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

Theorem brrelexi 4411
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 4409 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
31, 2mpan 408 1 (𝐴𝑅𝐵𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  Vcvv 2574   class class class wbr 3791  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379
This theorem is referenced by:  nprrel  4413  vtoclr  4415  opeliunxp2  4503  ideqg  4514  issetid  4517  fvmptss2  5274  brtpos2  5896  brdomg  6259  isfi  6271  en1uniel  6314  xpdom2  6335  xpdom1g  6337  climcl  10033  climi  10038  climrecl  10074
  Copyright terms: Public domain W3C validator