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Metamath Proof Explorer

Theorem usgrnloopv

Description: In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

		Ref	Expression
	Hypothesis	usgrnloopv.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
	Assertion	usgrnloopv	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑊 ) → ( ( 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	umgrnloopv	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑀 ∈ 𝑊 ) → ( ( 𝐸 ‘ 𝑋 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )
4	2 3	sylan	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑊 ) → ( ( 𝐸 ‘ 𝑋 ) = { 𝑀 , 𝑁 } → 𝑀 ≠ 𝑁 ) )