Theorem sspwuni

Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006)

	Ref	Expression
Assertion	sspwuni	$$⊢A\subseteq 𝒫B\leftrightarrow \bigcup A\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		velpw	$$⊢x\in 𝒫B\leftrightarrow x\subseteq B$$
2	1	ralbii	$$⊢\forall x\in Ax\in 𝒫B\leftrightarrow \forall x\in Ax\subseteq B$$
3		dfss3	$$⊢A\subseteq 𝒫B\leftrightarrow \forall x\in Ax\in 𝒫B$$
4		unissb	$$⊢\bigcup A\subseteq B\leftrightarrow \forall x\in Ax\subseteq B$$
5	2 3 4	3bitr4i	$$⊢A\subseteq 𝒫B\leftrightarrow \bigcup A\subseteq B$$