1. ## proof

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

$A \in \mathcal{P}(A) \subseteq \mathcal{P}(B)\; \Rightarrow \;A \in \mathcal{P}(B)\; \Rightarrow \;A \subseteq B$