Intersection with Complement is Empty iff Subset

Theorem

 * $$S \subseteq T \iff S \cap \complement \left({T}\right) = \varnothing$$

where:
 * $$S \subseteq T$$ denotes that $$S$$ is a subset of $$T$$;
 * $$S \cap T$$ denotes the intersection of $$S$$ and $$T$$;
 * $$\varnothing$$ denotes the empty set;
 * $$\complement$$ denotes set complement.

Proof
$$ $$ $$