**Proof**: *(related to "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.

^{1} For the $\bigcup$ notation see axiom of union.

q.e.d

| | | | created: 2019-01-29 23:48:46 | modified: 2019-01-29 23:54:09 | by: *bookofproofs* | references: [8055]

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[8055] **Hoffmann, D.**: “Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise”, Hoffmann, D., 2018

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