| Step |
Hyp |
Ref |
Expression |
| 1 |
|
stgrvtx0.g |
|- G = ( StarGr ` N ) |
| 2 |
|
stgrvtx0.v |
|- V = ( Vtx ` G ) |
| 3 |
1 2
|
stgrvtx0 |
|- ( N e. NN0 -> 0 e. V ) |
| 4 |
2
|
dfclnbgr4 |
|- ( 0 e. V -> ( G ClNeighbVtx 0 ) = ( { 0 } u. ( G NeighbVtx 0 ) ) ) |
| 5 |
3 4
|
syl |
|- ( N e. NN0 -> ( G ClNeighbVtx 0 ) = ( { 0 } u. ( G NeighbVtx 0 ) ) ) |
| 6 |
1 2
|
stgrnbgr0 |
|- ( N e. NN0 -> ( G NeighbVtx 0 ) = ( V \ { 0 } ) ) |
| 7 |
6
|
uneq2d |
|- ( N e. NN0 -> ( { 0 } u. ( G NeighbVtx 0 ) ) = ( { 0 } u. ( V \ { 0 } ) ) ) |
| 8 |
3
|
snssd |
|- ( N e. NN0 -> { 0 } C_ V ) |
| 9 |
|
undif |
|- ( { 0 } C_ V <-> ( { 0 } u. ( V \ { 0 } ) ) = V ) |
| 10 |
8 9
|
sylib |
|- ( N e. NN0 -> ( { 0 } u. ( V \ { 0 } ) ) = V ) |
| 11 |
5 7 10
|
3eqtrd |
|- ( N e. NN0 -> ( G ClNeighbVtx 0 ) = V ) |