Existence of Class Union

Theorem

For any classes $X, Y$, the union $X \cup Y$ exists and is unique.

That is:

$\forall X, Y: \exists! Z: \forall u: u \in Z \iff u \in X \lor u \in Y$

where $X \cup Y := Z$.

Proof

By Existence of Class Complement, there exist unique classes $\overline X, \overline Y$ such that:

$u \in \overline X \iff u \notin X$

$u \in \overline Y \iff u \notin Y$

for all sets $u$.

Then, by Existence of Class Intersection, there is a unique class $\overline X \cap \overline Y$ such that:

$u \in \overline X \cap \overline Y \iff u \in \overline X \land u \in \overline Y$

By the equivalences above:

$u \in \overline X \cap \overline Y \iff u \notin X \land u \notin Y$

Finally, by Existence of Class Complement, there is a unique class $\overline {\overline X \cap \overline Y}$ such that:

$u \in \overline {\overline X \cap \overline Y} \iff u \notin \overline X \cap \overline Y$

By the equivalence above:

$u \in \overline {\overline X \cap \overline Y} \iff \neg \paren {u \notin X \land u \notin Y}$

But then, by De Morgan's Laws:

$u \in \overline {\overline X \cap \overline Y} \iff u \in X \lor u \in Y$

Therefore, $Z := \overline {\overline X \cap \overline Y}$ exists and is unique.

$\blacksquare$