Theorem ncanth 6609

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4796). Specifically, the identity function maps the universe onto its power class. Compare canth 6608 that works for sets. See also the remark in ru 3434 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 6175	. . 3 ⊢ I :V–1-1-onto→V
2		pwv 4433	. . . 4 ⊢ 𝒫 V = V
3		f1oeq3 6129	. . . 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 6144	. 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 1483  Vcvv 3200  𝒫 cpw 4158   I cid 5023  –onto→wfo 5886  –1-1-onto→wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by: (None)