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Theorem brrelex1i 4666
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 4662 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
31, 2mpan 424 1 (𝐴𝑅𝐵𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  Vcvv 2737   class class class wbr 4000  Rel wrel 4628
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 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630
This theorem is referenced by:  nprrel  4668  vtoclr  4671  opeliunxp2  4763  ideqg  4774  issetid  4777  fvmptss2  5587  opeliunxp2f  6233  brtpos2  6246  brdomg  6742  ctex  6747  isfi  6755  en1uniel  6798  xpdom2  6825  xpdom1g  6827  xpen  6839  isbth  6960  djudom  7086  cc3  7255  aprcl  8590  climcl  11271  climi  11276  climrecl  11313  structex  12454
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