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Theorem iundifdif 24013
Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 24012 (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 4158 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4145 . . . . 5  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3462 . . . 4  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2459 . . 3  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3462 . 2  |-  ( O 
\  U_ x  e.  A  ( O  \  x
) )  =  ( O  \  ( O 
\  |^| A ) )
6 iundifdif.2 . . . . 5  |-  A  C_  ~P O
76jctl 526 . . . 4  |-  ( A  =/=  (/)  ->  ( A  C_ 
~P O  /\  A  =/=  (/) ) )
8 intssuni2 4075 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
9 unipw 4414 . . . . . 6  |-  U. ~P O  =  O
109sseq2i 3373 . . . . 5  |-  ( |^| A  C_  U. ~P O  <->  |^| A  C_  O )
1110biimpi 187 . . . 4  |-  ( |^| A  C_  U. ~P O  ->  |^| A  C_  O
)
127, 8, 113syl 19 . . 3  |-  ( A  =/=  (/)  ->  |^| A  C_  O )
13 dfss4 3575 . . 3  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
1412, 13sylib 189 . 2  |-  ( A  =/=  (/)  ->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
155, 14syl5req 2481 1  |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   U.cuni 4015   |^|cint 4050   U_ciun 4093   |^|_ciin 4094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096
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