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Theorem difin 3171
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 545 . . . . . . . 8 (¬ (𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝑥𝐵) → ⊥))
21expd 245 . . . . . . 7 (¬ (𝑥𝐴𝑥𝐵) → (𝑥𝐴 → (𝑥𝐵 → ⊥)))
3 dfnot 1262 . . . . . . 7 𝑥𝐵 ↔ (𝑥𝐵 → ⊥))
42, 3syl6ibr 151 . . . . . 6 (¬ (𝑥𝐴𝑥𝐵) → (𝑥𝐴 → ¬ 𝑥𝐵))
54com12 27 . . . . 5 (𝑥𝐴 → (¬ (𝑥𝐴𝑥𝐵) → ¬ 𝑥𝐵))
65imdistani 419 . . . 4 ((𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)) → (𝑥𝐴 ∧ ¬ 𝑥𝐵))
7 simpr 103 . . . . . 6 ((𝑥𝐴𝑥𝐵) → 𝑥𝐵)
87con3i 562 . . . . 5 𝑥𝐵 → ¬ (𝑥𝐴𝑥𝐵))
98anim2i 324 . . . 4 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
106, 9impbii 117 . . 3 ((𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
11 eldif 2924 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)))
12 elin 3123 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1312notbii 594 . . . . 5 𝑥 ∈ (𝐴𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
1413anbi2i 430 . . . 4 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
1511, 14bitri 173 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
16 eldif 2924 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1710, 15, 163bitr4i 201 . 2 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2037 1 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97   = wceq 1243  wfal 1248  wcel 1393  cdif 2911  cin 2913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-in 2921
This theorem is referenced by:  inssddif  3175  symdif1  3199  notrab  3211
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