Theorem maxidln1

Description: One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011)

	Ref	Expression
Hypotheses	maxidln1.1	|- H = ( 2nd ` R )
	maxidln1.2	|- U = ( GId ` H )
Assertion	maxidln1	|- ( ( R e. RingOps /\ M e. ( MaxIdl ` R ) ) -> -. U e. M )

Proof

Step	Hyp	Ref	Expression
1		maxidln1.1	|- H = ( 2nd ` R )
2		maxidln1.2	|- U = ( GId ` H )
3		eqid	|- ( 1st ` R ) = ( 1st ` R )
4		eqid	|- ran ( 1st ` R ) = ran ( 1st ` R )
5	3 4	maxidlnr	|- ( ( R e. RingOps /\ M e. ( MaxIdl ` R ) ) -> M =/= ran ( 1st ` R ) )
6		maxidlidl	|- ( ( R e. RingOps /\ M e. ( MaxIdl ` R ) ) -> M e. ( Idl ` R ) )
7	3 1 4 2	1idl	|- ( ( R e. RingOps /\ M e. ( Idl ` R ) ) -> ( U e. M <-> M = ran ( 1st ` R ) ) )
8	7	necon3bbid	|- ( ( R e. RingOps /\ M e. ( Idl ` R ) ) -> ( -. U e. M <-> M =/= ran ( 1st ` R ) ) )
9	6 8	syldan	|- ( ( R e. RingOps /\ M e. ( MaxIdl ` R ) ) -> ( -. U e. M <-> M =/= ran ( 1st ` R ) ) )
10	5 9	mpbird	|- ( ( R e. RingOps /\ M e. ( MaxIdl ` R ) ) -> -. U e. M )