ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  issetid GIF version

Theorem issetid 4780
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 4779 . 2 (𝐴 ∈ V → 𝐴 I 𝐴)
2 reli 4755 . . 3 Rel I
32brrelex1i 4668 . 2 (𝐴 I 𝐴𝐴 ∈ V)
41, 3impbii 126 1 (𝐴 ∈ V ↔ 𝐴 I 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2148  Vcvv 2737   class class class wbr 4002   I cid 4287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator