Theorem ruv 4465

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 2688	. 2 |- _V = { x | x = x }
2		equid 1677	. . . 4 |- x = x
3		elirrv 4463	. . . . 5 |- -. x e. x
4	3	nelir 2406	. . . 4 |- x e/ x
5	2, 4	2th 173	. . 3 |- ( x = x <-> x e/ x )
6	5	abbii 2255	. 2 |- { x | x = x } = { x | x e/ x }
7	1, 6	eqtr2i 2161	1 |- { x | x e/ x } = _V

Syntax hints:   = wceq 1331   {cab 2125   e/ wnel 2403   _Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-v 2688  df-dif 3073  df-sn 3533

This theorem is referenced by:  ruALT  4466