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	$$⊢A\in \mathrm{V}$$
Assertion	elpw	$$⊢A\in 𝒫B\leftrightarrow A\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		elpw.1	$$⊢A\in \mathrm{V}$$
2		elpwg	$$⊢A\in \mathrm{V}\to \left(A\in 𝒫B\leftrightarrow A\subseteq B\right)$$
3	1 2	ax-mp	$$⊢A\in 𝒫B\leftrightarrow A\subseteq B$$