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Theorem difdifdir 4386
 Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
Assertion
Ref Expression
difdifdir ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Proof of Theorem difdifdir
StepHypRef Expression
1 dif32 4198 . . . . 5 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
2 invdif 4174 . . . . 5 ((𝐴𝐶) ∩ (V ∖ 𝐵)) = ((𝐴𝐶) ∖ 𝐵)
31, 2eqtr4i 2785 . . . 4 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
4 un0 4287 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
53, 4eqtr4i 2785 . . 3 ((𝐴𝐵) ∖ 𝐶) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6 indi 4179 . . . 4 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
7 disjdif 4369 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ∅
8 incom 4107 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐶)
97, 8eqtr3i 2784 . . . . 5 ∅ = ((𝐴𝐶) ∩ 𝐶)
109uneq2i 4066 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
116, 10eqtr4i 2785 . . 3 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
125, 11eqtr4i 2785 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13 ddif 4043 . . . . 5 (V ∖ (V ∖ 𝐶)) = 𝐶
1413uneq2i 4066 . . . 4 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ 𝐶)
15 indm 4194 . . . . 5 (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶)))
16 invdif 4174 . . . . . 6 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
1716difeq2i 4026 . . . . 5 (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵𝐶))
1815, 17eqtr3i 2784 . . . 4 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = (V ∖ (𝐵𝐶))
1914, 18eqtr3i 2784 . . 3 ((V ∖ 𝐵) ∪ 𝐶) = (V ∖ (𝐵𝐶))
2019ineq2i 4115 . 2 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶)))
21 invdif 4174 . 2 ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶))) = ((𝐴𝐶) ∖ (𝐵𝐶))
2212, 20, 213eqtri 2786 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539  Vcvv 3410   ∖ cdif 3856   ∪ cun 3857   ∩ cin 3858  ∅c0 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227 This theorem is referenced by: (None)
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