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Theorem ruv 7310
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 2792 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1646 . . . 4  |-  x  =  x
3 elirrv 7307 . . . . 5  |-  -.  x  e.  x
4 df-nel 2451 . . . . 5  |-  ( x  e/  x  <->  -.  x  e.  x )
53, 4mpbir 202 . . . 4  |-  x  e/  x
62, 52th 232 . . 3  |-  ( x  =  x  <->  x  e/  x )
76abbii 2397 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
81, 7eqtr2i 2306 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 5    = wceq 1624    e. wcel 1685   {cab 2271    e/ wnel 2449   _Vcvv 2790
This theorem is referenced by:  ruALT  7311
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-reg 7302
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-v 2792  df-dif 3157  df-un 3159  df-nul 3458  df-sn 3648  df-pr 3649
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