Subset of Itself

From ProofWiki

Jump to: navigation, search

Theorem

Every set is a subset of itself:

$\forall S: S \subseteq S$

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

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x \in S$	$\implies$	$\displaystyle x \in S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Law of Identity	A statement implies itself.
	$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle S$	$\subseteq$	$\displaystyle S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of subset

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x \in S\)	\(\implies\)	\(\displaystyle x \in S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Law of Identity	A statement implies itself.
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle S\)	\(\subseteq\)	\(\displaystyle S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of subset