(related to Proposition: Contained Relation is a Strict Order)

- In order to show that $(X,\in_X)$ is a strictly ordered set, we have to show that the contained relation $\in_X$ is a strict total order, i.e. that it is irreflexive, asymmetric, transitive, and connex.
- Since we require that it is transitive and connex, it remains to be shown that it is irreflexive ($x\not\in_X x$ for all $x\in X$) and asymmetric (if $x\in_X y$, then $y\not\in_X x$ for all $x,y\in X$).
- But these two properties follow immediately from the axiom of foundation.∎

**Hoffmann, D.**: "Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise", Hoffmann, D., 2018