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Theorem unixpss 4929
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 4927 . . . . 5  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
21unissi 3981 . . . 4  |-  U. ( A  X.  B )  C_  U. ~P ~P ( A  u.  B )
3 unipw 4356 . . . 4  |-  U. ~P ~P ( A  u.  B
)  =  ~P ( A  u.  B )
42, 3sseqtri 3324 . . 3  |-  U. ( A  X.  B )  C_  ~P ( A  u.  B
)
54unissi 3981 . 2  |-  U. U. ( A  X.  B
)  C_  U. ~P ( A  u.  B )
6 unipw 4356 . 2  |-  U. ~P ( A  u.  B
)  =  ( A  u.  B )
75, 6sseqtri 3324 1  |-  U. U. ( A  X.  B
)  C_  ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    u. cun 3262    C_ wss 3264   ~Pcpw 3743   U.cuni 3958    X. cxp 4817
This theorem is referenced by:  relfld  5336  filnetlem3  26101
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-opab 4209  df-xp 4825
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