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 )