Membership Relation is Antisymmetric

Theorem
Let $\Bbb S$ be a set of sets in the context of pure set theory

Let $\RR$ denote the membership relation on $\Bbb S$:
 * $\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$

$\RR$ is an antisymmetric relation.