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

Theorem brrelexi 4438
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
brrelexi (𝐴𝑅𝐵𝐴 ∈ V)

Proof of Theorem brrelexi
StepHypRef Expression
1 brrelexi.1 . 2 Rel 𝑅
2 brrelex 4436 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
31, 2mpan 415 1 (𝐴𝑅𝐵𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  Vcvv 2612   class class class wbr 3811  Rel wrel 4404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4405  df-rel 4406
This theorem is referenced by:  nprrel  4440  vtoclr  4442  opeliunxp2  4532  ideqg  4543  issetid  4546  fvmptss2  5322  brtpos2  5946  brdomg  6393  isfi  6406  en1uniel  6449  xpdom2  6475  xpdom1g  6477  xpen  6489  djudom  6692  climcl  10493  climi  10498  climrecl  10534
  Copyright terms: Public domain W3C validator