Let $A$ be a commutative unitary ring, $P_1,\dots , P_r$ prime ideals, $I$ an ideal, and $x$ an element of $A$. If $xA+I\not \subset P_1\cup \cdots \cup P_r$, then there is $y\in I$ such that $x+y\not \in P_1\cup \cdots \cup P_r$.
I have tried several methods but nothing works. For example, I tried to apply the following theorem:
Let $\mathfrak{a}$ be an ideal, $P_1,\dots ,P_r$ prime ideals. If $\mathfrak{a}\subset P_1\cup \cdots \cup P_r$, there exists $1\leq k \leq r$ such that $\mathfrak{a}\subset P_k$.
And of course, I tried to show contraposition but failed. If you know how to solve, please help me!!