Theorems · Theorem · order theory
sSup_le_iff
∀ {α : Type u_1} [inst : CompleteSemilatticeSup α] {s : Set α} {a : α}, sSup s ≤ a ↔ ∀ b ∈ s, b ≤ a- Defined in
- Mathlib.Order.CompleteLattice.Defs
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses propext, Quot.sound
- Assumes
- CompleteSemilatticeSup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- SupSet.sSupstatement · cited by 954
- isLUB_le_iffproof · cited by 24
- isLUB_sSupproof · cited by 21
- CompleteSemilatticeSupstatement and proof · cited by 18
Cited by8
Results whose statement or proof uses this declaration.
- Set.sUnion_subset_iffproof · cited by 4
- exists_sSupIndep_disjoint_sSup_atomsproof · cited by 2
- sSup_lt_iffproof · cited by 1
- gc_sSup_Iicproof · cited by 1
- Topology.IsLower.isTopologicalSpace_basisproof · cited by 1
- Submodule.isOrtho_sSup_leftproof · cited by 1
- DedekindCut.principal_sSup_leftproof · cited by 0
- Submodule.submodule_eq_sSup_le_nonzero_spansproof · cited by 0