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Metamath Proof Explorer

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	$⊢ G \in USGraph \to (\exists x \in dom E E (x) = \{M, N\} \to M \neq N)$

Proof

Step	Hyp	Ref	Expression
1		usgrnloopv.e	$⊢ E = iEdg (G)$
2		usgrumgr	$⊢ G \in USGraph \to G \in UMGraph$
3	1	umgrnloop	$⊢ G \in UMGraph \to (\exists x \in dom E E (x) = \{M, N\} \to M \neq N)$
4	2 3	syl	$⊢ G \in USGraph \to (\exists x \in dom E E (x) = \{M, N\} \to M \neq N)$