Theorems · Theorem · order theory
lowerClosure_min
∀ {α : Type u_1} [inst : Preorder α] {s t : Set α}, s ⊆ t → IsLowerSet t → ↑(lowerClosure s) ⊆ t- Defined in
- Mathlib.Order.UpperLower.Closure
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 11 from the axioms · uses no axioms
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- SetLike.coestatement and proof · cited by 8,199
- Preorderstatement and proof · cited by 7,952
- LowerSetstatement · cited by 230
- IsLowerSetstatement and proof · cited by 167
- lowerClosurestatement and proof · cited by 83
Cited by6
Results whose statement or proof uses this declaration.
- lowerClosure_leproof · cited by 3
- Topology.IsUpperSet.closure_eq_lowerClosureproof · cited by 3
- IsLowerSet.lowerClosureproof · cited by 2
- closure_lowerClosure_comm_piproof · cited by 0
- lowerClosure_interior_subsetproof · cited by 0
- lowerClosure_interior_subset'proof · cited by 0