Theorem ncanth 3899

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by nvelv 2708). Specifically, the identity function maps the universe onto its power class. Compare canth 3898 that works for sets. See also the remark in ru 1934 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 3709	. . 3 |- I:V-1-1-onto->V
2		pwv 2497	. . . 4 |- P~V = V
3		f1oeq3 3677	. . . 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 3686	. 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 954  Vcvv 1807  P~cpw 2397  Icid 2826  -onto->wfo 3175  -1-1-onto->wf1o 3176

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192

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