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Theorem ruv 4356
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 2621 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1634 . . . 4  |-  x  =  x
3 elirrv 4354 . . . . 5  |-  -.  x  e.  x
43nelir 2353 . . . 4  |-  x  e/  x
52, 42th 172 . . 3  |-  ( x  =  x  <->  x  e/  x )
65abbii 2203 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
71, 6eqtr2i 2109 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1289   {cab 2074    e/ wnel 2350   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-v 2621  df-dif 2999  df-sn 3447
This theorem is referenced by:  ruALT  4357
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