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Metamath Proof Explorer

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	$⊢ \{x \| x \notin x\} = V$

Step	Hyp	Ref	Expression
1		vex	$⊢ x \in V$
2		elirr	$⊢ \neg x \in x$
3	2	nelir	$⊢ x \notin x$
4	1 3	2th	$⊢ x \in V \leftrightarrow x \notin x$
5	4	eqabi	$⊢ V = \{x \| x \notin x\}$
6	5	eqcomi	$⊢ \{x \| x \notin x\} = V$