ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brrelex2i GIF version

Theorem brrelex2i 4590
Description: The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
brrelexi.1 Rel 𝑅
Assertion
Ref Expression
brrelex2i (𝐴𝑅𝐵𝐵 ∈ V)

Proof of Theorem brrelex2i
StepHypRef Expression
1 brrelexi.1 . 2 Rel 𝑅
2 brrelex2 4587 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
31, 2mpan 421 1 (𝐴𝑅𝐵𝐵 ∈ 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:  vtoclr  4594  brdomi  6650  xpdom2  6732  xpdom1g  6734  mapdom1g  6748  djudom  6985  difinfsn  6992  enomnilem  7017  enmkvlem  7042  enwomnilem  7049  djuenun  7084  aprcl  8431  hashinfom  10555  clim  11081  ntrivcvgap0  11349
  Copyright terms: Public domain W3C validator