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Theorem issetid 4917
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 4916 . 2  |-  ( A  e.  _V  ->  A  _I  A )
2 reli 4892 . . 3  |-  Rel  _I
32brrelexi 4808 . 2  |-  ( A  _I  A  ->  A  e.  _V )
41, 3impbii 180 1  |-  ( A  e.  _V  <->  A  _I  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1710   _Vcvv 2864   class class class wbr 4102    _I cid 4383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775
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