We have seen in an example above that the contained relation $\in_X$ is well-founded. The following proposition shows that it can be used to create a strict order on a set $X.$

**Proposition**: Contained Relation is a Strict Order

Let $X$ be a set and let $\in_X$ be the contained relation defined on it. Then $(X,\in_X)$ is a strictly ordered set, if all $x,y,z\in X$ the relation $\in_X$ fulfills the following properties:

- transitive: If $x\in_X y$ and $y\in_X z$, then $x\in_X z,$
- connex: $x\in_X y$ or $y\in_X x.$

| | | | | created: 2019-02-03 14:38:43 | modified: 2019-02-03 15:03:40 | by: *bookofproofs* | references: [8055]

## 1.**Proof**: *(related to "Contained Relation is a Strict Order")*

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

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