Theorem unissel

Description: Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006)

	Ref	Expression
Assertion	unissel	$$⊢\left(\bigcup A\subseteq B\wedge B\in A\right)\to \bigcup A=B$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(\bigcup A\subseteq B\wedge B\in A\right)\to \bigcup A\subseteq B$$
2		elssuni	$$⊢B\in A\to B\subseteq \bigcup A$$
3	2	adantl	$$⊢\left(\bigcup A\subseteq B\wedge B\in A\right)\to B\subseteq \bigcup A$$
4	1 3	eqssd	$$⊢\left(\bigcup A\subseteq B\wedge B\in A\right)\to \bigcup A=B$$