Metamath Proof Explorer


Theorem upgr0eop

Description: The empty graph, with vertices but no edges, is a pseudograph. The empty graph is actually a simple graph, see usgr0eop , and therefore also a multigraph ( G e. UMGraph ). (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 11-Oct-2020)

Ref Expression
Assertion upgr0eop
|- ( V e. W -> <. V , (/) >. e. UPGraph )

Proof

Step Hyp Ref Expression
1 opex
 |-  <. V , (/) >. e. _V
2 1 a1i
 |-  ( V e. W -> <. V , (/) >. e. _V )
3 0ex
 |-  (/) e. _V
4 opiedgfv
 |-  ( ( V e. W /\ (/) e. _V ) -> ( iEdg ` <. V , (/) >. ) = (/) )
5 3 4 mpan2
 |-  ( V e. W -> ( iEdg ` <. V , (/) >. ) = (/) )
6 2 5 upgr0e
 |-  ( V e. W -> <. V , (/) >. e. UPGraph )