Theorem 2idl1

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

	Ref	Expression
Hypotheses	2idl0.u	$$⊢I=\mathrm{2Ideal}\left(R\right)$$
	2idl1.b	$$⊢B={\mathrm{Base}}_R$$
Assertion	2idl1	$$⊢R\in \mathrm{Ring}\to B\in I$$

Proof

Step	Hyp	Ref	Expression
1		2idl0.u	$$⊢I=\mathrm{2Ideal}\left(R\right)$$
2		2idl1.b	$$⊢B={\mathrm{Base}}_R$$
3		eqid	$$⊢\mathrm{LIdeal}\left(R\right)=\mathrm{LIdeal}\left(R\right)$$
4	3 2	lidl1	$$⊢R\in \mathrm{Ring}\to B\in \mathrm{LIdeal}\left(R\right)$$
5		eqid	$$⊢\mathrm{LIdeal}\left({opp}_r\left(R\right)\right)=\mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
6	5 2	ridl1	$$⊢R\in \mathrm{Ring}\to B\in \mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
7	4 6	elind	$$⊢R\in \mathrm{Ring}\to B\in \left(\mathrm{LIdeal}\left(R\right)\cap \mathrm{LIdeal}\left({opp}_r\left(R\right)\right)\right)$$
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	$$⊢R\in \mathrm{Ring}\to B\in I$$