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Theorem unixpss 4564
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 4563 . . . . 5  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
21unissi 3682 . . . 4  |-  U. ( A  X.  B )  C_  U. ~P ~P ( A  u.  B )
3 unipw 4053 . . . 4  |-  U. ~P ~P ( A  u.  B
)  =  ~P ( A  u.  B )
42, 3sseqtri 3059 . . 3  |-  U. ( A  X.  B )  C_  ~P ( A  u.  B
)
54unissi 3682 . 2  |-  U. U. ( A  X.  B
)  C_  U. ~P ( A  u.  B )
6 unipw 4053 . 2  |-  U. ~P ( A  u.  B
)  =  ( A  u.  B )
75, 6sseqtri 3059 1  |-  U. U. ( A  X.  B
)  C_  ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    u. cun 2998    C_ wss 3000   ~Pcpw 3433   U.cuni 3659    X. cxp 4449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-opab 3906  df-xp 4457
This theorem is referenced by:  relfld  4972
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