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	|- ( ph -> A e. X )
	1hegrvtxdg1.b	|- ( ph -> B e. V )
	1hegrvtxdg1.c	|- ( ph -> C e. V )
	1hegrvtxdg1.n	|- ( ph -> B =/= C )
	1hegrvtxdg1.x	|- ( ph -> E e. ~P V )
	1hegrvtxdg1.i	|- ( ph -> ( iEdg ` G ) = { <. A , E >. } )
	1hegrvtxdg1.e	|- ( ph -> { B , C } C_ E )
	1hegrvtxdg1.v	|- ( ph -> ( Vtx ` G ) = V )
Assertion	1hegrvtxdg1r	|- ( ph -> ( ( VtxDeg ` G ) ` C ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		1hegrvtxdg1.a	|- ( ph -> A e. X )
2		1hegrvtxdg1.b	|- ( ph -> B e. V )
3		1hegrvtxdg1.c	|- ( ph -> C e. V )
4		1hegrvtxdg1.n	|- ( ph -> B =/= C )
5		1hegrvtxdg1.x	|- ( ph -> E e. ~P V )
6		1hegrvtxdg1.i	|- ( ph -> ( iEdg ` G ) = { <. A , E >. } )
7		1hegrvtxdg1.e	|- ( ph -> { B , C } C_ E )
8		1hegrvtxdg1.v	|- ( ph -> ( Vtx ` G ) = V )
9	4	necomd	|- ( ph -> C =/= B )
10		prcom	|- { C , B } = { B , C }
11	10 7	eqsstrid	|- ( ph -> { C , B } C_ E )
12	1 3 2 9 5 6 11 8	1hegrvtxdg1	|- ( ph -> ( ( VtxDeg ` G ) ` C ) = 1 )