$f:\mathbb{R}\to\mathbb{R}$ is increasing. $\{y\in\mathbb{R}:\#f^{-1}(\{y\})\geq 2\}$ can contain at most countably many points. Why?

Question

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following exercise is Exercise 24 on p.40 in Exercises 2B in this book.

Exercise 24
Suppose $B\subset\mathbb{R}$ is a Borel set and $f:B\to\mathbb{R}$ is an increasing function. Prove that $f(B)$ is a Borel set.

I found an answer to Exercise 24 by Sangchul Lee.

Let $D:=\{y\in\mathbb{R}:\#f^{-1}(\{y\})\geq 2\}.$
Sangchul Lee wrote as follows:

$D$ can contain at most countably many points.

Why?

Is the following proposition true?

If $S$ is a set of disjoint nondegenerate intervals in $\mathbb{R}$, then $S$ is at most countable.