Theorem elpwgOLD

Description: Obsolete proof of elpwg as of 31-Dec-2023. (Contributed by NM, 6-Aug-2000) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		eleq1	$$⊢x=A\to \left(x\in 𝒫B\leftrightarrow A\in 𝒫B\right)$$
2		sseq1	$$⊢x=A\to \left(x\subseteq B\leftrightarrow A\subseteq B\right)$$
3		velpw	$$⊢x\in 𝒫B\leftrightarrow x\subseteq B$$
4	1 2 3	vtoclbg	$$⊢A\in V\to \left(A\in 𝒫B\leftrightarrow A\subseteq B\right)$$