Theorem xpss1 4721

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 3167	. 2 |- C C_ C
2		xpss12 4718	. 2 |- ( ( A C_ B /\ C C_ C ) -> ( A X. C ) C_ ( B X. C ) )
3	1, 2	mpan2 423	1 |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) )

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:  ssres2  4918  ssxp1  5047  funssxp  5367  tposssxp  6228  tpostpos2  6244  tfrlemibfn  6307  tfr1onlembfn  6323  tfrcllembfn  6336  enq0enq  7393  tx1cn  13063