Theorem xpss 4801

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 3223	. 2 |- A C_ _V
2		ssv 3223	. 2 |- B C_ _V
3		xpss12 4800	. 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 2776   C_ wss 3174   X. cxp 4691

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-opab 4122  df-xp 4699

This theorem is referenced by:  relxp  4802  eqbrrdva  4866  relrelss  5228  funinsn  5342  eqopi  6281  op1steq  6288  dfoprab4  6301  f1od2  6344  frecuzrdgtcl  10594  frecuzrdgfunlem  10601  reldvdsrsrg  13969  upxp  14859