Question about partitions in intervals of the real numbers. I have to prove the following:

Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any
  kind, except intervals containing a single element. Prove
  $ \mathcal{D} $ is countable.

Now if $\mathcal{D}$ is finite the result is trivial; my problem is when it is infinite. In this case there is an inyective function $ f: \mathbb{N} \rightarrow \mathcal{D}, $ so $ \left| \mathbb{N} \right| \leq \left| \mathcal{D} \right|. $ But I don't know how to construct an injective function in the other direction to, then, apply the Cantor-Bernstein Theorem. In fact, I'm not even sure this is the right way to prove the result. 
I would like to read some suggestions on how to tackle this problem.

HINT: There is a countable dense subset to the real numbers. Recall the definition of being dense.
