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Theorem elrel 5007
Description: A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
elrel  |-  ( ( Rel  R  /\  A  e.  R )  ->  E. x E. y  A  =  <. x ,  y >.
)
Distinct variable group:    x, y, A
Allowed substitution hints:    R( x, y)

Proof of Theorem elrel
StepHypRef Expression
1 df-rel 4914 . . . 4  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 188 . . 3  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 3334 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  ( _V  X.  _V ) )
4 elvv 4965 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
53, 4sylib 190 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  E. x E. y  A  =  <. x ,  y >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1727   _Vcvv 2962    C_ wss 3306   <.cop 3841    X. cxp 4905   Rel wrel 4912
This theorem is referenced by:  eliunxp  5041  elres  5210  unielrel  5423  frxp  6485  rntpos  6521  gsum2d2lem  15578  dfpo2  25409  fundmpss  25421  sscoid  25789  elfuns  25791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-opab 4292  df-xp 4913  df-rel 4914
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