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	$$⊢\left\{x|x\notin x\right\}=\mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		elirr	$$⊢\neg x\in x$$
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	$$⊢\left\{x|x\notin x\right\}=\mathrm{V}$$