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Theorem opelxp1 4645
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 4641 . 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 2141   <.cop 3586    X. cxp 4609
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617
This theorem is referenced by:  otelxp1  4647  dmxpss  5041  nfvres  5529  ressnop0  5677  swoord1  6542  swoord2  6543  txlm  13073
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