Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  brrelex1i GIF version

Theorem brrelex1i 4589
 Description: The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
Hypothesis
Ref Expression
brrelexi.1 Rel 𝑅
Assertion
Ref Expression
brrelex1i (𝐴𝑅𝐵𝐴 ∈ V)

Proof of Theorem brrelex1i
StepHypRef Expression
1 brrelexi.1 . 2 Rel 𝑅
2 brrelex1 4585 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
31, 2mpan 421 1 (𝐴𝑅𝐵𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1481  Vcvv 2689   class class class wbr 3936  Rel wrel 4551 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 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553 This theorem is referenced by:  nprrel  4591  vtoclr  4594  opeliunxp2  4686  ideqg  4697  issetid  4700  fvmptss2  5503  opeliunxp2f  6142  brtpos2  6155  brdomg  6649  ctex  6654  isfi  6662  en1uniel  6705  xpdom2  6732  xpdom1g  6734  xpen  6746  isbth  6862  djudom  6985  cc3  7099  aprcl  8431  climcl  11082  climi  11087  climrecl  11124  structex  12008
 Copyright terms: Public domain W3C validator