# Class Equality is Transitive

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## Theorem

Let $A$, $B$, and $C$ be classes.

Let $=$ denote class equality.

Then

- $\left({ A = B \land B = C }\right) \implies B = A$

## Proof

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By Universal Generalisation and Biconditional is Transitive:

- $\forall x: \left({ \left({ x \in A \iff x \in B }\right) \land \left({ x \in B \iff x \in C }\right) }\right) \implies \forall x: \left({ x \in A \iff x \in C }\right)$ by Universal Generalisation and Biconditional is Transitive

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 4.7 \ (3)$