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Proof: By Contradiction

(related to Proposition: Strict Orders are Extensional)

• Let $(X,\prec)$ be a strictly ordered set.
• To see this, assume that for $x,y\in X$ the following subsets are equal: $\{z\mid z\prec x\}=\{z\mid z\prec y\}\quad ( * )$.
• Further assume that the strict order "$\prec$" is not extensional.
• This means that $x\neq y.$
• Therefore, we have either $x\prec y$ or $y\prec x$.
• Therefore, we have either $x\in \{z\mid z\prec y\}$ or $y\in \{z\mid z\prec x\}$.
• This means by $( * )$ that either $x\in \{z\mid z\prec x\}$ or $y\in \{z\mid z\prec y\}$.
• This means that either $x\prec x$ or $y\prec y$ which contradicts the irreflexivity of the strict order "$\prec$".
• Therefore, $x=y$ and "$\prec$" is extensional.
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References

Bibliography

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