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	⊢ { 𝑥 ∣ 𝑥 ∉ 𝑥 } = V

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2		elirrv	⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir	⊢ 𝑥 ∉ 𝑥
4	1 3	2th	⊢ ( 𝑥 ∈ V ↔ 𝑥 ∉ 𝑥 )
5	4	eqabi	⊢ V = { 𝑥 ∣ 𝑥 ∉ 𝑥 }
6	5	eqcomi	⊢ { 𝑥 ∣ 𝑥 ∉ 𝑥 } = V