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Proof

(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.
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References

Bibliography

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