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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
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