User:Dfeuer/Binary Union of Sets

Theorem
Let $a$ and $b$ be sets.

Then $a \cup b = \bigcup \{a, b\}$.

Proof
First suppose that $x \in a \cup b$.

By the definition of binary union, $x \in a \lor x \in b$.

By the definition of unordered pair, $a \in \{a, b\}$ and $b \in \{a, b\}$.

If $x \in a$, then since $a \in \{a,b\}$, $x \in \bigcup \{a, b\}$ by the definition of union.

If $x \in b$, then since $b \in \{a,b\}$, $x \in \bigcup \{a, b\}$ by the definition of union.

Suppose instead that $x \in \bigcup \{a, b\}$.

By the definition of union, $\exists p: p \in \{a, b\} \land x \in p$.

By the definition of unordered pair, $p = a$ or $p = b$.

Thus $x \in a$ or $x \in b$.

Thus by the definition of binary union, $x \in a \cup b$.