Transitive set

In set theory, a branch of mathematics, a set ${\displaystyle A}$ is called transitive if either of the following equivalent conditions hold:

• whenever ${\displaystyle x\in A}$, and ${\displaystyle y\in x}$, then ${\displaystyle y\in A}$.
• whenever ${\displaystyle x\in A}$, and ${\displaystyle x}$ is not an urelement, then ${\displaystyle x}$ is a subset of ${\displaystyle A}$.

Similarly, a class ${\displaystyle M}$ is transitive if every element of ${\displaystyle M}$ is a subset of ${\displaystyle M}$.