Theorems · Theorem · order theory
eq_singleton_bot_of_sSup_eq_bot_of_nonempty
∀ {α : Type u_1} [inst : CompleteLattice α] {s : Set α}, sSup s = ⊥ → s.Nonempty → s = {⊥}- Defined in
- Mathlib.Order.CompleteLattice.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 24 from the axioms · uses propext, Quot.sound
- Assumes
- CompleteLattice
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 and proof · cited by 53,352
- Bot.botstatement and proof · cited by 4,720
- Set.Nonemptystatement and proof · cited by 2,627
- CompleteLatticestatement and proof · cited by 1,048
- SupSet.sSupstatement and proof · cited by 954
- sSup_eq_botproof · cited by 7
- Set.eq_singleton_iff_nonempty_unique_memproof · cited by 5
Cited by1
Results whose statement or proof uses this declaration.
- CompleteLattice.IsSupFiniteCompact.isSupClosedCompactproof · cited by 1