# Class Equality is Symmetric

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## Theorem

Let $A$ and $B$ be classes.

Let $=$ denote class equality.

Then:

- $A = B \implies B = A$

## Proof

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From Biconditional is Commutative:

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

Hence the result by definition of class equality.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 4.7 \ (2)$