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Theorem unixpss 4652
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 4651 . . . . 5  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
21unissi 3759 . . . 4  |-  U. ( A  X.  B )  C_  U. ~P ~P ( A  u.  B )
3 unipw 4139 . . . 4  |-  U. ~P ~P ( A  u.  B
)  =  ~P ( A  u.  B )
42, 3sseqtri 3131 . . 3  |-  U. ( A  X.  B )  C_  ~P ( A  u.  B
)
54unissi 3759 . 2  |-  U. U. ( A  X.  B
)  C_  U. ~P ( A  u.  B )
6 unipw 4139 . 2  |-  U. ~P ( A  u.  B
)  =  ( A  u.  B )
75, 6sseqtri 3131 1  |-  U. U. ( A  X.  B
)  C_  ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    u. cun 3069    C_ wss 3071   ~Pcpw 3510   U.cuni 3736    X. cxp 4537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-opab 3990  df-xp 4545
This theorem is referenced by:  relfld  5067
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