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Axiomatic Wiki
Axiom·A06

Separation

Subsets carved out by a property exist.

The set is a subset of ; its uniqueness follows from A01.
In words
For any set A, there is a set B where for every x, x belongs to B exactly when x belongs to A and x satisfies the property φ.
This is a ZFC axiom: it is assumed, not proven. Everything below it in a proof chain ultimately rests here.

Remarks

Separation (Specification): from any set and property there is a set of exactly those members of satisfying . In first-order logic this is a schema, one axiom per formula ; in second-order logic it collapses to a single axiom quantifying over all properties. It cannot create new sets, only thin an existing one; together with the existence clause of Infinity it yields the empty set (T01).
Wikipedia ↗Sep.sep ↗

Used by

L06L07L08L09L10L11L12L13L14L15L16L17L20L21L24L25L26L27L29L30L35L40L41L55L60L65L66T01T02T03T07T08T09T10T12T13T19T25T38
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