Theorem elpwb

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

	Ref	Expression
Assertion	elpwb	$$⊢A\in 𝒫B\leftrightarrow \left(A\in \mathrm{V}\wedge A\subseteq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in 𝒫B\to A\in \mathrm{V}$$
2		elpwg	$$⊢A\in \mathrm{V}\to \left(A\in 𝒫B\leftrightarrow A\subseteq B\right)$$
3	1 2	biadanii	$$⊢A\in 𝒫B\leftrightarrow \left(A\in \mathrm{V}\wedge A\subseteq B\right)$$