Class Equal to All its Elements

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Let $A$ be a class.


$A = \left\{{x : x \in A }\right\}$



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We have:

$x \in \left\{{x : x \in A}\right\} \iff x \in A$

by definition of class membership (applied to $\left\{{x : x \in A}\right\}$).

By Universal Generalisation, it follows that:

$\forall x: \left({ x \in A \iff x \in \left\{{x : x \in A}\right\} }\right)$

Hence the result, by definition of class equality.