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Theorem opelxp1 4722
Description: The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelxp1  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  ->  A  e.  C )

Proof of Theorem opelxp1
StepHypRef Expression
1 opelxp 4719 . 2  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  <->  ( A  e.  C  /\  B  e.  D ) )
21simplbi 446 1  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   <.cop 3643    X. cxp 4687
This theorem is referenced by:  otelxp1  4724  dff3  5673  ressnop0  5703  swoord1  6689  swoord2  6690  canthp1lem2  8275  txcmplem1  17335  txlm  17342  dvbsss  19252  vcoprnelem  21134  nvvcop  21150  nvvop  21165  linedegen  24766  opelopab3  26373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
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