280: No Set Has Itself as Member

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A description/proof of that no set has itself as member

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of set.

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Target Context

• The reader will have a description and a proof of the proposition that no set has itself as a member.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

For any set, $S$, $S\notin S$.

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2: Proof

This proposition stems from the regularity axiom.

Suppose that $S\in S$. By the subset axiom, $S^{\prime }:=\{s\in S|s=S\}=\{S\}$ would be a set. As $S\in S$, $S^{\prime }\cap S=\{S\}\cap S=\{S\}\ne \mathrm{\varnothing }$, a contradiction, being against the regularity axiom.

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References

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