Theorems · Theorem · logic and foundations

FirstOrder.Language.DirectLimit.cg

∀ {L : FirstOrder.Language} {ι : Type u_1} [Countable ι] [inst : Preorder ι] [inst_1 : IsDirectedOrder ι]
  [inst_2 : Nonempty ι] {G : ι → Type w} [inst_3 : (i : ι) → L.Structure (G i)]
  (f : (i j : ι) → i ≤ j → L.Embedding (G i) (G j)),
  (∀ (i : ι), FirstOrder.Language.Structure.CG L (G i)) →
    ∀ [inst_4 : DirectedSystem G fun i j h => ⇑(f i j h)],
      FirstOrder.Language.Structure.CG L (FirstOrder.Language.DirectLimit G f)

The direct limit of countably many countably generated structures is countably generated.

Defined in

Mathlib.ModelTheory.DirectLimit

Cited by

1 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CountablePreorderIsDirectedOrderNonemptyFirstOrder.Language.StructureDirectedSystem