Prove that every complete irreflexive and transitive

Question

1. Prove that every complete, irreflexive, and transitive relation is asymmetric: if

x ≠ y, then x > y if and only if yx.

2. Prove that the Gale-Shapley algorithm for finding a stable matching satisfies the following property: if Cleopatra asks Mark to stay in front of her house at stage k, and at a later stage she asks Julius to stay in front of her house, then she prefers Julius to Mark.