Why does my book use $\subseteq$ instead of $\subset$ to describe a transitive set? The following FOL statement is used in my book to describe a transitive set (which ultimately sets the stage for the definition of ordinals):

$z$ is a transitive set iff $\forall y \in z [ y \subseteq z]$

Given that $y\subseteq z \leftrightarrow y \subset z \ \lor y=z $, I am a little confused as to why $y \subseteq z$ is used.
In the context of ZFC set theory, it is my understanding that $z \in z$ is always false.
Therefore if $y \in z$, then clearly $y \neq z$. So why not just use $y \subset z$?
Am I missing something? (There is clearly no logical issue either way...it just seemed odd).
Edit: After realizing that, perhaps, there are different definitions for the various symbols $\subset$, $\subseteq$, and $\subseteqq$, I figured I would add that my book (The Foundations of Mathematics by Kenneth Kunen) asserts the following definition:

$x \subseteq y \iff \forall z ( z \in x \rightarrow z \in y)$

Also, it appears as though the only two subset symbols that are used in my book are $\subseteq$ and $\subsetneqq$ (the latter has no formal definition)...there is also $\nsubseteq$, but that is obviously not applicable here.
 A: In the presence of the axiom of foundation, it is true as you indicate that no set belongs to itself, and so the definition of transitive set can be written with $\subset$ (or $\subsetneq$, whichever symbol you prefer).
However, one may study also set theories where foundation fails, and then it is natural to define transitive sets in a way that allows self-membership. Indeed, any set $\Omega$ such that $\Omega=\{\Omega\}$, for instance, should be considered transitive.
One minor thing is that the natural translation of the natural language statement describing transitive sets gives us the rendering used by Kunen: A set $t$ is transitive if and only if any element of $t$ is a subset of $t$. This description is agnostic as to whether containment is proper or not. A somewhat more mathematical observation is that, even if Kunen is working here in a context where foundation is assumed, the definition he gives can be ported as is to a setting where foundation fails, there is no need to redefine things in this new context. This "portability" is particularly useful if one finds oneself in a situation where only fragments of ZFC hold, which is typically the case when studying independence proofs (or using countable structures in combinatorial arguments).
