Cantor's theorem in the context of Inclusion (set theory)

In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set ${\displaystyle A}$, the set of all subsets of ${\displaystyle A,}$ known as the power set of ${\displaystyle A,}$ has a strictly greater cardinality than ${\displaystyle A}$ itself.

For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with ${\displaystyle n}$ elements has a total of ${\displaystyle 2^{n}}$ subsets, and the theorem holds because ${\displaystyle 2^{n}>n}$ for all non-negative integers.