Theorems · Theorem · combinatorics
Graph.vertexSet_nontrivial_of_edgeSet_nonempty_of_loopless
∀ {α : Type u_1} {β : Type u_2} {G : Graph α β} [G.Loopless], G.edgeSet.Nonempty → G.vertexSet.Nontrivial- Defined in
- Mathlib.Combinatorics.Graph.Simple
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses no axioms
- Assumes
- Graph.Loopless
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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Set.Nonemptystatement and proof · cited by 2,627
- Graphstatement and proof · cited by 242
- Set.Nontrivialstatement · cited by 145
- Graph.IsLinkproof · cited by 124
- Graph.vertexSetstatement · cited by 87
- Graph.edgeSetstatement and proof · cited by 79
- Graph.IsLink.left_memproof · cited by 14
- Graph.Looplessstatement and proof · cited by 12
- Graph.exists_isLink_of_mem_edgeSetproof · cited by 6
- Graph.IsLink.right_memproof · cited by 6
- Graph.IsLink.adjproof · cited by 5
- Graph.Adj.neproof · cited by 3
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