Theorem xpss12 4718

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 3141	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3141	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 593	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4263	. 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 4617	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4617	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3190	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   C_ wss 3121   {copab 4049   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-opab 4051  df-xp 4617

This theorem is referenced by:  xpss  4719  xpss1  4721  xpss2  4722  djussxp  4756  ssxpbm  5046  ssrnres  5053  cossxp  5133  cossxp2  5134  cocnvss  5136  relrelss  5137  fssxp  5365  oprabss  5939  pmss12g  6653  caserel  7064  casef  7065  dmaddpi  7287  dmmulpi  7288  rexpssxrxp  7964  ltrelxr  7980  dfz2  9284  phimullem  12179  txuni2  13050  txbas  13052  neitx  13062  txcnp  13065  cnmpt2res  13091  psmetres2  13127  xmetres2  13173  metres2  13175  xmetresbl  13234  xmettx  13304  qtopbasss  13315  tgqioo  13341  resubmet  13342  limccnp2lem  13439  limccnp2cntop  13440