 We need to prove that \(\wp(S)\) has greater cardinality than \(S\). Assume to obtain a contradiction that this is not the case. Then there must exist a map \(f\) from \(S\) onto \(\wp(S)\). Now consider the set \(C = \{ x \in S \mid x \not\in f(x) \}\). Since \(f\) is onto \(\wp(S)\), there must exist a set \(c \in S\) such that \(f(c)=C\). However, we now obtain a contradiction, since the following holds: 