Theorem ncanth 6563

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4756). Specifically, the identity function maps the universe onto its power class. Compare canth 6562 that works for sets. See also the remark in ru 3416 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 6132	. . 3 ⊢ I :V–1-1-onto→V
2		pwv 4401	. . . 4 ⊢ 𝒫 V = V
3		f1oeq3 6086	. . . 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 6101	. 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 1480  Vcvv 3186  𝒫 cpw 4130   I cid 4984  –onto→wfo 5845  –1-1-onto→wf1o 5846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854

This theorem is referenced by: (None)