Metamath Proof Explorer


Theorem usgr0eop

Description: The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 16-Oct-2020)

Ref Expression
Assertion usgr0eop
|- ( V e. W -> <. V , (/) >. e. USGraph )

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 usgr0e
 |-  ( V e. W -> <. V , (/) >. e. USGraph )