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Theorem issetid 4663
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 4662 . 2 (𝐴 ∈ V → 𝐴 I 𝐴)
2 reli 4638 . . 3 Rel I
32brrelex1i 4552 . 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 1465  Vcvv 2660   class class class wbr 3899   I cid 4180
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516
This theorem is referenced by: (None)
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