Theorem ruv 4692

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

Assertion

Ref	Expression
ruv	|- { x | x e/ x } = _V

Proof of Theorem ruv

Step	Hyp	Ref	Expression
1		df-v 2823	. 2 |- _V = { x | x = x }
2		equid 1753	. . . 4 |- x = x
3		elirrv 4690	. . . . 5 |- -. x e. x
4	3	nelir 2518	. . . 4 |- x e/ x
5	2, 4	2th 174	. . 3 |- ( x = x <-> x e/ x )
6	5	abbii 2354	. 2 |- { x | x = x } = { x | x e/ x }
7	1, 6	eqtr2i 2260	1 |- { x | x e/ x } = _V

Syntax hints:   = wceq 1402   {cab 2224   e/ wnel 2515   _Vcvv 2821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-v 2823  df-dif 3222  df-sn 3711

This theorem is referenced by:  ruALT  4693