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Theorem indifundif 29874
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 4082 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
2 difundir 4081 . . . . 5 (((𝐴𝐵) ∪ (𝐴𝐵)) ∖ 𝐶) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
3 inundif 4240 . . . . . 6 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
43difeq1i 3922 . . . . 5 (((𝐴𝐵) ∪ (𝐴𝐵)) ∖ 𝐶) = (𝐴𝐶)
5 uncom 3955 . . . . 5 (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
62, 4, 53eqtr3i 2829 . . . 4 (𝐴𝐶) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
76uneq2i 3962 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)))
8 unass 3968 . . 3 (((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) ∪ ((𝐴𝐵) ∖ 𝐶)) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)))
9 undifabs 4239 . . . 4 ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) = (𝐴𝐵)
109uneq1i 3961 . . 3 (((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) ∪ ((𝐴𝐵) ∖ 𝐶)) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶))
117, 8, 103eqtr2i 2827 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶))
12 uncom 3955 . 2 ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) = (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵))
131, 11, 123eqtrri 2826 1 (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  cdif 3766  cun 3767  cin 3768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776
This theorem is referenced by:  inelcarsg  30889
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