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Theorem issetid 5861
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 5860 . 2 (𝐴 ∈ V → 𝐴 I 𝐴)
2 reli 5832 . . 3 Rel I
32brrelex1i 5738 . 2 (𝐴 I 𝐴𝐴 ∈ V)
41, 3impbii 208 1 (𝐴 ∈ V ↔ 𝐴 I 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  Vcvv 3473   class class class wbr 5152   I cid 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689
This theorem is referenced by: (None)
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