Theorem difin 4218

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		pm4.61 410	. . 3 ⊢ (¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		anclb 555	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
3		elin 3915	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
4	3	imbi2i 339	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
5		iman 407	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)))
6	2, 4, 5	3bitr2i 302	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)))
7	6	con2bii 360	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
8		eldif 3909	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
9	1, 7, 8	3bitr4i 306	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
10	9	difeqri 4076	1 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-in 3906

This theorem is used by:  dfin4  4224  indif  4226  dfsymdif3  4252  notrab  4268  disjdif2  4436  dfsdom2  9112  hashdif  14551  isercolllem3  15827  iuncld  23356  llycmpkgen2  23862  1stckgen  23866  txkgen  23964  cmmbl  25848  indifbi  33109  disjdifprg2  33163  ldgenpisyslem1  34789  onint1  37217  nonrel  44569  nzprmdif  45288