Theorem ringidcld

Description: The unity element of a ring belongs to the base set of the ring, deduction version. (Contributed by SN, 16-Oct-2025)

	Ref	Expression
Hypotheses	ringidcl.b	|- B = ( Base ` R )
	ringidcl.u	|- .1. = ( 1r ` R )
	ringidcld.r	|- ( ph -> R e. Ring )
Assertion	ringidcld	|- ( ph -> .1. e. B )

Proof

Step	Hyp	Ref	Expression
1		ringidcl.b	|- B = ( Base ` R )
2		ringidcl.u	|- .1. = ( 1r ` R )
3		ringidcld.r	|- ( ph -> R e. Ring )
4	1 2	ringidcl	|- ( R e. Ring -> .1. e. B )
5	3 4	syl	|- ( ph -> .1. e. B )