Theorems · Theorem · order theory
lowerClosure_singleton
∀ {α : Type u_1} [inst : Preorder α] (a : α), lowerClosure {a} = LowerSet.Iic a- Defined in
- Mathlib.Order.UpperLower.Closure
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 11 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Preorderstatement and proof · cited by 7,952
- Set.extproof · cited by 2,266
- SetLike.mem_coeproof · cited by 302
- LowerSetstatement · cited by 230
- Set.mem_singleton_iffproof · cited by 172
- lowerClosurestatement · cited by 83
- LowerSet.Iicstatement · cited by 37
- Set.mem_Iicproof · cited by 37
- LowerSet.extproof · cited by 29
- mem_lowerClosureproof · cited by 5
Cited by8
Results whose statement or proof uses this declaration.
- Topology.IsUpperSet.closure_singletonproof · cited by 3
- UpperSet.sdiff_singletonproof · cited by 2
- LowerSet.erase_sup_Iicproof · cited by 2
- Finset.truncatedSup_singletonproof · cited by 1
- Topology.IsLowerSet.nhdsKer_singletonproof · cited by 1
- Topology.IsScott.closure_singletonproof · cited by 0
- lowerClosure_oneproof · cited by 0
- lowerClosure_zeroproof · cited by 0