Immediate Predecessor under Total Ordering is Unique

Theorem
Let $\left({S, \preceq}\right)$ be a toset.

Let $a \in S$.

Then $a$ has at most one immediate predecessor.

Proof
Let $b, b' \in S$ be immediate predecessors of $a$.

Because $\preceq$ is a total ordering, WLOG:


 * $b \preceq b'$

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


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

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


 * $b' \prec a$

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

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

Hence the result.

Also see

 * Immediate Successor in Toset is Unique