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Theorem difdifdir 4491
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 4292 . . . . 5 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
2 invdif 4268 . . . . 5 ((𝐴𝐶) ∩ (V ∖ 𝐵)) = ((𝐴𝐶) ∖ 𝐵)
31, 2eqtr4i 2762 . . . 4 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
4 un0 4390 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = ((𝐴𝐶) ∩ (V ∖ 𝐵))
53, 4eqtr4i 2762 . . 3 ((𝐴𝐵) ∖ 𝐶) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
6 indi 4273 . . . 4 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
7 disjdif 4471 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ∅
8 incom 4201 . . . . . 6 (𝐶 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐶)
97, 8eqtr3i 2761 . . . . 5 ∅ = ((𝐴𝐶) ∩ 𝐶)
109uneq2i 4160 . . . 4 (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ((𝐴𝐶) ∩ 𝐶))
116, 10eqtr4i 2762 . . 3 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = (((𝐴𝐶) ∩ (V ∖ 𝐵)) ∪ ∅)
125, 11eqtr4i 2762 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶))
13 ddif 4136 . . . . 5 (V ∖ (V ∖ 𝐶)) = 𝐶
1413uneq2i 4160 . . . 4 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ 𝐶)
15 indm 4288 . . . . 5 (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶)))
16 invdif 4268 . . . . . 6 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
1716difeq2i 4119 . . . . 5 (V ∖ (𝐵 ∩ (V ∖ 𝐶))) = (V ∖ (𝐵𝐶))
1815, 17eqtr3i 2761 . . . 4 ((V ∖ 𝐵) ∪ (V ∖ (V ∖ 𝐶))) = (V ∖ (𝐵𝐶))
1914, 18eqtr3i 2761 . . 3 ((V ∖ 𝐵) ∪ 𝐶) = (V ∖ (𝐵𝐶))
2019ineq2i 4209 . 2 ((𝐴𝐶) ∩ ((V ∖ 𝐵) ∪ 𝐶)) = ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶)))
21 invdif 4268 . 2 ((𝐴𝐶) ∩ (V ∖ (𝐵𝐶))) = ((𝐴𝐶) ∖ (𝐵𝐶))
2212, 20, 213eqtri 2763 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3473  cdif 3945  cun 3946  cin 3947  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323
This theorem is referenced by: (None)
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