Theorem difin 1670

Description: Difference with intersection. Theorem 33 of [Suppes] p. 29.

Assertion

Ref	Expression
difin	⊢ (A ∖ (A ∩ B)) = (A ∖ B)

Proof of Theorem difin

Step	Hyp	Ref	Expression
1		abai 366	. . . 4 ⊢ ((x ∈ A ∧ ¬ x ∈ B) ↔ (x ∈ A ∧ (x ∈ A → ¬ x ∈ B)))
2		imnan 207	. . . . 5 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ ¬ (x ∈ A ∧ x ∈ B))
3	2	anbi2i 367	. . . 4 ⊢ ((x ∈ A ∧ (x ∈ A → ¬ x ∈ B)) ↔ (x ∈ A ∧ ¬ (x ∈ A ∧ x ∈ B)))
4	1, 3	bitr 151	. . 3 ⊢ ((x ∈ A ∧ ¬ x ∈ B) ↔ (x ∈ A ∧ ¬ (x ∈ A ∧ x ∈ B)))
5		eldif 1496	. . 3 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
6		eldif 1496	. . . 4 ⊢ (x ∈ (A ∖ (A ∩ B)) ↔ (x ∈ A ∧ ¬ x ∈ (A ∩ B)))
7		elin 1635	. . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
8	7	negbii 162	. . . . 5 ⊢ (¬ x ∈ (A ∩ B) ↔ ¬ (x ∈ A ∧ x ∈ B))
9	8	anbi2i 367	. . . 4 ⊢ ((x ∈ A ∧ ¬ x ∈ (A ∩ B)) ↔ (x ∈ A ∧ ¬ (x ∈ A ∧ x ∈ B)))
10	6, 9	bitr 151	. . 3 ⊢ (x ∈ (A ∖ (A ∩ B)) ↔ (x ∈ A ∧ ¬ (x ∈ A ∧ x ∈ B)))
11	4, 5, 10	3bitr4r 159	. 2 ⊢ (x ∈ (A ∖ (A ∩ B)) ↔ x ∈ (A ∖ B))
12	11	cleqri 1101	1 ⊢ (A ∖ (A ∩ B)) = (A ∖ B)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ∩ cin 1486

This theorem is referenced by:  dfin4 1673  indif 1675  symdif1 1689  dfsdom2 3362

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491

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