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Metamath Proof Explorer

Theorem upgr0e

Description: The empty graph, with vertices but no edges, is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 11-Oct-2020) (Proof shortened by AV, 25-Nov-2020)

	Ref	Expression
Hypotheses	umgr0e.g	$⊢ φ \to G \in W$
	umgr0e.e	$⊢ φ \to iEdg (G) = \emptyset$
Assertion	upgr0e	$⊢ φ \to G \in UPGraph$

Proof

Step	Hyp	Ref	Expression
1		umgr0e.g	$⊢ φ \to G \in W$
2		umgr0e.e	$⊢ φ \to iEdg (G) = \emptyset$
3	1 2	umgr0e	$⊢ φ \to G \in UMGraph$
4		umgrupgr	$⊢ G \in UMGraph \to G \in UPGraph$
5	3 4	syl	$⊢ φ \to G \in UPGraph$