Theorem uvtxnbgrss

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

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

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		dfss3	⊢ ( ( 𝑉 ∖ { 𝑁 } ) ⊆ ( 𝐺 NeighbVtx 𝑁 ) ↔ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) )
4	2 3	sylibr	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝑉 ∖ { 𝑁 } ) ⊆ ( 𝐺 NeighbVtx 𝑁 ) )