Theorems · Theorem · order theory
csSup_le_iff
∀ {α : Type u_1} [inst : ConditionallyCompleteLattice α] {s : Set α} {a : α},
BddAbove s → s.Nonempty → (sSup s ≤ a ↔ ∀ b ∈ s, b ≤ a)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 10 from the axioms · uses propext, Quot.sound
- Assumes
- ConditionallyCompleteLattice
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
- Set.Nonemptystatement and proof · cited by 2,627
- SupSet.sSupstatement · cited by 954
- BddAbovestatement and proof · cited by 620
- ConditionallyCompleteLatticestatement and proof · cited by 364
- isLUB_csSupproof · cited by 34
- isLUB_le_iffproof · cited by 24
Cited by2
Results whose statement or proof uses this declaration.
- Finset.Nonempty.csSup_eq_max'proof · cited by 5
- Finset.sup'_eq_csSup_imageproof · cited by 4