Theorem 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	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	vtxdusgradjvtx.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
Assertion	usgrvd00	$$⊢G\in \mathrm{USGraph}\to \left(\forall v\in V\mathrm{VtxDeg}\left(G\right)\left(v\right)=0\to E=\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		vtxdusgradjvtx.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		vtxdusgradjvtx.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
3		usgruhgr	$$⊢G\in \mathrm{USGraph}\to G\in \mathrm{UHGraph}$$
4	1 2	uhgrvd00	$$⊢G\in \mathrm{UHGraph}\to \left(\forall v\in V\mathrm{VtxDeg}\left(G\right)\left(v\right)=0\to E=\varnothing \right)$$
5	3 4	syl	$$⊢G\in \mathrm{USGraph}\to \left(\forall v\in V\mathrm{VtxDeg}\left(G\right)\left(v\right)=0\to E=\varnothing \right)$$