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Theorem issetid 4518
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 4517 . 2  |-  ( A  e.  _V  ->  A  _I  A )
2 reli 4493 . . 3  |-  Rel  _I
32brrelexi 4412 . 2  |-  ( A  _I  A  ->  A  e.  _V )
41, 3impbii 121 1  |-  ( A  e.  _V  <->  A  _I  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 102    e. wcel 1409   _Vcvv 2574   class class class wbr 3792    _I cid 4053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380
This theorem is referenced by: (None)
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