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Theorem iundifdif 28226
Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 28225. (Contributed by Thierry Arnoux, 4-Sep-2016.)
Hypotheses
Ref Expression
iundifdif.o  |-  O  e. 
_V
iundifdif.2  |-  A  C_  ~P O
Assertion
Ref Expression
iundifdif  |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
Distinct variable groups:    x, A    x, O

Proof of Theorem iundifdif
StepHypRef Expression
1 iundif2 4358 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4345 . . . . 5  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3559 . . . 4  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2486 . . 3  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3559 . 2  |-  ( O 
\  U_ x  e.  A  ( O  \  x
) )  =  ( O  \  ( O 
\  |^| A ) )
6 iundifdif.2 . . . . 5  |-  A  C_  ~P O
76jctl 548 . . . 4  |-  ( A  =/=  (/)  ->  ( A  C_ 
~P O  /\  A  =/=  (/) ) )
8 intssuni2 4273 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
9 unipw 4663 . . . . . 6  |-  U. ~P O  =  O
109sseq2i 3468 . . . . 5  |-  ( |^| A  C_  U. ~P O  <->  |^| A  C_  O )
1110biimpi 199 . . . 4  |-  ( |^| A  C_  U. ~P O  ->  |^| A  C_  O
)
127, 8, 113syl 18 . . 3  |-  ( A  =/=  (/)  ->  |^| A  C_  O )
13 dfss4 3688 . . 3  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
1412, 13sylib 201 . 2  |-  ( A  =/=  (/)  ->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
155, 14syl5req 2508 1  |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897    =/= wne 2632   _Vcvv 3056    \ cdif 3412    C_ wss 3415   (/)c0 3742   ~Pcpw 3962   U.cuni 4211   |^|cint 4247   U_ciun 4291   |^|_ciin 4292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-pw 3964  df-sn 3980  df-pr 3982  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294
This theorem is referenced by: (None)
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