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Theorem altxpexg 28009
Description: The alternate Cartesian 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 28008 . 2  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
2 pwexg 4476 . . . 4  |-  ( B  e.  W  ->  ~P B  e.  _V )
3 unexg 6381 . . . 4  |-  ( ( A  e.  V  /\  ~P B  e.  _V )  ->  ( A  u.  ~P B )  e.  _V )
42, 3sylan2 474 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  ~P B )  e.  _V )
5 pwexg 4476 . . 3  |-  ( ( A  u.  ~P B
)  e.  _V  ->  ~P ( A  u.  ~P B )  e.  _V )
6 pwexg 4476 . . 3  |-  ( ~P ( A  u.  ~P B )  e.  _V  ->  ~P ~P ( A  u.  ~P B )  e.  _V )
74, 5, 63syl 20 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ~P ( A  u.  ~P B )  e.  _V )
8 ssexg 4438 . 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 663 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  XX.  B
)  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   _Vcvv 2972    u. cun 3326    C_ wss 3328   ~Pcpw 3860    XX. caltxp 27988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-pw 3862  df-sn 3878  df-pr 3880  df-uni 4092  df-altop 27989  df-altxp 27990
This theorem is referenced by: (None)
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