Theorem xpss1 4773

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 3203	. 2 |- C C_ C
2		xpss12 4770	. 2 |- ( ( A C_ B /\ C C_ C ) -> ( A X. C ) C_ ( B X. C ) )
3	1, 2	mpan2 425	1 |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) )

Syntax hints:   -> wi 4   C_ wss 3157   X. cxp 4661

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-opab 4095  df-xp 4669

This theorem is referenced by:  ssres2  4973  ssxp1  5106  funssxp  5427  tposssxp  6307  tpostpos2  6323  tfrlemibfn  6386  tfr1onlembfn  6402  tfrcllembfn  6415  enq0enq  7498  tx1cn  14505