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Theorem issetid 4697
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 ∈ V ↔ 𝐴 I 𝐴)

Proof of Theorem issetid
StepHypRef Expression
1 ididg 4696 . 2 (𝐴 ∈ V → 𝐴 I 𝐴)
2 reli 4672 . . 3 Rel I
32brrelex1i 4586 . 2 (𝐴 I 𝐴𝐴 ∈ V)
41, 3impbii 125 1 (𝐴 ∈ V ↔ 𝐴 I 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1481  Vcvv 2687   class class class wbr 3933   I cid 4214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-id 4219  df-xp 4549  df-rel 4550
This theorem is referenced by: (None)
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