Subset of Set Difference iff Disjoint Set

Theorem
Let $S, T$ be sets.

Let $A \subseteq S$

Then:
 * $A \cap T = \O \iff A \subseteq S \setminus T$

where:
 * $A \cap T$ denotes set intersection
 * $\O$ denotes the empty set
 * $S \setminus T$ denotes set difference.

Proof
We have:

Then: