Theorem elpwi2

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021) (Proof shortened by Wolf Lammen, 26-May-2024)

	Ref	Expression
Hypotheses	elpwi2.1	$$⊢B\in V$$
	elpwi2.2	$$⊢A\subseteq B$$
Assertion	elpwi2	$$⊢A\in 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		elpwi2.1	$$⊢B\in V$$
2		elpwi2.2	$$⊢A\subseteq B$$
3	1	elexi	$$⊢B\in \mathrm{V}$$
4	3	elpw2	$$⊢A\in 𝒫B\leftrightarrow A\subseteq B$$
5	2 4	mpbir	$$⊢A\in 𝒫B$$