Theorem ncanth 6881

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5034). Specifically, the identity function maps the universe onto its power class. Compare canth 6880 that works for sets. See also the remark in ru 3651 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 6429	. . 3 ⊢ I :V–1-1-onto→V
2		pwv 4668	. . . 4 ⊢ 𝒫 V = V
3		f1oeq3 6382	. . . 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 223	. 2 ⊢ I :V–1-1-onto→𝒫 V
6		f1ofo 6398	. 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 198   = wceq 1601  Vcvv 3398  𝒫 cpw 4379   I cid 5260  –onto→wfo 6133  –1-1-onto→wf1o 6134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142

This theorem is referenced by: (None)