# Proof

(related to Corollary: Properties of Transitive Sets)

• Let $$Y\in X\cup\{X\}$$.
• We have to show that $$Y\subseteq X\cup\{X\}$$, if $$X$$ is a transitive set.
• It follows $Y\in X\cup\{X\}\Longrightarrow\cases{ Y=X\overset{\text{trivially}}{\Longrightarrow} Y\subseteq X\\ \text{or}\\ Y\in X\overset{X\text{ is transitive}}{\Longrightarrow} Y\subseteq X }$ Altogether, it follows that $$Y\subseteq X\cup\{X\}$$.

• Let $$Y\in \mathcal P(X)$$.
• We have to show that $$Y\subseteq \mathcal P(X)$$, if $$X$$ is a transitive set.
• Since $$Y\in \mathcal P(X)$$, it follows from the definitions of the power set that $$Y\subseteq X$$.
• This means that for all elements $$Y^\prime\in Y$$ it is also $$Y^\prime\in X$$ and, because of its transitivity $$Y^\prime \subseteq X$$.
• Now it follows from the definition of power set again that $$Y^\prime \subseteq \mathcal P(X)$$.

• Let $$Y\in \bigcup X$$.
• We have to show that $$Y\subseteq \bigcup X$$, if $$X$$ is a set of transitive sets.
• $$Y\in \bigcup X$$ means that $$Y$$ is the element of the union of all elements $$Z\in X$$, i.e. $$Y\in Z$$ for at least of one such $$Z$$.
• Because of the transitivity of $$Z$$ it is $$Y\subseteq Z$$ and so $$Y\subseteq \bigcup X$$.

• Let $$Y\in \bigcap X$$.
• We have to show that $$Y\subseteq \bigcap X$$, if $$X$$ is a set of transitive sets.
• $$Y\in \bigcap X$$ means that $$Y$$ is the element of all elements $$Z\in X$$, i.e. $$Y\in Z$$ for all $$Z\in X$$.
• Because each such $$Z$$ is transitive, it follows $$Y\subseteq Z$$ for all $$Z$$ and so $$Y\subseteq \bigcap X$$.

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### References

#### Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011