Theorem uhgr0v0e

Description: The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020)

	Ref	Expression
Hypotheses	uhgr0v0e.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	uhgr0v0e.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	uhgr0v0e	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑉 = ∅ ) → 𝐸 = ∅ )

Proof

Step	Hyp	Ref	Expression
1		uhgr0v0e.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		uhgr0v0e.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3	1	eqeq1i	⊢ ( 𝑉 = ∅ ↔ ( Vtx ‘ 𝐺 ) = ∅ )
4		uhgr0vb	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ UHGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
5	4	biimpd	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) = ∅ ) )
6	5	ex	⊢ ( 𝐺 ∈ UHGraph → ( ( Vtx ‘ 𝐺 ) = ∅ → ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) = ∅ ) ) )
7	3 6	syl5bi	⊢ ( 𝐺 ∈ UHGraph → ( 𝑉 = ∅ → ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) = ∅ ) ) )
8	7	pm2.43a	⊢ ( 𝐺 ∈ UHGraph → ( 𝑉 = ∅ → ( iEdg ‘ 𝐺 ) = ∅ ) )
9	8	imp	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑉 = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ )
10	2	eqeq1i	⊢ ( 𝐸 = ∅ ↔ ( Edg ‘ 𝐺 ) = ∅ )
11		uhgriedg0edg0	⊢ ( 𝐺 ∈ UHGraph → ( ( Edg ‘ 𝐺 ) = ∅ ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
12	10 11	syl5bb	⊢ ( 𝐺 ∈ UHGraph → ( 𝐸 = ∅ ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
13	12	adantr	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑉 = ∅ ) → ( 𝐸 = ∅ ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
14	9 13	mpbird	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑉 = ∅ ) → 𝐸 = ∅ )