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Theorem elrel 5093
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 4995 . . . 4  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 194 . . 3  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 3489 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  ( _V  X.  _V ) )
4 elvv 5047 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
53, 4sylib 196 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  E. x E. y  A  =  <. x ,  y >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106    C_ wss 3461   <.cop 4022    X. cxp 4986   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-xp 4994  df-rel 4995
This theorem is referenced by:  eliunxp  5129  elres  5297  unielrel  5515  frxp  6883  rntpos  6960  gsum2d2lem  17200  dfpo2  29428  fundmpss  29440  sscoid  29794  elfuns  29796  eliunxp2  33196
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