Theorem xpss1 4466

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 3018	. 2 |- C C_ C
2		xpss12 4463	. 2 |- ( ( A C_ B /\ C C_ C ) -> ( A X. C ) C_ ( B X. C ) )
3	1, 2	mpan2 415	1 |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) )

Syntax hints:   -> wi 4   C_ wss 2973   X. cxp 4361

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-opab 3840  df-xp 4369

This theorem is referenced by:  ssres2  4656  ssxp1  4777  funssxp  5080  tposssxp  5887  tpostpos2  5903  tfrlemibfn  5965  enq0enq  6621