Powerset is not Subset of its Set/Proof 3

Theorem

Let $A$ be a set.

Then:

$\powerset A \not \subseteq A$

Proof

Aiming for a contradiction, suppose that $\powerset A \subseteq A$.

Since $A \in \powerset A$, this implies:

$A \in A$

But this contradicts Set is Not Element of Itself.

$\blacksquare$

Axiom of Foundation

This theorem depends on the Axiom of Foundation.

Most mathematicians accept the Axiom of Foundation, but theories that reject it, or negate it, have found applications in Computer Science and Linguistics.