Theorem xpss2 4835

Description: Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Assertion

Ref	Expression
xpss2	|- ( A C_ B -> ( C X. A ) C_ ( C X. B ) )

Proof of Theorem xpss2

Step	Hyp	Ref	Expression
1		ssid 3245	. 2 |- C C_ C
2		xpss12 4831	. 2 |- ( ( C C_ C /\ A C_ B ) -> ( C X. A ) C_ ( C X. B ) )
3	1, 2	mpan 424	1 |- ( A C_ B -> ( C X. A ) C_ ( C X. B ) )

Syntax hints:   -> wi 4   C_ wss 3198   X. cxp 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3204  df-ss 3211  df-opab 4149  df-xp 4729

This theorem is referenced by:  ssxp2  5172  xpdom3m  7013  axresscn  8070  tx2cn  14984  dvfvalap  15395