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Theorem unixpss 4815
Description: The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
unixpss  |-  U. U. ( A  X.  B
)  C_  ( A  u.  B )

Proof of Theorem unixpss
StepHypRef Expression
1 xpsspw 4813 . . . . 5  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
2 uniss 3864 . . . . 5  |-  ( ( A  X.  B ) 
C_  ~P ~P ( A  u.  B )  ->  U. ( A  X.  B
)  C_  U. ~P ~P ( A  u.  B
) )
31, 2ax-mp 8 . . . 4  |-  U. ( A  X.  B )  C_  U. ~P ~P ( A  u.  B )
4 unipw 4240 . . . 4  |-  U. ~P ~P ( A  u.  B
)  =  ~P ( A  u.  B )
53, 4sseqtri 3223 . . 3  |-  U. ( A  X.  B )  C_  ~P ( A  u.  B
)
6 uniss 3864 . . 3  |-  ( U. ( A  X.  B
)  C_  ~P ( A  u.  B )  ->  U. U. ( A  X.  B )  C_  U. ~P ( A  u.  B ) )
75, 6ax-mp 8 . 2  |-  U. U. ( A  X.  B
)  C_  U. ~P ( A  u.  B )
8 unipw 4240 . 2  |-  U. ~P ( A  u.  B
)  =  ( A  u.  B )
97, 8sseqtri 3223 1  |-  U. U. ( A  X.  B
)  C_  ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    u. cun 3163    C_ wss 3165   ~Pcpw 3638   U.cuni 3843    X. cxp 4703
This theorem is referenced by:  relfld  5214  inposet  25381  filnetlem3  26432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-opab 4094  df-xp 4711
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