Metamath Proof Explorer


Theorem nbgrnself

Description: A vertex in a graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 3-Nov-2020) (Revised by AV, 21-Mar-2021)

Ref Expression
Hypothesis nbgrnself.v V = Vtx G
Assertion nbgrnself v V v G NeighbVtx v

Proof

Step Hyp Ref Expression
1 nbgrnself.v V = Vtx G
2 neldifsnd v V ¬ v V v
3 2 intnanrd v V ¬ v V v e Edg G v v e
4 df-nel v n V v | e Edg G v n e ¬ v n V v | e Edg G v n e
5 preq2 n = v v n = v v
6 5 sseq1d n = v v n e v v e
7 6 rexbidv n = v e Edg G v n e e Edg G v v e
8 7 elrab v n V v | e Edg G v n e v V v e Edg G v v e
9 4 8 xchbinx v n V v | e Edg G v n e ¬ v V v e Edg G v v e
10 3 9 sylibr v V v n V v | e Edg G v n e
11 eqidd v V v = v
12 eqid Edg G = Edg G
13 1 12 nbgrval v V G NeighbVtx v = n V v | e Edg G v n e
14 11 13 neleq12d v V v G NeighbVtx v v n V v | e Edg G v n e
15 10 14 mpbird v V v G NeighbVtx v
16 15 rgen v V v G NeighbVtx v