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Theorem difin 3234
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 ax-in2 580 . . . . . . . 8 (¬ (𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝑥𝐵) → ⊥))
21expd 254 . . . . . . 7 (¬ (𝑥𝐴𝑥𝐵) → (𝑥𝐴 → (𝑥𝐵 → ⊥)))
3 dfnot 1307 . . . . . . 7 𝑥𝐵 ↔ (𝑥𝐵 → ⊥))
42, 3syl6ibr 160 . . . . . 6 (¬ (𝑥𝐴𝑥𝐵) → (𝑥𝐴 → ¬ 𝑥𝐵))
54com12 30 . . . . 5 (𝑥𝐴 → (¬ (𝑥𝐴𝑥𝐵) → ¬ 𝑥𝐵))
65imdistani 434 . . . 4 ((𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)) → (𝑥𝐴 ∧ ¬ 𝑥𝐵))
7 simpr 108 . . . . . 6 ((𝑥𝐴𝑥𝐵) → 𝑥𝐵)
87con3i 597 . . . . 5 𝑥𝐵 → ¬ (𝑥𝐴𝑥𝐵))
98anim2i 334 . . . 4 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
106, 9impbii 124 . . 3 ((𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
11 eldif 3006 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)))
12 elin 3181 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1312notbii 629 . . . . 5 𝑥 ∈ (𝐴𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
1413anbi2i 445 . . . 4 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
1511, 14bitri 182 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
16 eldif 3006 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1710, 15, 163bitr4i 210 . 2 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2085 1 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1289  wfal 1294  wcel 1438  cdif 2994  cin 2996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-in 3003
This theorem is referenced by:  inssddif  3238  symdif1  3262  notrab  3274  unfiin  6616
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