Theorem xpss12 4825

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 3218	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3218	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 600	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4366	. 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 4724	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4724	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3267	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   C_ wss 3197   {copab 4143   X. cxp 4716

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210  df-opab 4145  df-xp 4724

This theorem is referenced by:  xpss  4826  xpss1  4828  xpss2  4829  djussxp  4866  ssxpbm  5163  ssrnres  5170  cossxp  5250  cossxp2  5251  cocnvss  5253  relrelss  5254  fssxp  5490  oprabss  6089  pmss12g  6820  caserel  7250  casef  7251  dmaddpi  7508  dmmulpi  7509  rexpssxrxp  8187  ltrelxr  8203  dfz2  9515  phimullem  12742  znleval  14611  txuni2  14924  txbas  14926  neitx  14936  txcnp  14939  cnmpt2res  14965  psmetres2  15001  xmetres2  15047  metres2  15049  xmetresbl  15108  xmettx  15178  qtopbasss  15189  tgqioo  15223  resubmet  15224  limccnp2lem  15344  limccnp2cntop  15345  mpodvdsmulf1o  15658  fsumdvdsmul  15659