Theorem indifundif 31516

Description: A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)

Assertion

Ref	Expression
indifundif	⊢ (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐵 ∩ 𝐶))

Proof of Theorem indifundif

Step	Hyp	Ref	Expression
1		difindi 4246	. 2 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))
2		difundir 4245	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ∖ 𝐶) = (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶))
3		inundif 4443	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴
4	3	difeq1i 4083	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ∖ 𝐶) = (𝐴 ∖ 𝐶)
5		uncom 4118	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶)) = (((𝐴 ∖ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶))
6	2, 4, 5	3eqtr3i 2767	. . . 4 ⊢ (𝐴 ∖ 𝐶) = (((𝐴 ∖ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶))
7	6	uneq2i 4125	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (((𝐴 ∖ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)))
8		unass 4131	. . 3 ⊢ (((𝐴 ∖ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶)) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (((𝐴 ∖ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)))
9		undifabs 4442	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶)) = (𝐴 ∖ 𝐵)
10	9	uneq1i 4124	. . 3 ⊢ (((𝐴 ∖ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶)) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶))
11	7, 8, 10	3eqtr2i 2765	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶))
12		uncom 4118	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)) = (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ (𝐴 ∖ 𝐵))
13	1, 11, 12	3eqtrri 2764	1 ⊢ (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1541   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920

This theorem is referenced by:  inelcarsg  33000