Immediate Predecessor under Total Ordering is Unique

Theorem
Let $\preceq$ be a total ordering.

Let $a$ be an immediate predecessor to $b$.

Then $a$ is unique.

That is, if $a$ and $a'$ are both immediate predecessors to $b$, then $a = a'$.

Proof
Let $a$ and $a'$ both be immediate predecessors of $b$.

We have that $\preceq$ is a total ordering.

, suppose:


 * $a \preceq a'$

By virtue of $a$ being a immediate predecessor of $b$:


 * $\neg \exists c \in S: a \prec c \prec b$

However, since $a'$ is also an immediate predecessor:


 * $a' \prec b$

Hence, it cannot be the case that $a \prec a'$.

Since $a \preceq a'$, it follows that $a = a'$.

Hence the result.

Also see

 * Immediate Successor under Total Ordering is Unique