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Theorem ruv 4274
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 2559 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1589 . . . 4  |-  x  =  x
3 elirrv 4272 . . . . 5  |-  -.  x  e.  x
43nelir 2300 . . . 4  |-  x  e/  x
52, 42th 163 . . 3  |-  ( x  =  x  <->  x  e/  x )
65abbii 2153 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
71, 6eqtr2i 2061 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1243   {cab 2026    e/ wnel 2205   _Vcvv 2557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-v 2559  df-dif 2920  df-sn 3381
This theorem is referenced by:  ruALT  4275
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