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## Lemma: Comparing the Elements of Strictly Ordered Sets

In a strictly ordered set $(V,\prec)$, for all elements $$a,b\in V$$ exactly one of the following cases holds:

• Either $a \prec b$ ($a$ is smaller than $b$),
• or $a \succ b$ ($a$ is greater than $b$).
• else $a=b$ ($a$ equals $b$).

Because of these three possibilities, the strict order $”\prec”$ is sometimes also called a trichotomous order.

| | | | | created: 2019-02-03 08:53:38 | modified: 2019-02-03 09:34:29 | by: bookofproofs | references: [979], [8055]

## 1.Proof: (related to "Comparing the Elements of Strictly Ordered Sets")

(none)

### Bibliography (further reading)

[8055] Hoffmann, D.: “Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise”, Hoffmann, D., 2018

[979] Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10