(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 BY-SA 4.0!
- Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition
- Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018