Theorem xpss 4642

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 3114	. 2 |- A C_ _V
2		ssv 3114	. 2 |- B C_ _V
3		xpss12 4641	. 2 |- ( ( A C_ _V /\ B C_ _V ) -> ( A X. B ) C_ ( _V X. _V ) )
4	1, 2, 3	mp2an 422	1 |- ( A X. B ) C_ ( _V X. _V )

Syntax hints:   _Vcvv 2681   C_ wss 3066   X. cxp 4532

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-opab 3985  df-xp 4540

This theorem is referenced by:  relxp  4643  eqbrrdva  4704  relrelss  5060  funinsn  5167  eqopi  6063  op1steq  6070  dfoprab4  6083  f1od2  6125  frecuzrdgtcl  10178  frecuzrdgfunlem  10185  upxp  12430