Theorems · Theorem · order theory
Set.subsingleton_iff_singleton
∀ {α : Type u} {s : Set α} {x : α}, x ∈ s → (s.Subsingleton ↔ s = {x})- Defined in
- Mathlib.Data.Set.Subsingleton
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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.Subsingletonstatement and proof · cited by 276
- Set.subsingleton_singletonproof · cited by 36
- Set.Subsingleton.eq_singleton_of_memproof · cited by 16
Cited by11
Results whose statement or proof uses this declaration.
- Set.eq_singleton_or_nontrivialproof · cited by 5
- totallyDisconnectedSpace_iff_connectedComponent_singletonproof · cited by 4
- HolderOnWith.hausdorffMeasure_image_leproof · cited by 2
- AffineSubspace.subsingleton_of_subsingleton_span_eq_topproof · cited by 2
- AntilipschitzWith.hausdorffMeasure_preimage_leproof · cited by 2
- MeasureTheory.Measure.hausdorffMeasure_le_one_of_subsingletonproof · cited by 1
- AffineSubspace.eq_univ_of_subsingleton_span_eq_topproof · cited by 1
- AddAction.IsPreprimitive.of_isTrivialBlock_of_notMem_fixedPointsproof · cited by 1
- MulAction.IsPreprimitive.of_isTrivialBlock_of_notMem_fixedPointsproof · cited by 1
- SimpleGraph.exists_adj_isEdgeReachable_twoproof · cited by 1
- Continuous.image_connectedComponent_eq_singletonproof · cited by 1