There are uncountably many planes containing the two points a and b. They are "almost disjoint", in the sense that the intersection of any two of them is just the line joining a to b. Since there are only countably many lines in A, at least one of these planes does not contain any of the lines.

Choose such a plane. It contains at most one point from each line in A. So that reduces the problem from three dimensions to two. The two-dimensional problem is this: Given a countable set of points B in a plane, show that the complement of B is connected.

The proof of that is a similar counting argument. Given two points a and b in the plane, there are uncountably many disjoint circular arcs connecting a and b, with the property that any two of them are almost disjoint (in the sense that their intersection consists only of the points a and b). Only countably many of these can contain a point in B. Choose an arc containing no such points. That demonstrates that the complement of B is connected (in fact, path-connected).