Theorem indifundif 32451

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 4267	. 2 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))
2		difundir 4266	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ∖ 𝐶) = (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶))
3		inundif 4454	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴
4	3	difeq1i 4097	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ∖ 𝐶) = (𝐴 ∖ 𝐶)
5		uncom 4133	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶)) = (((𝐴 ∖ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶))
6	2, 4, 5	3eqtr3i 2766	. . . 4 ⊢ (𝐴 ∖ 𝐶) = (((𝐴 ∖ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶))
7	6	uneq2i 4140	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (((𝐴 ∖ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)))
8		unass 4147	. . 3 ⊢ (((𝐴 ∖ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶)) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (((𝐴 ∖ 𝐵) ∖ 𝐶) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)))
9		undifabs 4453	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶)) = (𝐴 ∖ 𝐵)
10	9	uneq1i 4139	. . 3 ⊢ (((𝐴 ∖ 𝐵) ∪ ((𝐴 ∖ 𝐵) ∖ 𝐶)) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶))
11	7, 8, 10	3eqtr2i 2764	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶))
12		uncom 4133	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ ((𝐴 ∩ 𝐵) ∖ 𝐶)) = (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ (𝐴 ∖ 𝐵))
13	1, 11, 12	3eqtrri 2763	1 ⊢ (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1540   ∖ cdif 3923   ∪ cun 3924   ∩ cin 3925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933

This theorem is referenced by:  inelcarsg  34289