Theorem uvtxnbgr

Description: A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017) (Revised by AV, 3-Nov-2020) (Revised by AV, 23-Mar-2021)

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

Proof

Step	Hyp	Ref	Expression
1		uvtxnbgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	nbgrssovtx	⊢ ( 𝐺 NeighbVtx 𝑁 ) ⊆ ( 𝑉 ∖ { 𝑁 } )
3	2	a1i	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝐺 NeighbVtx 𝑁 ) ⊆ ( 𝑉 ∖ { 𝑁 } ) )
4	1	uvtxnbgrss	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝑉 ∖ { 𝑁 } ) ⊆ ( 𝐺 NeighbVtx 𝑁 ) )
5	3 4	eqssd	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )