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Theorem unixpss 4940
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 4939 . . . . 5  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
21unissi 4216 . . . 4  |-  U. ( A  X.  B )  C_  U. ~P ~P ( A  u.  B )
3 unipw 4643 . . . 4  |-  U. ~P ~P ( A  u.  B
)  =  ~P ( A  u.  B )
42, 3sseqtri 3476 . . 3  |-  U. ( A  X.  B )  C_  ~P ( A  u.  B
)
54unissi 4216 . 2  |-  U. U. ( A  X.  B
)  C_  U. ~P ( A  u.  B )
6 unipw 4643 . 2  |-  U. ~P ( A  u.  B
)  =  ( A  u.  B )
75, 6sseqtri 3476 1  |-  U. U. ( A  X.  B
)  C_  ( A  u.  B )
Colors of variables: wff setvar class
Syntax hints:    u. cun 3414    C_ wss 3416   ~Pcpw 3957   U.cuni 4193    X. cxp 4823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-opab 4456  df-xp 4831  df-rel 4832
This theorem is referenced by:  relfld  5351  filnetlem3  30621
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