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Theorem issetid 4994
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 4993 . 2  |-  ( A  e.  _V  ->  A  _I  A )
2 reli 4969 . . 3  |-  Rel  _I
32brrelexi 4885 . 2  |-  ( A  _I  A  ->  A  e.  _V )
41, 3impbii 181 1  |-  ( A  e.  _V  <->  A  _I  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1721   _Vcvv 2924   class class class wbr 4180    _I cid 4461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852
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