Set is Subset of Itself

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Theorem

Every set is a subset of itself:

$\forall S: S \subseteq S$


Thus, by definition, the relation is a subset of is reflexive.


Proof

\(\ds \forall x: \, \) \(\ds \leftparen {x \in S}\) \(\implies\) \(\ds \rightparen {x \in S}\) Law of Identity: a statement implies itself
\(\ds \leadsto \ \ \) \(\ds S\) \(\subseteq\) \(\ds S\) Definition of Subset

$\blacksquare$


Sources