Theorem uvtxnbgrvtx

Description: A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017) (Revised by AV, 30-Oct-2020)

	Ref	Expression
Hypothesis	uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	uvtxnbgrvtx	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ∀ 𝑣 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑁 ∈ ( 𝐺 NeighbVtx 𝑣 ) )

Proof

Step	Hyp	Ref	Expression
1		uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	vtxnbuvtx	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) )
3		eleq1w	⊢ ( 𝑛 = 𝑣 → ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) ↔ 𝑣 ∈ ( 𝐺 NeighbVtx 𝑁 ) ) )
4	3	rspcva	⊢ ( ( 𝑣 ∈ ( 𝑉 ∖ { 𝑁 } ) ∧ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) ) → 𝑣 ∈ ( 𝐺 NeighbVtx 𝑁 ) )
5		nbgrsym	⊢ ( 𝑣 ∈ ( 𝐺 NeighbVtx 𝑁 ) ↔ 𝑁 ∈ ( 𝐺 NeighbVtx 𝑣 ) )
6	5	a1i	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝑣 ∈ ( 𝐺 NeighbVtx 𝑁 ) ↔ 𝑁 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )
7	4 6	syl5ibcom	⊢ ( ( 𝑣 ∈ ( 𝑉 ∖ { 𝑁 } ) ∧ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) ) → ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → 𝑁 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )
8	7	expcom	⊢ ( ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) → ( 𝑣 ∈ ( 𝑉 ∖ { 𝑁 } ) → ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → 𝑁 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) )
9	8	com23	⊢ ( ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) → ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝑣 ∈ ( 𝑉 ∖ { 𝑁 } ) → 𝑁 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) )
10	2 9	mpcom	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝑣 ∈ ( 𝑉 ∖ { 𝑁 } ) → 𝑁 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )
11	10	ralrimiv	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ∀ 𝑣 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑁 ∈ ( 𝐺 NeighbVtx 𝑣 ) )