Theorem elpw

Description: Membership in a power class. Theorem 86 of Suppes p. 47. (Contributed by NM, 31-Dec-1993) (Proof shortened by BJ, 31-Dec-2023)

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

Proof

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