Theorem 1hegrvtxdg1r

Description: The vertex degree of a graph with one hyperedge, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by Alexander van der Vekens, 22-Dec-2017) (Revised by AV, 23-Feb-2021)

	Ref	Expression
Hypotheses	1hegrvtxdg1.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
	1hegrvtxdg1.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
	1hegrvtxdg1.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
	1hegrvtxdg1.n	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
	1hegrvtxdg1.x	⊢ ( 𝜑 → 𝐸 ∈ 𝒫 𝑉 )
	1hegrvtxdg1.i	⊢ ( 𝜑 → ( iEdg ‘ 𝐺 ) = { ⟨ 𝐴 , 𝐸 ⟩ } )
	1hegrvtxdg1.e	⊢ ( 𝜑 → { 𝐵 , 𝐶 } ⊆ 𝐸 )
	1hegrvtxdg1.v	⊢ ( 𝜑 → ( Vtx ‘ 𝐺 ) = 𝑉 )
Assertion	1hegrvtxdg1r	⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝐶 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		1hegrvtxdg1.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
2		1hegrvtxdg1.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
3		1hegrvtxdg1.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
4		1hegrvtxdg1.n	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
5		1hegrvtxdg1.x	⊢ ( 𝜑 → 𝐸 ∈ 𝒫 𝑉 )
6		1hegrvtxdg1.i	⊢ ( 𝜑 → ( iEdg ‘ 𝐺 ) = { ⟨ 𝐴 , 𝐸 ⟩ } )
7		1hegrvtxdg1.e	⊢ ( 𝜑 → { 𝐵 , 𝐶 } ⊆ 𝐸 )
8		1hegrvtxdg1.v	⊢ ( 𝜑 → ( Vtx ‘ 𝐺 ) = 𝑉 )
9	4	necomd	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
10		prcom	⊢ { 𝐶 , 𝐵 } = { 𝐵 , 𝐶 }
11	10 7	eqsstrid	⊢ ( 𝜑 → { 𝐶 , 𝐵 } ⊆ 𝐸 )
12	1 3 2 9 5 6 11 8	1hegrvtxdg1	⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝐶 ) = 1 )