Theorem xpss12 4877

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 3242	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3242	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 606	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4416	. 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 4775	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4775	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3291	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   C_ wss 3220   {copab 4186   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-opab 4188  df-xp 4775

This theorem is referenced by:  xpss  4878  xpss1  4880  xpss2  4881  djussxp  4920  ssxpbm  5218  ssrnres  5225  cossxp  5305  cossxp2  5306  cocnvss  5308  relrelss  5309  fssxp  5550  oprabss  6164  pmss12g  6946  caserel  7417  casef  7418  dmaddpi  7682  dmmulpi  7683  rexpssxrxp  8360  ltrelxr  8376  dfz2  9696  phimullem  12981  znleval  14960  txuni2  15280  txbas  15282  neitx  15292  txcnp  15295  cnmpt2res  15321  psmetres2  15357  xmetres2  15403  metres2  15405  xmetresbl  15464  xmettx  15534  qtopbasss  15545  tgqioo  15579  resubmet  15580  limccnp2lem  15700  limccnp2cntop  15701  mpodvdsmulf1o  16018  fsumdvdsmul  16019