Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Alexander van der Vekens
Graph theory (extension)
Closed neighborhood of a vertex
clnbupgr
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clnbupgrel
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Ascii
Unicode
Theorem
clnbupgr
Description:
The closed neighborhood of a vertex in a pseudograph.
(Contributed by
AV
, 10-May-2025)
Ref
Expression
Hypotheses
clnbuhgr.v
⊢
V
=
Vtx
⁡
G
clnbuhgr.e
⊢
E
=
Edg
⁡
G
Assertion
clnbupgr
⊢
G
∈
UPGraph
∧
N
∈
V
→
G
ClNeighbVtx
N
=
N
∪
n
∈
V
|
N
n
∈
E
Proof
Step
Hyp
Ref
Expression
1
clnbuhgr.v
⊢
V
=
Vtx
⁡
G
2
clnbuhgr.e
⊢
E
=
Edg
⁡
G
3
1
dfclnbgr4
⊢
N
∈
V
→
G
ClNeighbVtx
N
=
N
∪
G
NeighbVtx
N
4
3
adantl
⊢
G
∈
UPGraph
∧
N
∈
V
→
G
ClNeighbVtx
N
=
N
∪
G
NeighbVtx
N
5
1
2
nbupgr
⊢
G
∈
UPGraph
∧
N
∈
V
→
G
NeighbVtx
N
=
n
∈
V
∖
N
|
N
n
∈
E
6
5
uneq2d
⊢
G
∈
UPGraph
∧
N
∈
V
→
N
∪
G
NeighbVtx
N
=
N
∪
n
∈
V
∖
N
|
N
n
∈
E
7
rabdif
⊢
n
∈
V
|
N
n
∈
E
∖
N
=
n
∈
V
∖
N
|
N
n
∈
E
8
7
eqcomi
⊢
n
∈
V
∖
N
|
N
n
∈
E
=
n
∈
V
|
N
n
∈
E
∖
N
9
8
uneq2i
⊢
N
∪
n
∈
V
∖
N
|
N
n
∈
E
=
N
∪
n
∈
V
|
N
n
∈
E
∖
N
10
undif2
⊢
N
∪
n
∈
V
|
N
n
∈
E
∖
N
=
N
∪
n
∈
V
|
N
n
∈
E
11
9
10
eqtri
⊢
N
∪
n
∈
V
∖
N
|
N
n
∈
E
=
N
∪
n
∈
V
|
N
n
∈
E
12
11
a1i
⊢
G
∈
UPGraph
∧
N
∈
V
→
N
∪
n
∈
V
∖
N
|
N
n
∈
E
=
N
∪
n
∈
V
|
N
n
∈
E
13
4
6
12
3eqtrd
⊢
G
∈
UPGraph
∧
N
∈
V
→
G
ClNeighbVtx
N
=
N
∪
n
∈
V
|
N
n
∈
E