Theorem elpwb

Description: Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021)

	Ref	Expression
Assertion	elpwb	⊢ ( 𝐴 ∈ 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V )
2		elpwg	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
3	1 2	biadanii	⊢ ( 𝐴 ∈ 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵 ) )