# Set is Subset of Itself

## 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: $\quad$ a statement implies itself $\ds \leadsto \ \$ $\ds S$ $\subseteq$ $\ds S$ Definition of Subset

$\blacksquare$