If $f$ is monotonically increasing, $f(x)=y$ and $f(y)=x$, then is $x=y$?

Question

If I have a function $f:\Bbb{R} \to \Bbb{R}$ which is monotonically increasing, and also $f(x)=y$ and $f(y)=x$. Is it true that $x=y$?

I think this is true because (if I understood correctly) it was used in the solution of problem 4 here, but I'm failing to prove. I know that monotonically increasing implies $f$ is injective, then, by definition, $f(x)=f(y) \implies x=y$. But I don't have $f(x)=f(y)$, so I don't know what should I do.

Answer 1

If $x<y$, then $y=f(x)\leq f(y)=x$, so...