Reflexive Closure of Strict Total Ordering is Total Ordering

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Theorem

Let $S$ be a set.

Let $\prec$ be a strict total ordering of $S$.

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


Then $\preceq$ is an total ordering of $S$.


Theorem

By the definition of strict total ordering, $\prec$ is a strict ordering which connects $S$.

By Reflexive Closure of Strict Ordering is Ordering, $\preceq$ is a ordering.

Since $\prec$ connects $S$, for each $a, b \in S$, either $a = b$, $a \prec b$ or $b \prec a$.

If $a = b$, then $a \preceq b$.

If $a \prec b$, then $a \preceq b$.

If $b \prec a$, then $b \preceq a$.

Thus in all cases either $a \preceq b$ or $b \preceq a$.

Thus $\preceq$ is a total ordering of $S$.

$\blacksquare$