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Theorem ruv 7528
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 2922 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1684 . . . 4  |-  x  =  x
3 elirrv 7525 . . . . 5  |-  -.  x  e.  x
4 df-nel 2574 . . . . 5  |-  ( x  e/  x  <->  -.  x  e.  x )
53, 4mpbir 201 . . . 4  |-  x  e/  x
62, 52th 231 . . 3  |-  ( x  =  x  <->  x  e/  x )
76abbii 2520 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
81, 7eqtr2i 2429 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649   {cab 2394    e/ wnel 2572   _Vcvv 2920
This theorem is referenced by:  ruALT  7529
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-reg 7520
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-v 2922  df-dif 3287  df-un 3289  df-nul 3593  df-sn 3784  df-pr 3785
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