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

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

$\blacksquare$


Sources