Powerset is not Subset of its Set

Theorem
Let $A$ be a set.

$A \not \subseteq \powerset A$

Proof
that $A \subseteq \powerset A$ and define:
 * $C = \set { x \in \powerset A : x \notin x }$

We have that $C \subseteq \powerset A$.

Since $\powerset A \subseteq A$, we have:
 * $C \subseteq A$

and thus
 * $C \in \powerset A$

We can derive a similar contradiction to Russell's Paradox.

If $C \in C$, then it must meet $C$'s condition that $C \notin C$.

If $C \notin C$, then it meets $C$'s condition for $C \in C$.

Also see

 * No Injection from Power Set to Set