  It is easily shown that Δ is a subset of A which is definable in A by a formula of the language of set theory. Also, each subset Δ′ ⊆ Δ such that Δ′ ∈ A has a model. For the set C of all a ∈ A for which a occurs in Δ′ belongs to A — since Δ′ does — and so, if we interpret c as any member of the (necessarily nonempty) set A − C, then A is a model of Δ′. Accordingly, (5.5) implies that Δ has a model ⟨B, E⟩. If we interpret each constant a as the element a ∈ A, then ⟨B, E⟩ is a proper end-extension of A. The proof is complete.