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

Theorem iundifdif 26057
Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 26056 (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 4338 . . . 4  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^|_ x  e.  A  x )
2 intiin 4325 . . . . 5  |-  |^| A  =  |^|_ x  e.  A  x
32difeq2i 3572 . . . 4  |-  ( O 
\  |^| A )  =  ( O  \  |^|_ x  e.  A  x )
41, 3eqtr4i 2483 . . 3  |-  U_ x  e.  A  ( O  \  x )  =  ( O  \  |^| A
)
54difeq2i 3572 . 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 4254 . . . 4  |-  ( ( A  C_  ~P O  /\  A  =/=  (/) )  ->  |^| A  C_  U. ~P O
)
9 unipw 4643 . . . . . 6  |-  U. ~P O  =  O
109sseq2i 3482 . . . . 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 3685 . . 3  |-  ( |^| A  C_  O  <->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
1412, 13sylib 196 . 2  |-  ( A  =/=  (/)  ->  ( O  \  ( O  \  |^| A ) )  = 
|^| A )
155, 14syl5req 2505 1  |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071    \ cdif 3426    C_ wss 3429   (/)c0 3738   ~Pcpw 3961   U.cuni 4192   |^|cint 4229   U_ciun 4272   |^|_ciin 4273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-pw 3963  df-sn 3979  df-pr 3981  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator