A prime ideal in the intersection of powers of another ideal 
Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which   lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$?

My try is to choose the prime ideal generated by $x$. I wanted to show that $x$ can not be in the ideal $(x,y)^2$. But, I am stuck in comparing the powers of $x$ in the two sides of the pop-up equality. Thanks for help!

By Krull's Intersection Theorem we have $⋂_{n≥0}(x,y)^n=0$. I leave you the pleasure to draw the conclusion.
