theorem ncanth 6753

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4932). Specifically, the identity function maps the universe onto its power class. Compare canth 6752 that works for sets. See also the remark in ru 3587 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 6317	. . 3 ⊢ I :V–1-1-onto→V
2		pwv 4572	. . . 4 ⊢ 𝒫 V = V
3		f1oeq3 6271	. . . 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 6286	. 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 1631  Vcvv 3351  𝒫 cpw 4298   I cid 5157  –onto→wfo 6030  –1-1-onto→wf1o 6031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039

This theorem is referenced by: (None)