Theorem pwssb

Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006)

	Ref	Expression
Assertion	pwssb	$$⊢A\subseteq 𝒫B\leftrightarrow \forall x\in Ax\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		sspwuni	$$⊢A\subseteq 𝒫B\leftrightarrow \bigcup A\subseteq B$$
2		unissb	$$⊢\bigcup A\subseteq B\leftrightarrow \forall x\in Ax\subseteq B$$
3	1 2	bitri	$$⊢A\subseteq 𝒫B\leftrightarrow \forall x\in Ax\subseteq B$$