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Theorem iundifdifd 26064
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 4346 . . . . 5  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4333 . . . . . 6  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3580 . . . . 5  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2486 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3580 . . 3  |-  ( O 
\  U_ x  e.  A  ( O  \  x
) )  =  ( O  \  ( O 
\  |^| A ) )
6 intssuni2 4262 . . . . 5  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
7 unipw 4651 . . . . 5  |-  U. ~P O  =  O
86, 7syl6sseq 3511 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  O )
9 dfss4 3693 . . . 4  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
108, 9sylib 196 . . 3  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  -> 
( O  \  ( O  \  |^| A ) )  =  |^| A
)
115, 10syl5req 2508 . 2  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  =  ( O 
\  U_ x  e.  A  ( O  \  x
) ) )
1211ex 434 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 369    = wceq 1370    =/= wne 2648    \ cdif 3434    C_ wss 3437   (/)c0 3746   ~Pcpw 3969   U.cuni 4200   |^|cint 4237   U_ciun 4280   |^|_ciin 4281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-pw 3971  df-sn 3987  df-pr 3989  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283
This theorem is referenced by:  sigaclci  26721
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