Theorem lidl1

Description: Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Proof shortened by AV, 18-Apr-2025)

	Ref	Expression
Hypotheses	rnglidl0.u	$$⊢U=\mathrm{LIdeal}\left(R\right)$$
	rnglidl1.b	$$⊢B={\mathrm{Base}}_R$$
Assertion	lidl1	$$⊢R\in \mathrm{Ring}\to B\in U$$

Proof

Step	Hyp	Ref	Expression
1		rnglidl0.u	$$⊢U=\mathrm{LIdeal}\left(R\right)$$
2		rnglidl1.b	$$⊢B={\mathrm{Base}}_R$$
3		ringrng	$$⊢R\in \mathrm{Ring}\to R\in \mathrm{Rng}$$
4	1 2	rnglidl1	$$⊢R\in \mathrm{Rng}\to B\in U$$
5	3 4	syl	$$⊢R\in \mathrm{Ring}\to B\in U$$