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.

Proof

This theorem requires a proof. In particular: Applies in the case where the Axiom of Foundation applies, but I'm not so sure about non-standard sets. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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 $1.5.1$