Theorem ncanth 5113

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 3449). Specifically, the identity function maps the universe onto its power class. Compare canth 5112 that works for sets. See also the remark in ru 2451 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 4673	. . 3 |- _I :_V-1-1-onto->_V
2		pwv 3176	. . . 4 |- ~P_V = _V
3		f1oeq3 4632	. . . 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 207	. 2 |- _I :_V-1-1-onto->~P_V
6		f1ofo 4643	. 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 163   = wceq 1298  _Vcvv 2292  ~Pcpw 3032   _I cid 3582  -onto->wfo 3996  -1-1-onto->wf1o 3997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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