Theorem xpss 4733

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 3177	. 2 |- A C_ _V
2		ssv 3177	. 2 |- B C_ _V
3		xpss12 4732	. 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 2737   C_ wss 3129   X. cxp 4623

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-opab 4064  df-xp 4631

This theorem is referenced by:  relxp  4734  eqbrrdva  4796  relrelss  5154  funinsn  5264  eqopi  6170  op1steq  6177  dfoprab4  6190  f1od2  6233  frecuzrdgtcl  10407  frecuzrdgfunlem  10414  reldvdsrsrg  13192  upxp  13643