Theorem ncanth 3893

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by nvelv 2703). Specifically, the identity function maps the universe onto its power class. Compare canth 3892 that works for sets. See also the remark in ru 1928 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 3703	. . 3 |- I:V-1-1-onto->V
2		pwv 2492	. . . 4 |- P~V = V
3		f1oeq3 3671	. . . 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 190	. 2 |- I:V-1-1-onto->P~V
6		f1ofo 3680	. 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 146   = wceq 953  Vcvv 1802  P~cpw 2391  Icid 2820  -onto->wfo 3170  -1-1-onto->wf1o 3171

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187

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