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Theorem ruv 7314
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 2790 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1644 . . . 4  |-  x  =  x
3 elirrv 7311 . . . . 5  |-  -.  x  e.  x
4 df-nel 2449 . . . . 5  |-  ( x  e/  x  <->  -.  x  e.  x )
53, 4mpbir 200 . . . 4  |-  x  e/  x
62, 52th 230 . . 3  |-  ( x  =  x  <->  x  e/  x )
76abbii 2395 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
81, 7eqtr2i 2304 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   {cab 2269    e/ wnel 2447   _Vcvv 2788
This theorem is referenced by:  ruALT  7315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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