Theorem xpss2 4722

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 3167	. 2 |- C C_ C
2		xpss12 4718	. 2 |- ( ( C C_ C /\ A C_ B ) -> ( C X. A ) C_ ( C X. B ) )
3	1, 2	mpan 422	1 |- ( A C_ B -> ( C X. A ) C_ ( C X. B ) )

Syntax hints:   -> wi 4   C_ wss 3121   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-opab 4051  df-xp 4617

This theorem is referenced by:  ssxp2  5048  xpdom3m  6812  axresscn  7822  tx2cn  13064  dvfvalap  13444