Using this definition of ordinals, do I need foundation?

Definition. A set is an ordinal if it is a transitive set of transitive sets.

This is the simplest definition of an ordinal I have ever encountered, and I happen to like it a lot for this reason. However, it seems to have one downfall: I am having a lot of trouble showing that the class of ordinals is well-ordered without using the axiom of regularity (foundation). Do I need to use regularity to show this? Are there models of $$ZFC-(\text{regularity})$$ in which the transitive sets of transitive sets are not well-founded?

If you remove the axiom of foundation from ZF, then you cannot exclude the existence of, for instance, sets satisfying $$x = \{x\}$$. Such a set would be transitive, and have only transitive sets as elements, so it would fall under your definition of an ordinal.
• Huh! This makes me wonder: Why would anyone define ordinal'' the way I have above? This can't possibly be a useful definition in ill-founded set theory. – fauxefox Oct 8 '18 at 23:54