exists_countable_separating

theorem exists_countable_separating : ∀ (α : Type u_1) (p : Set α → Prop) (t : Set α) [h : HasCountableSeparatingOn α p t], ∃ S, S.Countable ∧ (∀ s ∈ S, p s) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y

The theorem `exists_countable_separating` states that for any type `α` and any property `p` on sets of `α`, along with any set `t` of `α` that has a countable separating sequence based on `p`, there exists a countable set `S` such that every element in `S` satisfies `p` and for any two elements `x` and `y` in `t`, if they belong to the same sets in `S`, then `x` equals `y`. In other words, `S` is a countable collection of sets that separates elements of `t` based on their membership in these sets and each set in `S` satisfies the property `p`.

More concisely: Given a type `α`, a property `p` on sets of `α`, and a countable separating sequence `t` for `p`, there exists a countable collection `S` of sets satisfying `p` such that elements in the same sets of `S` are equal.