Theorem pwv 3795

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 3169	. . . 4 |- x C_ _V
2		vex 2733	. . . . 5 |- x e. _V
3	2	elpw 3572	. . . 4 |- ( x e. ~P _V <-> x C_ _V )
4	1, 3	mpbir 145	. . 3 |- x e. ~P _V
5	4, 2	2th 173	. 2 |- ( x e. ~P _V <-> x e. _V )
6	5	eqriv 2167	1 |- ~P _V = _V

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  univ  4461