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Theorem iundifdif 28174
Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 28173 (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 4364 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4351 . . . . 5  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3581 . . . 4  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2455 . . 3  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3581 . 2  |-  ( O 
\  U_ x  e.  A  ( O  \  x
) )  =  ( O  \  ( O 
\  |^| A ) )
6 iundifdif.2 . . . . 5  |-  A  C_  ~P O
76jctl 544 . . . 4  |-  ( A  =/=  (/)  ->  ( A  C_ 
~P O  /\  A  =/=  (/) ) )
8 intssuni2 4279 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
9 unipw 4669 . . . . . 6  |-  U. ~P O  =  O
109sseq2i 3490 . . . . 5  |-  ( |^| A  C_  U. ~P O  <->  |^| A  C_  O )
1110biimpi 198 . . . 4  |-  ( |^| A  C_  U. ~P O  ->  |^| A  C_  O
)
127, 8, 113syl 18 . . 3  |-  ( A  =/=  (/)  ->  |^| A  C_  O )
13 dfss4 3708 . . 3  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
1412, 13sylib 200 . 2  |-  ( A  =/=  (/)  ->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
155, 14syl5req 2477 1  |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   _Vcvv 3082    \ cdif 3434    C_ wss 3437   (/)c0 3762   ~Pcpw 3980   U.cuni 4217   |^|cint 4253   U_ciun 4297   |^|_ciin 4298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-pw 3982  df-sn 3998  df-pr 4000  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300
This theorem is referenced by: (None)
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