Subset Relation is Transitive

From ProofWiki

Jump to: navigation, search

Theorem

$\left({R \subseteq S}\right) \land \left({S \subseteq T}\right) \implies R \subseteq T$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle \left({R \subseteq S}\right) \land \left({S \subseteq T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle \left({x \in R \implies x \in S}\right) \land \left({x \in S \implies x \in T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of subset
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle \left({x \in R \implies x \in T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Hypothetical Syllogism
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle R \subseteq T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of subset

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \left({R \subseteq S}\right) \land \left({S \subseteq T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left({x \in R \implies x \in S}\right) \land \left({x \in S \implies x \in T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of subset
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left({x \in R \implies x \in T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Hypothetical Syllogism
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle R \subseteq T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of subset