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Theorem altxpexg 25339
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 25338 . 2  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
2 pwexg 4296 . . . 4  |-  ( B  e.  W  ->  ~P B  e.  _V )
3 unexg 4624 . . . 4  |-  ( ( A  e.  V  /\  ~P B  e.  _V )  ->  ( A  u.  ~P B )  e.  _V )
42, 3sylan2 460 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  ~P B )  e.  _V )
5 pwexg 4296 . . 3  |-  ( ( A  u.  ~P B
)  e.  _V  ->  ~P ( A  u.  ~P B )  e.  _V )
6 pwexg 4296 . . 3  |-  ( ~P ( A  u.  ~P B )  e.  _V  ->  ~P ~P ( A  u.  ~P B )  e.  _V )
74, 5, 63syl 18 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ~P ( A  u.  ~P B )  e.  _V )
8 ssexg 4262 . 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 644 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  XX.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1715   _Vcvv 2873    u. cun 3236    C_ wss 3238   ~Pcpw 3714    XX. caltxp 25318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-pw 3716  df-sn 3735  df-pr 3736  df-uni 3930  df-altop 25319  df-altxp 25320
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