The following lemma shows that transitive sets play a prominent role in the set theory. They are omnipresent.

Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull

For every set $X$ there exists a transitive set $Y$, such that $X$ is a subset of $Y,$ formally $X\subseteq Y.$ This set is called the transitive hull of the set $X$.

Proofs: 1

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  1. Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018