# Subset Relation is Transitive/Proof 1

## Theorem

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

## Proof

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

$\blacksquare$