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