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		df-v	$$⊢\mathrm{V}=\left\{x|x=x\right\}$$
2		equid	$$⊢x=x$$
3		elirrv	$$⊢\neg x\in x$$
4	3	nelir	$$⊢x\notin x$$
5	2 4	2th	$$⊢x=x\leftrightarrow x\notin x$$
6	5	abbii	$$⊢\left\{x|x=x\right\}=\left\{x|x\notin x\right\}$$
7	1 6	eqtr2i	$$⊢\left\{x|x\notin x\right\}=\mathrm{V}$$