Let $R$ be a commutative ring. The Prime Avoidance lemma says that if an ideal $I\subseteq R$ is not contained within the prime ideals $p_1,p_2$, then it is not contained within $p_1\cup p_2$ either.
Let $R$ be the ring $\Bbb{C}[x,y]$, and let $I=(x,y), p_1=(x+y)$ and $p_2=(x-y)$. Then clearly, $(x,y)\not\subseteq (x+y),(x-y)$. However, it is included in the union of $(x+y)$ and $(x-y)$, which is precisely $(x,y)$. Isn't this a contradiction? Where am I going wrong?