Theorem cusgr0v

Description: A graph with no vertices and no edges is a complete simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017) (Revised by AV, 1-Nov-2020)

	Ref	Expression
Hypothesis	cplgr0v.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	cusgr0v	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ ComplUSGraph )

Proof

Step	Hyp	Ref	Expression
1		cplgr0v.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	eqeq1i	⊢ ( 𝑉 = ∅ ↔ ( Vtx ‘ 𝐺 ) = ∅ )
3		usgr0v	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ USGraph )
4	2 3	syl3an2b	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ USGraph )
5	1	cplgr0v	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ) → 𝐺 ∈ ComplGraph )
6	5	3adant3	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ ComplGraph )
7		iscusgr	⊢ ( 𝐺 ∈ ComplUSGraph ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph ) )
8	4 6 7	sylanbrc	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ ComplUSGraph )