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Theorem difin 3281
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 587 . . . . . . . 8 (¬ (𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝑥𝐵) → ⊥))
21expd 256 . . . . . . 7 (¬ (𝑥𝐴𝑥𝐵) → (𝑥𝐴 → (𝑥𝐵 → ⊥)))
3 dfnot 1332 . . . . . . 7 𝑥𝐵 ↔ (𝑥𝐵 → ⊥))
42, 3syl6ibr 161 . . . . . 6 (¬ (𝑥𝐴𝑥𝐵) → (𝑥𝐴 → ¬ 𝑥𝐵))
54com12 30 . . . . 5 (𝑥𝐴 → (¬ (𝑥𝐴𝑥𝐵) → ¬ 𝑥𝐵))
65imdistani 439 . . . 4 ((𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)) → (𝑥𝐴 ∧ ¬ 𝑥𝐵))
7 simpr 109 . . . . . 6 ((𝑥𝐴𝑥𝐵) → 𝑥𝐵)
87con3i 604 . . . . 5 𝑥𝐵 → ¬ (𝑥𝐴𝑥𝐵))
98anim2i 337 . . . 4 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
106, 9impbii 125 . . 3 ((𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
11 eldif 3048 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)))
12 elin 3227 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1312notbii 640 . . . . 5 𝑥 ∈ (𝐴𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
1413anbi2i 450 . . . 4 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
1511, 14bitri 183 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐴𝑥𝐵)))
16 eldif 3048 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1710, 15, 163bitr4i 211 . 2 (𝑥 ∈ (𝐴 ∖ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2112 1 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1314  wfal 1319  wcel 1463  cdif 3036  cin 3038
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045
This theorem is referenced by:  inssddif  3285  symdif1  3309  notrab  3321  disjdif2  3409  unfiin  6780
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