Let's start with a small discrete example: . Then, the power set of would be the set . So, it contains all subsets with zero elements (the empty set), all subsets with 1 element, and all subsets with two elements. How do you show that two sets are equal? You show that the first set is a subset or equal to the second set. Then, you show that the second set is a subset or equal to the first. So, since , we know that (the power set of S) since contains all subsets of . But, since no set can contain itself (that is one of the axioms of set theory). So, contains at least one element that does not contain. Hence, the two sets cannot be equal.