Initial Segment Determined by Smallest Element is Empty

Theorem

Let $\left({S, \preceq}\right)$ be a well-ordered set, where $S$ is non-empty.

Let $s_0 = \min S$, the smallest element of $S$.

Then the initial segment determined by $s_0$, $S_{s_0}$, is empty.

Proof

By the definition of initial segment:

$S_{s_0} := \left\{{b \in S: b \prec s_0}\right\}$

By the definition of smallest element:

$\forall b \in S: s_0 \preceq b$

Thus $S_{s_0}$ is empty.

$\blacksquare$