Theorem xpss 4826

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 3246	. 2 |- A C_ _V
2		ssv 3246	. 2 |- B C_ _V
3		xpss12 4825	. 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 2799   C_ wss 3197   X. cxp 4716

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-opab 4145  df-xp 4724

This theorem is referenced by:  relxp  4827  eqbrrdva  4891  relrelss  5254  funinsn  5369  eqopi  6316  op1steq  6323  dfoprab4  6336  f1od2  6379  frecuzrdgtcl  10629  frecuzrdgfunlem  10636  reldvdsrsrg  14050  upxp  14940