Class is Extensional

Theorem
Let $A$ be a class.

Then:


 * $A = \set {x : x \in A}$

Proof
We have:


 * $x \in \set {x : x \in A} \iff x \in A$

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

By Universal Generalisation, it follows that:


 * $\forall x: \paren {x \in A \iff x \in \set {x : x \in A} }$

Hence the result, by definition of class equality.