Theorem xpss1 4769

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 3199	. 2 |- C C_ C
2		xpss12 4766	. 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 3153   X. cxp 4657

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 3159  df-ss 3166  df-opab 4091  df-xp 4665

This theorem is referenced by:  ssres2  4969  ssxp1  5102  funssxp  5423  tposssxp  6302  tpostpos2  6318  tfrlemibfn  6381  tfr1onlembfn  6397  tfrcllembfn  6410  enq0enq  7491  tx1cn  14437