Choose an element $\displaystyle \begin{align*} x \end{align*}$. If $\displaystyle \begin{align*} x \in A \end{align*}$ and $\displaystyle \begin{align*} A \subset \mathcal{I} \end{align*}$ then $\displaystyle \begin{align*} x \in \mathcal{I} \end{align*}$.

But if $\displaystyle \begin{align*} x \in B \end{align*}$ and $\displaystyle \begin{align*} B \subset \mathcal{I} \end{align*}$, then $\displaystyle \begin{align*} x \in \mathcal{I} \end{align*}$.

Since x is in both A and B, that means $\displaystyle \begin{align*} x \in A \cap B \end{align*}$, and since we know $\displaystyle \begin{align*} x \in \mathcal{I} \end{align*}$, that means $\displaystyle \begin{align*} A \cap B \subset \mathcal{I} \end{align*}$.