TITLE: If $x$ is an element of $y$ and $y$ is an element of $z$, is $x$ an element of $z$?
QUESTION [5 upvotes]: Let $x\in y$ and $y\in z$. Does this imply that $x\in z$?
For example: Let $y=\{A,B\}$ and $z=\{\{A,B\},C\}$.
If $x=A$, then $x\in y$. My understanding, however, is that $x$ is not an element of $z$ since $A$ is not an element of $z$.

REPLY [2 votes]: You have described the transitive property of $z$.
$z$ is transitive if and only if:

$y \in z \land x \in y \implies x \in z$

In your example, $z$ is therefore not transitive.