Taken from Miles Reid "Undergraduate Commutative Algebra" p.35 ex. 1.12 b)
Let $I,J_1,J_2 \subset A$ be ideals of a commutative ring $A$. Let $P$ be a prime ideal, then if $I \subset J_1 \cup J_2 \cup P$ then $I \subset J_1$ or $J_2$ or $P$.
I've handled the case with two ideals, but I can't manage to generalize the reasoning. Moreover, I don't know how to use the hypothesis of P prime ideal. Can someone provide me some advices?