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Theorem inindif 28150
Description: See inundif 3875. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
inindif  |-  ( ( A  i^i  C )  i^i  ( A  \  C ) )  =  (/)

Proof of Theorem inindif
StepHypRef Expression
1 inss2 3683 . . . 4  |-  ( A  i^i  C )  C_  C
21orci 391 . . 3  |-  ( ( A  i^i  C ) 
C_  C  \/  A  C_  C )
3 inss 3691 . . 3  |-  ( ( ( A  i^i  C
)  C_  C  \/  A  C_  C )  -> 
( ( A  i^i  C )  i^i  A ) 
C_  C )
42, 3ax-mp 5 . 2  |-  ( ( A  i^i  C )  i^i  A )  C_  C
5 inssdif0 3864 . 2  |-  ( ( ( A  i^i  C
)  i^i  A )  C_  C  <->  ( ( A  i^i  C )  i^i  ( A  \  C
) )  =  (/) )
64, 5mpbi 211 1  |-  ( ( A  i^i  C )  i^i  ( A  \  C ) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    \/ wo 369    = wceq 1437    \ cdif 3433    i^i cin 3435    C_ wss 3436   (/)c0 3761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-v 3082  df-dif 3439  df-in 3443  df-ss 3450  df-nul 3762
This theorem is referenced by:  resf1o  28322  gsummptres  28556  measunl  29047  carsgclctun  29162  probdif  29262
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