Theorem usgr1v

Description: A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017) (Revised by AV, 18-Oct-2020)

	Ref	Expression
Assertion	usgr1v	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		usgr1vr	⊢ ( ( 𝐴 ∈ V ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) = ∅ ) )
2	1	adantrl	⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) = ∅ ) )
3		simplrl	⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ 𝑊 )
4		simpr	⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ )
5	3 4	usgr0e	⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ USGraph )
6	5	ex	⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( ( iEdg ‘ 𝐺 ) = ∅ → 𝐺 ∈ USGraph ) )
7	2 6	impbid	⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
8	7	ex	⊢ ( 𝐴 ∈ V → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) )
9		snprc	⊢ ( ¬ 𝐴 ∈ V ↔ { 𝐴 } = ∅ )
10		simpl	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → 𝐺 ∈ 𝑊 )
11		simprr	⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( Vtx ‘ 𝐺 ) = { 𝐴 } )
12		simpl	⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → { 𝐴 } = ∅ )
13	11 12	eqtrd	⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( Vtx ‘ 𝐺 ) = ∅ )
14		usgr0vb	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
15	10 13 14	syl2an2	⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
16	15	ex	⊢ ( { 𝐴 } = ∅ → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) )
17	9 16	sylbi	⊢ ( ¬ 𝐴 ∈ V → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) )
18	8 17	pm2.61i	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )