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Proof

(related to Corollary: Properties of Transitive Sets)

Ad 1.1

• 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 \(Y\subseteq X\cup\{X\}\).

Ad 1.2

• 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 \(Y^\prime \subseteq \mathcal P(X)\).

Ad 2.1

• 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 \(Y\subseteq \bigcup X\).

Ad 2.2

• 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