Theorem vtxdgfusgr

Description: In a finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018) (Revised by AV, 12-Dec-2020)

	Ref	Expression
Hypothesis	vtxdgfusgrf.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	vtxdgfusgr	⊢ ( 𝐺 ∈ FinUSGraph → ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		vtxdgfusgrf.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	vtxdgfusgrf	⊢ ( 𝐺 ∈ FinUSGraph → ( VtxDeg ‘ 𝐺 ) : 𝑉 ⟶ ℕ0 )
3	2	ffvelcdmda	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) ∈ ℕ0 )
4	3	ralrimiva	⊢ ( 𝐺 ∈ FinUSGraph → ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) ∈ ℕ0 )