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Theorem unixpss 4955
Description: The double class union of a Cartesian 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 4953 . . . . 5  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
21unissi 4114 . . . 4  |-  U. ( A  X.  B )  C_  U. ~P ~P ( A  u.  B )
3 unipw 4542 . . . 4  |-  U. ~P ~P ( A  u.  B
)  =  ~P ( A  u.  B )
42, 3sseqtri 3388 . . 3  |-  U. ( A  X.  B )  C_  ~P ( A  u.  B
)
54unissi 4114 . 2  |-  U. U. ( A  X.  B
)  C_  U. ~P ( A  u.  B )
6 unipw 4542 . 2  |-  U. ~P ( A  u.  B
)  =  ( A  u.  B )
75, 6sseqtri 3388 1  |-  U. U. ( A  X.  B
)  C_  ( A  u.  B )
Colors of variables: wff setvar class
Syntax hints:    u. cun 3326    C_ wss 3328   ~Pcpw 3860   U.cuni 4091    X. cxp 4838
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
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-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-opab 4351  df-xp 4846
This theorem is referenced by:  relfld  5363  filnetlem3  28601
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