Theorem ridl0

Description: Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025)

	Ref	Expression
Hypotheses	ridl0.u	|- U = ( LIdeal ` ( oppR ` R ) )
	ridl0.z	|- .0. = ( 0g ` R )
Assertion	ridl0	|- ( R e. Ring -> { .0. } e. U )

Proof

Step	Hyp	Ref	Expression
1		ridl0.u	|- U = ( LIdeal ` ( oppR ` R ) )
2		ridl0.z	|- .0. = ( 0g ` R )
3		eqid	|- ( oppR ` R ) = ( oppR ` R )
4	3	opprring	|- ( R e. Ring -> ( oppR ` R ) e. Ring )
5	3 2	oppr0	|- .0. = ( 0g ` ( oppR ` R ) )
6	1 5	lidl0	|- ( ( oppR ` R ) e. Ring -> { .0. } e. U )
7	4 6	syl	|- ( R e. Ring -> { .0. } e. U )