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
 Hoffmann, D.: "Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise", Hoffmann, D., 2018