Theorem 2idl0

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

	Ref	Expression
Hypotheses	2idl0.u	$$⊢I=\mathrm{2Ideal}\left(R\right)$$
	2idl0.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	2idl0	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in I$$

Proof

Step	Hyp	Ref	Expression
1		2idl0.u	$$⊢I=\mathrm{2Ideal}\left(R\right)$$
2		2idl0.z	$$⊢\underset{\dot{}}{0}=0_R$$
3		eqid	$$⊢\mathrm{LIdeal}\left(R\right)=\mathrm{LIdeal}\left(R\right)$$
4	3 2	lidl0	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{LIdeal}\left(R\right)$$
5		eqid	$$⊢\mathrm{LIdeal}\left({opp}_r\left(R\right)\right)=\mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
6	5 2	ridl0	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
7	4 6	elind	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in \left(\mathrm{LIdeal}\left(R\right)\cap \mathrm{LIdeal}\left({opp}_r\left(R\right)\right)\right)$$
8		eqid	$$⊢{opp}_r\left(R\right)={opp}_r\left(R\right)$$
9	3 8 5 1	2idlval	$$⊢I=\mathrm{LIdeal}\left(R\right)\cap \mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
10	7 9	eleqtrrdi	$$⊢R\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in I$$