Why empty set is a proper subset of every set?

Answer 1

It isn't. The empty set is not a proper subset of the empty set.

The empty set is a subset of every set, for the following reason. "A" is a subset of "B" means that if an item "x" is an element of "A", then it is also an element of "B".

(x is element of A) implies (x is element of B)

Since the empty set has no elements, the left part of the implication is false. Therefore, the entire implication is true. If you have trouble grasping this last part, look up the Wikipedia article, or other sources, for "trivially true" or "vacuous truth". (Briefly, an implication, like "F implies G", can only be false if "F" is true, and "G" is false - in all other cases, it is true.)