Union of Unordered Tuples

Theorem

Let $x_1, \dots, x_n, x_{n+1}, \dots, x_m$ be arbitrary.

Then

$\set {x_1, \dots, x_n} \cup \set {x_{n + 1}, \dots, x_m} = \set {x_1, \dots, x_n, x_{n + 1}, \dots, x_m}$

Proof

Let $a$ be arbitrary.

$a \in \set {x_1, \dots, x_n} \cup \set {x_{n + 1}, \dots, x_m}$

if and only if

$a \in \set {x_1, \dots, x_n}$ or $a \in \set {x_{n + 1}, \dots, x_m}$ by definition of union

if and only if

$a = x_1 \lor \dots \lor a = x_n$ or $a = x_{n + 1} \lor \dots \lor a = x_m$ by definition of unordered tuple

if and only if

$a = x_1 \lor \dots \lor a = x_n \lor a = x_{n + 1} \lor \dots \lor a = x_m$

if and only if

$a \in \set {x_1, \dots, x_n, x_{n + 1}, \dots, x_m}$ by definition of unordered tuple

Thus result follows by definition of set equality.

$\blacksquare$