Theorem pwv 4433

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 3625	. . . 4 |- x C_ _V
2		selpw 4165	. . . 4 |- ( x e. ~P _V <-> x C_ _V )
3	1, 2	mpbir 221	. . 3 |- x e. ~P _V
4		vex 3203	. . 3 |- x e. _V
5	3, 4	2th 254	. 2 |- ( x e. ~P _V <-> x e. _V )
6	5	eqriv 2619	1 |- ~P _V = _V

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  univ  4919  ncanth  6609