Theorems · Theorem · general topology
pathComponentIn_nonempty_iff
∀ {X : Type u_1} [inst : TopologicalSpace X] {x : X} {F : Set X}, (pathComponentIn F x).Nonempty ↔ x ∈ F- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
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Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.Elemproof · cited by 7,166
- Set.Nonemptystatement and proof · cited by 2,627
- unitIntervalproof · cited by 607
- Pathproof · cited by 318
- Path.sourceproof · cited by 26
- pathComponentInstatement and proof · cited by 19
- mem_pathComponentIn_selfproof · cited by 7
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