Full Idea
If, for every b∈d, a∈b entails that a∈d, the d is said to be 'transitive'. In other words, d is transitive if it contains every member of each of its members.
Gist of Idea
A set is 'transitive' if contains every member of each of its members
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 4.2)
Book Reference
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.85
A Reaction
The alternative would be that the members of the set are subsets, but the members of those subsets are not themselves members of the higher-level set.