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Theorem otelxp1 4740
Description: The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)
Assertion
Ref Expression
otelxp1  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( R  X.  S
)  X.  T )  ->  A  e.  R
)

Proof of Theorem otelxp1
StepHypRef Expression
1 opelxp1 4738 . 2  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( R  X.  S
)  X.  T )  ->  <. A ,  B >.  e.  ( R  X.  S ) )
2 opelxp1 4738 . 2  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  ->  A  e.  R )
31, 2syl 15 1  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( R  X.  S
)  X.  T )  ->  A  e.  R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   <.cop 3656    X. cxp 4703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711
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