statistics - CDF from joint PDF

My joint PDF function of (x,y) is given
$$\begin{cases} 2e^{-x-y}, & 0\leq x\leq y\leq \infty\\0, & otherwise \end{cases}$$
How do I find joint CDF from this PDF? I know that my CDF is zero when both x and y are less then zero and that I'm supposed to integrate PDF on a given region, but there is a solution in my book that says I'm supposed to calculate CDF on this area too:

$$0\leq y\leq x, y>0$$

and it equals $$e^{-2y} - 1$$

Why do I do this? Shouldn't CDF be zero in this case? Also, isn't it redudant to say that y>0 in that area?