Theorems · Theorem · order theory
sSup_empty
∀ {α : Type u_1} [inst : CompleteLattice α], sSup ∅ = ⊥- Defined in
- Mathlib.Order.CompleteLattice.Basic
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
- Assumes
- CompleteLattice
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Bot.botstatement · cited by 4,720
- CompleteLatticestatement and proof · cited by 1,048
- SupSet.sSupstatement · cited by 954
- IsLUB.sSup_eqproof · cited by 17
- isLUB_emptyproof · cited by 1
Cited by19
Results whose statement or proof uses this declaration.
- Set.sUnion_emptyproof · cited by 17
- iSup_of_emptyproof · cited by 5
- MonotoneOn.map_sSup_of_continuousWithinAtproof · cited by 3
- DirectedOn.inf_sSup_eqproof · cited by 3
- sSup_ne_of_notMemproof · cited by 3
- sSupIndep_singletonproof · cited by 3
- IsCompactlyGenerated.BooleanGenerators.mem_of_isAtom_of_le_sSup_atomsproof · cited by 3
- TopologicalSpace.NoetherianSpace.exists_finite_set_closeds_irreducibleproof · cited by 2
- SimpleGraph.isAcyclic_sSup_of_isAcyclic_directedOnproof · cited by 1
- Subspace.dualAnnihilator_iInf_eqproof · cited by 1
- Set.exists_seq_iSup_eq_top_iff_countableproof · cited by 1
- CompleteSublattice.bot_memproof · cited by 0