Can we use Bayes' Theorem on CDFs

Question

I'm learning probability on my own, so I'd like to know when I can use Bayes' Rule.

From my understanding it's only valid if our probability axioms hold:

1. $P(x) \geq 0$

2. $P(S) = 1$ where S is the entire sample space

3. $P(A \cup B) = P(A) + P(B)$ if A and B are disjoint events.

So the first axiom and the last one are definitely true for CDFs and PDFs. But the second one is a little harder to prove.

I guess my question really boils down to can I use Bayes' Theorem for CDFs and PDFs.