Metamath Proof Explorer


Theorem mxidlnr

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 )

Proof

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 )