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Proof: By Contradiction

(related to Proposition: Partial Orders are Extensional)

• Let $(X,\preceq )$ be a poset.
• Assume that for $x,y\in X$ the following subsets of a given are equal: $\{z\mid z\preceq x\}=\{z\mid z\preceq y\}\quad( * ).$
• Assume that the partial order "$\preceq$" is not extensional.
• This means that $x\neq y.$
• Therefore, we have either $x\npreceq y$ or $y\npreceq x.$
• Therefore, we have either $x\not\in \{z\mid z\preceq y\}$ or $y\not\in \{z\mid z\preceq x\}.$
• This means by $( * )$ that either $x\not\in \{z\mid z\preceq x\}$ or $y\not\in \{z\mid z\preceq y\}.$
• This means that either $x\npreceq x$ or $y\npreceq y.$
• But this contradicts the reflexivity of the partial order $"\preceq".$
• Therefore, $x=y$ and "$\preceq$" is extensional.
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References

Bibliography

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