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Theorem brrelexi 4452
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 4450 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
31, 2mpan 415 1 (𝐴𝑅𝐵𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1436  Vcvv 2615   class class class wbr 3822  Rel wrel 4418
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420
This theorem is referenced by:  nprrel  4454  vtoclr  4456  opeliunxp2  4546  ideqg  4557  issetid  4560  fvmptss2  5344  brtpos2  5972  brdomg  6419  isfi  6432  en1uniel  6475  xpdom2  6501  xpdom1g  6503  xpen  6515  isbth  6623  djudom  6734  climcl  10568  climi  10573  climrecl  10609
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