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 $\RR$ 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 $1.5.1$