Immediate Successor under Total Ordering is Unique

Theorem
Let $\preceq$ be a total ordering.

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

Then $b$ is unique.

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

Proof
Let $b$ and $b'$ both be immediate successors to $a$.

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