Hoeffding's Covariance Identity

Question

I want to prove that theorem. I saw this post. And it would suffice to prove that $$\text{Cov}\left(X,Y\right)=\int\limits _{-\infty}^{\infty}\int\limits _{-\infty}^{\infty}\text{Cov}\left(1_{\left\{ X\leq x\right\} },1_{\left\{ Y\leq y\right\} }\right) \, dx \, dy$$ Could you give me a hint?