# Element of Toset has at most One Immediate Successor

## 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.

$\blacksquare$