Theorem uvtxnbgrb

Description: A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018) (Revised by AV, 3-Nov-2020) (Revised by AV, 23-Mar-2021) (Proof shortened by AV, 14-Feb-2022)

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

Proof

Step	Hyp	Ref	Expression
1		uvtxnbgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	uvtxnbgr	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )
3		simpl	⊢ ( ( 𝑁 ∈ 𝑉 ∧ ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) ) → 𝑁 ∈ 𝑉 )
4		raleleq	⊢ ( ( 𝑉 ∖ { 𝑁 } ) = ( 𝐺 NeighbVtx 𝑁 ) → ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) )
5	4	eqcoms	⊢ ( ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) → ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) )
6	5	adantl	⊢ ( ( 𝑁 ∈ 𝑉 ∧ ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) ) → ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) )
7	1	uvtxel	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑁 ∈ 𝑉 ∧ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) ) )
8	3 6 7	sylanbrc	⊢ ( ( 𝑁 ∈ 𝑉 ∧ ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) ) → 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) )
9	8	ex	⊢ ( 𝑁 ∈ 𝑉 → ( ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) → 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) )
10	2 9	impbid2	⊢ ( 𝑁 ∈ 𝑉 → ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) ) )