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.

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.

$\blacksquare$