Description: The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | 0ringdif.b | |- B = ( Base ` R ) |
|
0ringdif.0 | |- .0. = ( 0g ` R ) |
||
Assertion | 0ringbas | |- ( R e. ( Ring \ NzRing ) -> B = { .0. } ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ringdif.b | |- B = ( Base ` R ) |
|
2 | 0ringdif.0 | |- .0. = ( 0g ` R ) |
|
3 | 1 2 | 0ringdif | |- ( R e. ( Ring \ NzRing ) <-> ( R e. Ring /\ B = { .0. } ) ) |
4 | 3 | simprbi | |- ( R e. ( Ring \ NzRing ) -> B = { .0. } ) |