Theorem xpss 4883

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 3270	. 2 |- A C_ _V
2		ssv 3270	. 2 |- B C_ _V
3		xpss12 4882	. 2 |- ( ( A C_ _V /\ B C_ _V ) -> ( A X. B ) C_ ( _V X. _V ) )
4	1, 2, 3	mp2an 430	1 |- ( A X. B ) C_ ( _V X. _V )

This proof depends on syntax axioms:   _Vcvv 2821   C_ wss 3220   X. cxp 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-opab 4193  df-xp 4780

This theorem is used by:  relxp  4884  eqbrrdva  4950  relrelss  5314  funinsn  5430  eqopi  6406  op1steq  6413  dfoprab4  6426  f1od2  6471  frecuzrdgtcl  10849  frecuzrdgfunlem  10856  upxp  15373