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Metamath Proof Explorer

Theorem usgr0vb

Description: The null graph, with no vertices, is a simple graph iff the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017) (Revised by AV, 16-Oct-2020)

Ref Expression
Assertion usgr0vb ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )

Proof

Step Hyp Ref Expression
1 usgruhgr ⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UHGraph )
2 uhgr0vb ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ UHGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
3 1 2 imbitrid ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) = ∅ ) )
4 simpl ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ 𝑊 )
5 simpr ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ )
6 4 5 usgr0e ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ USGraph )
7 6 ex ⊢ ( 𝐺 ∈ 𝑊 → ( ( iEdg ‘ 𝐺 ) = ∅ → 𝐺 ∈ USGraph ) )
8 7 adantr ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( ( iEdg ‘ 𝐺 ) = ∅ → 𝐺 ∈ USGraph ) )
9 3 8 impbid ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )