Theorem xpss2 4786

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 3213	. 2 |- C C_ C
2		xpss12 4782	. 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 3166   X. cxp 4673

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-opab 4106  df-xp 4681

This theorem is referenced by:  ssxp2  5120  xpdom3m  6929  axresscn  7973  tx2cn  14742  dvfvalap  15153