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Theorem iinvdif 4397
Description: The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)
Assertion
Ref Expression
iinvdif  |-  |^|_ x  e.  A  ( _V  \  B )  =  ( _V  \  U_ x  e.  A  B )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem iinvdif
StepHypRef Expression
1 dif0 3897 . . . 4  |-  ( _V 
\  (/) )  =  _V
2 0iun 4382 . . . . 5  |-  U_ x  e.  (/)  B  =  (/)
32difeq2i 3619 . . . 4  |-  ( _V 
\  U_ x  e.  (/)  B )  =  ( _V 
\  (/) )
4 0iin 4383 . . . 4  |-  |^|_ x  e.  (/)  ( _V  \  B )  =  _V
51, 3, 43eqtr4ri 2507 . . 3  |-  |^|_ x  e.  (/)  ( _V  \  B )  =  ( _V  \  U_ x  e.  (/)  B )
6 iineq1 4340 . . 3  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( _V  \  B )  =  |^|_ x  e.  (/)  ( _V  \  B ) )
7 iuneq1 4339 . . . 4  |-  ( A  =  (/)  ->  U_ x  e.  A  B  =  U_ x  e.  (/)  B )
87difeq2d 3622 . . 3  |-  ( A  =  (/)  ->  ( _V 
\  U_ x  e.  A  B )  =  ( _V  \  U_ x  e.  (/)  B ) )
95, 6, 83eqtr4a 2534 . 2  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( _V  \  B )  =  ( _V  \  U_ x  e.  A  B )
)
10 iindif2 4394 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( _V  \  B )  =  ( _V  \  U_ x  e.  A  B )
)
119, 10pm2.61ine 2780 1  |-  |^|_ x  e.  A  ( _V  \  B )  =  ( _V  \  U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3113    \ cdif 3473   (/)c0 3785   U_ciun 4325   |^|_ciin 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-iun 4327  df-iin 4328
This theorem is referenced by: (None)
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