Partitioning Pairs of Countable Sets of Ordinals

Lemma 1. There are ω1 disjoint stationary subsets of ω1. Proof. Let Cub(ω1) = {X ⊆ ω1 : ∃C ⊆ X(C is club on ω1)}. Then Cub(ω1) is countably complete filter. Its dual ideal Cub (ω1) = {X ⊆ ω1 : ∃X ′ ∈ Cub(ω1)(X = ω1\X ′)} is countably complete and contains all singletons and so all countable subsets of ω1. Recall also that X ⊆ ω1 is stationary if and only if X / ∈ Cub(ω1). For every ρ < ω1 let fρ : ρ → ω be an injective mapping. Then ∀α < ω1 ∀n ∈ ω let X α = {ρ < ω1 : α < ρ and fρ(α) = n}. Note that if α 6= β then for every n ∈ ω we have X α ∩ X β = ∅ (otherwise ∃ρ < κ greater than α, β such that fρ(α) = fρ(β) = n which is a contradiction to fρ being injective). Also for every α < ω1 ∪n∈ωX α = {ρ < ω1 : α < ρ} ∈ Cub(ω1).