Sorting by Exchanging

We adopt the following convention: α, β, γ, δ denote ordinal numbers, k denotes a natural number, and x, y, z, t, X, Y , Z denote sets. The following propositions are true: (1) x ∈ (α+ β) \ α iff there exists γ such that x = α+ γ and γ ∈ β. (2) Suppose α ∈ β and γ ∈ δ. Then γ 6= α and γ 6= β and δ 6= α and δ 6= β or γ ∈ α and δ = α or γ ∈ α and δ = β or γ = α and δ ∈ β or γ = α and δ = β or γ = α and β ∈ δ or α ∈ γ and δ = β or γ = β and β ∈ δ. (3) If x 6∈ y, then (y ∪ {x}) \ y = {x}. (4) succx \ x = {x}. (5) Let f be a function, r be a binary relation, and given x. Then x ∈ f◦r if and only if there exist y, z such that 〈y, z〉 ∈ r and 〈y, z〉 ∈ dom f and f(y, z) = x. (6) If α \β 6= ∅, then inf(α \β) = β and sup(α \β) = α and ⋃ (α \β) = ⋃ α.