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Theorem altxpexg 24512
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 24511 . 2  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
2 pwexg 4194 . . . 4  |-  ( B  e.  W  ->  ~P B  e.  _V )
3 unexg 4521 . . . 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 4194 . . 3  |-  ( ( A  u.  ~P B
)  e.  _V  ->  ~P ( A  u.  ~P B )  e.  _V )
6 pwexg 4194 . . 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 4160 . 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 1684   _Vcvv 2788    u. cun 3150    C_ wss 3152   ~Pcpw 3625    XX. caltxp 24491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-altop 24492  df-altxp 24493
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