Subset in Subsets

Theorem
Let $S, B$ be sets.

Let $A$ be subset of $S$.

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

Sufficient Condition
Follows by definition of subset.

Necessary Condition
Let:
 * $\forall x \in S: x \in A \implies x \in B$

Let $x \in A$.

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

Thus by assumption:
 * $x \in B$

The result follows by definition of subset.