Inverse of Order Isomorphism is Order Isomorphism

Theorem
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi$ be a bijection from $\left({S, \preceq_1}\right)$ to $\left({T, \preceq_2}\right)$.

Then:
 * $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$

is an order isomorphism iff:
 * $\phi^{-1}: \left({T, \preceq_2}\right) \to \left({S, \preceq_1}\right)$

is also an order isomorphism.

Proof 1
Follows directly from the definition of order isomorphism.

Proof 2
A poset is a relational structure where order isomorphism is a special case of relation isomorphism.

The result follows directly from Inverse of Relation Isomorphism.