Theorem 0edg0rgr

Description: A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018) (Revised by AV, 26-Dec-2020)

	Ref	Expression
Assertion	0edg0rgr	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 RegGraph 0 )

Proof

Step	Hyp	Ref	Expression
1		simpr	⊢ ( ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) ∧ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ) → 𝑣 ∈ ( Vtx ‘ 𝐺 ) )
2		simplr	⊢ ( ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) ∧ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ) → ( iEdg ‘ 𝐺 ) = ∅ )
3		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
4		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
5	3 4	vtxdg0e	⊢ ( ( 𝑣 ∈ ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = 0 )
6	1 2 5	syl2anc	⊢ ( ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) ∧ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = 0 )
7	6	ralrimiva	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = 0 )
8		0xnn0	⊢ 0 ∈ ℕ0*
9	7 8	jctil	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ( 0 ∈ ℕ0* ∧ ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = 0 ) )
10	8	a1i	⊢ ( ( iEdg ‘ 𝐺 ) = ∅ → 0 ∈ ℕ0* )
11		eqid	⊢ ( VtxDeg ‘ 𝐺 ) = ( VtxDeg ‘ 𝐺 )
12	3 11	isrgr	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 0 ∈ ℕ0* ) → ( 𝐺 RegGraph 0 ↔ ( 0 ∈ ℕ0* ∧ ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = 0 ) ) )
13	10 12	sylan2	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ( 𝐺 RegGraph 0 ↔ ( 0 ∈ ℕ0* ∧ ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = 0 ) ) )
14	9 13	mpbird	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 RegGraph 0 )