Description: In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | 0ring.b | |
|
0ring.0 | |
||
0ring01eq.1 | |
||
Assertion | 0ring01eq | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ring.b | |
|
2 | 0ring.0 | |
|
3 | 0ring01eq.1 | |
|
4 | 1 2 | 0ring | |
5 | 1 3 | ringidcl | |
6 | eleq2 | |
|
7 | elsni | |
|
8 | 7 | eqcomd | |
9 | 6 8 | syl6bi | |
10 | 5 9 | syl5com | |
11 | 10 | adantr | |
12 | 4 11 | mpd | |