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Theorem iundifdif 27089
Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 27088 (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 4385 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4372 . . . . 5  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3612 . . . 4  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2492 . . 3  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3612 . 2  |-  ( O 
\  U_ x  e.  A  ( O  \  x
) )  =  ( O  \  ( O 
\  |^| A ) )
6 iundifdif.2 . . . . 5  |-  A  C_  ~P O
76jctl 541 . . . 4  |-  ( A  =/=  (/)  ->  ( A  C_ 
~P O  /\  A  =/=  (/) ) )
8 intssuni2 4300 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
9 unipw 4690 . . . . . 6  |-  U. ~P O  =  O
109sseq2i 3522 . . . . 5  |-  ( |^| A  C_  U. ~P O  <->  |^| A  C_  O )
1110biimpi 194 . . . 4  |-  ( |^| A  C_  U. ~P O  ->  |^| A  C_  O
)
127, 8, 113syl 20 . . 3  |-  ( A  =/=  (/)  ->  |^| A  C_  O )
13 dfss4 3725 . . 3  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
1412, 13sylib 196 . 2  |-  ( A  =/=  (/)  ->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
155, 14syl5req 2514 1  |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106    \ cdif 3466    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   U.cuni 4238   |^|cint 4275   U_ciun 4318   |^|_ciin 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-pw 4005  df-sn 4021  df-pr 4023  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321
This theorem is referenced by: (None)
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