Immediate Successor in Toset is Unique

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Theorem

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

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

$b \preceq b'$

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.

$\blacksquare$


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