◀ ▲ ▶Branches / Set-theory / Proposition: Contained Relation is a Strict Order
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 $y\in_X x.$
Table of Contents
Proofs: 1
Thank you to the contributors under CC BY-SA 4.0! 
- Github:
-
References
Bibliography
- Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018
