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Metamath Proof Explorer

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 } ) ) )