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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	\|- ( ph -> G e. W )
	umgr0e.e	\|- ( ph -> ( iEdg ` G ) = (/) )
Assertion	upgr0e	\|- ( ph -> G e. UPGraph )

Proof

Step	Hyp	Ref	Expression
1		umgr0e.g	\|- ( ph -> G e. W )
2		umgr0e.e	\|- ( ph -> ( iEdg ` G ) = (/) )
3	1 2	umgr0e	\|- ( ph -> G e. UMGraph )
4		umgrupgr	\|- ( G e. UMGraph -> G e. UPGraph )
5	3 4	syl	\|- ( ph -> G e. UPGraph )