Subset in Subsets

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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:

$x \in B$

$\blacksquare$