Theorem xpss12 4734

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 3150	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3150	. . . 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 4279	. 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 4633	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4633	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3199	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   C_ wss 3130   {copab 4064   X. cxp 4625

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3136  df-ss 3143  df-opab 4066  df-xp 4633

This theorem is referenced by:  xpss  4735  xpss1  4737  xpss2  4738  djussxp  4773  ssxpbm  5065  ssrnres  5072  cossxp  5152  cossxp2  5153  cocnvss  5155  relrelss  5156  fssxp  5384  oprabss  5961  pmss12g  6675  caserel  7086  casef  7087  dmaddpi  7324  dmmulpi  7325  rexpssxrxp  8002  ltrelxr  8018  dfz2  9325  phimullem  12225  txuni2  13759  txbas  13761  neitx  13771  txcnp  13774  cnmpt2res  13800  psmetres2  13836  xmetres2  13882  metres2  13884  xmetresbl  13943  xmettx  14013  qtopbasss  14024  tgqioo  14050  resubmet  14051  limccnp2lem  14148  limccnp2cntop  14149