I have a continious function $f$ that is strictly increasing. And a continious function $g$ that is strictly decreasing. How to I rigorously prove that $f(x)=g(x)$ has a unique solution?

Intuitively, I understand that if I take limits to infinity, then $f$ grows really large and $g$ grows very small so the difference is less than zero. If I take the limits to negative infinity then the opposite happens. Using the intermediate value theorem, there must be an intersection, and since they are strictly increasing/decreasing, only one intersection happens.

My question is how do I apply the intermediate value theorem here? I don't have the interval to apply it on. I don't know when $f$ crosses $g$ and therefore can't take any interval. Or is there some sort of axiom applied here that I am missing?

Intuitively, I understand that if I take limits to infinity, then $f$ grows really large and $g$ grows very small so the difference is less than zero. If I take the limits to negative infinity then the opposite happens. Using the intermediate value theorem, there must be an intersection, and since they are strictly increasing/decreasing, only one intersection happens.

My question is how do I apply the intermediate value theorem here? I don't have the interval to apply it on. I don't know when $f$ crosses $g$ and therefore can't take any interval. Or is there some sort of axiom applied here that I am missing?