Special cases of prime avoidance theorem

Question

Let $\{p_i\}$ be a family of minimal prime ideals in a commutative ring $R$ with $1$, and let $I$ be a finitely generated ideal of $R$ such that $I\subseteq \cup p_i$. Can we deduce that there exist $i_1, i_2,..., i_n$ such that $I\subseteq p_{i_1}\cup p_{i_1}\cup ...\cup p_{i_n}$? And so $I$ is contained in one of them.