Description: If the number of neighbors of a vertex in a finite simple graph is the number of vertices of the graph minus 1, the vertex is universal. (Contributed by Alexander van der Vekens, 14-Jul-2018) (Revised by AV, 16-Dec-2020) (Proof shortened by AV, 13-Feb-2022)
Ref | Expression | ||
---|---|---|---|
Hypothesis | uvtxnm1nbgr.v | |
|
Assertion | nbusgrvtxm1uvtx | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvtxnm1nbgr.v | |
|
2 | 1 | nbgrssovtx | |
3 | 2 | sseli | |
4 | eldifsn | |
|
5 | 1 | nbusgrvtxm1 | |
6 | 5 | imp | |
7 | 4 6 | syl5bi | |
8 | 3 7 | impbid2 | |
9 | 8 | eqrdv | |
10 | 1 | uvtxnbgrb | |
11 | 10 | ad2antlr | |
12 | 9 11 | mpbird | |
13 | 12 | ex | |