Class is Extensional

Theorem
Let $A$ be a class.

Then:


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

Proof
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.