This paper gives the first proof that the subtyping relation of a higher-order lambda calculus, ${\cal F}^{\omega}_{\leq}$, is anti-symmetric, establishing in the process that the subtyping relation is a partial order – reflexive, transitive, and anti-symmetric up to $\beta$-equality. While a subtyping relation is reflexive and transitive by definition, anti-symmetry is a derived property. The result, which may seem obvious to the non-expert, is technically challenging, and had been an open problem for almost a decade. In this context, typed operational semantics for subtyping, and the logical relation used to prove its equivalence with the declarative presentation of ${\cal F}^{\omega}_{\leq}$, offers a powerful new technology to solve the problem: of particular importance is our extended rule for the well-formedness of types with head variables. The paper also gives a presentation of ${\cal F}^{\omega}_{\leq}$ without a relation for $\beta$-equality, which is apparently the first such, and shows its equivalence with the traditional presentation.