Theorems · Theorem · order theory
DirectedOn.csSup_le
∀ {α : Type u_2} [inst : ConditionallyCompletePartialOrderSup α] {s : Set α} {a : α},
DirectedOn (fun x1 x2 => x1 ≤ x2) s → s.Nonempty → (∀ b ∈ s, b ≤ a) → sSup s ≤ a- Cited by
- 4 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- SupSet.sSupstatement · cited by 954
- DirectedOnstatement and proof · cited by 271
- ConditionallyCompletePartialOrderSupstatement and proof · cited by 52
- DirectedOn.isLUB_csSupproof · cited by 9
Cited by4
Results whose statement or proof uses this declaration.
- Directed.ciSup_leproof · cited by 2
- DirectedOn.csSup_eq_of_forall_le_of_forall_lt_exists_gtproof · cited by 1
- DirectedOn.csSup_le_csSupproof · cited by 0
- DirectedOn.le_csSup_iffproof · cited by 0