if the power set of A is a subset of the power set of B then A is a subset of B
proof: if X is an element of the power set of A then X is an element of the power set of B. In other words, if X is a subset of A then X is a subset of B. If y is an element of X then y is also an element of A and of B. since y is an element of A and of B, A is a subset of B.
is this proof correct and clear enough