Theorems · Theorem · order theory
LowerSet.sdiff_sup_lowerClosure
∀ {α : Type u_1} [inst : Preorder α] {s : LowerSet α} {t : Set α},
t ⊆ ↑s → (∀ b ∈ s, ∀ c ∈ t, c ≤ b → b ∈ t) → s.sdiff t ⊔ lowerClosure t = s- Defined in
- Mathlib.Order.UpperLower.Closure
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 63 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.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- SetLike.coestatement and proof · cited by 8,199
- Preorderstatement and proof · cited by 7,952
- le_antisymmproof · cited by 2,068
- OrderDual.toDualproof · cited by 481
- LowerSetstatement and proof · cited by 230
- sup_leproof · cited by 159
- Set.subset_union_leftproof · cited by 142
- Set.subset_union_rightproof · cited by 123
- emproof · cited by 115
- upperClosureproof · cited by 84
Cited by2
Results whose statement or proof uses this declaration.
- LowerSet.erase_sup_Iicproof · cited by 2
- LowerSet.lowerClosure_sup_sdiffproof · cited by 0