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So defining $x=y$ as $\forall z\left(z\in x\leftrightarrow z\in y\right)$ is not enough? Should I say that two classes are equal if they also belong to the same classes, too?
Thank you, I finally managed to write down a proof in the forward direction. But now, what about the other way around when $x_0$ and $x_1$ are two sets?
