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Theorem eqv 3274
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 2076 . 2  |-  ( A  =  _V  <->  A. x
( x  e.  A  <->  x  e.  _V ) )
2 vex 2605 . . . 4  |-  x  e. 
_V
32tbt 245 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  <->  x  e.  _V ) )
43albii 1400 . 2  |-  ( A. x  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  _V ) )
51, 4bitr4i 185 1  |-  ( A  =  _V  <->  A. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   _Vcvv 2602
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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by:  setindel  4289  dmi  4578
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