Intersection of irreducible sets in $\mathbb A_{\mathbb C}^3$ is not irreducible I am looking for a counterexample in order to answer to the following: 

Is the intersection of two closed irreducible sets in $\mathbb
A_{\mathbb C}^3$ still irreducible?

The topology on $\mathbb A_{\mathbb C}^3$ is clearly the Zariski one; by irreducible set, I mean a set which cannot be written as a union of two proper closed subsets (equivalently, every open subset is dense).
I think the answer to the question is "No", but I do not manage to find a counterexample. I think I would be happy if I found two prime ideals (in $\mathbb C[x,y,z]$) s.t. their sum is not prime. Am I right? Is there an easier way? 
Thanks.

Choose any two irreducible plane curves, they will intersect in a finite number of points.
