Union of Class is Subclass implies Class is Transitive

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Theorem

Let $A$ be a class.

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


Let:

$\bigcup A \subseteq A$

Then $A$ is transitive.


Proof

Let $\bigcup A \subseteq A$.


Let $x \in \bigcup A$.

Then by definition:

$\exists y \in A: x \in y$

By definition of subclass:

$x \in A$

Thus we have that:

$x \in y \land y \in A \implies x \in A$

It follows by definition that $A$ is a transitive class.

$\blacksquare$


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