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Theorem iundifdifd 28226
Description: The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)
Assertion
Ref Expression
iundifdifd  |-  ( A 
C_  ~P O  ->  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) ) )
Distinct variable groups:    x, A    x, O

Proof of Theorem iundifdifd
StepHypRef Expression
1 iundif2 4359 . . . . 5  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4346 . . . . . 6  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3560 . . . . 5  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2487 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3560 . . 3  |-  ( O 
\  U_ x  e.  A  ( O  \  x
) )  =  ( O  \  ( O 
\  |^| A ) )
6 intssuni2 4274 . . . . 5  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
7 unipw 4664 . . . . 5  |-  U. ~P O  =  O
86, 7syl6sseq 3490 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  O )
9 dfss4 3689 . . . 4  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
108, 9sylib 201 . . 3  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  -> 
( O  \  ( O  \  |^| A ) )  =  |^| A
)
115, 10syl5req 2509 . 2  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  =  ( O 
\  U_ x  e.  A  ( O  \  x
) ) )
1211ex 440 1  |-  ( A 
C_  ~P O  ->  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    =/= wne 2633    \ cdif 3413    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   U.cuni 4212   |^|cint 4248   U_ciun 4292   |^|_ciin 4293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-pw 3965  df-sn 3981  df-pr 3983  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295
This theorem is referenced by:  sigaclci  29003
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