I don't think we can say the the same about the edge with respect to the cube:
The edge does not have other points than those contained in the cube.
Do you agree?
The subset operator would still show asymetry here.
The less-than operator would not show asymmetry between the cardinalities of the two sets.
Our intuition tells us, if A is a strict subset of B, then |A| < |B|.
Our conclusion from this discussion is that this transformation, from a statement of sets, to a statement of cardinalities, does not hold when sets are infinite.
Is there some other commonly accepted mathematical construct, e.g. denoted for set A by [A], that evaluates to a number, such that if A is a strict subset of B, then [A] < even for infinite sets?