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Theorem indifundif 30271
 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 4233 . 2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
2 difundir 4232 . . . . 5 (((𝐴𝐵) ∪ (𝐴𝐵)) ∖ 𝐶) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
3 inundif 4400 . . . . . 6 ((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴
43difeq1i 4071 . . . . 5 (((𝐴𝐵) ∪ (𝐴𝐵)) ∖ 𝐶) = (𝐴𝐶)
5 uncom 4105 . . . . 5 (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
62, 4, 53eqtr3i 2852 . . . 4 (𝐴𝐶) = (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶))
76uneq2i 4112 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)))
8 unass 4118 . . 3 (((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) ∪ ((𝐴𝐵) ∖ 𝐶)) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∖ 𝐶) ∪ ((𝐴𝐵) ∖ 𝐶)))
9 undifabs 4399 . . . 4 ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) = (𝐴𝐵)
109uneq1i 4111 . . 3 (((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) ∪ ((𝐴𝐵) ∖ 𝐶)) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶))
117, 8, 103eqtr2i 2850 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶))
12 uncom 4105 . 2 ((𝐴𝐵) ∪ ((𝐴𝐵) ∖ 𝐶)) = (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵))
131, 11, 123eqtrri 2849 1 (((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917 This theorem is referenced by:  inelcarsg  31576
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