Theorems · Theorem · order theory
DirectedOn.le_csInf_iff
∀ {α : Type u_1} [inst : ConditionallyCompletePartialOrderInf α] {s : Set α} {a : α},
DirectedOn (fun x1 x2 => x2 ≤ x1) s → BddBelow s → s.Nonempty → (a ≤ sInf s ↔ ∀ b ∈ s, a ≤ b)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses propext, Quot.sound
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Cites8
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
- InfSet.sInfstatement · cited by 935
- BddBelowstatement and proof · cited by 401
- DirectedOnstatement and proof · cited by 271
- ConditionallyCompletePartialOrderInfstatement and proof · cited by 51
- le_isGLB_iffproof · cited by 17
- DirectedOn.isGLB_csInfproof · cited by 8
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