Axiom·A08
Regularity
Every non-empty set has a member disjoint from it.
Here
is D002.
In words
For any set A, if A is not empty then there is some x with x belonging to A and for every y, if y belongs to x then y does not belong to A.
This is a ZFC axiom: it is assumed, not proven. Everything below it in a proof chain ultimately rests here.
Remarks
Regularity (Foundation): every non-empty set
has an element
that shares no member with
. This rules out infinite descending membership chains
and forbids self-membership
; the shortest cases (no set is a member of itself, and no two sets are members of each other) are proved from this axiom in L05.
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