Theorems · Theorem · order theory
CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
∀ {α : Type u_3} [inst : CompleteLattice α] {k : α},
IsCompactElement k → ∀ {s : Set α}, s.Nonempty → DirectedOn (fun x1 x2 => x1 ≤ x2) s → (∀ x ∈ s, x < k) → sSup s < kA compact element k has the property that any directed set lying strictly below k has
its sSup strictly below k.
- Defined in
- Mathlib.Order.CompactlyGenerated.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CompleteLattice
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- Set.Nonemptystatement and proof · cited by 2,627
- LT.lt.leproof · cited by 2,189
- CompleteLatticestatement and proof · cited by 1,048
- SupSet.sSupstatement and proof · cited by 954
- LT.lt.neproof · cited by 872
- Eq.leproof · cited by 605
- LE.le.antisymmproof · cited by 507
- DirectedOnstatement and proof · cited by 271
- sSup_leproof · cited by 35
- IsCompactElementstatement and proof · cited by 34
- eq_iff_le_not_ltproof · cited by 11
Cited by1
Results whose statement or proof uses this declaration.
- CompleteLattice.Iic_coatomic_of_compact_elementproof · cited by 1