Realizability for Constructive Zermelo-Fraenkel Set Theory

2. If B is a regular set, then B ∩ V(A) is a set.Proof: (1): Let x, y ∈ B. Then f : 2 → B, where f(0) = x, f(1) = y. Hence, byregularity of B, the range of f is in B, that is {x, y} ∈ B.As 〈x, y〉 = {{x}, {x, y}}, 〈x, y〉 ∈ B follows from closure under unordered pairs.(ii): To see this let κ = rank(B), where the function rank is defined by rank(x) :=⋃ {rank(y)+1 : y ∈ x} with z +1 := z ∪{z}. One easily shows that for all sets x, rank(x)is an ordinal. Let Φ be the inductive definition withxΦ iff ∀u∈a (u ∈ |A| × x).aInvoking Lemma 3.4, let J be the class such that V(A) = ⋃ α J α , and for each α,J α = Γ Φ ( ⋃ β∈αJ β ).Moreover, define the operation Υ byΥ(X) := {u ∈ B : u ⊆ |A| × X}and by recursion on α setLetΥ α = Υ( ⋃ β∈αΥ β ).Υ