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 )