Theorem usgrnloop

Description: In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017) (Proof shortened by Alexander van der Vekens, 20-Mar-2018) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

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

Proof

Step	Hyp	Ref	Expression
1		usgrnloopv.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2		usgrumgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
3	1	umgrnloop	⊢ ( 𝐺 ∈ UMGraph → ( ∃ 𝑥 ∈ dom 𝐸 ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )
4	2 3	syl	⊢ ( 𝐺 ∈ USGraph → ( ∃ 𝑥 ∈ dom 𝐸 ( 𝐸 ‘ 𝑥 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )