Theorem xpss12 4826

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 4367	. 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 4725	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4725	. 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 4144   X. cxp 4717

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 4146  df-xp 4725

This theorem is referenced by:  xpss  4827  xpss1  4829  xpss2  4830  djussxp  4867  ssxpbm  5164  ssrnres  5171  cossxp  5251  cossxp2  5252  cocnvss  5254  relrelss  5255  fssxp  5493  oprabss  6096  pmss12g  6830  caserel  7265  casef  7266  dmaddpi  7523  dmmulpi  7524  rexpssxrxp  8202  ltrelxr  8218  dfz2  9530  phimullem  12762  znleval  14632  txuni2  14945  txbas  14947  neitx  14957  txcnp  14960  cnmpt2res  14986  psmetres2  15022  xmetres2  15068  metres2  15070  xmetresbl  15129  xmettx  15199  qtopbasss  15210  tgqioo  15244  resubmet  15245  limccnp2lem  15365  limccnp2cntop  15366  mpodvdsmulf1o  15679  fsumdvdsmul  15680