Theorem usgr0e

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) (Proof shortened by AV, 25-Nov-2020)

	Ref	Expression
Hypotheses	usgr0e.g	|- ( ph -> G e. W )
	usgr0e.e	|- ( ph -> ( iEdg ` G ) = (/) )
Assertion	usgr0e	|- ( ph -> G e. USGraph )

Proof

Step	Hyp	Ref	Expression
1		usgr0e.g	|- ( ph -> G e. W )
2		usgr0e.e	|- ( ph -> ( iEdg ` G ) = (/) )
3	2	f10d	|- ( ph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
4		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
5		eqid	|- ( iEdg ` G ) = ( iEdg ` G )
6	4 5	isusgr	|- ( G e. W -> ( G e. USGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
7	1 6	syl	|- ( ph -> ( G e. USGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
8	3 7	mpbird	|- ( ph -> G e. USGraph )