Theorem elpwg

Description: Membership in a power class. Theorem 86 of Suppes p. 47. See also elpw2g . (Contributed by NM, 6-Aug-2000) (Proof shortened by BJ, 31-Dec-2023)

	Ref	Expression
Assertion	elpwg	$$⊢A\in V\to \left(A\in 𝒫B\leftrightarrow A\subseteq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sseq1	$$⊢x=y\to \left(x\subseteq B\leftrightarrow y\subseteq B\right)$$
2		sseq1	$$⊢y=A\to \left(y\subseteq B\leftrightarrow A\subseteq B\right)$$
3		df-pw	$$⊢𝒫B=\left\{x|x\subseteq B\right\}$$
4	1 2 3	elab2gw	$$⊢A\in V\to \left(A\in 𝒫B\leftrightarrow A\subseteq B\right)$$