◀ ▲ ▶Branches / Settheory / Proposition: Contained Relation is a Strict Order
We have seen in an example above that the contained relation $\in_X$ is wellfounded. 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 $y\in_X x.$
Table of Contents
Proofs: 1
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References
Bibliography
 Hoffmann, D.: "Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise", Hoffmann, D., 2018