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Metamath Proof Explorer

Theorem vtxdumgr0nedg

Description: If a vertex in a multigraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017) (Revised by AV, 12-Dec-2020) (Proof shortened by AV, 15-Dec-2020)

Ref Expression
Hypotheses vtxdushgrfvedg.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
vtxdushgrfvedg.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
vtxdushgrfvedg.d ⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
Assertion vtxdumgr0nedg ( ( 𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉 ∧ ( 𝐷 ‘ 𝑈 ) = 0 ) → ¬ ∃ 𝑣 ∈ 𝑉 { 𝑈 , 𝑣 } ∈ 𝐸 )

Proof

Step Hyp Ref Expression
1 vtxdushgrfvedg.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 vtxdushgrfvedg.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
3 vtxdushgrfvedg.d ⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
4 umgruhgr ⊢ ( 𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
5 1 2 3 vtxduhgr0nedg ⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉 ∧ ( 𝐷 ‘ 𝑈 ) = 0 ) → ¬ ∃ 𝑣 ∈ 𝑉 { 𝑈 , 𝑣 } ∈ 𝐸 )
6 4 5 syl3an1 ⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉 ∧ ( 𝐷 ‘ 𝑈 ) = 0 ) → ¬ ∃ 𝑣 ∈ 𝑉 { 𝑈 , 𝑣 } ∈ 𝐸 )