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Metamath Proof Explorer

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 ffvelcdmda 
 |-  ( ( ( 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 ) =/= (/) )