# 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

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations: Exercise $1.5.1$