Powerset is not Subset of its Set/Proof 1

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Theorem

Let $A$ be a set.

Then:

$\powerset A \not \subseteq A$


Proof

Aiming for a contradiction, suppose 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 $C \in C$.

$\blacksquare$


Sources