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Theorem ruv 7330
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
StepHypRef Expression
1 df-v 2803 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1662 . . . 4  |-  x  =  x
3 elirrv 7327 . . . . 5  |-  -.  x  e.  x
4 df-nel 2462 . . . . 5  |-  ( x  e/  x  <->  -.  x  e.  x )
53, 4mpbir 200 . . . 4  |-  x  e/  x
62, 52th 230 . . 3  |-  ( x  =  x  <->  x  e/  x )
76abbii 2408 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
81, 7eqtr2i 2317 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   {cab 2282    e/ wnel 2460   _Vcvv 2801
This theorem is referenced by:  ruALT  7331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660
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