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Metamath Proof Explorer

Theorem mxidln1

Description: One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011) (Revised by Thierry Arnoux, 19-Jan-2024)

Ref Expression
Hypotheses mxidlval.1 
|- B = ( Base ` R )
mxidln1.1 
|- .1. = ( 1r ` R )
Assertion mxidln1 
|- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> -. .1. e. M )

Proof

Step Hyp Ref Expression
1 mxidlval.1 
 |-  B = ( Base ` R )
2 mxidln1.1 
 |-  .1. = ( 1r ` R )
3 1 mxidlnr 
 |-  ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M =/= B )
4 1 mxidlidl 
 |-  ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M e. ( LIdeal ` R ) )
5 eqid 
 |-  ( LIdeal ` R ) = ( LIdeal ` R )
6 5 1 2 lidl1el 
 |-  ( ( R e. Ring /\ M e. ( LIdeal ` R ) ) -> ( .1. e. M <-> M = B ) )
7 4 6 syldan 
 |-  ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> ( .1. e. M <-> M = B ) )
8 7 necon3bbid 
 |-  ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> ( -. .1. e. M <-> M =/= B ) )
9 3 8 mpbird 
 |-  ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> -. .1. e. M )