# Restriction of Ordering is Ordering

## Theorem

Let $S$ be a set.

Let $\preceq$ be an ordering on $S$.

Let $T \subseteq S$ be a subset of $S$.

Let $\preceq \restriction_T$ be the restriction of $\preceq$ to $T$.

Then $\preceq \restriction_T$ is an ordering on $T$.

## Proof

Let $\preceq$ be an ordering on $S$.

Then, by definition:

$\preceq$ is a reflexive relation on $S$
$\preceq$ is an antisymmetric relation on $S$
$\preceq$ is a transitive relation on $S$.

Then:

from Restriction of Reflexive Relation is Reflexive, $\preceq \restriction_T$ is a reflexive relation on $T$
from Restriction of Antisymmetric Relation is Antisymmetric, $\preceq \restriction_T$ is an antisymmetric relation on $T$
from Restriction of Transitive Relation is Transitive, $\preceq \restriction_T$ is a transitive relation on $T$

and so it follows by definition that $\preceq \restriction_T$ is an ordering on $T$.

$\blacksquare$