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Theorem ruv 7503
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 2903 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1683 . . . 4  |-  x  =  x
3 elirrv 7500 . . . . 5  |-  -.  x  e.  x
4 df-nel 2555 . . . . 5  |-  ( x  e/  x  <->  -.  x  e.  x )
53, 4mpbir 201 . . . 4  |-  x  e/  x
62, 52th 231 . . 3  |-  ( x  =  x  <->  x  e/  x )
76abbii 2501 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
81, 7eqtr2i 2410 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649   {cab 2375    e/ wnel 2553   _Vcvv 2901
This theorem is referenced by:  ruALT  7504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-reg 7495
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-v 2903  df-dif 3268  df-un 3270  df-nul 3574  df-sn 3765  df-pr 3766
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