Theorem hashnbusgrnn0

Description: The number of neighbors of a vertex in a finite simple graph is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Jul-2018) (Revised by AV, 15-Dec-2020)

	Ref	Expression
Hypothesis	hashnbusgrnn0.v	|- V = ( Vtx ` G )
Assertion	hashnbusgrnn0	|- ( ( G e. FinUSGraph /\ U e. V ) -> ( # ` ( G NeighbVtx U ) ) e. NN0 )

Proof

Step	Hyp	Ref	Expression
1		hashnbusgrnn0.v	|- V = ( Vtx ` G )
2	1	eleq2i	|- ( U e. V <-> U e. ( Vtx ` G ) )
3		nbfiusgrfi	|- ( ( G e. FinUSGraph /\ U e. ( Vtx ` G ) ) -> ( G NeighbVtx U ) e. Fin )
4	2 3	sylan2b	|- ( ( G e. FinUSGraph /\ U e. V ) -> ( G NeighbVtx U ) e. Fin )
5		hashcl	|- ( ( G NeighbVtx U ) e. Fin -> ( # ` ( G NeighbVtx U ) ) e. NN0 )
6	4 5	syl	|- ( ( G e. FinUSGraph /\ U e. V ) -> ( # ` ( G NeighbVtx U ) ) e. NN0 )