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:

Proofs: 1

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  1. Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018