**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)

[8055] **Hoffmann, D.**: “Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise”, Hoffmann, D., 2018

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

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