Theorem difin 4222

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		pm4.61 408	. . 3 ⊢ (¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		anclb 553	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
3		elin 3918	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
4	3	imbi2i 338	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
5		iman 405	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)))
6	2, 4, 5	3bitr2i 301	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)))
7	6	con2bii 359	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
8		eldif 3912	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
9	1, 7, 8	3bitr4i 305	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
10	9	difeqri 4080	1 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∖ cdif 3899   ∩ cin 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3905  df-in 3909

This theorem is referenced by:  dfin4  4228  indif  4230  dfsymdif3  4256  notrab  4272  disjdif2  4431  dfsdom2  9066  hashdif  14420  isercolllem3  15685  iuncld  23093  llycmpkgen2  23598  1stckgen  23602  txkgen  23700  cmmbl  25584  indifbi  32679  disjdifprg2  32736  ldgenpisyslem1  34421  onint1  36770  nonrel  44121  nzprmdif  44856