A property of minimal prime ideals

Question

Let $R$ be a commutative ring with $1$, and let $\frak{p}$ be a minimal prime ideal of $R$. If $\mathfrak{p}\subseteq I_1\cap I_2$, where $I_1$ and $I_2$ are two ideals of $R$, can we deduce that $ \mathfrak{p}\subseteq I_1 $ or $ \mathfrak{p}\subseteq I_2 $?

Answer 1

It's obvious, and $\mathfrak p$ doesn't have to be prime for that: $I_1\cap I_2\subset I_1$, for instance.