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Proof

(related to Proposition: The Contained Relation is Extensional)

• Let $X$ be a set.
• Assume that in a contained relation $\in_X\subset X\times X$ for any two elements $x,y\in X$ the following sets are equal: $\{z\in X\mid z\in x\}=\{z\in X\mid z\in y\}.$
• Assume that $\in_X$ is not extensional.
• Then $x\neq y.$
• But this contradicts the axiom of extensionality, because the sets $x$ and $y$ are determined by their elements $z.$
• Therefore, $x=y$ and $\in_X$ is extensional. .
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References

Bibliography

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