Theorems · Theorem · order theory
Directed.exists_mem_subset_of_finset_subset_biUnion
∀ {α : Type u_1} {ι : Type u_2} [Nonempty ι] {f : ι → Set α},
Directed (fun x1 x2 => x1 ⊆ x2) f → ∀ {s : Finset α}, ↑s ⊆ ⋃ i, f i → ∃ i, ↑s ⊆ f i- Defined in
- Mathlib.Data.Set.Finite.Lattice
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Nonempty
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Finsetstatement and proof · cited by 13,712
- SetLike.coestatement and proof · cited by 8,199
- LE.le.transproof · cited by 3,151
- Set.iUnionstatement and proof · cited by 2,483
- Finset.consproof · cited by 221
- Directedstatement and proof · cited by 213
- Finset.coe_emptyproof · cited by 109
- Finset.cons_inductionproof · cited by 85
- Finset.coe_consproof · cited by 18
Cited by3
Results whose statement or proof uses this declaration.
- DirectedOn.exists_mem_subset_of_finset_subset_biUnionproof · cited by 3
- FirstOrder.Language.Theory.isSatisfiable_directed_union_iffproof · cited by 1