Suppose I have two continuous, non-negative random variables, $X$ and $Y$ and I have that $$ P(X) = P(X|Y)\cdot P(Y). $$
Can I go on and say that $$ P(X\gt z) = P(X\gt z | Y\gt z)\cdot P(Y\gt z) = P(X\gt z, Y\gt z)?$$
I'd also appreciate it if someone could come up with a better title.
There is a slight misstep in notation often given with the conditional probability rule. In particular,
$$P(X) = P(X|Y)P(Y)$$
intuitively assumes that we are conditioning upon the entire support of $Y$ ($S_Y$), and assuming $Y$ is a discrete random variable (the continuous case can dealt with similarly), we can see that the more appropriate notation would be
$$P(X) = \sum_{Y \in S_Y} P(X,Y) = \sum_{Y \in S_Y} P(X|Y)P(Y).$$
Hence, you will need to be more careful with defining what values $z$ can take on in the following statement.