Theorems · Theorem · order theory
IsUpperSet.Ici_subset
∀ {α : Type u_1} [inst : Preorder α] {s : Set α}, IsUpperSet s → ∀ ⦃a : α⦄, a ∈ s → Set.Ici a ⊆ sAlias of the forward direction of isUpperSet_iff_Ici_subset.
- Defined in
- Mathlib.Order.UpperLower.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- Preorderstatement and proof · cited by 7,952
- Set.Icistatement · cited by 1,070
- IsUpperSetstatement · cited by 148
- isUpperSet_iff_Ici_subsetproof · cited by 1
Cited by6
Results whose statement or proof uses this declaration.
- IsUpperSet.ordConnectedproof · cited by 4
- Topology.IsScott.isOpen_iff_isUpperSet_and_dirSupInaccOnproof · cited by 2
- UpperSet.le_Iciproof · cited by 1
- Topology.IsLower.closure_singletonproof · cited by 1
- IsUpperSet.Ioi_subsetproof · cited by 1
- OrderEmbedding.image_Iciproof · cited by 1