Theorems · Theorem · order theory
Set.Subsingleton.eq_empty_or_singleton
∀ {α : Type u} {s : Set α}, s.Subsingleton → s = ∅ ∨ ∃ x, s = {x}- Defined in
- Mathlib.Data.Set.Subsingleton
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
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
- Set.Nonemptyproof · cited by 2,627
- Set.Subsingletonstatement and proof · cited by 276
- Set.eq_empty_or_nonemptyproof · cited by 248
- Set.Subsingleton.eq_singleton_of_memproof · cited by 16
Cited by16
Results whose statement or proof uses this declaration.
- Set.Subsingleton.induction_onproof · cited by 12
- Set.exists_eq_singleton_iff_nonempty_subsingletonproof · cited by 9
- Set.Subsingleton.isClosedproof · cited by 3
- Set.eq_empty_or_singleton_of_subsingletonproof · cited by 2
- Filter.exists_mem_singleton_mem_of_mem_of_nonempty_of_forall_separatingproof · cited by 2
- vectorSpan_of_subsingletonproof · cited by 2
- ProbabilityTheory.Kernel.iIndepSets.of_subsingletonproof · cited by 1
- Set.Subsingleton.minimal_mem_iffproof · cited by 0
- TopologicalGroup.IsSES.inducedMeasure_lt_of_injOnproof · cited by 0
- Convexity.IsConvexSet.of_subsingletonproof · cited by 0
- Projectivization.isCollinear_subsingletonproof · cited by 0
- Set.subsingleton_iff_eq_empty_or_singletonproof · cited by 0