  The subtyping rules for intersection types traditionally employ a
transitivity rule (Barendregt et al. 1983), which means that subtyping does not
satisfy the subformula property, making it more difficult to use in filter
models for compiler verification. Laurent develops a sequent-style subtyping
system, without transitivity, and proves transitivity via a sequence of six
lemmas that culminate in cut-elimination (2018). This article develops a
subtyping system in regular style that omits transitivity and provides a direct
proof of transitivity, significantly reducing the length of the proof,
exchanging the six lemmas for just one. Inspired by Laurent's system, the rule
for function types is essentially the $\beta$-soundness property. The new
system satisfies the "subformula conjunction property": every type occurring in
the derivation of $A <: B$ is a subformula of $A$ or $B$, or an intersection of
such subformulas. The article proves that the new subtyping system is
equivalent to that of Barendregt, Coppo, and Dezani-Ciancaglini.
