Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iundifdifd Structured version   Unicode version

Theorem iundifdifd 27642
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 4382 . . . . 5  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4369 . . . . . 6  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3605 . . . . 5  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2486 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3605 . . 3  |-  ( O 
\  U_ x  e.  A  ( O  \  x
) )  =  ( O  \  ( O 
\  |^| A ) )
6 intssuni2 4297 . . . . 5  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
7 unipw 4687 . . . . 5  |-  U. ~P O  =  O
86, 7syl6sseq 3535 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  O )
9 dfss4 3729 . . . 4  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
108, 9sylib 196 . . 3  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  -> 
( O  \  ( O  \  |^| A ) )  =  |^| A
)
115, 10syl5req 2508 . 2  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  =  ( O 
\  U_ x  e.  A  ( O  \  x
) ) )
1211ex 432 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 367    = wceq 1398    =/= wne 2649    \ cdif 3458    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   |^|cint 4271   U_ciun 4315   |^|_ciin 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001  df-sn 4017  df-pr 4019  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318
This theorem is referenced by:  sigaclci  28365
  Copyright terms: Public domain W3C validator