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Theorem ruv 7560
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 2950 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1688 . . . 4  |-  x  =  x
3 elirrv 7557 . . . . 5  |-  -.  x  e.  x
4 df-nel 2601 . . . . 5  |-  ( x  e/  x  <->  -.  x  e.  x )
53, 4mpbir 201 . . . 4  |-  x  e/  x
62, 52th 231 . . 3  |-  ( x  =  x  <->  x  e/  x )
76abbii 2547 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
81, 7eqtr2i 2456 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652   {cab 2421    e/ wnel 2599   _Vcvv 2948
This theorem is referenced by:  ruALT  7561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-reg 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-nul 3621  df-sn 3812  df-pr 3813
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