Theorem xpss12 4857

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 3232	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3232	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 602	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4397	. 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 4755	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4755	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3281	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   C_ wss 3211   {copab 4170   X. cxp 4747

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224  df-opab 4172  df-xp 4755

This theorem is referenced by:  xpss  4858  xpss1  4860  xpss2  4861  djussxp  4900  ssxpbm  5198  ssrnres  5205  cossxp  5285  cossxp2  5286  cocnvss  5288  relrelss  5289  fssxp  5530  oprabss  6139  pmss12g  6909  caserel  7378  casef  7379  dmaddpi  7640  dmmulpi  7641  rexpssxrxp  8318  ltrelxr  8334  dfz2  9650  phimullem  12922  znleval  14801  txuni2  15121  txbas  15123  neitx  15133  txcnp  15136  cnmpt2res  15162  psmetres2  15198  xmetres2  15244  metres2  15246  xmetresbl  15305  xmettx  15375  qtopbasss  15386  tgqioo  15420  resubmet  15421  limccnp2lem  15541  limccnp2cntop  15542  mpodvdsmulf1o  15858  fsumdvdsmul  15859