Jech's Definition of an Ordinal Number

Question

Jech, in his SET THEORY, defines ordinals as sets that are both transitive and well-ordered by the relation of set membership. But it seems to me that a set well-ordered by membership is already transitive. So, to include transitivity in the definition seems to be superfluous. If it really is as it seems to be, I think to include transitivity in the definition might be merely for the sake of emphasis. What do you guys think of it?

Answer 1

Sets well ordered by membership needn’t be transitive for instance consider: $\{\{\varnothing\}\}$ it isn’t transitive but is clearly well ordered by $\in$.