Immediate Successor is Unique in Toset

Theorem

Let $(S, \preceq)$ be a totally ordered set.

Let $x, y \in S$.

Suppose that $y$ is an immediate successor of $x$.

Then $y$ is the only immediate successor of $x$.

Aiming for a contradiction, suppose that $x$ has another immediate successor, $z ≠ y$.

Then by definition:

$x \prec y$

$x \prec z$

Since $\preceq$ is a total ordering and $y ≠ z$:

$y \prec z$ or $z \prec y$.

If $y \prec z$ then $x \prec y \prec z$, contradicting the fact that $z$ is an immediate successor of $x$.

If $z \prec y$ then $x \prec z \prec y$, contradicting the fact that $y$ is an immediate successor of $x$.

Thus we conclude that there can be no such $z$.

$\blacksquare$