Subsets of Disjoint Sets are Disjoint

Theorem
Let $S$ and $T$ be disjoint sets.

Let $S' \subseteq S$ and $T' \subseteq T$.

Then $S'$ and $T'$ are disjoint.

Proof
Let $S \cap T = \O$.

Let $S' \subseteq S$ and $T' \subseteq T$.

$S' \cap T' \ne \O$.

Then:

From this contradiction:
 * $S' \cap T' = \O$

Hence the result by definition of disjoint sets.