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Theorem xpss1 4736
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 3175 . 2  |-  C  C_  C
2 xpss12 4733 . 2  |-  ( ( A  C_  B  /\  C  C_  C )  -> 
( A  X.  C
)  C_  ( B  X.  C ) )
31, 2mpan2 425 1  |-  ( A 
C_  B  ->  ( A  X.  C )  C_  ( B  X.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3129    X. cxp 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3135  df-ss 3142  df-opab 4065  df-xp 4632
This theorem is referenced by:  ssres2  4934  ssxp1  5065  funssxp  5385  tposssxp  6249  tpostpos2  6265  tfrlemibfn  6328  tfr1onlembfn  6344  tfrcllembfn  6357  enq0enq  7429  tx1cn  13739
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