Theorem umgrnloop2

Description: A multigraph has no loops. (Contributed by AV, 27-Oct-2020) (Revised by AV, 30-Nov-2020)

	Ref	Expression
Assertion	umgrnloop2	⊢ ( 𝐺 ∈ UMGraph → { 𝑁 , 𝑁 } ∉ ( Edg ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
3	1 2	umgrpredgv	⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝑁 , 𝑁 } ∈ ( Edg ‘ 𝐺 ) ) → ( 𝑁 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) )
4	3	simpld	⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝑁 , 𝑁 } ∈ ( Edg ‘ 𝐺 ) ) → 𝑁 ∈ ( Vtx ‘ 𝐺 ) )
5		eqid	⊢ 𝑁 = 𝑁
6	2	umgredgne	⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝑁 , 𝑁 } ∈ ( Edg ‘ 𝐺 ) ) → 𝑁 ≠ 𝑁 )
7		eqneqall	⊢ ( 𝑁 = 𝑁 → ( 𝑁 ≠ 𝑁 → ¬ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) )
8	5 6 7	mpsyl	⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝑁 , 𝑁 } ∈ ( Edg ‘ 𝐺 ) ) → ¬ 𝑁 ∈ ( Vtx ‘ 𝐺 ) )
9	4 8	pm2.65da	⊢ ( 𝐺 ∈ UMGraph → ¬ { 𝑁 , 𝑁 } ∈ ( Edg ‘ 𝐺 ) )
10		df-nel	⊢ ( { 𝑁 , 𝑁 } ∉ ( Edg ‘ 𝐺 ) ↔ ¬ { 𝑁 , 𝑁 } ∈ ( Edg ‘ 𝐺 ) )
11	9 10	sylibr	⊢ ( 𝐺 ∈ UMGraph → { 𝑁 , 𝑁 } ∉ ( Edg ‘ 𝐺 ) )