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 ) )