Definition:Class Equality

Definition
Let $A$ and $B$ be classes.

Definition 2
When $x$ is a set variable, equality of $x$ and $A$ is defined using the same formula:


 * $x = A$ $\forall y: \paren {y \in x \iff y \in A}$
 * $A = x$ $\forall y: \paren {y \in A \iff y \in x}$

Axiom of Extension
The concept of class equality is axiomatised as the Axiom of Extension:

Equality as applied to Sets
In the context of set theory, the same definition applies:

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

 * Equivalence of Definitions of Class Equality


 * Class Equality Extension of Set Equality


 * Definition:Small Class