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Theorem eqv 3778
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 2415 . 2  |-  ( A  =  _V  <->  A. x
( x  e.  A  <->  x  e.  _V ) )
2 vex 3083 . . . 4  |-  x  e. 
_V
32tbt 345 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  <->  x  e.  _V ) )
43albii 1685 . 2  |-  ( A. x  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  _V ) )
51, 4bitr4i 255 1  |-  ( A  =  _V  <->  A. x  x  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435    = wceq 1437    e. wcel 1872   _Vcvv 3080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-12 1909  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3082
This theorem is referenced by:  dmi  5068  dfac10  8574  dfac10c  8575  dfac10b  8576  uniwun  9172  fnsingle  30691  bj-abtru  31477  ttac  35861  nev  36332
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