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Theorem ruv 4433
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 2660 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1660 . . . 4  |-  x  =  x
3 elirrv 4431 . . . . 5  |-  -.  x  e.  x
43nelir 2381 . . . 4  |-  x  e/  x
52, 42th 173 . . 3  |-  ( x  =  x  <->  x  e/  x )
65abbii 2231 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
71, 6eqtr2i 2137 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1314   {cab 2101    e/ wnel 2378   _Vcvv 2658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-v 2660  df-dif 3041  df-sn 3501
This theorem is referenced by:  ruALT  4434
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