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Theorem ruv 7282
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 2765 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1818 . . . 4  |-  x  =  x
3 elirrv 7279 . . . . 5  |-  -.  x  e.  x
4 df-nel 2424 . . . . 5  |-  ( x  e/  x  <->  -.  x  e.  x )
53, 4mpbir 202 . . . 4  |-  x  e/  x
62, 52th 232 . . 3  |-  ( x  =  x  <->  x  e/  x )
76abbii 2370 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
81, 7eqtr2i 2279 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 5    = wceq 1619    e. wcel 1621   {cab 2244    e/ wnel 2422   _Vcvv 2763
This theorem is referenced by:  ruALT  7283
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-reg 7274
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-v 2765  df-dif 3130  df-un 3132  df-nul 3431  df-sn 3620  df-pr 3621
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