Union of Transitive Class is Subclass

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Theorem

Let $A$ be a transitive class.

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


Then:

$\ds \bigcup A \subseteq A$


Proof

Let $A$ be transitive.

Let $x \in \ds \bigcup A$.

Then by definition:

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

By definition of transitive class:

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

and so:

$x \in A$

Hence the result by definition of subclass.

$\blacksquare$


Sources