Class is Extensional

Theorem

Let $A$ be a class.

Then:

$A = \set {x : x \in A}$

That is, $A$ is extensional.

Proof

We have:

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

by Characterization 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.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 4.9$