Properties of Transitive Sets

Proof

1a) 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 \(Y\subseteq X\cup\{X\}\).

1b) 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 \(Y^\prime \subseteq \mathcal P(X)\).

2a) 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 \(Y\subseteq \bigcup X\).

2b) 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 \(Z\) it transitive, it follows \(Y\subseteq Z\) for all \(Z\) and so \(Y\subseteq \bigcap X\).

References

[656] Hoffmann, Dirk W.: “Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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