Exists Subset which is not Element/Proof 2

Theorem

Let $S$ be a set.

Then there exists at least one subset of $S$ which is not an element of $S$.

Proof

Consider the power set $\powerset S$ of $S$.

From Cantor's Theorem, there is no surjection $f: S \to \powerset S$.

That is, there are more subsets of $S$ than there are elements of $S$.

So there must be at least one subset of $S$ which is not an element of $S$.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 9$ Zermelo set theory