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Theorem altxpexg 25828
Description: The alternate cross product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
Assertion
Ref Expression
altxpexg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  XX.  B
)  e.  _V )

Proof of Theorem altxpexg
StepHypRef Expression
1 altxpsspw 25827 . 2  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
2 pwexg 4386 . . . 4  |-  ( B  e.  W  ->  ~P B  e.  _V )
3 unexg 4713 . . . 4  |-  ( ( A  e.  V  /\  ~P B  e.  _V )  ->  ( A  u.  ~P B )  e.  _V )
42, 3sylan2 462 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  ~P B )  e.  _V )
5 pwexg 4386 . . 3  |-  ( ( A  u.  ~P B
)  e.  _V  ->  ~P ( A  u.  ~P B )  e.  _V )
6 pwexg 4386 . . 3  |-  ( ~P ( A  u.  ~P B )  e.  _V  ->  ~P ~P ( A  u.  ~P B )  e.  _V )
74, 5, 63syl 19 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ~P ( A  u.  ~P B )  e.  _V )
8 ssexg 4352 . 2  |-  ( ( ( A  XX.  B
)  C_  ~P ~P ( A  u.  ~P B )  /\  ~P ~P ( A  u.  ~P B )  e.  _V )  ->  ( A  XX.  B )  e.  _V )
91, 7, 8sylancr 646 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  XX.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   _Vcvv 2958    u. cun 3320    C_ wss 3322   ~Pcpw 3801    XX. caltxp 25807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-pr 3823  df-uni 4018  df-altop 25808  df-altxp 25809
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