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Theorem opelxp1 4568
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 4564 . 2  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  <->  ( A  e.  C  /\  B  e.  D ) )
21simplbi 272 1  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480   <.cop 3525    X. cxp 4532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-xp 4540
This theorem is referenced by:  otelxp1  4570  dmxpss  4964  nfvres  5447  ressnop0  5594  swoord1  6451  swoord2  6452  txlm  12437
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