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Theorem eqv 3382
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)
Assertion
Ref Expression
eqv  |-  ( A  =  _V  <->  A. x  x  e.  A )
Distinct variable group:    x, A

Proof of Theorem eqv
StepHypRef Expression
1 dfcleq 2133 . 2  |-  ( A  =  _V  <->  A. x
( x  e.  A  <->  x  e.  _V ) )
2 vex 2689 . . . 4  |-  x  e. 
_V
32tbt 246 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  <->  x  e.  _V ) )
43albii 1446 . 2  |-  ( A. x  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  _V ) )
51, 4bitr4i 186 1  |-  ( A  =  _V  <->  A. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   _Vcvv 2686
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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688
This theorem is referenced by:  setindel  4453  dmi  4754
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