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Theorem difin 3345
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 605 . . . . . . . 8 (¬ (𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝑥𝐵) → ⊥))
21expd 256 . . . . . . 7 (¬ (𝑥𝐴𝑥𝐵) → (𝑥𝐴 → (𝑥𝐵 → ⊥)))
3 dfnot 1353 . . . . . . 7 𝑥𝐵 ↔ (𝑥𝐵 → ⊥))
42, 3syl6ibr 161 . . . . . 6 (¬ (𝑥𝐴𝑥𝐵) → (𝑥𝐴 → ¬ 𝑥𝐵))
54com12 30 . . . . 5 (𝑥𝐴 → (¬ (𝑥𝐴𝑥𝐵) → ¬ 𝑥𝐵))
65imdistani 442 . . . 4 ((𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)) → (𝑥𝐴 ∧ ¬ 𝑥𝐵))
7 simpr 109 . . . . . 6 ((𝑥𝐴𝑥𝐵) → 𝑥𝐵)
87con3i 622 . . . . 5 𝑥𝐵 → ¬ (𝑥𝐴𝑥𝐵))
98anim2i 340 . . . 4 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
106, 9impbii 125 . . 3 ((𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
11 eldif 3111 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)))
12 elin 3291 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1312notbii 658 . . . . 5 𝑥 ∈ (𝐴𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
1413anbi2i 453 . . . 4 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
1511, 14bitri 183 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
16 eldif 3111 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1710, 15, 163bitr4i 211 . 2 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2154 1 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1335  wfal 1340  wcel 2128  cdif 3099  cin 3101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108
This theorem is referenced by:  inssddif  3349  symdif1  3373  notrab  3385  disjdif2  3473  unfiin  6872  bj-charfundcALT  13455
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