Theorem subrngid

Description: Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypothesis	subrngss.1	|- B = ( Base ` R )
Assertion	subrngid	|- ( R e. Rng -> B e. ( SubRng ` R ) )

Proof

Step	Hyp	Ref	Expression
1		subrngss.1	|- B = ( Base ` R )
2		id	|- ( R e. Rng -> R e. Rng )
3	1	ressid	|- ( R e. Rng -> ( R |`s B ) = R )
4	3 2	eqeltrd	|- ( R e. Rng -> ( R |`s B ) e. Rng )
5		ssidd	|- ( R e. Rng -> B C_ B )
6	1	issubrng	|- ( B e. ( SubRng ` R ) <-> ( R e. Rng /\ ( R |`s B ) e. Rng /\ B C_ B ) )
7	2 4 5 6	syl3anbrc	|- ( R e. Rng -> B e. ( SubRng ` R ) )