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.
| | | | created: 2019-02-03 14:44:54 | modified: 2019-02-03 14:49:27 | by: bookofproofs | references: 
This work is a derivative of:
Bibliography (further reading)
 Hoffmann, D.: “Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise”, Hoffmann, D., 2018