Joint probability with inequality

Question

Suppose that $x$ and $y$ are two independent random variables that have the same CDF $F(\cdot)$ and $a$ is a constant. What is the probability that -a< x<-y?

Answer 1

It equals:$$\int\int[-a<x<-y]dF(x)dF(y)=\int F_-(-y)-F(-a)dF(y)=\int F_-(-y)dF(y)-F(-a)$$where $[-a<x<-y]$ takes value $1$ if $-a<x<-y$ and takes value $0$ otherwise.

Here $F_-(y):=\lim_{z\uparrow y}F(z)$.