Theorem elpwuni

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

	Ref	Expression
Assertion	elpwuni	$$⊢B\in A\to \left(A\subseteq 𝒫B\leftrightarrow \bigcup A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sspwuni	$$⊢A\subseteq 𝒫B\leftrightarrow \bigcup A\subseteq B$$
2		unissel	$$⊢\left(\bigcup A\subseteq B\wedge B\in A\right)\to \bigcup A=B$$
3	2	expcom	$$⊢B\in A\to \left(\bigcup A\subseteq B\to \bigcup A=B\right)$$
4		eqimss	$$⊢\bigcup A=B\to \bigcup A\subseteq B$$
5	3 4	impbid1	$$⊢B\in A\to \left(\bigcup A\subseteq B\leftrightarrow \bigcup A=B\right)$$
6	1 5	bitrid	$$⊢B\in A\to \left(A\subseteq 𝒫B\leftrightarrow \bigcup A=B\right)$$