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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.

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References

Bibliography

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

Footnotes