Assume we have a set $X$ which is a (closed) line segment. Prove that if we split $X$ into 2 parts $X_1$ and $X_2$ then at least one of those sets would have the same cardinality as $X$.
My attempt: 1) Let's assume that $X_1$ contains some line segment itself, let's call it $Y$.
2) We know that there is a bijection between $X$ and $Y$ and therefore $|X| = |Y|$.
3) And since $X_1$ is embedded in $X$ we can use Cantor-Bernstein theorem to conclude that $|X_1| = |X|$.
Now, I am not sure how to prove the second case: when $X_1$ (or $X_2$) does not contain a line segment. Is there a way to use Cantor-Bernstein there?