Subset in Subsets

Theorem
Let $S, B$ be sets.

Let $A$ be subset of $S$.

Then $A \subseteq B \iff \forall x \in S: x \in A \implies x \in B$

Proof
Sufficient condition follows by definition of subset.

For necessary condition assume that
 * $\forall x \in S: x \in A \implies x \in B$

Let $x \in A$.

By definition of subset:
 * $x \in S$

Thus by assumption:
 * $x \in B$