Description: If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019) (Proof shortened by SN, 23-Feb-2025)
Ref | Expression | ||
---|---|---|---|
Hypotheses | 0ring.b | |
|
0ring.0 | |
||
0ring01eq.1 | |
||
Assertion | 01eq0ring | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ring.b | |
|
2 | 0ring.0 | |
|
3 | 0ring01eq.1 | |
|
4 | eqcom | |
|
5 | 1 2 | ring0cl | |
6 | 5 | ne0d | |
7 | 5 | adantr | |
8 | 1 3 2 | ring1eq0 | |
9 | 7 8 | mpd3an3 | |
10 | 9 | impancom | |
11 | 10 | ralrimiv | |
12 | eqsn | |
|
13 | 12 | biimpar | |
14 | 6 11 13 | syl2an2r | |
15 | 4 14 | sylan2b | |