Theorem vtxduhgr0edgnel

Description: A vertex in a hypergraph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 24-Dec-2020)

	Ref	Expression
Hypotheses	vtxdushgrfvedg.v	|- V = ( Vtx ` G )
	vtxdushgrfvedg.e	|- E = ( Edg ` G )
	vtxdushgrfvedg.d	|- D = ( VtxDeg ` G )
Assertion	vtxduhgr0edgnel	|- ( ( G e. UHGraph /\ U e. V ) -> ( ( D ` U ) = 0 <-> -. E. e e. E U e. e ) )

Proof

Step	Hyp	Ref	Expression
1		vtxdushgrfvedg.v	|- V = ( Vtx ` G )
2		vtxdushgrfvedg.e	|- E = ( Edg ` G )
3		vtxdushgrfvedg.d	|- D = ( VtxDeg ` G )
4		eqid	|- ( iEdg ` G ) = ( iEdg ` G )
5	1 4 3	vtxd0nedgb	|- ( U e. V -> ( ( D ` U ) = 0 <-> -. E. i e. dom ( iEdg ` G ) U e. ( ( iEdg ` G ) ` i ) ) )
6	5	adantl	|- ( ( G e. UHGraph /\ U e. V ) -> ( ( D ` U ) = 0 <-> -. E. i e. dom ( iEdg ` G ) U e. ( ( iEdg ` G ) ` i ) ) )
7	4 2	uhgrvtxedgiedgb	|- ( ( G e. UHGraph /\ U e. V ) -> ( E. i e. dom ( iEdg ` G ) U e. ( ( iEdg ` G ) ` i ) <-> E. e e. E U e. e ) )
8	7	notbid	|- ( ( G e. UHGraph /\ U e. V ) -> ( -. E. i e. dom ( iEdg ` G ) U e. ( ( iEdg ` G ) ` i ) <-> -. E. e e. E U e. e ) )
9	6 8	bitrd	|- ( ( G e. UHGraph /\ U e. V ) -> ( ( D ` U ) = 0 <-> -. E. e e. E U e. e ) )