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

\(\displaystyle \forall x: \ \ \) \(\displaystyle (x \in S\) \(\implies\) \(\displaystyle x \in S)\) Law of Identity: a statement implies itself
\(\displaystyle \leadsto \ \ \) \(\displaystyle S\) \(\subseteq\) \(\displaystyle S\) Definition of Subset

$\blacksquare$


Sources