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Theorem indifundif 28740
Description: A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
Assertion
Ref Expression
indifundif (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))

Proof of Theorem indifundif
StepHypRef Expression
1 difindi 3840 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
2 difundir 3839 . . . . 5 (((𝐴𝐵) ∪ (𝐴𝐵)) ∖ 𝐶) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
3 inundif 3998 . . . . . 6 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
43difeq1i 3686 . . . . 5 (((𝐴𝐵) ∪ (𝐴𝐵)) ∖ 𝐶) = (𝐴𝐶)
5 uncom 3719 . . . . 5 (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
62, 4, 53eqtr3i 2640 . . . 4 (𝐴𝐶) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
76uneq2i 3726 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)))
8 unass 3732 . . 3 (((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) ∪ ((𝐴𝐵) ∖ 𝐶)) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)))
9 undifabs 3997 . . . 4 ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) = (𝐴𝐵)
109uneq1i 3725 . . 3 (((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) ∪ ((𝐴𝐵) ∖ 𝐶)) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶))
117, 8, 103eqtr2i 2638 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶))
12 uncom 3719 . 2 ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) = (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵))
131, 11, 123eqtrri 2637 1 (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  cdif 3537  cun 3538  cin 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547
This theorem is referenced by:  inelcarsg  29700
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