Proof
(related to Lemma: Any Set is Subset of Some Transitive Set  Its Transitive Hull)
 For a given set $X,$ construct a set $Y$ as follows: $$Y:=\bigcup\{ y_n \mid n\in\mathbb N \}$$ with the elements $y_n$ being sets recursively defined by $y_0:=X,$ $y_{n+1}:=\bigcup y_n.$^{1}
 $X$ is subset of $Y$ by construction, since $X=y_0\subset Y.$
 It remains to be shown that $Y$ is transitive:
 Let $w\in Y.$
 It follows that $w\in y_n$ for some $n\in\mathbb N.$
 Let $u\in w$.
 It follows $u\in \bigcup y_n.$
 By definition $u\in y_{n+1},$ and $u\in Y.$
 Therefore $w\subseteq Y.$
 This means that $Y$ is transitive.
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Hoffmann, D.: "Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise", Hoffmann, D., 2018
Footnotes