Minimum CDF of non-independent random variables

How can I find the CDF for F_{X,Y}={x,x} if X,Y are not-independent. In other words, what will be Pr{min(X,Y)<x} if X,Y are not-independent?

If I already know the individual CDF of both X and Y, i.e. F_{X}(x) and F_{Y}(x), can they be useful to compute the Pr{min(X,Y)<x}, where X,Y are not-independent variables?