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

Step	Hyp	Ref	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 ) )
3	1, 2	mpan2 416	1 |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) )

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