Powerset is not Subset of its Set

Theorem
Let $A$ be a set.

Then:


 * $\powerset A \not \subseteq A$

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

We have that $C \subseteq \powerset A$, as it contains only the $x \in \powerset A$ meeting the condition $x \notin x$.

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