Proof
(related to Lemma: Comparing the Elements of Strictly Ordered Sets)
 Let $(V,\prec)$ be a strictly ordered set.
 By definition, "$\prec$" is a total order. Therefore it is ensured that at least one of the possibilities $a \prec b,$ $a \succ b,$ or $a=b$ is fulfilled. It remains to be shown that exactly one is possible:
 The possibilities $a \prec b$ and $a=b$ cannot occur simultaneously, because "$\prec$" is irreflexive, by definition.
 The same argument holds for the possibilities $a \succ b$ and $a=b$ occurring simultaneously.
 Neither can the possibilities $a \succ b$ and $a \prec b$ occur simultaneously, because "$\prec$" is asymmetric, by definition.
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Reinhardt F., Soeder H.: "dtvAtlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition
 Hoffmann, D.: "Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise", Hoffmann, D., 2018