**Proof**: *(related to "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.

q.e.d

| | | | created: 2019-02-03 14:44:54 | modified: 2019-02-03 14:49:27 | by: *bookofproofs* | references: [8055]

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

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