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.
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.
$\blacksquare$