Theorem is0trl

Description: A pair of an empty set (of edges) and a sequence of one vertex is a trail (of length 0). (Contributed by AV, 7-Jan-2021) (Revised by AV, 23-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypothesis	0wlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	is0trl	⊢ ( ( 𝑃 = { ⟨ 0 , 𝑁 ⟩ } ∧ 𝑁 ∈ 𝑉 ) → ∅ ( Trails ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		0wlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		1fv	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑃 = { ⟨ 0 , 𝑁 ⟩ } ) → ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) )
3	2	ancoms	⊢ ( ( 𝑃 = { ⟨ 0 , 𝑁 ⟩ } ∧ 𝑁 ∈ 𝑉 ) → ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) )
4	3	simpld	⊢ ( ( 𝑃 = { ⟨ 0 , 𝑁 ⟩ } ∧ 𝑁 ∈ 𝑉 ) → 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 )
5	1	1vgrex	⊢ ( 𝑁 ∈ 𝑉 → 𝐺 ∈ V )
6	5	adantl	⊢ ( ( 𝑃 = { ⟨ 0 , 𝑁 ⟩ } ∧ 𝑁 ∈ 𝑉 ) → 𝐺 ∈ V )
7	1	0trl	⊢ ( 𝐺 ∈ V → ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
8	6 7	syl	⊢ ( ( 𝑃 = { ⟨ 0 , 𝑁 ⟩ } ∧ 𝑁 ∈ 𝑉 ) → ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
9	4 8	mpbird	⊢ ( ( 𝑃 = { ⟨ 0 , 𝑁 ⟩ } ∧ 𝑁 ∈ 𝑉 ) → ∅ ( Trails ‘ 𝐺 ) 𝑃 )