Restriction to Subset of Strict Total Ordering is Strict Total Ordering

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Theorem

Let $S$ be a set or class.

Let $\prec$ be a strict total ordering on $A$.

Let $T$ be a subset or subclass of $A$.

Then the restriction of $\prec$ to $B$ is a strict total ordering of $B$.

Proof

Follows from:

Restriction of Transitive Relation is Transitive

Restriction of Antireflexive Relation is Antireflexive

Restriction of Connected Relation is Connected

$\blacksquare$