Theorem pwv

Description: The power class of the universe is the universe. Exercise 4.12(d) of Mendelson p. 235.

The collection of all classes is of course larger than _V , which is the collection of all sets. But ~PV , being a class, cannot contain proper classes, so ~P V is actually no larger than _V . This fact is exploited in ncanth . (Contributed by NM, 14-Sep-2003)

	Ref	Expression
Assertion	pwv	|- ~P _V = _V

Proof

Step	Hyp	Ref	Expression
1		ssv	|- x C_ _V
2		velpw	|- ( x e. ~P _V <-> x C_ _V )
3	1 2	mpbir	|- x e. ~P _V
4		vex	|- x e. _V
5	3 4	2th	|- ( x e. ~P _V <-> x e. _V )
6	5	eqriv	|- ~P _V = _V