elementary set theory - A question regarding joint cumulative probability distribution functions

Consider two random variables $X$ and $Y$ and their joint cumulative probability distribution function $F$. I'm attempting to show that $P\{a_1

My attempt
$$P\{a_1$$P(\{X>a_1\} \cap \{Xb_1\} \cap \{Y$$1 - P((\{X>a_1\} \cap \{Xb_1\} \cap \{Y$$1 - P(\{X>a_1\}^c \cup \{Xb_1\}^c \cup \{Y $$1 - P(\{X\leq a_1\} \cup \{X\geq a_2\} \cup \{Y\leq b_1\} \cup \{Y\geq b_2\})$$

So here I suppose I must apply the Inclusion–exclusion principle and becuase
$\{X\leq a_1\}$, $\{X\geq a_2\}$ and $\{Y\leq b_1\}$, $\{Y\geq b_2\}$ are respectively mutually exclusive we have that the terms will only be probabilities defined on single sets and on the intersection of two non-mutually exclusive sets. Despite of this it still seems a bit too cumbersome and I am not sure how to simplify the probabilities of intersections.

$$1 - P(\{X\leq a_1\}) - P(\{X\geq a_2\}) - P(\{X\leq b_1\}) - P(\{X\geq b_2\})$$
$$+ \,P(\{X\leq a_1\}\cap \{Y\leq b_1\}) + P(\{X\leq a_1\}\cap \{Y\geq b_2\})$$
$$+ \,P(\{X\geq a_2\}\cap \{Y\leq b_1\}) + P(\{X\geq a_2\}\cap \{Y\geq b_2\})$$

It just seems messy and I don't know how to continue. Am going in the right direction at least or am I completely off?

Answer

\begin{equation}
P(a_1\end{equation}

Also

\begin{equation}
P(X\end{equation}

\begin{equation}
P(X\end{equation}

And your result follows.