Theorem xpss 4782

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 3214	. 2 |- A C_ _V
2		ssv 3214	. 2 |- B C_ _V
3		xpss12 4781	. 2 |- ( ( A C_ _V /\ B C_ _V ) -> ( A X. B ) C_ ( _V X. _V ) )
4	1, 2, 3	mp2an 426	1 |- ( A X. B ) C_ ( _V X. _V )

Syntax hints:   _Vcvv 2771   C_ wss 3165   X. cxp 4672

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-opab 4105  df-xp 4680

This theorem is referenced by:  relxp  4783  eqbrrdva  4847  relrelss  5208  funinsn  5322  eqopi  6257  op1steq  6264  dfoprab4  6277  f1od2  6320  frecuzrdgtcl  10555  frecuzrdgfunlem  10562  reldvdsrsrg  13825  upxp  14715