MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difundi Structured version   Visualization version   GIF version

Theorem difundi 4296
Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundi (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem difundi
StepHypRef Expression
1 dfun3 4282 . . 3 (𝐵𝐶) = (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
21difeq2i 4133 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
3 inindi 4243 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶)))
4 dfin2 4277 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
5 invdif 4285 . . . 4 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
6 invdif 4285 . . . 4 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
75, 6ineq12i 4226 . . 3 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∩ (𝐴𝐶))
83, 4, 73eqtr3i 2771 . 2 (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) = ((𝐴𝐵) ∩ (𝐴𝐶))
92, 8eqtri 2763 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cdif 3960  cun 3961  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970
This theorem is referenced by:  undm  4303  uncld  23065  inmbl  25591  difuncomp  32574  clsun  36311  poimirlem8  37615  ntrclskb  44059  ntrclsk3  44060  ntrclsk13  44061
  Copyright terms: Public domain W3C validator