Immediate Successor under Total Ordering is Unique

Theorem
Let $\struct {S, \preceq}$ be a toset.

Let $a \in S$.

Then $a$ has at most one immediate successor.

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

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




 * $b \preceq b'$

By virtue of $b'$ being a immediate successor of $a$:


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

However, since $b$ is also an immediate successor:


 * $a \prec b$

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 Predecessor under Total Ordering is Unique