Theorem upgrn0

Description: An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	|- V = ( Vtx ` G )
	isupgr.e	|- E = ( iEdg ` G )
Assertion	upgrn0	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) =/= (/) )

Proof

Step	Hyp	Ref	Expression
1		isupgr.v	|- V = ( Vtx ` G )
2		isupgr.e	|- E = ( iEdg ` G )
3		ssrab2	|- { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } C_ ( ~P V \ { (/) } )
4	1 2	upgrfn	|- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
5	4	ffvelrnda	|- ( ( ( G e. UPGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
6	5	3impa	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
7	3 6	sselid	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. ( ~P V \ { (/) } ) )
8		eldifsni	|- ( ( E ` F ) e. ( ~P V \ { (/) } ) -> ( E ` F ) =/= (/) )
9	7 8	syl	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) =/= (/) )