Systematic way to show equality between ideals of $K[x,y]$

Question

I am studying primary decompositions.

I am struggling to show that $$I = \left<xy, x^3-x^2, x^2y-xy\right> = \left<x\right>\cap \left<x-1,y\right> \cap \left<x^2,y\right>$$ where $I$ is an ideal of $K[x,y]$.

I tried get an generic element $f$ in $\left<x\right>\cap \left<x-1,y\right> \cap \left<x^2,y\right>$ and show that $f$ is in $I$, but I got lost in so many possibilities.

How can I show this equality?

Thanks