Metamath Proof Explorer


Theorem acycgr0v

Description: A null graph (with no vertices) is an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023)

Ref Expression
Hypothesis acycgr0v.1 𝑉 = ( Vtx ‘ 𝐺 )
Assertion acycgr0v ( ( 𝐺𝑊𝑉 = ∅ ) → 𝐺 ∈ AcyclicGraph )

Proof

Step Hyp Ref Expression
1 acycgr0v.1 𝑉 = ( Vtx ‘ 𝐺 )
2 br0 ¬ 𝑓𝑝
3 df-cycls Cycles = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Paths ‘ 𝑔 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) } )
4 3 relmptopab Rel ( Cycles ‘ 𝐺 )
5 cycliswlk ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
6 df-br ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ↔ ⟨ 𝑓 , 𝑝 ⟩ ∈ ( Cycles ‘ 𝐺 ) )
7 df-br ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ↔ ⟨ 𝑓 , 𝑝 ⟩ ∈ ( Walks ‘ 𝐺 ) )
8 5 6 7 3imtr3i ( ⟨ 𝑓 , 𝑝 ⟩ ∈ ( Cycles ‘ 𝐺 ) → ⟨ 𝑓 , 𝑝 ⟩ ∈ ( Walks ‘ 𝐺 ) )
9 4 8 relssi ( Cycles ‘ 𝐺 ) ⊆ ( Walks ‘ 𝐺 )
10 1 eqeq1i ( 𝑉 = ∅ ↔ ( Vtx ‘ 𝐺 ) = ∅ )
11 g0wlk0 ( ( Vtx ‘ 𝐺 ) = ∅ → ( Walks ‘ 𝐺 ) = ∅ )
12 10 11 sylbi ( 𝑉 = ∅ → ( Walks ‘ 𝐺 ) = ∅ )
13 9 12 sseqtrid ( 𝑉 = ∅ → ( Cycles ‘ 𝐺 ) ⊆ ∅ )
14 ss0 ( ( Cycles ‘ 𝐺 ) ⊆ ∅ → ( Cycles ‘ 𝐺 ) = ∅ )
15 breq ( ( Cycles ‘ 𝐺 ) = ∅ → ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝𝑓𝑝 ) )
16 15 notbid ( ( Cycles ‘ 𝐺 ) = ∅ → ( ¬ 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ↔ ¬ 𝑓𝑝 ) )
17 13 14 16 3syl ( 𝑉 = ∅ → ( ¬ 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ↔ ¬ 𝑓𝑝 ) )
18 2 17 mpbiri ( 𝑉 = ∅ → ¬ 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 )
19 18 intnanrd ( 𝑉 = ∅ → ¬ ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝𝑓 ≠ ∅ ) )
20 19 nexdv ( 𝑉 = ∅ → ¬ ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝𝑓 ≠ ∅ ) )
21 20 nexdv ( 𝑉 = ∅ → ¬ ∃ 𝑓𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝𝑓 ≠ ∅ ) )
22 isacycgr ( 𝐺𝑊 → ( 𝐺 ∈ AcyclicGraph ↔ ¬ ∃ 𝑓𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝𝑓 ≠ ∅ ) ) )
23 22 biimpar ( ( 𝐺𝑊 ∧ ¬ ∃ 𝑓𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝𝑓 ≠ ∅ ) ) → 𝐺 ∈ AcyclicGraph )
24 21 23 sylan2 ( ( 𝐺𝑊𝑉 = ∅ ) → 𝐺 ∈ AcyclicGraph )