Theorem elpw2

Description: Membership in a power class. Theorem 86 of Suppes p. 47. (Contributed by NM, 11-Oct-2007)

	Ref	Expression
Hypothesis	elpw2.1	⊢ 𝐵 ∈ V
Assertion	elpw2	⊢ ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elpw2.1	⊢ 𝐵 ∈ V
2		elpw2g	⊢ ( 𝐵 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 )