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Theorem xpss1 4476
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 3019 . 2  |-  C  C_  C
2 xpss12 4473 . 2  |-  ( ( A  C_  B  /\  C  C_  C )  -> 
( A  X.  C
)  C_  ( B  X.  C ) )
31, 2mpan2 416 1  |-  ( A 
C_  B  ->  ( A  X.  C )  C_  ( B  X.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 2974    X. cxp 4369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-opab 3848  df-xp 4377
This theorem is referenced by:  ssres2  4666  ssxp1  4787  funssxp  5091  tposssxp  5898  tpostpos2  5914  tfrlemibfn  5977  tfr1onlembfn  5993  tfrcllembfn  6006  enq0enq  6683
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