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Theorem iinvdif 4374
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 3871 . . . 4  |-  ( _V 
\  (/) )  =  _V
2 0iun 4359 . . . . 5  |-  U_ x  e.  (/)  B  =  (/)
32difeq2i 3586 . . . 4  |-  ( _V 
\  U_ x  e.  (/)  B )  =  ( _V 
\  (/) )
4 0iin 4360 . . . 4  |-  |^|_ x  e.  (/)  ( _V  \  B )  =  _V
51, 3, 43eqtr4ri 2469 . . 3  |-  |^|_ x  e.  (/)  ( _V  \  B )  =  ( _V  \  U_ x  e.  (/)  B )
6 iineq1 4317 . . 3  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( _V  \  B )  =  |^|_ x  e.  (/)  ( _V  \  B ) )
7 iuneq1 4316 . . . 4  |-  ( A  =  (/)  ->  U_ x  e.  A  B  =  U_ x  e.  (/)  B )
87difeq2d 3589 . . 3  |-  ( A  =  (/)  ->  ( _V 
\  U_ x  e.  A  B )  =  ( _V  \  U_ x  e.  (/)  B ) )
95, 6, 83eqtr4a 2496 . 2  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( _V  \  B )  =  ( _V  \  U_ x  e.  A  B )
)
10 iindif2 4371 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( _V  \  B )  =  ( _V  \  U_ x  e.  A  B )
)
119, 10pm2.61ine 2744 1  |-  |^|_ x  e.  A  ( _V  \  B )  =  ( _V  \  U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   _Vcvv 3087    \ cdif 3439   (/)c0 3767   U_ciun 4302   |^|_ciin 4303
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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-in 3449  df-ss 3456  df-nul 3768  df-iun 4304  df-iin 4305
This theorem is referenced by: (None)
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