Being asked to prove that x and x' aren't shared or common between two frames is ridiculous, since the distinctness of a stationary and a moving frame is an a priori assumption, much like assuming you can define a point in 3-dimensions with a tuple like (x,y,z). It's an axiom of Euclidean geometry, moreover, that any two points in the plane are separated by a distance.
What we have in three dimensions is a stationary frame, which has Cartesian coordinates in 3-space: (x,y,z), and a moving frame with coordinates (x',y',z'); aligning any of the axes so they are collinear just means they point in the same direction and all their points coincide. With the fourth dimension included, the moving frame must slide all its points over the extended axis of the stationary frame; ergo the set of points x and the set of points x' are not the same set, but can 'initially' be at the same location.
The moving set of points doesn't take anything with it from the stationary set; likewise when you pass a fixed object, there isn't anything connecting you to it, that's just crazy shit, bro.