A is a subset of B iff for all elements a of A a is in B.

(Assume there exists a in the empty set, now a false proposition implies anything

so this assumption implies that a is in B, so for all a in the null set a is in B)

Also A is not a subset of B if there exists an element a of A which is not in B.

Now this is always true of the empty set as there are no elements of A which are not in B.

So the empty set is a subset of every set.

RonL