I don't think so. A joint distribution has domain $(-\infty, \infty) \times  (-\infty, \infty)$. If we partition each component in two by selecting some value $x$ and some value $y$, then we get $4$ subsets,

$$(-\infty, x] \times  (-\infty, y],\;\;(-\infty, x] \times  [y,\infty),\\
[x, \infty) \times  (-\infty, y],\;\;[x, \infty) \times  [y,\infty)$$

So

$$\begin{align}\Pr(X\le x, Y\le y) =& 1- \Pr(X\le x, Y\ge y)\\
&-\Pr(X\ge x, Y\le y)\\
&-\Pr(X\ge x, Y\ge y) \end{align}$$

and you should differentiate all three cdf's in the right-hand side. In general, There is nothing "special" about $\Pr(X\ge x, Y\ge y)$ that would permit you to derive from it the other two.