Union of Transitive Class is Transitive/Proof 1

Theorem

Let $A$ be a class.

Let $\bigcup A$ denote the union of $A$.


Let $A$ be transitive.

Then $\bigcup A$ is also transitive.


Proof

Let $A$ be transitive.

By Class is Transitive iff Union is Subclass:

$\bigcup A \subseteq A$

By Union of Subclass is Subclass of Union of Class:

$\map \bigcup {\bigcup A} \subseteq \bigcup A$

Then by Class is Transitive iff Union is Subclass:

$\bigcup A$ is transitive.

$\blacksquare$


Sources

  • 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 10$ Some useful facts about transitivity: Theorem $10.2$
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