Theorem xpss 4858

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 3260	. 2 |- A C_ _V
2		ssv 3260	. 2 |- B C_ _V
3		xpss12 4857	. 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 2813   C_ wss 3211   X. cxp 4747

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-opab 4172  df-xp 4755

This theorem is referenced by:  relxp  4859  eqbrrdva  4925  relrelss  5289  funinsn  5405  eqopi  6366  op1steq  6373  dfoprab4  6386  f1od2  6431  frecuzrdgtcl  10774  frecuzrdgfunlem  10781  upxp  15137