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Theorem indifundif 29694
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 4030 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
2 difundir 4029 . . . . 5 (((𝐴𝐵) ∪ (𝐴𝐵)) ∖ 𝐶) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
3 inundif 4188 . . . . . 6 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
43difeq1i 3875 . . . . 5 (((𝐴𝐵) ∪ (𝐴𝐵)) ∖ 𝐶) = (𝐴𝐶)
5 uncom 3908 . . . . 5 (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
62, 4, 53eqtr3i 2801 . . . 4 (𝐴𝐶) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
76uneq2i 3915 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)))
8 unass 3921 . . 3 (((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) ∪ ((𝐴𝐵) ∖ 𝐶)) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)))
9 undifabs 4187 . . . 4 ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) = (𝐴𝐵)
109uneq1i 3914 . . 3 (((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) ∪ ((𝐴𝐵) ∖ 𝐶)) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶))
117, 8, 103eqtr2i 2799 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶))
12 uncom 3908 . 2 ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) = (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵))
131, 11, 123eqtrri 2798 1 (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cdif 3720  cun 3721  cin 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730
This theorem is referenced by:  inelcarsg  30713
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