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 ⊆ V
2		velpw	⊢ ( 𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V )
3	1 2	mpbir	⊢ 𝑥 ∈ 𝒫 V
4		vex	⊢ 𝑥 ∈ V
5	3 4	2th	⊢ ( 𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V )
6	5	eqriv	⊢ 𝒫 V = V