Let's say we have a finite metric space $(S,d)$. I need to prove that $P(S)$ (the power set of $S$) is a metric space with the metric $\bar d:P(S)\times P(S)\to[0,\infty)$ defined as

$$\bar d(X,Y)=\sum\_{x,y\in X\cup Y}d(x,y)-\sum\_{x,y\in X\cap Y}d(x,y)$$

Non-negativity, identity of discernibles and symmetry are almost obvious, but what about the triangle inequality? I'm not able to prove that. I cannot find a value between $\bar d(X,Z)$ and $\bar d(X,Y)+\bar d(Y,Z)$ that sends me in the right track. Thanks.