Ordered Set with Order Type of Natural Numbers plus Dual has Minimum Element

Theorem
Let $\struct {S, \preccurlyeq}$ be an ordered structure such that:
 * $\map \ot {S, \preccurlyeq} = \omega + \omega^*$

where:
 * $\ot$ denotes order type
 * $\omega$ denotes the order type of the natural numbers $\N$
 * $\omega^*$ denotes the dual of $\omega$
 * $+$ denotes addition of order types.

Then $\struct {S, \preccurlyeq}$ has a smallest element.

Proof
By definition of order type addition:
 * $\struct {S, \preccurlyeq}$ is isomorphic to $\struct {\N, \le} \oplus \struct {\N, \ge}$

where:
 * $\cong$ denotes order isomorphism
 * $\oplus$ denotes order sum.

By the Well-Ordering Principle, $\struct {\N, \le}$ has a smallest element.

By definition of order sum, every element of $\struct {\N, \le}$ precedes every element of $\struct {\N, \ge}$.

Hence the smallest element of $\struct {\N, \le}$ is also the smallest element of $\struct {\N, \le} \oplus \struct {\N, \ge}$.

Hence the result.