Step |
Hyp |
Ref |
Expression |
1 |
|
nbgrel.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
csbfv |
⊢ ⦋ 𝐺 / 𝑔 ⦌ ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) |
3 |
1 2
|
eqtr4i |
⊢ 𝑉 = ⦋ 𝐺 / 𝑔 ⦌ ( Vtx ‘ 𝑔 ) |
4 |
|
neleq2 |
⊢ ( 𝑉 = ⦋ 𝐺 / 𝑔 ⦌ ( Vtx ‘ 𝑔 ) → ( 𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋ 𝐺 / 𝑔 ⦌ ( Vtx ‘ 𝑔 ) ) ) |
5 |
3 4
|
ax-mp |
⊢ ( 𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋ 𝐺 / 𝑔 ⦌ ( Vtx ‘ 𝑔 ) ) |
6 |
5
|
biimpi |
⊢ ( 𝑋 ∉ 𝑉 → 𝑋 ∉ ⦋ 𝐺 / 𝑔 ⦌ ( Vtx ‘ 𝑔 ) ) |
7 |
6
|
olcd |
⊢ ( 𝑋 ∉ 𝑉 → ( 𝐺 ∉ V ∨ 𝑋 ∉ ⦋ 𝐺 / 𝑔 ⦌ ( Vtx ‘ 𝑔 ) ) ) |
8 |
|
df-nbgr |
⊢ NeighbVtx = ( 𝑔 ∈ V , 𝑣 ∈ ( Vtx ‘ 𝑔 ) ↦ { 𝑛 ∈ ( ( Vtx ‘ 𝑔 ) ∖ { 𝑣 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝑔 ) { 𝑣 , 𝑛 } ⊆ 𝑒 } ) |
9 |
8
|
mpoxneldm |
⊢ ( ( 𝐺 ∉ V ∨ 𝑋 ∉ ⦋ 𝐺 / 𝑔 ⦌ ( Vtx ‘ 𝑔 ) ) → ( 𝐺 NeighbVtx 𝑋 ) = ∅ ) |
10 |
7 9
|
syl |
⊢ ( 𝑋 ∉ 𝑉 → ( 𝐺 NeighbVtx 𝑋 ) = ∅ ) |