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Database GRAPH THEORY Undirected graphs Vertex degree usgrvd00
Description: If every vertex in a simple graph has degree 0, there is no edge in the
graph. (Contributed by Alexander van der Vekens , 12-Jul-2018)
(Revised by AV , 17-Dec-2020) (Proof shortened by AV , 23-Dec-2020)
Ref
Expression
Hypotheses
vtxdusgradjvtx.v |- V = ( Vtx ` G )
vtxdusgradjvtx.e |- E = ( Edg ` G )
Assertion
usgrvd00 |- ( G e. USGraph -> ( A. v e. V ( ( VtxDeg ` G ) ` v ) = 0 -> E = (/) ) )
Proof
Step
Hyp
Ref
Expression
1
vtxdusgradjvtx.v |- V = ( Vtx ` G )
2
vtxdusgradjvtx.e |- E = ( Edg ` G )
3
usgruhgr |- ( G e. USGraph -> G e. UHGraph )
4
1 2
uhgrvd00 |- ( G e. UHGraph -> ( A. v e. V ( ( VtxDeg ` G ) ` v ) = 0 -> E = (/) ) )
5
3 4
syl |- ( G e. USGraph -> ( A. v e. V ( ( VtxDeg ` G ) ` v ) = 0 -> E = (/) ) )