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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.
    ∎

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References

Bibliography

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