Membership Relation is Antisymmetric

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Theorem

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

Let $\mathcal R$ denote the membership relation on $\Bbb S$:

$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \mathcal R \iff a \in b$


$\mathcal R$ is an antisymmetric relation.


Proof


Sources