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Metamath Proof Explorer

Theorem umgrnloop

Description: In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017) (Revised by AV, 11-Dec-2020)

Ref Expression
Hypothesis umgrnloopv.e ⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion umgrnloop ( 𝐺 ∈ UMGraph → ( ∃ 𝑥 ∈ dom 𝐸 ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 umgrnloopv.e ⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3 1 2 umgredgprv ⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) ) )
4 3 imp ⊢ ( ( ( 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) )
5 1 umgrnloopv ⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑀 ∈ ( Vtx ‘ 𝐺 ) ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )
6 5 ex ⊢ ( 𝐺 ∈ UMGraph → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) ) )
7 6 com23 ⊢ ( 𝐺 ∈ UMGraph → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → 𝑀 ≠ 𝑁 ) ) )
8 7 adantr ⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → 𝑀 ≠ 𝑁 ) ) )
9 8 imp ⊢ ( ( ( 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → 𝑀 ≠ 𝑁 ) )
10 9 com12 ⊢ ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) → ( ( ( 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → 𝑀 ≠ 𝑁 ) )
11 10 adantr ⊢ ( ( 𝑀 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → ( ( ( 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → 𝑀 ≠ 𝑁 ) )
12 4 11 mpcom ⊢ ( ( ( 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom 𝐸 ) ∧ ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } ) → 𝑀 ≠ 𝑁 )
13 12 rexlimdva2 ⊢ ( 𝐺 ∈ UMGraph → ( ∃ 𝑥 ∈ dom 𝐸 ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )