Theorem ruv

Description: The Russell class is equal to the universe _V . Exercise 5 of TakeutiZaring p. 22. (Contributed by Alan Sare, 4-Oct-2008)

	Ref	Expression
Assertion	ruv	⊢ { 𝑥 ∣ 𝑥 ∉ 𝑥 } = V

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2		elirr	⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir	⊢ 𝑥 ∉ 𝑥
4	1 3	2th	⊢ ( 𝑥 ∈ V ↔ 𝑥 ∉ 𝑥 )
5	4	abbi2i	⊢ V = { 𝑥 ∣ 𝑥 ∉ 𝑥 }
6	5	eqcomi	⊢ { 𝑥 ∣ 𝑥 ∉ 𝑥 } = V