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	$$⊢\phi \to G\in W$$
	umgr0e.e	$$⊢\phi \to \mathrm{iEdg}\left(G\right)=\varnothing$$
Assertion	upgr0e	$$⊢\phi \to G\in \mathrm{UPGraph}$$

Proof

Step	Hyp	Ref	Expression
1		umgr0e.g	$$⊢\phi \to G\in W$$
2		umgr0e.e	$$⊢\phi \to \mathrm{iEdg}\left(G\right)=\varnothing$$
3	1 2	umgr0e	$$⊢\phi \to G\in \mathrm{UMGraph}$$
4		umgrupgr	$$⊢G\in \mathrm{UMGraph}\to G\in \mathrm{UPGraph}$$
5	3 4	syl	$$⊢\phi \to G\in \mathrm{UPGraph}$$