Theorems · Theorem · order theory
isLowerSet_compl
∀ {α : Type u_1} [inst : LE α] {s : Set α}, IsLowerSet sᶜ ↔ IsUpperSet s- Defined in
- Mathlib.Order.UpperLower.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- LE
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Compl.complstatement and proof · cited by 2,925
- compl_complproof · cited by 229
- IsLowerSetstatement and proof · cited by 167
- IsUpperSetstatement and proof · cited by 148
- IsUpperSet.complproof · cited by 9
- IsLowerSet.complproof · cited by 7
Cited by6
Results whose statement or proof uses this declaration.
- IsUpperSet.interiorproof · cited by 4
- stableUnderSpecialization_compl_iffproof · cited by 2
- IsUpperSet.card_inter_le_finsetproof · cited by 2
- Topology.IsUpperSet.isClosed_iff_isLowerproof · cited by 1
- Topology.IsLower.isUpperSet_of_isClosedproof · cited by 1
- IsUpperSet.le_card_inter_finsetproof · cited by 0