Theorem lidl0

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

	Ref	Expression
Hypotheses	rnglidl0.u	$$⊢U=\mathrm{LIdeal}\left(R\right)$$
	rnglidl0.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	lidl0	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in U$$

Proof

Step	Hyp	Ref	Expression
1		rnglidl0.u	$$⊢U=\mathrm{LIdeal}\left(R\right)$$
2		rnglidl0.z	$$⊢\underset{\dot{}}{0}=0_R$$
3		ringrng	$$⊢R\in \mathrm{Ring}\to R\in \mathrm{Rng}$$
4	1 2	rnglidl0	$$⊢R\in \mathrm{Rng}\to \left\{\underset{\dot{}}{0}\right\}\in U$$
5	3 4	syl	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in U$$