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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 → 𝐺 ∈ 𝑊 )
	umgr0e.e	⊢ ( 𝜑 → ( iEdg ‘ 𝐺 ) = ∅ )
Assertion	upgr0e	⊢ ( 𝜑 → 𝐺 ∈ UPGraph )

Proof

Step	Hyp	Ref	Expression
1		umgr0e.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
2		umgr0e.e	⊢ ( 𝜑 → ( iEdg ‘ 𝐺 ) = ∅ )
3	1 2	umgr0e	⊢ ( 𝜑 → 𝐺 ∈ UMGraph )
4		umgrupgr	⊢ ( 𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph )
5	3 4	syl	⊢ ( 𝜑 → 𝐺 ∈ UPGraph )