Cantor's Theorem

Definition

$|A|<|2^A|$

The set of all subsets of $A$, the power set of $A$ has strictly greater cardinality than $A$ itself

Proof

Since $A\approx \{\{x\}|x\in A\}\subset \mathcal{P}(A)$ $\Rightarrow |A|\le |\mathcal{P}(A)|$

Let $B=\{x\in A|x\notin f(A)\}$ and $\exists$ a Bijective function $f:A\sim P(A)$ the definition of $B$ implies $\forall x\in A:x\in B\iff x\notin f(x)$

Then, since $f$ is Bijective and $B\subset \mathcal{A}$, $\exists \xi \in A\text{ s.t. }f(\xi )=B$. But by the definition of $B$, $\xi \in B\iff \xi \notin f(\xi )=B$. This is a contradiction. So $f$ can not be Bijective $\Rightarrow |A|\ne |\mathcal{P}(A)|$

$\therefore |A|<|\mathcal{P}(A)|$