Theorems · Theorem · order theory
sSup_eq_bot
∀ {α : Type u_1} [inst : CompleteLattice α] {s : Set α}, sSup s = ⊥ ↔ ∀ a ∈ s, a = ⊥- Defined in
- Mathlib.Order.CompleteLattice.Basic
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 10 from the axioms · uses no axioms
- Assumes
- CompleteLattice
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
- Bot.botstatement and proof · cited by 4,720
- CompleteLatticestatement and proof · cited by 1,048
- SupSet.sSupstatement and proof · cited by 954
- le_bot_iffproof · cited by 116
- le_sSupproof · cited by 79
- bot_uniqueproof · cited by 57
- sSup_leproof · cited by 35
Cited by7
Results whose statement or proof uses this declaration.
- iSup_eq_botproof · cited by 5
- LieAlgebra.hasTrivialRadical_of_no_solvable_idealsproof · cited by 1
- sSup_eq_bot'proof · cited by 1
- LieAlgebra.HasTrivialRadical.eq_bot_of_isSolvableproof · cited by 1
- eq_singleton_bot_of_sSup_eq_bot_of_nonemptyproof · cited by 1
- Set.sUnion_eq_emptyproof · cited by 0
- ENat.sSup_eq_zeroproof · cited by 0