# Subset Relation is Transitive

## Theorem

The relation "is a subset of" is transitive:

$\paren {R \subseteq S} \land \paren {S \subseteq T} \implies R \subseteq T$

## Proof

 $\displaystyle$  $\displaystyle \paren {R \subseteq S} \land \paren {S \subseteq T}$ $\displaystyle$ $\leadsto$ $\displaystyle \paren {x \in R \implies x \in S} \land \paren {x \in S \implies x \in T}$ Definition of Subset $\displaystyle$ $\leadsto$ $\displaystyle \paren {x \in R \implies x \in T}$ Hypothetical Syllogism $\displaystyle$ $\leadsto$ $\displaystyle R \subseteq T$ Definition of Subset

$\blacksquare$