Theorem xpss 4655

Description: A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Assertion

Ref	Expression
xpss	|- ( A X. B ) C_ ( _V X. _V )

Proof of Theorem xpss

Step	Hyp	Ref	Expression
1		ssv 3124	. 2 |- A C_ _V
2		ssv 3124	. 2 |- B C_ _V
3		xpss12 4654	. 2 |- ( ( A C_ _V /\ B C_ _V ) -> ( A X. B ) C_ ( _V X. _V ) )
4	1, 2, 3	mp2an 423	1 |- ( A X. B ) C_ ( _V X. _V )

Syntax hints:   _Vcvv 2689   C_ wss 3076   X. cxp 4545

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-opab 3998  df-xp 4553

This theorem is referenced by:  relxp  4656  eqbrrdva  4717  relrelss  5073  funinsn  5180  eqopi  6078  op1steq  6085  dfoprab4  6098  f1od2  6140  frecuzrdgtcl  10216  frecuzrdgfunlem  10223  upxp  12480