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
Follows directly from the definition of order isomorphism.