Description: The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rngen1zr0 , and semirings, see srgen1zr0 . (Contributed by FL, 15-Feb-2010) (Revised by AV, 25-Jan-2020) (Proof shortened by AV, 19-Jun-2026)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ring1zr.b | |- B = ( Base ` R ) |
|
| ring1zr.p | |- .+ = ( +g ` R ) |
||
| ring1zr.t | |- .* = ( .r ` R ) |
||
| ringen1zr0.0 | |- Z = ( 0g ` R ) |
||
| Assertion | ringen1zr0 | |- ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ring1zr.b | |- B = ( Base ` R ) |
|
| 2 | ring1zr.p | |- .+ = ( +g ` R ) |
|
| 3 | ring1zr.t | |- .* = ( .r ` R ) |
|
| 4 | ringen1zr0.0 | |- Z = ( 0g ` R ) |
|
| 5 | ringrng | |- ( R e. Ring -> R e. Rng ) |
|
| 6 | 1 2 3 4 | rngen1zr0 | |- ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) ) |
| 7 | 5 6 | syl3an1 | |- ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) ) |