Multiplicatively Closed Subset is Saturated iff Complement is Union of Prime Ideals

Definition
Let $A$ be a commutative ring with unity.

Let $S \subseteq A$ be a multiplicatively closed subset.

Then $S$ is saturated :
 * $\ds \exists \TT \subseteq \Spec A : \relcomp A S = \bigcup_{\mathfrak p \mathop \in \TT} \mathfrak p$

where:
 * $\Spec A$ denotes the prime spectrum of $A$
 * $\relcomp A S$ denotes the complement of $S$