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Theorem difin 3853
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 pm4.61 442 . . 3 (¬ (𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 anclb 569 . . . . 5 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 → (𝑥𝐴𝑥𝐵)))
3 elin 3788 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43imbi2i 326 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 → (𝑥𝐴𝑥𝐵)))
5 iman 440 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)))
62, 4, 53bitr2i 288 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)))
76con2bii 347 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)) ↔ ¬ (𝑥𝐴𝑥𝐵))
8 eldif 3577 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
91, 7, 83bitr4i 292 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
109difeqri 3722 1 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  cdif 3564  cin 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570  df-in 3574
This theorem is referenced by:  dfin4  3859  indif  3861  dfsymdif3  3885  notrab  3896  disjdif2  4038  dfsdom2  8068  hashdif  13184  isercolllem3  14378  iuncld  20830  llycmpkgen2  21334  1stckgen  21338  txkgen  21436  cmmbl  23283  disjdifprg2  29361  ldgenpisyslem1  30200  onint1  32423  nonrel  37709  nzprmdif  38338
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