Theorem ncanth 4206

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 2787). Specifically, the identity function maps the universe onto its power class. Compare canth 4205 that works for sets. See also the remark in ru 1984 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.

Assertion

Ref	Expression
ncanth	|- I:V-onto->P~V

Proof of Theorem ncanth

Step	Hyp	Ref	Expression
1		f1ovi 3829	. . 3 |- I:V-1-1-onto->V
2		pwv 2568	. . . 4 |- P~V = V
3		f1oeq3 3794	. . . 4 |- (P~V = V -> (I:V-1-1-onto->P~V <-> I:V-1-1-onto->V))
4	2, 3	ax-mp 7	. . 3 |- (I:V-1-1-onto->P~V <-> I:V-1-1-onto->V)
5	1, 4	mpbir 188	. 2 |- I:V-1-1-onto->P~V
6		f1ofo 3803	. 2 |- (I:V-1-1-onto->P~V -> I:V-onto->P~V)
7	5, 6	ax-mp 7	1 |- I:V-onto->P~V

Syntax hints:   <-> wb 144   = wceq 992  Vcvv 1857  P~cpw 2458  Icid 2909  -onto->wfo 3261  -1-1-onto->wf1o 3262

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278

Copyright terms: Public domain