Theorems · Theorem · order theory

DirectedOn.subset_Icc_csInf_csSup

∀ {α : Type u_1} [inst : ConditionallyCompletePartialOrder α] {s : Set α},
  DirectedOn (fun x1 x2 => x1 ≥ x2) s →
    DirectedOn (fun x1 x2 => x1 ≤ x2) s → BddBelow s → BddAbove s → s ⊆ Set.Icc (sInf s) (sSup s)

Defined in: Mathlib.Order.ConditionallyCompletePartialOrder.Basic
Cited by: 0 results in Mathlib
Foundations: Depth 10 from the axioms · uses no axioms
Assumes: ConditionallyCompletePartialOrder

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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.Iccstatement · cited by 1,702
SupSet.sSupstatement · cited by 954
InfSet.sInfstatement · cited by 935
BddAbovestatement and proof · cited by 620
BddBelowstatement and proof · cited by 401
DirectedOnstatement and proof · cited by 271
DirectedOn.csInf_leproof · cited by 9
DirectedOn.le_csSupproof · cited by 9
ConditionallyCompletePartialOrderstatement and proof · cited by 4

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