Theorem pwv 3579

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 2965	. . . 4 |- x C_ _V
2		vex 2560	. . . . 5 |- x e. _V
3	2	elpw 3365	. . . 4 |- ( x e. ~P _V <-> x C_ _V )
4	1, 3	mpbir 134	. . 3 |- x e. ~P _V
5	4, 2	2th 163	. 2 |- ( x e. ~P _V <-> x e. _V )
6	5	eqriv 2037	1 |- ~P _V = _V

Syntax hints:   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  univ  4207