Reflexive Closure of Strict Ordering is Ordering

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Theorem

Let $S$ be a set.

Let $\prec$ be a strict ordering on $S$.

Let $\preceq$ be the reflexive closure of $\prec$.


Then $\preceq$ is an ordering.


Proof

Since $\prec$ is a strict ordering, it is by definition transitive and asymmetric.

By Asymmetric Relation is Antisymmetric, $\prec$ is antisymmetric.

Thus by Reflexive Closure of Transitive Antisymmetric Relation is Ordering, $\preceq$ is an ordering.

$\blacksquare$


Sources