Theorem ridl0

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

	Ref	Expression
Hypotheses	ridl0.u	$$⊢U=\mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
	ridl0.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	ridl0	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in U$$

Proof

Step	Hyp	Ref	Expression
1		ridl0.u	$$⊢U=\mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
2		ridl0.z	$$⊢\underset{\dot{}}{0}=0_R$$
3		eqid	$$⊢{opp}_r\left(R\right)={opp}_r\left(R\right)$$
4	3	opprring	$$⊢R\in \mathrm{Ring}\to {opp}_r\left(R\right)\in \mathrm{Ring}$$
5	3 2	oppr0	$$⊢\underset{\dot{}}{0}=0_{{opp}_r\left(R\right)}$$
6	1 5	lidl0	$$⊢{opp}_r\left(R\right)\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in U$$
7	4 6	syl	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in U$$