Theorem difin 3493

Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem difin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm4.61 415	. . 3 ⊢ (¬ (x ∈ A → x ∈ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
2		anclb 530	. . . . 5 ⊢ ((x ∈ A → x ∈ B) ↔ (x ∈ A → (x ∈ A ∧ x ∈ B)))
3		elin 3220	. . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
4	3	imbi2i 303	. . . . 5 ⊢ ((x ∈ A → x ∈ (A ∩ B)) ↔ (x ∈ A → (x ∈ A ∧ x ∈ B)))
5		iman 413	. . . . 5 ⊢ ((x ∈ A → x ∈ (A ∩ B)) ↔ ¬ (x ∈ A ∧ ¬ x ∈ (A ∩ B)))
6	2, 4, 5	3bitr2i 264	. . . 4 ⊢ ((x ∈ A → x ∈ B) ↔ ¬ (x ∈ A ∧ ¬ x ∈ (A ∩ B)))
7	6	con2bii 322	. . 3 ⊢ ((x ∈ A ∧ ¬ x ∈ (A ∩ B)) ↔ ¬ (x ∈ A → x ∈ B))
8		eldif 3222	. . 3 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
9	1, 7, 8	3bitr4i 268	. 2 ⊢ ((x ∈ A ∧ ¬ x ∈ (A ∩ B)) ↔ x ∈ (A ∖ B))
10	9	difeqri 3388	1 ⊢ (A ∖ (A ∩ B)) = (A ∖ B)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by:  dfin4  3496  indif  3498  symdif1  3520  notrab  3533