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Theorem eqv 3483
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 2290 . 2  |-  ( A  =  _V  <->  A. x
( x  e.  A  <->  x  e.  _V ) )
2 vex 2804 . . . 4  |-  x  e. 
_V
32tbt 333 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  <->  x  e.  _V ) )
43albii 1556 . 2  |-  ( A. x  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  _V ) )
51, 4bitr4i 243 1  |-  ( A  =  _V  <->  A. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  dmi  4909  dfac10  7779  dfac10c  7780  dfac10b  7781  uniwun  8378  fnsingle  24529  dominc  25383  rninc  25384  ttac  27232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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