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Theorem eqv 3643
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 2430 . 2  |-  ( A  =  _V  <->  A. x
( x  e.  A  <->  x  e.  _V ) )
2 vex 2959 . . . 4  |-  x  e. 
_V
32tbt 334 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  <->  x  e.  _V ) )
43albii 1575 . 2  |-  ( A. x  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  _V ) )
51, 4bitr4i 244 1  |-  ( A  =  _V  <->  A. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2956
This theorem is referenced by:  dmi  5084  dfac10  8017  dfac10c  8018  dfac10b  8019  uniwun  8615  fnsingle  25764  ttac  27107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-v 2958
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