Membership Relation is Antisymmetric
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This theorem requires a proof.
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.
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
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.
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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 $1.5.1$
