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Theorem iundifdifd 28166
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 4363 . . . . 5  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4350 . . . . . 6  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3580 . . . . 5  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2454 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3580 . . 3  |-  ( O 
\  U_ x  e.  A  ( O  \  x
) )  =  ( O  \  ( O 
\  |^| A ) )
6 intssuni2 4278 . . . . 5  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
7 unipw 4667 . . . . 5  |-  U. ~P O  =  O
86, 7syl6sseq 3510 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  O )
9 dfss4 3707 . . . 4  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
108, 9sylib 199 . . 3  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  -> 
( O  \  ( O  \  |^| A ) )  =  |^| A
)
115, 10syl5req 2476 . 2  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  =  ( O 
\  U_ x  e.  A  ( O  \  x
) ) )
1211ex 435 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 370    = wceq 1437    =/= wne 2618    \ cdif 3433    C_ wss 3436   (/)c0 3761   ~Pcpw 3979   U.cuni 4216   |^|cint 4252   U_ciun 4296   |^|_ciin 4297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-pw 3981  df-sn 3997  df-pr 3999  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299
This theorem is referenced by:  sigaclci  28949
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