Category:Axioms/Axiom of Extension

This category contains axioms related to Axiom of Extension.

Set Theory

Let $A$ and $B$ be sets.

The Axiom of Extension states that:

$A$ and $B$ are equal

if and only if:

they contain the same elements.

That is, if and only if:

every element of $A$ is also an element of $B$

and:

every element of $B$ is also an element of $A$.

This can be formulated as follows:

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

Class Theory

The axiom of extension in the context of class theory has the same form:

Let $A$ and $B$ be classes.

Then:

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