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Axiom·A01

Extensionality

Sets with the same members are equal.

The converse, that equal sets have the same members, follows from substitutivity Eq4.
In words
For any A, B, if for every x, x belongs to A exactly when x belongs to B, then A equals B.
This is a ZFC axiom: it is assumed, not proven. Everything below it in a proof chain ultimately rests here.

Remarks

A set is determined entirely by its members. Combined with Eq4 this gives the iff , the engine behind uniqueness proofs such as that of the empty set (T01) and of Separation's witness (A06).
Wikipedia ↗Set.ext ↗

Used by

A06L01L24L25L27L29L34L38L60L64T01T04T05T07T08T09T14T16T32T35T36
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