Definition·D024
Transitive set
A set whose members are also subsets of it: membership does not escape.
In words
A set x is transitive exactly when for every y, if y is a member of x then y is a subset of x: elements of elements never lead outside the set.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Equivalently,
(D004, D001). Transitivity is the structural backbone of the von Neumann construction of numbers: each natural number is transitive (L07), and so is the set
of all naturals (L06). For the order on the naturals, transitivity of the sets is exactly what makes the order relation transitive.
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