Category:Axioms/Axiom of Extension

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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:

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