# Diagonal Relation is Transitive

The diagonal relation $\Delta_S$ on a set $S$ is a transitive relation in $S$.
 $\, \displaystyle \forall x, y, z \in S: \,$ $\displaystyle \tuple {x, y}$ $\in$ $\displaystyle \Delta_S \land \tuple {y, z} \in \Delta_S$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle y \land y = z$ Definition of Diagonal Relation $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle z$ Equality is Transitive $\displaystyle \leadsto \ \$ $\displaystyle \tuple {x, z}$ $\in$ $\displaystyle \Delta_S$ Definition of Diagonal Relation
So $\Delta_S$ is transitive.
$\blacksquare$