Definition:Transitive with Respect to a Relation

Definition
Let $A$ be a class.

Let $\prec$ be a foundational relation.

Furthermore, let every $\prec$-initial segment of $A$ be a small class.

Let $S$ be a set.

$S$ is transitive with respect to $\prec$ iff:


 * $\forall x \in A: \forall y \in S: ( x \prec y \implies x \in S )$