Theorems · Theorem · order theory
upperClosure_singleton
∀ {α : Type u_1} [inst : Preorder α] (a : α), upperClosure {a} = UpperSet.Ici a- Defined in
- Mathlib.Order.UpperLower.Closure
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
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 · cited by 53,352
- Preorderstatement and proof · cited by 7,952
- Set.extproof · cited by 2,266
- UpperSetstatement · cited by 245
- upperClosurestatement · cited by 84
- UpperSet.Icistatement · cited by 38
- UpperSet.extproof · cited by 29
Cited by8
Results whose statement or proof uses this declaration.
- LowerSet.sdiff_singletonproof · cited by 2
- Topology.IsLowerSet.closure_singletonproof · cited by 2
- UpperSet.erase_inf_Iciproof · cited by 2
- PrimitiveSpectrum.isTopologicalBasis_relativeLowerproof · cited by 1
- Topology.IsUpperSet.nhdsKer_singletonproof · cited by 1
- upperClosure_oneproof · cited by 0
- upperClosure_zeroproof · cited by 0
- Finset.truncatedInf_singletonproof · cited by 0