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Theorem unixpss 4799
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 4797 . . . . 5  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
2 uniss 3848 . . . . 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 4224 . . . 4  |-  U. ~P ~P ( A  u.  B
)  =  ~P ( A  u.  B )
53, 4sseqtri 3210 . . 3  |-  U. ( A  X.  B )  C_  ~P ( A  u.  B
)
6 uniss 3848 . . 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 4224 . 2  |-  U. ~P ( A  u.  B
)  =  ( A  u.  B )
97, 8sseqtri 3210 1  |-  U. U. ( A  X.  B
)  C_  ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    u. cun 3150    C_ wss 3152   ~Pcpw 3625   U.cuni 3827    X. cxp 4687
This theorem is referenced by:  relfld  5198  inposet  25278  filnetlem3  26329
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-opab 4078  df-xp 4695
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