Description: A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011) (Revised by Thierry Arnoux, 19-Jan-2024)
Ref | Expression | ||
---|---|---|---|
Hypothesis | mxidlval.1 | |- B = ( Base ` R ) |
|
Assertion | mxidlnr | |- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M =/= B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mxidlval.1 | |- B = ( Base ` R ) |
|
2 | 1 | ismxidl | |- ( R e. Ring -> ( M e. ( MaxIdeal ` R ) <-> ( M e. ( LIdeal ` R ) /\ M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) ) ) |
3 | 2 | biimpa | |- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> ( M e. ( LIdeal ` R ) /\ M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) ) |
4 | 3 | simp2d | |- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M =/= B ) |