Theorem xpss12 4751

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 3164	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3164	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 598	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4296	. 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 4650	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4650	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3213	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   C_ wss 3144   {copab 4078   X. cxp 4642

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-in 3150  df-ss 3157  df-opab 4080  df-xp 4650

This theorem is referenced by:  xpss  4752  xpss1  4754  xpss2  4755  djussxp  4790  ssxpbm  5082  ssrnres  5089  cossxp  5169  cossxp2  5170  cocnvss  5172  relrelss  5173  fssxp  5402  oprabss  5983  pmss12g  6702  caserel  7117  casef  7118  dmaddpi  7355  dmmulpi  7356  rexpssxrxp  8033  ltrelxr  8049  dfz2  9356  phimullem  12260  txuni2  14233  txbas  14235  neitx  14245  txcnp  14248  cnmpt2res  14274  psmetres2  14310  xmetres2  14356  metres2  14358  xmetresbl  14417  xmettx  14487  qtopbasss  14498  tgqioo  14524  resubmet  14525  limccnp2lem  14622  limccnp2cntop  14623