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Theorem difin 4237
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 ∖ (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem difin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm4.61 405 . . 3 (¬ (𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 anclb 546 . . . . 5 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 → (𝑥𝐴𝑥𝐵)))
3 elin 4168 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43imbi2i 337 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 → (𝑥𝐴𝑥𝐵)))
5 iman 402 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)))
62, 4, 53bitr2i 300 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)))
76con2bii 359 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)) ↔ ¬ (𝑥𝐴𝑥𝐵))
8 eldif 3945 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
91, 7, 83bitr4i 304 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
109difeqri 4100 1 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  cdif 3932  cin 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-dif 3938  df-in 3942
This theorem is referenced by:  dfin4  4243  indif  4245  dfsymdif3  4268  notrab  4279  disjdif2  4426  dfsdom2  8629  hashdif  13764  isercolllem3  15013  iuncld  21583  llycmpkgen2  22088  1stckgen  22092  txkgen  22190  cmmbl  24064  disjdifprg2  30255  ldgenpisyslem1  31322  onint1  33695  nonrel  39824  nzprmdif  40531
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