Worked on the leanprover-community/mathlib4 repository to formalize a triangle-free property for Hasse diagrams in preorder structures, strengthening the order-theory foundation. Developed a reusable theorem within the SimpleGraph namespace, rigorously proving that no three elements in such diagrams can be pairwise adjacent. The approach involved a comprehensive eight-case proof split, addressing both transitive and cyclic scenarios to ensure completeness. All work was implemented in Lean, leveraging formal proof techniques and graph theory abstractions. Detailed documentation and commit traceability were provided, supporting future formalizations in combinatorics and order theory while enhancing correctness guarantees for downstream mathematical applications.