Theorem usgr0v

Description: The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020)

	Ref	Expression
Assertion	usgr0v	$$⊢\left(G\in W\wedge \mathrm{Vtx}\left(G\right)=\varnothing \wedge \mathrm{iEdg}\left(G\right)=\varnothing \right)\to G\in \mathrm{USGraph}$$

Proof

Step	Hyp	Ref	Expression
1		usgr0vb	$$⊢\left(G\in W\wedge \mathrm{Vtx}\left(G\right)=\varnothing \right)\to \left(G\in \mathrm{USGraph}\leftrightarrow \mathrm{iEdg}\left(G\right)=\varnothing \right)$$
2	1	biimp3ar	$$⊢\left(G\in W\wedge \mathrm{Vtx}\left(G\right)=\varnothing \wedge \mathrm{iEdg}\left(G\right)=\varnothing \right)\to G\in \mathrm{USGraph}$$