Theorem difin 4217

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 4093	. . . . . 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 4049	. . 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 4065	1 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∖ cdif 4045   ∩ cin 4089

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047  df-in 4091

This theorem is used by:  dfin4  4223  indif  4225  dfsymdif3  4251  notrab  4267  disjdif2  4435  dfsdom2  9123  hashdif  14562  isercolllem3  15840  iuncld  23369  llycmpkgen2  23875  1stckgen  23879  txkgen  23977  cmmbl  25861  indifbi  33164  disjdifprg2  33218  ldgenpisyslem1  34843  onint1  37330  nonrel  44679  nzprmdif  45397