let formula_12, and let "t" be any constructible subset of "s". then there is some β with formula_13, so formula_14, for some formula φ and some formula_15 drawn from formula_16. by the downward löwenheim–skolem theorem and mostowski collapse, there must be some transitive set "k" containing formula_17 and some formula_18, and having the same first-order theory as formula_16 with the formula_18 substituted for the formula_15; and this "k" will have the same cardinal as formula_17. since formula_23 is true in formula_16, it is also true in "k", so formula_25 for some γ having the same cardinal as α. and formula_26 because formula_16 and formula_28 have the same theory. so "t" is in fact in formula_29.