Theorem xpss2 4771

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 3200	. 2 |- C C_ C
2		xpss12 4767	. 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 3154   X. cxp 4658

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3160  df-ss 3167  df-opab 4092  df-xp 4666

This theorem is referenced by:  ssxp2  5104  xpdom3m  6890  axresscn  7922  tx2cn  14449  dvfvalap  14860