Theorems · Theorem · order theory
lowerClosure_eq_bot_iff
∀ {α : Type u_1} [inst : Preorder α] {s : Set α}, lowerClosure s = ⊥ ↔ s = ∅- Defined in
- Mathlib.Order.UpperLower.Closure
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 64 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.
Cites8
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
- Preorderstatement and proof · cited by 7,952
- Bot.botstatement · cited by 4,720
- LowerSetstatement · cited by 230
- eq_bot_iffproof · cited by 159
- lowerClosurestatement · cited by 83
- Set.subset_empty_iffproof · cited by 40
- lowerClosure_leproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- lowerClosure_emptyproof · cited by 1