Immediate Successor under Total Ordering is Unique

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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 $b = b'$.


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

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

Without loss of generality:

$b \preceq b'$

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.


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