# Definition:Class Equality

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## Contents

## Definition

Let $A$ and $B$ be classes.

Then $A$ and $B$ are **equal**, denoted $A = B$, iff:

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

where $\in$ denotes class membership.

When $x$ is a set variable, **equality** of $x$ and $A$ is defined using the same formula:

- $x = A$ iff $\forall y: \left({y \in x \iff y \in A}\right)$
- $A = x$ iff $\forall y: \left({y \in A \iff y \in x}\right)$

## Comment

This definition "overloads" the $=$ sign, since $x = y$ could refer to either **class equality** or set equality.

However, this overloading is justified because for sets $x$ and $y$, $x = y$ is equal for either set equality or **class equality**.

This fact is proved on Class Equality Extension of Set Equality.

## Also see

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 4.5$