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Theorem issetid 5056
Description: Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
issetid  |-  ( A  e.  _V  <->  A  _I  A )

Proof of Theorem issetid
StepHypRef Expression
1 ididg 5055 . 2  |-  ( A  e.  _V  ->  A  _I  A )
2 reli 5031 . . 3  |-  Rel  _I
32brrelexi 4947 . 2  |-  ( A  _I  A  ->  A  e.  _V )
41, 3impbii 182 1  |-  ( A  e.  _V  <->  A  _I  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1727   _Vcvv 2962   class class class wbr 4237    _I cid 4522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914
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