Theorem pwv 3600

Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
pwv	|- ~P _V = _V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 3019	. . . 4 |- x C_ _V
2		vex 2604	. . . . 5 |- x e. _V
3	2	elpw 3388	. . . 4 |- ( x e. ~P _V <-> x C_ _V )
4	1, 3	mpbir 144	. . 3 |- x e. ~P _V
5	4, 2	2th 172	. 2 |- ( x e. ~P _V <-> x e. _V )
6	5	eqriv 2078	1 |- ~P _V = _V

Syntax hints:   = wceq 1284   e. wcel 1433   _Vcvv 2601   C_ wss 2973   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  univ  4225