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Theorem altxpexg 25735
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 25734 . 2  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
2 pwexg 4351 . . . 4  |-  ( B  e.  W  ->  ~P B  e.  _V )
3 unexg 4677 . . . 4  |-  ( ( A  e.  V  /\  ~P B  e.  _V )  ->  ( A  u.  ~P B )  e.  _V )
42, 3sylan2 461 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  ~P B )  e.  _V )
5 pwexg 4351 . . 3  |-  ( ( A  u.  ~P B
)  e.  _V  ->  ~P ( A  u.  ~P B )  e.  _V )
6 pwexg 4351 . . 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 4317 . 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 645 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  XX.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   _Vcvv 2924    u. cun 3286    C_ wss 3288   ~Pcpw 3767    XX. caltxp 25714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-pw 3769  df-sn 3788  df-pr 3789  df-uni 3984  df-altop 25715  df-altxp 25716
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