Theorem xpss12 4767

Description: Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
xpss12	|- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Proof of Theorem xpss12

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3174	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3174	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 598	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4310	. 2 |- ( ( A C_ B /\ C C_ D ) -> { <. x , y >. | ( x e. A /\ y e. C ) } C_ { <. x , y >. | ( x e. B /\ y e. D ) } )
5		df-xp 4666	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4666	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3223	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   C_ wss 3154   {copab 4090   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:  xpss  4768  xpss1  4770  xpss2  4771  djussxp  4808  ssxpbm  5102  ssrnres  5109  cossxp  5189  cossxp2  5190  cocnvss  5192  relrelss  5193  fssxp  5422  oprabss  6005  pmss12g  6731  caserel  7148  casef  7149  dmaddpi  7387  dmmulpi  7388  rexpssxrxp  8066  ltrelxr  8082  dfz2  9392  phimullem  12366  znleval  14152  txuni2  14435  txbas  14437  neitx  14447  txcnp  14450  cnmpt2res  14476  psmetres2  14512  xmetres2  14558  metres2  14560  xmetresbl  14619  xmettx  14689  qtopbasss  14700  tgqioo  14734  resubmet  14735  limccnp2lem  14855  limccnp2cntop  14856