Theorem ruvALT

Description: Alternate proof of ruv with one fewer syntax step thanks to using elirrv instead of elirr . However, it does not change the compressed proof size or the number of symbols in the generated display, so it is not considered a shortening according to conventions . (Contributed by SN, 1-Sep-2024) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	ruvALT	$$⊢\left\{x|x\notin x\right\}=\mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		elirrv	$$⊢\neg x\in x$$
3	2	nelir	$$⊢x\notin x$$
4	1 3	2th	$$⊢x\in \mathrm{V}\leftrightarrow x\notin x$$
5	4	abbi2i	$$⊢\mathrm{V}=\left\{x|x\notin x\right\}$$
6	5	eqcomi	$$⊢\left\{x|x\notin x\right\}=\mathrm{V}$$