Theorems · Theorem · order theory
DirectedOn.csInf_le_iff
∀ {α : Type u_1} [inst : ConditionallyCompletePartialOrderInf α] {s : Set α} {a : α},
DirectedOn (fun x1 x2 => x2 ≤ x1) s → BddBelow s → s.Nonempty → (sInf s ≤ a ↔ ∀ b ∈ lowerBounds s, b ≤ a)- Cited by
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- Foundations
- Depth 10 from the axioms · uses no axioms
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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
- Set.Nonemptystatement and proof · cited by 2,627
- InfSet.sInfstatement and proof · cited by 935
- BddBelowstatement and proof · cited by 401
- DirectedOnstatement and proof · cited by 271
- lowerBoundsstatement and proof · cited by 212
- ConditionallyCompletePartialOrderInfstatement and proof · cited by 51
- ge_transproof · cited by 34
- DirectedOn.csInf_leproof · cited by 9
- DirectedOn.le_csInfproof · cited by 4
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