D is the set of all finite lines, so it is not more than all finite lines. If you remove the collection of all finite lines then nothing remains.
You have agreed: If you remove any finite set E from D a non empty set (a different one for every E) remains.
You have agreed: If you remove any finite set E from D what remains contains every natural number.
Note that any line plus its predecessors is a finite set. So, there is no contradiction in saying that you can remove any one line (plus its predecessors) but you cannot remove the collection of all lines