Union of Subclass is Subclass of Union of Class

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Theorem

Let $A$ and $B$ be classes.

Let $\bigcup A$ and $\bigcup B$ denote the union of $A$ and union of $B$ respectively.


Let $A$ be a subclass of $B$:

$A \subseteq B$


Then $\bigcup A$ is a subclass of $\bigcup B$:

$\bigcup A \subseteq \bigcup B$


Proof

Let $x \in \bigcup A$.

Then:

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

But as $A \subseteq B$ it follows that $y \in B$.

That is:

$\exists y \in B: x \in y$

That is:

$x \in \bigcup B$

Hence the result by definition of subclass.

$\blacksquare$


Sources