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Theorem difindi 4082
Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difindi (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem difindi
StepHypRef Expression
1 dfin3 4067 . . 3 (𝐵𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))
21difeq2i 3923 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
3 indi 4074 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶)))
4 dfin2 4061 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
5 invdif 4069 . . . 4 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
6 invdif 4069 . . . 4 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
75, 6uneq12i 3963 . . 3 ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴𝐶))
83, 4, 73eqtr3i 2829 . 2 (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴𝐵) ∪ (𝐴𝐶))
92, 8eqtri 2821 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  Vcvv 3385  cdif 3766  cun 3767  cin 3768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776
This theorem is referenced by:  difdif2  4085  indm  4087  fndifnfp  6671  dprddisj2  18754  fctop  21137  cctop  21139  mretopd  21225  restcld  21305  cfinfil  22025  csdfil  22026  indifundif  29874  difres  29930  unelcarsg  30890  clsk3nimkb  39120  ntrclskb  39149  ntrclsk3  39150  ntrclsk13  39151  salincl  41286
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