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Theorem difdif 3171
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif (𝐴 ∖ (𝐵𝐴)) = 𝐴

Proof of Theorem difdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl 108 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) → 𝑥𝐴)
2 pm4.45im 332 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
3 imanim 662 . . . . . 6 ((𝑥𝐵𝑥𝐴) → ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
4 eldif 3050 . . . . . 6 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
53, 4sylnibr 651 . . . . 5 ((𝑥𝐵𝑥𝐴) → ¬ 𝑥 ∈ (𝐵𝐴))
65anim2i 339 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
72, 6sylbi 120 . . 3 (𝑥𝐴 → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
81, 7impbii 125 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
98difeqri 3166 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1316  wcel 1465  cdif 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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043
This theorem is referenced by:  dif0  3403
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