The intersection is contained in each set

Statement

Let $A$ and $B$ be sets. Prove that $A \cap B \subseteq A$.

Solution

By the definition of inclusion we must show that every element of $A \cap B$ belongs to $A$.

Let $x \in A \cap B$. By the definition of intersection, $x \in A$ and $x \in B$. In particular $x \in A$.

Since $x$ was an arbitrary element of $A \cap B$, we have shown that every element of $A \cap B$ lies in $A$, i.e. $A \cap B \subseteq A$. $\blacksquare$