Theorem pridln1

Description: A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024)

	Ref	Expression
Hypotheses	pridln1.1	|- B = ( Base ` R )
	pridln1.2	|- .1. = ( 1r ` R )
Assertion	pridln1	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> -. .1. e. I )

Proof

Step	Hyp	Ref	Expression
1		pridln1.1	|- B = ( Base ` R )
2		pridln1.2	|- .1. = ( 1r ` R )
3		eqid	|- ( LIdeal ` R ) = ( LIdeal ` R )
4	3 1 2	lidl1el	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> ( .1. e. I <-> I = B ) )
5	4	necon3bbid	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> ( -. .1. e. I <-> I =/= B ) )
6	5	biimp3ar	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> -. .1. e. I )