Theorems · Theorem · general topology
Metric.ne_of_mem_sphere
∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, y ∈ Metric.sphere x ε → ε ≠ 0 → y ≠ x- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
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 · cited by 53,352
- Realstatement and proof · cited by 25,697
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.spherestatement and proof · cited by 371
- dist_selfproof · cited by 116
Cited by5
Results whose statement or proof uses this declaration.
- Metric.sphere_eq_empty_of_subsingletonproof · cited by 3
- AnalyticAt.eventually_constant_or_nhds_le_map_nhds_auxproof · cited by 1
- AffineSpace.cobounded_eq_iSup_sphere_asymptoticNhdsproof · cited by 1
- Complex.norm_max_aux₁proof · cited by 1
- isBounded_iff_asymptoticCone_subset_singletonproof · cited by 1