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Theorem xpss1 4619
Description: Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
Assertion
Ref Expression
xpss1  |-  ( A 
C_  B  ->  ( A  X.  C )  C_  ( B  X.  C
) )

Proof of Theorem xpss1
StepHypRef Expression
1 ssid 3087 . 2  |-  C  C_  C
2 xpss12 4616 . 2  |-  ( ( A  C_  B  /\  C  C_  C )  -> 
( A  X.  C
)  C_  ( B  X.  C ) )
31, 2mpan2 421 1  |-  ( A 
C_  B  ->  ( A  X.  C )  C_  ( B  X.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3041    X. cxp 4507
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-in 3047  df-ss 3054  df-opab 3960  df-xp 4515
This theorem is referenced by:  ssres2  4816  ssxp1  4945  funssxp  5262  tposssxp  6114  tpostpos2  6130  tfrlemibfn  6193  tfr1onlembfn  6209  tfrcllembfn  6222  enq0enq  7207  tx1cn  12349
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