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Theorem issetid 5432
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 5431 . 2 (𝐴 ∈ V → 𝐴 I 𝐴)
2 reli 5405 . . 3 Rel I
32brrelexi 5315 . 2 (𝐴 I 𝐴𝐴 ∈ V)
41, 3impbii 199 1 (𝐴 ∈ V ↔ 𝐴 I 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2139  Vcvv 3340   class class class wbr 4804   I cid 5173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273
This theorem is referenced by: (None)
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