# Class Equal to All its Elements

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## Theorem

Let $A$ be a class.

Then:

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

## Proof

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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.

$\blacksquare$

## Sources

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