Axiom:Axiom of Extension

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Two sets are equal if and only if they contain the same elements:

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

The order of the elements in the sets is immaterial.

Also defined as

For set theories that define $=$ instead of admitting it as a primitive, the Axiom of Extension becomes:

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

where $A$ and $B$ denote sets.


This is the fundamental definition of what a set is: a set is determined by its elements.

Also known as

Otherwise known as the Axiom of Extensionality or Axiom of Extent.

Also see