How to prove if sets A and B satisfy $|A\cup B|=|\mathbb R|$, then $|A|=|\mathbb R|$ or $|B|=|\mathbb R|$?

Question

I saw this problem when I'm learning set theory and real analysis.

This problem can easily be solved if we acknowledge The continuum hypothesis(CH), because if the cardinality of A or B is not bigger than $\aleph_0$, then the cardinality of the union of A and B is not bigger than $\aleph_0$.

But this problem doesn't seem to rely on the CH. Without the CH, I don't know how to deal with this kind of set whose cardinality is between $\aleph_0$ and $\aleph_1$.

Help and thanks!

(We discuss this problem in ZFC set theory.)