Theorems · Theorem · order theory
subset_lowerClosure
∀ {α : Type u_1} [inst : Preorder α] {s : Set α}, s ⊆ ↑(lowerClosure s)- Defined in
- Mathlib.Order.UpperLower.Closure
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 10 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 · cited by 8,199
- Preorderstatement and proof · cited by 7,952
- le_rflproof · cited by 1,558
- LowerSetstatement · cited by 230
- lowerClosurestatement · cited by 83
Cited by12
Results whose statement or proof uses this declaration.
- Set.OrdConnected.upperClosure_inter_lowerClosureproof · cited by 5
- lowerClosure_leproof · cited by 3
- Topology.IsUpperSet.closure_eq_lowerClosureproof · cited by 3
- IsLowerSet.lowerClosureproof · cited by 2
- LowerSet.sdiff_sup_lowerClosureproof · cited by 2
- IsUpperSet.disjoint_lowerClosure_leftproof · cited by 1
- upperBounds_lowerClosureproof · cited by 1
- Finset.truncatedSup_of_isAntichainproof · cited by 0
- lowerClosure_interior_subsetproof · cited by 0
- lowerClosure_interior_subset'proof · cited by 0
- lowerClosure_univproof · cited by 0
- closure_lowerClosure_comm_piproof · cited by 0