Theorems · Theorem · order theory
lowerClosure_le
∀ {α : Type u_1} [inst : Preorder α] {s : Set α} {t : LowerSet α}, lowerClosure s ≤ t ↔ s ⊆ ↑t- Defined in
- Mathlib.Order.UpperLower.Closure
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- LE.le.transproof · cited by 3,151
- LowerSetstatement and proof · cited by 230
- lowerClosurestatement and proof · cited by 83
- LowerSet.lowerproof · cited by 13
- subset_lowerClosureproof · cited by 12
- lowerClosure_minproof · cited by 6
- LowerSet.coe_subset_coeproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- gc_lowerClosure_coeproof · cited by 3
- LowerSet.sdiff_sup_lowerClosureproof · cited by 2
- lowerClosure_eq_bot_iffproof · cited by 1