Metamath Proof Explorer


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 𝒫V=V

Proof

Step Hyp Ref Expression
1 ssv xV
2 velpw x𝒫VxV
3 1 2 mpbir x𝒫V
4 vex xV
5 3 4 2th x𝒫VxV
6 5 eqriv 𝒫V=V