I'm not really sure about this. I think the proof relies on the theorem: If $U$ is a subset of $A$, and $U$ is uncountable, then $A$ is uncountable. 

The problem statement says a collection of lines is countable. $\mathbb{R}^2$ is not countable so $\mathbb{R}^2$ cant be a subset of a countable collection of lines in $\mathbb{R}^2$.